Introduction to high-dimensional data

Overview

Teaching: 30 min
Exercises: 20 min

Questions

• What are high-dimensional data and what do these data look like in the biosciences?

• What are the challenges when analysing high-dimensional data?

• What statistical methods are suitable for analysing these data?

• How can Bioconductor be used to access high-dimensional data in the biosciences?

Objectives

• Explore examples of high-dimensional data in the biosciences.

• Appreciate challenges involved in analysing high-dimensional data.

• Explore different statistical methods used for analysing high-dimensional data.

• Work with example data created from biological studies.

What are high-dimensional data?

High-dimensional data are defined as data with many features (variables observed). In recent years, advances in information technology have allowed large amounts of data to be collected and stored with relative ease. As such, high-dimensional data have become more common in many scientific fields, including the biological sciences, where datasets in subjects like genomics and medical sciences often have a large numbers of features. For example, hospital data may record many variables, including symptoms, blood test results, behaviours, and general health. An example of what high-dimensional data might look like in a biomedical study is shown in the figure below.

Example of a high-dimensional data table with features in the columns and individual observations (patients) in rows.

Researchers often want to relate such features to specific patient outcomes (e.g. survival, length of time spent in hospital). However, analysing high-dimensional data can be extremely challenging since standard methods of analysis, such as individual plots of features and linear regression, are no longer appropriate when we have many features. In this lesson, we will learn alternative methods for dealing with high-dimensional data and discover how these can be applied for practical high-dimensional data analysis in the biological sciences.

Challenge 1

Descriptions of four research questions and their datasets are given below. Which of these scenarios use high-dimensional data?

1. Predicting patient blood pressure using: cholesterol level in blood, age, and BMI measurements, collected from 100 patients.
2. Predicting patient blood pressure using: cholesterol level in blood, age, and BMI, as well as information on 200,000 single nucleotide polymorphisms from 100 patients.
3. Predicting the length of time patients spend in hospital with pneumonia infection using: measurements on age, BMI, length of time with symptoms, number of symptoms, and percentage of neutrophils in blood, using data from 200 patients.
4. Predicting probability of a patient’s cancer progressing using gene expression data from 20,000 genes, as well as data associated with general patient health (age, weight, BMI, blood pressure) and cancer growth (tumour size, localised spread, blood test results).

Solution

1. No. The number of features is relatively small (4 including the response variable since this is an observed variable).
2. Yes, this is an example of high-dimensional data. There are 200,004 features.
3. No. The number of features is relatively small (6).
4. Yes. There are 20,008 features.

Now that we have an idea of what high-dimensional data look like we can think about the challenges we face in analysing them.

Why is dealing with high-dimensional data challenging?

Most classical statistical methods are set up for use on low-dimensional data (i.e. with a small number of features, $p$). This is because low-dimensional data were much more common in the past when data collection was more difficult and time consuming.

One challenge when analysing high-dimensional data is visualising the many variables. When exploring low-dimensional datasets, it is possible to plot the response variable against each of features to get an idea which of these are important predictors of the response. With high-dimensional data, the large number of features makes doing this difficult. In addition, in some high-dimensional datasets it can also be difficult to identify a single response variable, making standard data exploration and analysis techniques less useful.

Let’s have a look at a simple dataset with lots of features to understand some of the challenges we are facing when working with high-dimensional data. For reference, all data used throughout the lesson are described in the data page.

Challenge 2

For illustrative purposes, we start with a simple dataset that is not technically high-dimensional but contains many features. This will illustrate the general problems encountered when working with many features in a high-dimensional data set.

First, make sure you have completed the setup instructions here. Next, let’s load the prostate dataset as follows:

library("here")
prostate <- readRDS(here("data/prostate.rds"))

Examine the dataset (in which each row represents a single patient) to:

1. Determine how many observations ($n$) and features ($p$) are available (hint: see the dim() function).
2. Examine what variables were measured (hint: see the names() and head() functions).
3. Plot the relationship between the variables (hint: see the pairs() function). What problem(s) with high-dimensional data analysis does this illustrate?

Solution

dim(prostate)    # print the number of rows and columns

names(prostate)  # examine the variable names
head(prostate)   # print the first 6 rows

names(prostate)  # examine column names

[1] "lcavol"  "lweight" "age"     "lbph"    "svi"     "lcp"     "gleason"
[8] "pgg45"   "lpsa"   

pairs(prostate)  # plot each pair of variables against each other

Pairwise plots of the 'prostate' dataset.

The pairs() function plots relationships between each of the variables in the prostate dataset. This is possible for datasets with smaller numbers of variables, but for datasets in which $p$ is larger it becomes difficult (and time consuming) to visualise relationships between all variables in the dataset. Even where visualisation is possible, fitting models to datasets with many variables is difficult due to the potential for overfitting and difficulties in identifying a response variable.

Note that function documentation and information on function arguments will be useful throughout this lesson. We can access these easily in R by running ? followed by the package name. For example, the documentation for the dim function can be accessed by running ?dim.

Locating data with R - the here package

It is often desirable to access external datasets from inside R and to write code that does this reliably on different computers. While R has an inbulit function setwd() that can be used to denote where external datasets are stored, this usually requires the user to adjust the code to their specific system and folder structure. The here package is meant to be used in R projects. It allows users to specify the data location relative to the R project directory. This makes R code more portable and can contribute to improve the reproducibility of an analysis.

As well as many variables causing problems when working with high-dimensional data, having relatively few observations ($n$) compared to the number of features ($p$) causes additional challenges. To illustrate these challenges, imagine we are carrying out least squares regression on a dataset with 25 observations. Fitting a best fit line through these data produces a plot shown in the left-hand panel of the figure below.

However, imagine a situation in which the number of observations and features in a dataset are almost equal. In that situation the effective number of observations per feature is low. The result of fitting a best fit line through few observations can be seen in the right-hand panel below.

Scatter plot of two variables (x and y) from a data set with 25 observations (left) and 2 observations (right) with a fitted regression line (red).

In the first situation, the least squares regression line does not fit the data perfectly and there is some error around the regression line. But, when there are only two observations the regression line will fit through the points exactly, resulting in overfitting of the data. This suggests that carrying out least squares regression on a dataset with few data points per feature would result in difficulties in applying the resulting model to further datsets. This is a common problem when using regression on high-dimensional datasets.

Another problem in carrying out regression on high-dimensional data is dealing with correlations between explanatory variables. The large numbers of features in these datasets makes high correlations between variables more likely. Let’s explore why high correlations might be an issue in a Challenge.

Challenge 3

Use the cor() function to examine correlations between all variables in the prostate dataset. Are some pairs of variables highly correlated using a threshold of 0.75 for the correlation coefficients?

Use the lm() function to fit univariate regression models to predict patient age using two variables that are highly correlated as predictors. Which of these variables are statistically significant predictors of age? Hint: the summary() function can help here.

Fit a multiple linear regression model predicting patient age using both variables. What happened?

Solution

Create a correlation matrix of all variables in the prostate dataset

cor(prostate)

           lcavol      lweight       age         lbph         svi          lcp
lcavol  1.0000000  0.194128286 0.2249999  0.027349703  0.53884500  0.675310484
lweight 0.1941283  1.000000000 0.3075286  0.434934636  0.10877851  0.100237795
age     0.2249999  0.307528614 1.0000000  0.350185896  0.11765804  0.127667752
lbph    0.0273497  0.434934636 0.3501859  1.000000000 -0.08584324 -0.006999431
svi     0.5388450  0.108778505 0.1176580 -0.085843238  1.00000000  0.673111185
lcp     0.6753105  0.100237795 0.1276678 -0.006999431  0.67311118  1.000000000
gleason 0.4324171 -0.001275658 0.2688916  0.077820447  0.32041222  0.514830063
pgg45   0.4336522  0.050846821 0.2761124  0.078460018  0.45764762  0.631528245
lpsa    0.7344603  0.354120390 0.1695928  0.179809410  0.56621822  0.548813169
             gleason      pgg45      lpsa
lcavol   0.432417056 0.43365225 0.7344603
lweight -0.001275658 0.05084682 0.3541204
age      0.268891599 0.27611245 0.1695928
lbph     0.077820447 0.07846002 0.1798094
svi      0.320412221 0.45764762 0.5662182
lcp      0.514830063 0.63152825 0.5488132
gleason  1.000000000 0.75190451 0.3689868
pgg45    0.751904512 1.00000000 0.4223159
lpsa     0.368986803 0.42231586 1.0000000

round(cor(prostate), 2) # rounding helps to visualise the correlations

        lcavol lweight  age  lbph   svi   lcp gleason pgg45 lpsa
lcavol    1.00    0.19 0.22  0.03  0.54  0.68    0.43  0.43 0.73
lweight   0.19    1.00 0.31  0.43  0.11  0.10    0.00  0.05 0.35
age       0.22    0.31 1.00  0.35  0.12  0.13    0.27  0.28 0.17
lbph      0.03    0.43 0.35  1.00 -0.09 -0.01    0.08  0.08 0.18
svi       0.54    0.11 0.12 -0.09  1.00  0.67    0.32  0.46 0.57
lcp       0.68    0.10 0.13 -0.01  0.67  1.00    0.51  0.63 0.55
gleason   0.43    0.00 0.27  0.08  0.32  0.51    1.00  0.75 0.37
pgg45     0.43    0.05 0.28  0.08  0.46  0.63    0.75  1.00 0.42
lpsa      0.73    0.35 0.17  0.18  0.57  0.55    0.37  0.42 1.00

As seen above, some variables are highly correlated. In particular, the correlation between gleason and pgg45 is equal to 0.75.

Fitting univariate regression models to predict age using gleason and pgg45 as predictors.

model_gleason <- lm(age ~ gleason, data = prostate)
model_pgg45 <- lm(age ~ pgg45, data = prostate)

Check which covariates have a significant efffect

summary(model_gleason)

Call:
lm(formula = age ~ gleason, data = prostate)

Residuals:
    Min      1Q  Median      3Q     Max 
-20.780  -3.552   1.448   4.220  13.448 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   45.146      6.918   6.525 3.29e-09 ***
gleason        2.772      1.019   2.721  0.00774 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 7.209 on 95 degrees of freedom
Multiple R-squared:  0.0723,	Adjusted R-squared:  0.06254 
F-statistic: 7.404 on 1 and 95 DF,  p-value: 0.007741

summary(model_pgg45)

Call:
lm(formula = age ~ pgg45, data = prostate)

Residuals:
     Min       1Q   Median       3Q      Max 
-21.0889  -3.4533   0.9111   4.4534  15.1822 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 62.08890    0.96758   64.17  < 2e-16 ***
pgg45        0.07289    0.02603    2.80  0.00619 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 7.193 on 95 degrees of freedom
Multiple R-squared:  0.07624,	Adjusted R-squared:  0.06651 
F-statistic:  7.84 on 1 and 95 DF,  p-value: 0.006189

Based on these results we conclude that both gleason and pgg45 have a statistically significant univariate effect (also referred to as a marginal effect) as predictors of age (5% significance level).

Fitting a multivariate regression model using both both gleason and pgg45 as predictors

model_multivar <- lm(age ~ gleason + pgg45, data = prostate)
summary(model_multivar)

Call:
lm(formula = age ~ gleason + pgg45, data = prostate)

Residuals:
    Min      1Q  Median      3Q     Max 
-20.927  -3.677   1.323   4.323  14.420 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 52.95548    9.74316   5.435  4.3e-07 ***
gleason      1.45363    1.54299   0.942    0.349    
pgg45        0.04490    0.03951   1.137    0.259    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 7.198 on 94 degrees of freedom
Multiple R-squared:  0.08488,	Adjusted R-squared:  0.06541 
F-statistic: 4.359 on 2 and 94 DF,  p-value: 0.01547

Although gleason and pgg45 have statistically significant univariate effects, this is no longer the case when both variables are simultaneously included as covariates in a multivariate regression model.

Including highly correlated variables such as gleason and pgg45 simultaneously the same regression model can lead to problems in fitting a regression model and interpreting its output. Although each variable appears to be associated with the response individually, the model cannot distinguish the contribution of each variable to the model. This can also increase the risk of over-fitting since the model may fit redundant variables to noise rather than true relationships.

To allow the information from variables to be included in the same model despite high levels of correlation, we can use dimensionality reduction methods to collapse multiple variables into a single new variable (we will explore this dataset further in the dimensionality reduction lesson). We can also use modifications to linear regression like regularisation, which we will discuss in the lesson on high-dimensional regression.

What statistical methods are used to analyse high-dimensional data?

We have discussed so far that high-dimensional data analysis can be challenging since variables are difficult to visualise, leading to challenges identifying relationships between variables and suitable response variables; we may have relatively few observations compared to features, leading to over-fitting; and features may be highly correlated, leading to challenges interpreting models. We therefore require alternative approaches to examine whether, for example, groups of observations show similar characteristics and whether these groups may relate to other features in the data (e.g. phenotype in genetics data).

In this course, we will cover four methods that help in dealing with high-dimensional data: (1) regression with numerous outcome variables, (2) regularised regression, (3) dimensionality reduction, and (4) clustering. Here are some examples of when each of these approaches may be used:

1. Regression with numerous outcomes refers to situations in which there are many variables of a similar kind (expression values for many genes, methylation levels for many sites in the genome) and when one is interested in assessing whether these variables are associated with a specific covariate of interest, such as experimental condition or age. In this case, multiple univariate regression models (one per each outcome, using the covariate of interest as predictor) could be fitted independently. In the context of high-dimensional molecular data, a typical example are differential gene expression analyses. We will explore this type of analysis in the Regression with many outcomes episode.

2. Regularisation (also known as regularised regression or penalised regression) is typically used to fit regression models when there is a single outcome variable or interest but the number of potential predictors is large, e.g. there are more predictors than observations. Regularisation can help to prevent overfitting and may be used to identify a small subset of predictors that are associated with the outcome of interest. For example, regularised regression has been often used when building epigenetic clocks, where methylation values across several thousands of genomic sites are used to predict chronological age. We will explore this in more detail in the Regularised regression episode.

3. Dimensionality reduction is commonly used on high-dimensional datasets for data exploration or as a preprocessing step prior to other downstream analyses. For instance, a low-dimensional visualisation of a gene expression dataset may be used to inform quality control steps (e.g. are there any anomalous samples?). This course contains two episodes that explore dimensionality reduction techniques: Principal component analysis and Factor analysis.

4. Clustering methods can be used to identify potential grouping patterns within a dataset. A popular example is the identification of distinct cell types through clustering cells with similar gene expression patterns. The K-means episode will explore a specific method to perform clustering analysis.

Using Bioconductor to access high-dimensional data in the biosciences

In this workshop, we will look at statistical methods that can be used to visualise and analyse high-dimensional biological data using packages available from Bioconductor, open source software for analysing high throughput genomic data. Bioconductor contains useful packages and example datasets as shown on the website https://www.bioconductor.org/.

Bioconductor packages can be installed and used in R using the BiocManager package. Let’s load the minfi package from Bioconductor (a package for analysing Illumina Infinium DNA methylation arrays).

library("minfi")

browseVignettes("minfi")

We can explore these packages by browsing the vignettes provided in Bioconductor. Bioconductor has various packages that can be used to load and examine datasets in R that have been made available in Bioconductor, usually along with an associated paper or package.

Next, we load the methylation dataset which represents data collected using Illumina Infinium methylation arrays which are used to examine methylation across the human genome. These data include information collected from the assay as well as associated metadata from individuals from whom samples were taken.

library("here")
library("ComplexHeatmap")

methylation <- readRDS(here("data/methylation.rds"))
head(colData(methylation))

DataFrame with 6 rows and 14 columns
                    Sample_Well Sample_Name    purity         Sex       Age
                    <character> <character> <integer> <character> <integer>
201868500150_R01C01         A07     PCA0612        94           M        39
201868500150_R03C01         C07   NKpan2510        95           M        49
201868500150_R05C01         E07      WB1148        95           M        20
201868500150_R07C01         G07       B0044        97           M        49
201868500150_R08C01         H07   NKpan1869        95           F        33
201868590193_R02C01         B03   NKpan1850        93           F        21
                    weight_kg  height_m       bmi    bmi_clas Ethnicity_wide
                    <numeric> <numeric> <numeric> <character>    <character>
201868500150_R01C01   88.4505    1.8542   25.7269  Overweight          Mixed
201868500150_R03C01   81.1930    1.6764   28.8911  Overweight  Indo-European
201868500150_R05C01   80.2858    1.7526   26.1381  Overweight  Indo-European
201868500150_R07C01   82.5538    1.7272   27.6727  Overweight  Indo-European
201868500150_R08C01   87.5433    1.7272   29.3452  Overweight  Indo-European
201868590193_R02C01   87.5433    1.6764   31.1507       Obese          Mixed
                       Ethnic_self      smoker       Array       Slide
                       <character> <character> <character>   <numeric>
201868500150_R01C01       Hispanic          No      R01C01 2.01869e+11
201868500150_R03C01      Caucasian          No      R03C01 2.01869e+11
201868500150_R05C01        Persian          No      R05C01 2.01869e+11
201868500150_R07C01      Caucasian          No      R07C01 2.01869e+11
201868500150_R08C01      Caucasian          No      R08C01 2.01869e+11
201868590193_R02C01 Finnish/Creole          No      R02C01 2.01869e+11

methyl_mat <- t(assay(methylation))
## calculate correlations between cells in matrix
cor_mat <- cor(methyl_mat)

cor_mat[1:10, 1:10] # print the top-left corner of the correlation matrix

The assay() function creates a matrix-like object where rows represent probes for genes and columns represent samples. We calculate correlations between features in the methylation dataset and examine the first 100 cells of this matrix. The size of the dataset makes it difficult to examine in full, a common challenge in analysing high-dimensional genomics data.

Further reading

• Buhlman, P. & van de Geer, S. (2011) Statistics for High-Dimensional Data. Springer, London.
• Buhlman, P., Kalisch, M. & Meier, L. (2014) High-dimensional statistics with a view toward applications in biology. Annual Review of Statistics and Its Application.
• Johnstone, I.M. & Titterington, D.M. (2009) Statistical challenges of high-dimensional data. Philosophical Transactions of the Royal Society A 367:4237-4253.
• Bioconductor ethylation array analysis vignette.
• The Introduction to Machine Learning with Python course covers additional methods that could be used to analyse high-dimensional data. See Introduction to machine learning, Tree models and Neural networks. Some related (an important!) content is also available in Responsible machine learning.

Other resources suggested by former students

• Josh Starmer’s youtube channel.

Key Points

• High-dimensional data are data in which the number of features, $p$, are close to or larger than the number of observations, $n$.

• These data are becoming more common in the biological sciences due to increases in data storage capabilities and computing power.

• Standard statistical methods, such as linear regression, run into difficulties when analysing high-dimensional data.

• In this workshop, we will explore statistical methods used for analysing high-dimensional data using datasets available on Bioconductor.

Regression with many outcomes

Overview

Teaching: 70 min
Exercises: 50 min

Questions

• How can we apply linear regression in a high-dimensional setting?

• How can we benefit from the fact that we have many outcomes?

• How can we control for the fact that we do many tests?

Objectives

• Perform and critically analyse high-dimensional regression.

• Understand methods for shrinkage of noise parameters in high-dimensional regression.

• Perform multiple testing adjustment.

DNA methylation data

For the following few episodes, we will be working with human DNA methylation data from flow-sorted blood samples, described in data. DNA methylation assays measure, for each of many sites in the genome, the proportion of DNA that carries a methyl mark (a chemical modification that does not alter the DNA sequence). In this case, the methylation data come in the form of a matrix of normalised methylation levels (M-values), where negative values correspond to unmethylated DNA and positive values correspond to methylated DNA. Along with this, we have a number of sample phenotypes (eg, age in years, BMI).

Let’s read in the data for this episode:

library("here")
library("minfi")
methylation <- readRDS(here("data/methylation.rds"))

Note: the code that we used to download these data from its source is available here

methylation is actually a special Bioconductor SummarizedExperiment object that summarises lots of different information about the data. These objects are very useful for storing all of the information about a dataset in a high-throughput context. The structure of SummarizedExperiment objects is described in the vignettes on Bioconductor. Here, we show how to extract the information for analysis.

We can extract

• the dimensions of the dataset using dim(). Importantly, in these objects and data structures for computational biology in R generally, observations are stored as columns and features (in this case, sites in the genome) are stored as rows. This is in contrast to usual tabular data, where features or variables are stored as columns and observations are stored as rows;
• assays, (normalised methylation levels here), using assay();
• sample-level information using colData().

dim(methylation)

[1] 5000   37

You can see in this output that this object has a dim() of $5000 \times 37$, meaning it has 5000 features and 37 observations. To extract the matrix of methylation M-values, we can use the assay() function.

methyl_mat <- assay(methylation)

The distribution of these M-values looks like this:

hist(methyl_mat, xlab = "M-value")

Methylation levels are generally bimodally distributed.

You can see that there are two peaks in this distribution, corresponding to features which are largely unmethylated and methylated, respectively.

Similarly, we can examine the colData(), which represents the sample-level metadata we have relating to these data. In this case, the metadata, phenotypes, and groupings in the colData look like this for the first 6 samples:

head(colData(methylation))

In this episode, we will focus on the association between age and methylation. The following heatmap summarises age and methylation levels available in the methylation dataset:

library("ComplexHeatmap")

age <- methylation$Age

# sort methylation values by age 
order <- order(age)
age_ord <- age[order]
methyl_mat_ord <- methyl_mat[, order]

# plot heatmap
Heatmap(methyl_mat_ord,
        cluster_columns = FALSE,
        show_row_names = FALSE,
        show_column_names = FALSE,
        name = "M-value",
        row_title = "Feature", 
        column_title =  "Sample", 
        top_annotation = columnAnnotation(age = age_ord))

Heatmap of methylation values across all features.

Challenge 1

Why can we not just fit many linear regression models relating every combination of feature (colData and assays) and draw conclusions by associating all variables with significant model p-values?

Solution

There are a number of problems that this kind of approach presents. For example:

1. If we perform 5000 tests for each of 14 variables, even if there were no true associations in the data, we’d be likely to observe some strong spurious associations that arise just from random noise.
2. We may not have a representative sample for each of these covariates. For example, we may have very small sample sizes for some ethnicities, leading to spurious findings.
3. Without a research question in mind when creating a model, it’s not clear how we can interpret each model, and rationalising the results after the fact can be dangerous; it’s easy to make up a “story” that isn’t grounded in anything but the fact that we have significant findings.

In general, it is scientifically interesting to explore two modelling problems using the three types of data:

1. Predicting methylation levels using age as a predictor. In this case, we would have 5000 outcomes (methylation levels across the genome) and a single covariate (age).

2. Predicting age using methylation levels as predictors. In this case, we would have a single outcome (age) which will be predicted using 5000 covariates (methylation levels across the genome).

The examples in this episode will focus on the first type of problem, whilst the next episode will focus on the second.

Measuring DNA Methylation

DNA methylation is an epigenetic modification of DNA. Generally, we are interested in the proportion of methylation at many sites or regions in the genome. DNA methylation microarrays, as we are using here, measure DNA methylation using two-channel microarrays, where one channel captures signal from methylated DNA and the other captures unmethylated signal. These data can be summarised as “Beta values” ($\beta$ values), which is the ratio of the methylated signal to the total signal (methylated plus unmethylated). The $\beta$ value for site $i$ is calculated as

\[\beta_i = \frac{ m_i } { u_{i} + m_{i} }\]

where $m_i$ is the methylated signal for site $i$ and $u_i$ is the unmethylated signal for site $i$. $\beta$ values take on a value in the range $[0, 1]$, with 0 representing a completely unmethylated site and 1 representing a completely methylated site.

The M-values we use here are the $\log_2$ ratio of methylated versus unmethylated signal:

\[M_i = \log_2\left(\frac{m_i}{u_i}\right)\]

M-values are not bounded to an interval as Beta values are, and therefore can be easier to work with in statistical models.

Regression with many outcomes

In high-throughput studies, it is common to have one or more phenotypes or groupings that we want to relate to features of interest (eg, gene expression, DNA methylation levels). In general, we want to identify differences in the features of interest that are related to a phenotype or grouping of our samples. Identifying features of interest that vary along with phenotypes or groupings can allow us to understand how phenotypes arise or manifest. Analysis of this type is sometimes referred to using the term differential analysis.

For example, we might want to identify genes that are expressed at a higher level in mutant mice relative to wild-type mice to understand the effect of a mutation on cellular phenotypes. Alternatively, we might have samples from a set of patients, and wish to identify epigenetic features that are different in young patients relative to old patients, to help us understand how ageing manifests.

Using linear regression, it is possible to identify differences like these. However, high-dimensional data like the ones we’re working with require some special considerations. A first consideration, as we saw above, is that there are far too many features to fit each one-by-one as we might do when analysing low-dimensional datasets (for example using lm() on each feature and checking the linear model assumptions). A second consideration is that statistical approaches may behave slightly differently when applied to very high-dimensional data, compared to low-dimensional data. A third consideration is the speed at which we can actually compute statistics for data this large – methods optimised for low-dimensional data may be very slow when applied to high-dimensional data.

Ideally when performing regression, we want to identify cases like this, where there is a clear association, and we probably “don’t need” statistics:

A scatter plot of age and a feature of interest.

or equivalently for a discrete covariate:

A scatter plot of a grouping and a feature of interest.

However, often due to small differences and small sample sizes, the problem is more difficult:

A scatter plot of a grouping and a feature of interest.

And, of course, we often have an awful lot of features and need to prioritise a subset of them! We need a rigorous way to prioritise genes for further analysis.

Fitting a linear model

So, in the data we have read in, we have a matrix of methylation values $X$ and a vector of ages, $y$. One way to model this is to see if we can use age to predict the expected (average) methylation value for sample $j$ at a given locus $i$, which we can write as $X_{ij}$. We can write that model as:

[\mathbf{E}(X_{ij}) = \beta_0 + \beta_1 \text{Age}_j]

where $\text{Age}_j$ is the age of sample $j$. In this model, $\beta_1$ represents the unit change in mean methylation level for each unit (year) change in age. For a specific CpG, we can fit this model and get more information from the model object. For illustration purposes, here we arbitrarily select the first CpG in the methyl_mat matrix (the one on its first row).

age <- methylation$Age
# methyl_mat[1, ] indicates that the 1st CpG will be used as outcome variable
lm_age_methyl1 <- lm(methyl_mat[1, ] ~ age)
lm_age_methyl1

Call:
lm(formula = methyl_mat[1, ] ~ age)

Coefficients:
(Intercept)          age  
   0.902334     0.008911  

We now have estimates for the expected methylation level when age equals 0 (the intercept) and the change in methylation level for a unit change in age (the slope). We could plot this linear model:

plot(age, methyl_mat[1, ], xlab = "Age", ylab = "Methylation level", pch = 16)
abline(lm_age_methyl1)

A scatter plot of age versus the methylation level for an arbitrarily selected CpG side (the one stored as the first column of methyl_mat). Each dot represents an individual. The black line represents the estimated linear model.

For this feature, we can see that there is no strong relationship between methylation and age. We could try to repeat this for every feature in our dataset; however, we have a lot of features! We need an approach that allows us to assess associations between all of these features and our outcome while addressing the three considerations we outlined previously. Before we introduce this approach, let’s go into detail about how we generally check whether the results of a linear model are statistically significant.

Hypothesis testing in linear regression

Using the linear model we defined above, we can ask questions based on the estimated value for the regression coefficients. For example, do individuals with different age have different methylation values for a given CpG? We usually do this via hypothesis testing. This framework compares the results that we observed (here, estimated linear model coefficients) to the results you would expect under a null hypothesis associated to our question. In the example above, a suitable null hypothesis would test whether the regression coefficient associated to age ($\beta_1$) is equal to zero or not. If $\beta_1$ is equal to zero, the linear model indicates that there is no linear relationship between age and the methylation level for the CpG (remember: as its name suggests, linear regression can only be used to model linear relationships between predictors and outcomes!). In other words, the answer to our question would be: no!

The output of a linear model typically returns the results associated with the null hypothesis described above (this may not always be the most realistic or useful null hypothesis, but it is the one we have by default!). To be more specific, the test compares our observed results with a set of hypothetical counter-examples of what we would expect to observe if we repeated the same experiment and analysis over and over again under the null hypothesis.

For this linear model, we can use tidy() from the broom package to extract detailed information about the coefficients and the associated hypothesis tests in this model:

library("broom")
tidy(lm_age_methyl1)

# A tibble: 2 × 5
  term        estimate std.error statistic p.value
  <chr>          <dbl>     <dbl>     <dbl>   <dbl>
1 (Intercept)  0.902      0.344      2.62   0.0129
2 age          0.00891    0.0100     0.888  0.381 

The standard errors (std.error) represent the statistical uncertainty in our regression coefficient estimates (often referred to as effect size). The test statistics and p-values (statistic and p.value) represent measures of how (un)likely it would be to observe results like this under the “null hypothesis”. For coefficient $k$ in a linear model (in our case, it would be the slope), the test statistic calculated in statistic above is a t-statistic given by:

[t_{k} = \frac{\hat{\beta}{k}}{SE\left(\hat{\beta}{k}\right)}]

$SE\left(\hat{\beta}_{k}\right)$ measures the uncertainty we have in our effect size estimate. Knowing what distribution these t-statistics follow under the null hypothesis allows us to determine how unlikely it would be for us to observe what we have under those circumstances, if we repeated the experiment and analysis over and over again. To demonstrate how the t-statistics are calculated, we can compute them “by hand”:

table_age_methyl1 <- tidy(lm_age_methyl1)
tvals <- table_age_methyl1$estimate / table_age_methyl1$std.error
all.equal(tvals, table_age_methyl1$statistic)

[1] TRUE

We can see that the t-statistic is just the ratio between the coefficient estimate and the standard error.

Calculating the p-values is a bit more tricky. Specifically, it is the proportion of the distribution of the test statistic under the null hypothesis that is as extreme or more extreme than the observed value of the test statistic. This is easy to observe visually, by plotting the theoretical distribution of the test statistic under the null hypothesis (see next call-out box for more details about it):

The p-value for a regression coefficient represents how often it'd be observed under the null.

The red-ish shaded region represents the portion of the distribution of the test statistic under the null hypothesis that is equal or greater to the value we observe for the intercept term. As our null hypothesis relates to a 2-tailed test (as the null hypothesis states that the regression coefficient is equal to zero, we would reject it if the regression coefficient is substantially larger or smaller than zero), the p-value for the test is twice the value of the shaded region. In this case, the shaded region is small relative to the total area of the null distribution; therefore, the p-value is small ($p=0.013$). The blue-ish shaded region represents the same measure for the slope term; this is larger, relative to the total area of the distribution, therefore the p-value is larger than the one for the intercept term ($p=0.381$). The p-value is a function of the test statistic: the ratio between the effect size we’re estimating and the uncertainty we have in that effect. A large effect with large uncertainty may not lead to a small p-value, and a small effect with small uncertainty may lead to a small p-value.

Calculating p-values from a linear model

Manually calculating the p-value for a linear model is a little bit more complex than calculating the t-statistic. The intuition posted above is definitely sufficient for most cases, but for completeness, here is how we do it:

Since the statistic in a linear model is a t-statistic, it follows a student t distribution under the null hypothesis, with degrees of freedom (a parameter of the student t-distribution) given by the number of observations minus the number of coefficients fitted, in this case $37 - 2 = 35$. We want to know what portion of the distribution function of the test statistic is as extreme as, or more extreme than, the value we observed. The function pt()(similar topnorm(), etc) can give us this information.

Since we’re not sure if the coefficient will be larger or smaller than zero, we want to do a 2-tailed test. Therefore we take the absolute value of the t-statistic, and look at the upper rather than lower tail. In the figure above the shaded areas are only looking at “half” of the t-distribution (which is symmetric around zero), therefore we multiply the shaded area by 2 in order to calculate the p-value.

Combining all of this gives us:

pvals <- 2 * pt(abs(tvals), df = lm_age_methyl1$df, lower.tail = FALSE)
all.equal(table_age_methyl1$p.value, pvals)

[1] TRUE

Challenge 2

In the model we fitted, the estimate for the intercept is 0.902 and its associated p-value is 0.0129. What does this mean?

Solution

The first coefficient in a linear model like this is the intercept, which measures the mean of the outcome (in this case, the methylation value for the first CpG) when age is zero. In this case, the intercept estimate is 0.902. However, this is not a particularly noteworthy finding as we do not have any observations with age zero (nor even any with age < 20!).

The reported p-value is associated to the following null hypothesis: the intercept ($\beta_0$ above) is equal to zero. Using the usual significance threshold of 0.05, we reject the null hypothesis as the p-value is smaller than 0.05. However, it is not really interesting if this intercept is zero or not, since we probably do not care what the methylation level is when age is zero. In fact, this question does not even make much sense! In this example, we are more interested in the regression coefficient associated to age, as that can tell us whether there is a linear relationship between age and methylation for the CpG.

Fitting a lot of linear models

In the linear model above, we are generally interested in the second regression coefficient (often referred to as slope) which measures the linear relationship between age and methylation levels. For the first CpG, here is its estimate:

coef_age_methyl1 <- tidy(lm_age_methyl1)[2, ]
coef_age_methyl1

# A tibble: 1 × 5
  term  estimate std.error statistic p.value
  <chr>    <dbl>     <dbl>     <dbl>   <dbl>
1 age    0.00891    0.0100     0.888   0.381

In this case, the p-value is equal to 0.381 and therefore we cannot reject the null hypothesis: there is no statistical evidence to suggest that the regression coefficient associated to age is not equal to zero.

Now, we could do this for every feature (CpG) in the dataset and rank the results based on their test statistic or associated p-value. However, fitting models in this way to 5000 features is not very computationally efficient, and it would also be laborious to do programmatically. There are ways to get around this, but first let us talk about what exactly we are doing when we look at significance tests in this context.

Sharing information across outcome variables

We are going to introduce an idea that allows us to take advantage of the fact that we carry out many tests at once on structured data. We can leverage this fact to share information between model parameters. The insight that we use to perform information pooling or sharing is derived from our knowledge about the structure of the data. For example, in a high-throughput experiment like a DNA methylation assay, we know that all of the features were measured simultaneously, using the same technique. This means that generally, we expect the base-level variability for each feature to be broadly similar.

This can enable us to get a better estimate of the uncertainty of model parameters than we could get if we consider each feature in isolation. So, to share information between features allows us to get more robust estimators. Remember that the t-statistic for coefficient $\beta_k$ in a linear model is the ratio between the coefficient estimate and its standard error:

[t_{k} = \frac{\hat{\beta}{k}}{SE\left(\hat{\beta}{k}\right)}]

It is clear that large effect sizes will likely lead to small p-values, as long as the standard error for the coefficent is not large. However, the standard error is affected by the amount of noise, as we saw earlier. If we have a small number of observations, it is common for the noise for some features to be extremely small simply by chance. This, in turn, causes small p-values for these features, which may give us unwarranted confidence in the level of certainty we have in the results (false positives).

There are many statistical methods in genomics that use this type of approach to get better estimates by pooling information between features that were measured simultaneously using the same techniques. Here we will focus on the package limma, which is an established software package used to fit linear models, originally for the gene expression micro-arrays that were common in the 2000s, but which is still in use in RNAseq experiments, among others. The authors of limma made some assumptions about the distributions that these follow, and pool information across genes to get a better estimate of the uncertainty in effect size estimates. It uses the idea that noise levels should be similar between features to moderate the estimates of the test statistic by shrinking the estimates of standard errors towards a common value. This results in a moderated t-statistic.

The process of running a model in limma is somewhat different to what you may have seen when running linear models. Here, we define a model matrix or design matrix, which is a way of representing the coefficients that should be fit in each linear model. These are used in similar ways in many different modelling libraries.

What is a model matrix?

R fits a regression model by choosing the vector of regression coefficients that minimises the differences between outcome values and predicted values using the covariates (or predictor variables). To get predicted values, we multiply the matrix of predictors by the coefficients. The latter matrix is called the model matrix (or design matrix). It has one row for each observation and one column for each predictor, plus one additional column of ones (the intercept column). Many R libraries contruct the model matrix behind the scenes, but limma does not. Usually, the model matrix can be extracted from a model fit using the function model.matrix(). Here is an example of a model matrix for the methylation model:

design_age <- model.matrix(lm_age_methyl1) # model matrix
head(design_age)

                    (Intercept) age
201868500150_R01C01           1  39
201868500150_R03C01           1  49
201868500150_R05C01           1  20
201868500150_R07C01           1  49
201868500150_R08C01           1  33
201868590193_R02C01           1  21

dim(design_age)

[1] 37  2

As you can see, the model matrix has the same number of rows as our methylation data has samples. It also has two columns - one for the intercept (similar to the linear model we fit above) and one for age. This happens “under the hood” when fitting a linear model with lm(), but here we have to specify it directly. The limma user manual has more detail on how to make model matrices for different types of experimental design, but here we are going to stick with this simple two-variable case.

We pass our methylation data to lmFit(), specifying the model matrix. Internally, this function runs lm() on each row of the data in an efficient way. The function eBayes(), when applied to the output of lmFit(), performs the pooled estimation of standard errors that results in the moderated t-statistics and resulting p-values.

library("limma")
design_age <- model.matrix(lm_age_methyl1) # model matrix
fit_age <- lmFit(methyl_mat, design = design_age)
fit_age <- eBayes(fit_age)

To obtain the results of the linear models, we can use the topTable() function. By default, this returns results for the first coefficient in the model. As we saw above when using lm(), and when we defined design_age above, the first coefficient relates to the intercept term, which we are not particularly interested in here; therefore we specify coef = 2. Further, topTable() by default only returns the top 10 results. To see all of the results in the data, we specify number = nrow(fit_age) to ensure that it returns a row for every row of the input matrix.

toptab_age <- topTable(fit_age, coef = 2, number = nrow(fit_age))
head(toptab_age)

                 logFC    AveExpr         t      P.Value   adj.P.Val        B
cg08446924 -0.02571353 -0.4185868 -6.039068 5.595675e-07 0.002797837 5.131574
cg06493994  0.01550941 -2.1057877  5.593988 2.239813e-06 0.005599533 3.747986
cg17661642  0.02266668 -2.0527722  5.358739 4.658336e-06 0.006048733 3.019698
cg05168977  0.02276336 -2.2918472  5.346500 4.838987e-06 0.006048733 2.981904
cg24549277  0.01975577 -1.7466088  4.939242 1.708355e-05 0.011508818 1.731821
cg04436528 -0.01943612  0.7033503 -4.917179 1.828563e-05 0.011508818 1.664608

The output of topTable includes the coefficient, here termed a log fold change logFC, the average level (aveExpr), the t-statistic t, the p-value (P.Value), and the adjusted p-value (adj.P.Val). We’ll cover what an adjusted p-value is very shortly. The table also includes B, which represents the log-odds that a feature is signficantly different, which we won’t cover here, but which will generally be a 1-1 transformation of the p-value. The coefficient estimates here are termed logFC for legacy reasons relating to how microarray experiments were traditionally performed. There are more details on this topic in many places, for example this tutorial by Kasper D. Hansen

Now we have estimates of effect sizes and p-values for the association between methylation level at each locus and age for our 37 samples. It’s useful to create a plot of effect size estimates (model coefficients) against p-values for each of these linear models, to visualise the magnitude of effects and the statistical significance of each. These plots are often called “volcano plots”, because they resemble an eruption.

plot(toptab_age$logFC, -log10(toptab_age$P.Value),
    xlab = "Effect size", ylab = bquote(-log[10](p-value)),
    pch = 19
)

Plotting p-values against effect sizes using limma; the results are similar to a standard linear model.

In this figure, every point represents a feature of interest. The x-axis represents the effect size observed for that feature in a linear model, while the y-axis is the $-\log_{10}(\text{p-value})$, where larger values indicate increasing statistical evidence of a non-zero effect size. A positive effect size represents increasing methylation with increasing age, and a negative effect size represents decreasing methylation with increasing age. Points higher on the y-axis represent features for which we think the results we observed would be very unlikely under the null hypothesis.

Since we want to identify features that have different methylation levels in different age groups, in an ideal case there would be clear separation between “null” and “non-null” features. However, usually we observe results as we do here: there is a continuum of effect sizes and p-values, with no clear separation between these two classes of features. While statistical methods exist to derive insights from continuous measures like these, it is often convenient to obtain a list of features which we are confident have non-zero effect sizes. This is made more difficult by the number of tests we perform.

Challenge 3

The effect size estimates are very small, and yet many of the p-values are well below a usual significance level of p < 0.05. Why is this?

Solution

Because age has a much larger range than methylation levels, the unit change in methylation level even for a strong relationship is very small!

As we mentioned, the p-value is a function of both the effect size estimate and the uncertainty (standard error) of that estimate. Because the uncertainty in our estimates is much smaller than the estimates themselves, the p-values are also small.

If we predicted age using methylation level, it is likely we would see much larger coefficients, though broadly similar p-values!

It is worthwhile considering what exactly the effect of the moderation or information sharing that limma performs has on our results. To do this, let us compare the effect sizes estimates and p-values from the two approaches.

Plot of effect sizes using limma vs. those using lm.

These are exactly identical! This is because limma does not perform any sharing of information when estimating effect sizes. This is in contrast to similar packages that apply shrinkage to the effect size estimates, like DESeq2. These often use information sharing to shrink or moderate the effect size estimates, in the case of DESeq2 by again sharing information between features about sample-to-sample variability. In contrast, let us look at the p-values from limma and R’s built-in lm() function:

Plot of p-values using limma vs. those using lm.

We can see that for the vast majority of features, the results are broadly similar. There seems to be a minor general tendency for limma to produce smaller p-values, but for several features, the p-values from limma are considerably larger than the p-values from lm(). This is because the information sharing tends to shrink large standard error estimates downwards and small estimates upwards. When the degree of statistical significance is due to an abnormally small standard error rather than a large effect, this effect results in this prominent reduction in statistical significance, which has been shown to perform well in case studies. The degree of shrinkage generally depends on the amount of pooled information and the strength of the evidence independent of pooling. For example, with very few samples and many features, information sharing has a larger effect, because there are a lot of genes that can be used to provide pooled estimates, and the evidence from the data that this is weighed against is relatively sparse. In contrast, when there are many samples and few features, there is not much opportunity to generate pooled estimates, and the evidence of the data can easily outweigh the pooling.

Shrinkage methods like these ones can be complex to implement and understand, but it is useful to develop an intuition about why these approaches may be more precise and sensitive than the naive approach of fitting a model to each feature separately.

Challenge 4

1. Try to run the same kind of linear model with smoking status as covariate instead of age, and making a volcano plot. Note: smoking status is stored as methylation$smoker.
2. We saw in the example in the lesson that this information sharing can lead to larger p-values. Why might this be preferable?

Solution

1. The following code runs the same type of model with smoking status:

   design_smoke <- model.matrix(~methylation$smoker)
   fit_smoke <- lmFit(methyl_mat, design = design_smoke)
   fit_smoke <- eBayes(fit_smoke)
   toptab_smoke <- topTable(fit_smoke, coef = 2, number = nrow(fit_smoke))
   plot(toptab_smoke$logFC, -log10(toptab_smoke$P.Value),
       xlab = "Effect size", ylab = bquote(-log[10](p)),
       pch = 19
   )
   

   

   A plot of significance against effect size for a regression of smoking against methylation.

2. Being a bit more conservative when identifying features can help to avoid false discoveries. Furthermore, when rejecting the null hypothesis is based more on a small standard error resulting from abnormally low levels of variability for a given feature, we might want to be a bit more conservative in our expectations.

Shrinkage

Shrinkage is an intuitive term for an effect of information sharing, and is something observed in a broad range of statistical models. Often, shrinkage is induced by a multilevel modelling approach or by Bayesian methods.

The general idea is that these models incorporate information about the structure of the data into account when fitting the parameters. We can share information between features because of our knowledge about the data structure; this generally requires careful consideration about how the data were generated and the relationships within.

An example people often use is estimating the effect of attendance on grades in several schools. We can assume that this effect is similar in different schools (but maybe not identical), so we can share information about the effect size between schools and shrink our estimates towards a common value.

For example in DESeq2, the authors used the observation that genes with similar expression counts in RNAseq data have similar dispersion, and a better estimate of these dispersion parameters makes estimates of fold changes much more stable. Similarly, in limma the authors made the assumption that in the absence of biological effects, we can often expect the technical variation in the measurement of the expression of each of the genes to be broadly similar. Again, better estimates of variability allow us to prioritise genes in a more reliable way.

There are many good resources to learn about this type of approach, including:

• a blog post by TJ Mahr
• a book by David Robinson
• a (relatively technical) book by Gelman and Hill

The problem of multiple tests

With such a large number of features, it would be useful to decide which features are “interesting” or “significant” for further study. However, if we were to apply a normal significance threshold of 0.05, it would be likely we end up with a lot of false positives. This is because a p-value threshold like this represents a $\frac{1}{20}$ chance that we observe results as extreme or more extreme under the null hypothesis (that there is no assocation between age and methylation level). If we carry out many more than 20 such tests, we can expect to see situations where, despite the null hypothesis being true, we observe observe signifiant p-values due to random chance. To demonstrate this, it is useful to see what happens if we permute (scramble) the age values and run the same test again:

set.seed(123) 
age_perm <- age[sample(ncol(methyl_mat), ncol(methyl_mat))]
design_age_perm <- model.matrix(~age_perm)

fit_age_perm <- lmFit(methyl_mat, design = design_age_perm)
fit_age_perm <- eBayes(fit_age_perm)
toptab_age_perm <- topTable(fit_age_perm, coef = 2, number = nrow(fit_age_perm))

plot(toptab_age_perm$logFC, -log10(toptab_age_perm$P.Value),
    xlab = "Effect size", ylab = bquote(-log[10](p)),
    pch = 19
)
abline(h = -log10(0.05), lty = "dashed", col = "red")

Plotting p-values against effect sizes for a randomised outcome shows we still observe 'significant' results.

Since we have generated a random sequence of ages, we have no reason to suspect that there is a true association between methylation levels and this sequence of random numbers. However, you can see that the p-value for many features is still lower than a traditional significance level of $p=0.05$. In fact, here 226 features are significant at p < 0.05. If we were to use this fixed threshold in a real experiment, it is likely that we would identify many features as associated with age, when the results we are observing are simply due to chance.

Challenge 5

1. If we run 5000 tests, even if there are no true differences, how many of them (on average) will be statistically significant at a threshold of $p < 0.05$?
2. Why would we want to be conservative in labelling features as significantly different? By conservative, we mean to err towards labelling true differences as “not significant” rather than vice versa.
3. How could we account for a varying number of tests to ensure “significant” changes are truly different?

Solution

1. By default we expect $5000 \times 0.05 = 250$ features to be statistically significant under the null hypothesis, because p-values should always be uniformly distributed under the null hypothesis.
2. Features that we label as “significantly different” will often be reported in manuscripts. We may also spend time and money investigating them further, computationally or in the lab. Therefore, spurious results have a real cost for ourselves and for others.
3. One approach to controlling for the number of tests is to divide our significance threshold by the number of tests performed. This is termed “Bonferroni correction” and we’ll discuss this further now.

Adjusting for multiple tests

When performing many statistical tests to categorise features, we are effectively classifying features as “non-significant” or “significant”, that latter meaning those for which we reject the null hypothesis. We also generally hope that there is a subset of features for which the null hypothesis is truly false, as well as many for which the null truly does hold. We hope that for all features for which the null hypothesis is true, we accept it, and for all features for which the null hypothesis is not true, we reject it. As we showed in the example with permuted age, with a large number of tests it is inevitable that we will get some of these wrong.

We can think of these features as being “truly different” or “not truly different”¹. Using this idea, we can see that each categorisation we make falls into four categories:

If the null hypothesis was true for every feature, then as we perform more and more tests we’d tend to correctly categorise most results as negative. However, since p-values are uniformly distributed under the null, at a significance level of 5%, 5% of all results will be “significant” even though we would expect to see these results, given the null hypothesis is true, simply by chance. These would fall under the label “false positives” in the table above, and are also termed “false discoveries.”

There are two common ways of controlling these false discoveries. The first is to say, when we’re doing $n$ tests, that we want to have the same certainty of making one false discovery with $n$ tests as we have if we’re only doing one test. This is “Bonferroni” correction,² which divides the significance level by the number of tests performed, $n$. Equivalently, we can use the non-transformed p-value threshold but multiply our p-values by the number of tests. This is often very conservative, especially with a lot of features!

p_raw <- toptab_age$P.Value
p_fwer <- p.adjust(p_raw, method = "bonferroni")
plot(p_raw, p_fwer, pch = 16, log="xy")
abline(0:1, lty = "dashed")
abline(v = 0.05, lty = "dashed", col = "red")
abline(h = 0.05, lty = "dashed", col = "red")

Bonferroni correction often produces very large p-values, especially with low sample sizes.

You can see that the p-values are exactly one for the vast majority of tests we performed! This is not ideal sometimes, because unfortunately we usually don’t have very large sample sizes in health sciences.

The second main way of controlling for multiple tests is to control the false discovery rate (FDR).³ This is the proportion of false positives, or false discoveries, we’d expect to get each time if we repeated the experiment over and over.

1. Rank the p-values
2. Assign each a rank (1 is smallest)
3. Calculate the critical value \(q = \left(\frac{i}{m}\right)Q\), where $i$ is rank, $m$ is the number of tests, and $Q$ is the false discovery rate we want to target.⁴
4. Find the largest p-value less than the critical value. All smaller than this are significant.

Challenge 6

1. At a significance level of 0.05, with 100 tests performed, what is the Bonferroni significance threshold?
2. In a gene expression experiment, after FDR correction with an FDR-adjusted p-value threshold of 0.05, we observe 500 significant genes. What proportion of these genes are truly different?
3. Try running FDR correction on the p_raw vector. Hint: check help("p.adjust") to see what the method is called.
   Compare these values to the raw p-values and the Bonferroni p-values.

Solution

1. The Bonferroni threshold for this significance threshold is \(\frac{0.05}{100} = 0.0005\)

2. We can’t say what proportion of these genes are truly different. However, if we repeated this experiment and statistical test over and over, on average 5% of the results from each run would be false discoveries.

3. The following code runs FDR correction and compares it to non-corrected values and to Bonferroni:

   p_fdr <- p.adjust(p_raw, method = "BH")
   plot(p_raw, p_fdr, pch = 16, log="xy")
   abline(0:1, lty = "dashed")
   abline(v = 0.05, lty = "dashed", col = "red")
   abline(h = 0.05, lty = "dashed", col = "red")
   

   

   Benjamini-Hochberg correction is less conservative than Bonferroni

   plot(p_fwer, p_fdr, pch = 16, log="xy")
   abline(0:1, lty = "dashed")
   abline(v = 0.05, lty = "dashed", col = "red")
   abline(h = 0.05, lty = "dashed", col = "red")
   

   

   plot of chunk plot-fdr-fwer

Feature selection

In this episode, we have focussed on regression in a setting where there are more features than observations. This approach is relevant if we are interested in the association of each feature with some outcome or if we want to screen for features that have a strong association with an outcome. If, however, we are interested in predicting an outcome or if we want to know which features explain the variation in the outcome, we may want to restrict ourselves to a subset of relevant features. One way of doing this is called regularisation, and this is the topic of the next episode. An alternative is called feature selection. This is covered in the subsequent (optional) episode.

Further reading

• limma tutorial by Kasper D. Hansen
• limma user manual.
• The VariancePartition package has similar functionality to limma but allows the inclusion of random effects.

Footnotes

Key Points

• Performing linear regression in a high-dimensional setting requires us to perform hypothesis testing in a way that low-dimensional regression may not.

• Sharing information between features can increase power and reduce false positives.

• When running a lot of null hypothesis tests for high-dimensional data, multiple testing correction allows retain power and avoid making costly false discoveries.

• Multiple testing methods can be more conservative or more liberal, depending on our goals.

Regularised regression

Overview

Teaching: 110 min
Exercises: 60 min

Questions

• What is regularisation?

• How does regularisation work?

• How can we select the level of regularisation for a model?

Objectives

• Understand the benefits of regularised models.

• Understand how different types of regularisation work.

• Apply and critically analyse regularised regression models.

Introduction

This episode is about regularisation, also called regularised regression or penalised regression. This approach can be used for prediction and for feature selection and it is particularly useful when dealing with high-dimensional data.

One reason that we need special statistical tools for high-dimensional data is that standard linear models cannot handle high-dimensional data sets – one cannot fit a linear model where there are more features (predictor variables) than there are observations (data points). In the previous lesson, we dealt with this problem by fitting individual models for each feature and sharing information among these models. Now we will take a look at an alternative approach that can be used to fit models with more features than observations by stabilising coefficient estimates. This approach is called regularisation. Compared to many other methods, regularisation is also often very fast and can therefore be extremely useful in practice.

First, let us check out what happens if we try to fit a linear model to high-dimensional data! We start by reading in the methylation data from the last lesson:

library("here")
library("minfi")
methylation <- readRDS(here("data/methylation.rds"))

## here, we transpose the matrix to have features as rows and samples as columns
methyl_mat <- t(assay(methylation))
age <- methylation$Age

Then, we try to fit a model with outcome age and all 5,000 features in this dataset as predictors (average methylation levels, M-values, across different sites in the genome).

# by using methyl_mat in the formula below, R will run a multivariate regression
# model in which each of the columns in methyl_mat is used as a predictor. 
fit <- lm(age ~ methyl_mat)
summary(fit)

Call:
lm(formula = age ~ methyl_mat)

Residuals:
ALL 37 residuals are 0: no residual degrees of freedom!

Coefficients: (4964 not defined because of singularities)
                          Estimate Std. Error t value Pr(>|t|)
(Intercept)               2640.474        NaN     NaN      NaN
methyl_matcg00075967      -108.216        NaN     NaN      NaN
methyl_matcg00374717      -139.637        NaN     NaN      NaN
methyl_matcg00864867        33.102        NaN     NaN      NaN
methyl_matcg00945507        72.250        NaN     NaN      NaN
 [ reached getOption("max.print") -- omitted 4996 rows ]

Residual standard error: NaN on 0 degrees of freedom
Multiple R-squared:      1,	Adjusted R-squared:    NaN 
F-statistic:   NaN on 36 and 0 DF,  p-value: NA

You can see that we’re able to get some effect size estimates, but they seem very high! The summary also says that we were unable to estimate effect sizes for 4,964 features because of “singularities”. We clarify what singularities are in the note below but this means that R couldn’t find a way to perform the calculations necessary to fit the model. Large effect sizes and singularities are common when naively fitting linear regression models with a large number of features (i.e., to high-dimensional data), often since the model cannot distinguish between the effects of many, correlated features or when we have more features than observations.

Singularities

The message that lm() produced is not necessarily the most intuitive. What are “singularities” and why are they an issue? A singular matrix is one that cannot be inverted. R uses inverse operations to fit linear models (find the coefficients) using:

\[(X^TX)^{-1}X^Ty,\]

where $X$ is a matrix of predictor features and $y$ is the outcome vector. Thus, if the matrix $X^TX$ cannot be inverted to give $(X^TX)^{-1}$, R cannot fit the model and returns the error that there are singularities.

Why might R be unable to calculate $(X^TX)^{-1}$ and return the error that there are singularities? Well, when the determinant of the matrix is zero, we are unable to find its inverse. The determinant of the matrix is zero when there are more features than observations or often when the features are highly correlated.

xtx <- t(methyl_mat) %*% methyl_mat
det(xtx)

[1] 0

Correlated features – common in high-dimensional data

In high-dimensional datasets, there are often multiple features that contain redundant information (correlated features). If we visualise the level of correlation between sites in the methylation dataset, we can see that many of the features represent the same information - there are many off-diagonal cells, which are deep red or blue. For example, the following heatmap visualises the correlations for the first 500 features in the methylation dataset (we selected 500 features only as it can be hard to visualise patterns when there are too many features!).

library("ComplexHeatmap")
small <- methyl_mat[, 1:500]
cor_mat <- cor(small)
Heatmap(cor_mat,
    column_title = "Feature-feature correlation in methylation data",
    name = "Pearson correlation",
    show_row_dend = FALSE, show_column_dend = FALSE,
    show_row_names = FALSE, show_column_names = FALSE
)

Heatmap of pairwise feature-feature correlations between CpG sites in DNA methylation data

Challenge 1

Consider or discuss in groups:

1. Why would we observe correlated features in high-dimensional biological data?
2. Why might correlated features be a problem when fitting linear models?
3. What issue might correlated features present when selecting features to include in a model one at a time?

Solution

1. Many of the features in biological data represent very similar information biologically. For example, sets of genes that form complexes are often expressed in very similar quantities. Similarly, methylation levels at nearby sites are often very highly correlated.
2. Correlated features can make inference unstable or even impossible mathematically.
3. When we are selecting features one at a time we want to pick the most predictive feature each time. When a lot of features are very similar but encode slightly different information, which of the correlated features we select to include can have a huge impact on the later stages of model selection!

Regularisation can help us to deal with correlated features, as well as effectively reduce the number of features in our model. Before we describe regularisation, let’s recap what’s going on when we fit a linear model.

Coefficient estimates of a linear model

When we fit a linear model, we’re finding the line through our data that minimises the sum of the squared residuals. We can think of that as finding the slope and intercept that minimises the square of the length of the dashed lines. In this case, the red line in the left panel is the line that accomplishes this objective, and the red dot in the right panel is the point that represents this line in terms of its slope and intercept among many different possible models, where the background colour represents how well different combinations of slope and intercept accomplish this objective.

Illustrative example demonstrated how regression coefficients are inferred under a linear model framework.

Mathematically, we can write the sum of squared residuals as

[\sum_{i=1}^N ( y_i-x’_i\beta)^2]

where $\beta$ is a vector of (unknown) covariate effects which we want to learn by fitting a regression model: the $j$-th element of $\beta$, which we denote as $\beta_j$ quantifies the effect of the $j$-th covariate. For each individual $i$, $x_i$ is a vector of $j$ covariate values and $y_i$ is the true observed value for the outcome. The notation $x’_i\beta$ indicates matrix multiplication. In this case, the result is equivalent to multiplying each element of $x_i$ by its corresponding element in $\beta$ and then calculating the sum across all of those values. The result of this product (often denoted by $\hat{y}_i$) is the predicted value of the outcome generated by the model. As such, $y_i-x’_i\beta$ can be interpreted as the prediction error, also referred to as model residual. To quantify the total error across all individuals, we sum the square residuals $( y_i-x’_i\beta)^2$ across all the individuals in our data.

Finding the value of $\beta$ that minimises the sum above is the line of best fit through our data when considering this goal of minimising the sum of squared error. However, it is not the only possible line we could use! For example, we might want to err on the side of caution when estimating effect sizes (coefficients). That is, we might want to avoid estimating very large effect sizes.

Challenge 2

Discuss in groups:

1. What are we minimising when we fit a linear model?
2. Why are we minimising this objective? What assumptions are we making about our data when we do so?

Solution

1. When we fit a linear model we are minimising the squared error. In fact, the standard linear model estimator is often known as “ordinary least squares”. The “ordinary” really means “original” here, to distinguish between this method, which dates back to ~1800, and some more “recent” (think 1940s…) methods.
2. Squared error is useful because it ignores the sign of the residuals (whether they are positive or negative). It also penalises large errors much more than small errors. On top of all this, it also makes the solution very easy to compute mathematically. Least squares assumes that, when we account for the change in the mean of the outcome based on changes in the income, the data are normally distributed. That is, the residuals of the model, or the error left over after we account for any linear relationships in the data, are normally distributed, and have a fixed variance.

Model selection using training and test sets

Sets of models are often compared using statistics such as adjusted $R^2$, AIC or BIC. These show us how well the model is learning the data used in fitting that same model ¹. However, these statistics do not really tell us how well the model will generalise to new data. This is an important thing to consider – if our model doesn’t generalise to new data, then there’s a chance that it’s just picking up on a technical or batch effect in our data, or simply some noise that happens to fit the outcome we’re modelling. This is especially important when our goal is prediction – it’s not much good if we can only predict well for samples where the outcome is already known, after all!

To get an idea of how well our model generalises, we can split the data into two - a “training” and a “test” set. We use the “training” data to fit the model, and then see its performance on the “test” data.

Schematic representation of how a dataset can be divided into a training and a test set.

One thing that often happens in this context is that large coefficient values minimise the training error, but they don’t minimise the test error on unseen data. First, we’ll go through an example of what exactly this means.

To compare the training and test errors for a model of methylation features and age, we’ll split the data into training and test sets, fit a linear model and calculate the errors. First, let’s calculate the training error. Let’s start by splitting the data into training and test sets:

methylation <- readRDS(here::here("data/methylation.rds"))

library("SummarizedExperiment")
age <- methylation$Age
methyl_mat <- t(assay(methylation))

We will then subset the data to a set of CpG markers that are known to be associated with age from a previous study by Horvath et al.², described in data.

cpg_markers <- c("cg16241714", "cg14424579", "cg22736354", "cg02479575", "cg00864867", 
    "cg25505610", "cg06493994", "cg04528819", "cg26297688", "cg20692569", 
    "cg04084157", "cg22920873", "cg10281002", "cg21378206", "cg26005082", 
    "cg12946225", "cg25771195", "cg26845300", "cg06144905", "cg27377450"
)

horvath_mat <- methyl_mat[, cpg_markers]

## Generate an index to split the data
set.seed(42)
train_ind <- sample(nrow(methyl_mat), 25)

## Split the data 
train_mat <- horvath_mat[train_ind, ]
train_age <- age[train_ind]
test_mat <- horvath_mat[-train_ind, ]
test_age <- age[-train_ind]

Now let’s fit a linear model to our training data and calculate the training error. Here we use the mean of the squared difference between our predictions and the observed data, or “mean squared error” (MSE).

## Fit a linear model
# as.data.frame() converts train_mat into a data.frame
# Using the `.` syntax above together with a `data` argument will lead to
# the same result as using `train_age ~ train_mat`: R will fit a multivariate 
# regression model in which each of the columns in `train_mat` is used as 
# a predictor. We opted to use the `.` syntax because it will help us to 
# obtain model predictions using the `predict()` function. 

fit_horvath <- lm(train_age ~ ., data = as.data.frame(train_mat))

## Function to calculate the (mean squared) error
mse <- function(true, prediction) { 
    mean((true - prediction)^2) 
} 

## Calculate the training error 
err_lm_train <- mse(train_age, fitted(fit_horvath)) 
err_lm_train

[1] 1.319628

The training error appears very low here – on average we’re only off by about a year!

Challenge 3

For the fitted model above, calculate the mean squared error for the test set.

Solution

First, let’s find the predicted values for the ‘unseen’ test data:

pred_lm <- predict(fit_horvath, newdata = as.data.frame(test_mat)) 

The mean squared error for the test set is the mean of the squared error between the predicted values and true test data.

err_lm <- mse(test_age, pred_lm)
err_lm

[1] 223.3571

Unfortunately, the test error is a lot higher than the training error. If we plot true age against predicted age for the samples in the test set, we can gain more insight into the performance of the model on the test set. Ideally, the predicted values should be close to the test data.

par(mfrow = c(1, 1))
plot(test_age, pred_lm, pch = 19)
abline(coef = 0:1, lty = "dashed")

A scatter plot of observed age versus predicted age for individuals in the test set. Each dot represents one individual. Dashed line is used as a reference to indicate how perfect predictions would look (observed = predicted).

This figure shows the predicted ages obtained from a linear model fit plotted against the true ages, which we kept in the test dataset. If the prediction were good, the dots should follow a line. Regularisation can help us to make the model more generalisable, improving predictions for the test dataset (or any other dataset that is not used when fitting our model).

Regularisation

Regularisation can be used to reduce correlation between predictors, the number of features, and improve generalisability by restricting model parameter estimates. Essentially, we add another condition to the problem we’re solving with linear regression that controls the total size of the coefficients that come out and shrinks many coefficients to zero (or near zero).

For example, we might say that the point representing the slope and intercept must fall within a certain distance of the origin, $(0, 0)$. For the 2-parameter model (slope and intercept), we could visualise this constraint as a circle with a given radius. We want to find the “best” solution (in terms of minimising the residuals) that also falls within a circle of a given radius (in this case, 2).

Illustrative example demonstrated how regression coefficients are inferred under a linear model framework, with (blue line) and without (red line) regularisation. A ridge penalty is used in this example

Ridge regression

There are multiple ways to define the distance that our solution must fall in. The one we’ve plotted above controls the squared sum of the coefficients, $\beta$. This is also sometimes called the $L^2$ norm. This is defined as

[\left\lVert \beta\right\lVert_2 = \sqrt{\sum_{j=1}^p \beta_j^2}]

To control this, we specify that the solution for the equation above also has to have an $L^2$ norm smaller than a certain amount. Or, equivalently, we try to minimise a function that includes our $L^2$ norm scaled by a factor that is usually written $\lambda$.

[\sum_{i=1}^N \biggl( y_i - x’_i\beta\biggr)^2 + \lambda \left\lVert \beta \right\lVert_2 ^2]

This type of regularisation is called ridge regression. Another way of thinking about this is that when finding the best model, we’re weighing up a balance of the ordinary least squares objective and a “penalty” term that punishes models with large coefficients. The balance between the penalty and the ordinary least squares objective is controlled by $\lambda$ - when $\lambda$ is large, we want to penalise large coefficients. In other words, we care a lot about the size of the coefficients and we want to reduce the complexity of our model. When $\lambda$ is small, we don’t really care a lot about shrinking our coefficients and we opt for a more complex model. When it’s zero, we’re back to just using ordinary least squares. We see how a penalty term, $\lambda$, might be chosen later in this episode.

For now, to see how regularisation might improve a model, let’s fit a model using the same set of 20 features (stored as cpg_markers) selected earlier in this episode (these are a subset of the features identified by Horvarth et al), using both regularised and ordinary least squares. To fit regularised regression models, we will use the glmnet package.

library("glmnet")

## glmnet() performs scaling by default, supply un-scaled data:
horvath_mat <- methyl_mat[, cpg_markers] # select the same 20 sites as before
train_mat <- horvath_mat[train_ind, ] # use the same individuals as selected before
test_mat <- horvath_mat[-train_ind, ]

ridge_fit <- glmnet(x = train_mat, y = train_age, alpha = 0)
plot(ridge_fit, xvar = "lambda")
abline(h = 0, lty = "dashed")

A line plot of coefficient estimates against log lambda for a ridge regression model.

This plot shows how the estimated coefficients for each CpG site change as we increase the penalty, $\lambda$. That is, as we decrease the size of the region that solutions can fall into, the values of the coefficients that we get back tend to decrease. In this case, coefficients tend towards zero but generally don’t reach it until the penalty gets very large. We can see that initially, some parameter estimates are really, really large, and these tend to shrink fairly rapidly.

We can also notice that some parameters “flip signs”; that is, they start off positive and become negative as lambda grows. This is a sign of collinearity, or correlated predictors. As we reduce the importance of one feature, we can “make up for” the loss in accuracy from that one feature by adding a bit of weight to another feature that represents similar information.

Since we split the data into test and training data, we can prove that ridge regression predicts the test data better than the model with no regularisation. Let’s generate our predictions under the ridge regression model and calculate the mean squared error in the test set:

# Obtain a matrix of predictions from the ridge model,
# where each column corresponds to a different lambda value
pred_ridge <- predict(ridge_fit, newx = test_mat)

# Calculate MSE for every column of the prediction matrix against the vector of true ages
err_ridge <- apply(pred_ridge, 2, function(col) mse(test_age, col)) 
min_err_ridge <- min(err_ridge)

# Identify the lambda value that results in the lowest MSE (ie, the "best" lambda value)
which_min_err <- which.min(err_ridge)
pred_min_ridge <- pred_ridge[, which_min_err]

## Return errors
min_err_ridge

[1] 46.76802

This is much lower than the test error for the model without regularisation:

err_lm  

[1] 223.3571

We can see where on the continuum of lambdas we’ve picked a model by plotting the coefficient paths again. In this case, we’ve picked a model with fairly modest coefficient shrinking.

chosen_lambda <- ridge_fit$lambda[which.min(err_ridge)]
plot(ridge_fit, xvar = "lambda")
abline(v = log(chosen_lambda), lty = "dashed")

A line plot of coefficient estimates against log lambda for a ridge regression model, showing the optimal value based on the minimal test error.

Challenge 4

1. Which performs better, ridge or OLS?
2. Plot predicted ages for each method against the true ages. How do the predictions look for both methods? Why might ridge be performing better?

Solution

1. Ridge regression performs significantly better on unseen data, despite being “worse” on the training data.

   min_err_ridge
   

   [1] 46.76802
   

   err_lm
   

   [1] 223.3571
   

2. The ridge ones are much less spread out with far fewer extreme predictions.

   all <- c(pred_lm, test_age, pred_min_ridge)
   lims <- range(all)
   par(mfrow = 1:2)
   plot(test_age, pred_lm,
       xlim = lims, ylim = lims,
       pch = 19
   )
   abline(coef = 0:1, lty = "dashed")
   plot(test_age, pred_min_ridge,
       xlim = lims, ylim = lims,
       pch = 19
   )
   abline(coef = 0:1, lty = "dashed")
   

   

   Two plots showing OLS predictions (left) and ridge regression predictions (right) of age (y) against true age (x).

LASSO regression

LASSO is another type of regularisation. In this case we use the $L^1$ norm, or the sum of the absolute values of the coefficients.

This tends to produce sparse models; that is to say, it tends to remove features from the model that aren’t necessary to produce accurate predictions. This is because the region we’re restricting the coefficients to has sharp edges. So, when we increase the penalty (reduce the norm), it’s more likely that the best solution that falls in this region will be at the corner of this diagonal (i.e., one or more coefficient is exactly zero).

Illustrative example demonstrated how regression coefficients are inferred under a linear model framework, with (blue line) and without (red line) regularisation. A LASSO penalty is used in this example.

Challenge 5

1. Use glmnet() to fit a LASSO model (hint: set alpha = 1).
2. Plot the model object. Remember that for ridge regression, we set xvar = "lambda". What if you don’t set this? What’s the relationship between the two plots?
3. How do the coefficient paths differ to the ridge case?

Solution

1. Fitting a LASSO model is very similar to a ridge model, we just need to change the alpha setting.

   fit_lasso <- glmnet(x = methyl_mat, y = age, alpha = 1)
   

2. When xvar = "lambda", the x-axis represents increasing model sparsity from left to right, because increasing lambda increases the penalty added to the coefficients. When we instead plot the L1 norm on the x-axis, increasing L1 norm means that we are allowing our coefficients to take on increasingly large values.

   par(mfrow = c(1, 2))
   plot(fit_lasso, xvar = "lambda")
   plot(fit_lasso)
   

   

   Line plots showing coefficient values from a LASSO model against log lambda (left) and L1 norm (right).

3. When using LASSO, the paths tend to go to exactly zero much more as the penalty ($ \lambda $) increases. In the ridge case, the paths tend towards zero but less commonly reach exactly zero.

Cross-validation to find the best value of $\lambda$

Ideally, we want $\lambda$ to be large enough to reduce the complexity of our model, thus reducing the number of and correlations between the features, and improving generalisability. However, we don’t want the value of $\lambda$ to be so large that we lose a lot of the valuable information in the features.

Various methods can be used to balance this trade-off and thus select the “best” value for $\lambda$. One method splits the data into $K$ chunks. We then use $K-1$ of these as the training set, and the remaining $1$ chunk as the test set. We can repeat this until we’ve rotated through all $K$ chunks, giving us a good estimate of how well each of the lambda values work in our data. This is called cross-validation, and doing this repeated test/train split gives us a better estimate of how generalisable our model is.

Schematic representiation of a $K$-fold cross-validation procedure.

We can use this new idea to choose a lambda value by finding the lambda that minimises the error across each of the test and training splits. In R:

# fit lasso model with cross-validation across a range of lambda values
lasso <- cv.glmnet(methyl_mat, age, alpha = 1)
plot(lasso)

Cross-validated mean squared error for different values of lambda under a LASSO penalty.

# Extract the coefficients from the model with the lowest mean squared error from cross-validation
coefl <- coef(lasso, lasso$lambda.min)
# select only non-zero coefficients
selection <- which(coefl != 0)
# and convert to a normal matrix
selected_coefs <- as.matrix(coefl)[selection, 1]
selected_coefs

  (Intercept)    cg02388150    cg06493994    cg22449114    cg22736354 
-8.4133296328  0.6966503392  0.1615535465  6.4255580409 12.0507794749 
   cg03330058    cg09809672    cg11299964    cg19761273    cg26162695 
-0.0002362055 -0.7487594618 -2.0399663416 -5.2538055304 -0.4486970332 
   cg09502015    cg24771684    cg08446924    cg13215762    cg24549277 
 1.0787003366  4.5743800395 -0.5960137381  0.1481402638  0.6290915767 
   cg12304482    cg13131095    cg17962089    cg13842639    cg04080666 
-1.0167896196  2.8860222552  6.3065284096  0.1590465147  2.4889065761 
   cg06147194    cg03669936    cg14230040    cg19848924    cg23964682 
-0.6838637838 -0.0352696698  0.1280760909 -0.0006938337  1.3378854603 
   cg13578928    cg02745847    cg17410295    cg17459023    cg06223736 
-0.8601170264  2.2346315955 -2.3008028295  0.0370389967  1.6158734083 
   cg06717750    cg20138604    cg12851161    cg20972027    cg23878422 
 2.3401693309  0.0084327521 -3.3033355652  0.2442751682  1.1059030593 
   cg16612298    cg03762081    cg14428146    cg16908369    cg16271524 
 0.0050053190 -6.5228858163  0.3167227488  0.2302773154 -1.3787104336 
   cg22071651    cg04262805    cg24969251    cg11233105    cg03156032 
 0.3480551279  1.1841804186  8.3024629942  0.6130598151 -1.1121959544 

We can see that cross-validation has selected a value of $\lambda$ resulting in 44 features and the intercept.

Elastic nets: blending ridge regression and the LASSO

So far, we’ve used ridge regression (where alpha = 0) and LASSO regression, (where alpha = 1). What if alpha is set to a value between zero and one? Well, this actually lets us blend the properties of ridge and LASSO regression. This allows us to have the nice properties of the LASSO, where uninformative variables are dropped automatically, and the nice properties of ridge regression, where our coefficient estimates are a bit more conservative, and as a result our predictions are a bit better.

Formally, the objective function of elastic net regression is to optimise the following function:

[\left(\sum_{i=1}^N y_i - x’_i\beta\right) + \lambda \left( \alpha \left\lVert \beta \right\lVert_1 + (1-\alpha) \left\lVert \beta \right\lVert_2 \right)]

You can see that if alpha = 1, then the L1 norm term is multiplied by one, and the L2 norm is multiplied by zero. This means we have pure LASSO regression. Conversely, if alpha = 0, the L2 norm term is multiplied by one, and the L1 norm is multiplied by zero, meaning we have pure ridge regression. Anything in between gives us something in between.

The contour plots we looked at previously to visualise what this penalty looks like for different values of alpha.

For an elastic net, the panels show the effect of the regularisation across different values of alpha

Challenge 6

1. Fit an elastic net model (hint: alpha = 0.5) without cross-validation and plot the model object.
2. Fit an elastic net model with cross-validation and plot the error. Compare with LASSO.
3. Select the lambda within one standard error of the minimum cross-validation error (hint: lambda.1se). Compare the coefficients with the LASSO model.
4. Discuss: how could we pick an alpha in the range (0, 1)? Could we justify choosing one a priori?

Solution

1. Fitting an elastic net model is just like fitting a LASSO model. You can see that coefficients tend to go exactly to zero, but the paths are a bit less extreme than with pure LASSO; similar to ridge.

   elastic <- glmnet(methyl_mat[, -1], age, alpha = 0.5)
   plot(elastic)
   

   

   Line plot showing coefficient values from an elastic net model against L1 norm.

2. The process of model selection is similar for elastic net models as for LASSO models.

   elastic_cv <- cv.glmnet(methyl_mat[, -1], age, alpha = 0.5)
   plot(elastic_cv)
   

   

   A plot depicting mean squared error (MSE) against discrete values of lambda, with red points showing the average mean squared error and grey error bars showing the 1 standard error interval around them. The MSE rises as log lambda increases, indicating a larger prediction error. Two dashed lines indicate the lambda value corresponding to the lowest overall MSE value and the lambda value corresponding to the lambda value with MSE with one standard error of the minimum.

3. You can see that the coefficients from these two methods are broadly similar, but the elastic net coefficients are a bit more conservative. Further, more coefficients are exactly zero in the LASSO model.

   coefe <- coef(elastic_cv, elastic_cv$lambda.1se)
   sum(coefe[, 1] == 0)
   

   [1] 4973
   

   sum(coefl[, 1] == 0)
   

   [1] 4956
   

   plot(
       coefl[, 1], coefe[, 1],
       pch = 16,
       xlab = "LASSO coefficients",
       ylab = "Elastic net coefficients"
   )
   

   Error in xy.coords(x, y, xlabel, ylabel, log): 'x' and 'y' lengths differ
   

   abline(0:1, lty = "dashed")
   

   Error in int_abline(a = a, b = b, h = h, v = v, untf = untf, ...): plot.new has not been called yet
   

4. You could pick an arbitrary value of alpha, because arguably pure ridge regression or pure LASSO regression are also arbitrary model choices. To be rigorous and to get the best-performing model and the best inference about predictors, it’s usually best to find the best combination of alpha and lambda using a grid search approach in cross-validation. However, this can be very computationally demanding.

The bias-variance tradeoff

When we make predictions in statistics, there are two sources of error that primarily influence the (in)accuracy of our predictions. these are bias and variance.

The total expected error in our predictions is given by the following equation:

\[E(y - \hat{y}) = \text{Bias}^2 + \text{Variance} + \sigma^2\]

Here, $\sigma^2$ represents the irreducible error, that we can never overcome. Bias results from erroneous assumptions in the model used for predictions. Fundamentally, bias means that our model is mis-specified in some way, and fails to capture some components of the data-generating process (which is true of all models). If we have failed to account for a confounding factor that leads to very inaccurate predictions in a subgroup of our population, then our model has high bias.

Variance results from sensitivity to particular properties of the input data. For example, if a tiny change to the input data would result in a huge change to our predictions, then our model has high variance.

Linear regression is an unbiased model under certain conditions. In fact, the Gauss-Markov theorem shows that under the right conditions, OLS is the best possible type of unbiased linear model.

Introducing penalties to means that our model is no longer unbiased, meaning that the coefficients estimated from our data will systematically deviate from the ground truth. Why would we do this? As we saw, the total error is a function of bias and variance. By accepting a small amount of bias, it’s possible to achieve huge reductions in the total expected error.

This bias-variance tradeoff is also why people often favour elastic net regression over pure LASSO regression.

Other types of outcomes

You may have noticed that glmnet is written as glm, not lm. This means we can actually model a variety of different outcomes using this regularisation approach. For example, we can model binary variables using logistic regression, as shown below. The type of outcome can be specified using the family argument, which specifies the family of the outcome variable.

In fact, glmnet is somewhat cheeky as it also allows you to model survival using Cox proportional hazards models, which aren’t GLMs, strictly speaking.

For example, in the current dataset we can model smoking status as a binary variable in logistic regression by setting family = "binomial".

The package documentation explains this in more detail.

smoking <- as.numeric(factor(methylation$smoker)) - 1
# binary outcome
table(smoking)

smoking
 0  1 
30  7 

fit <- cv.glmnet(x = methyl_mat, nfolds = 5, y = smoking, family = "binomial")

Warning in lognet(xd, is.sparse, ix, jx, y, weights, offset, alpha, nobs, : one
multinomial or binomial class has fewer than 8 observations; dangerous ground
Warning in lognet(xd, is.sparse, ix, jx, y, weights, offset, alpha, nobs, : one
multinomial or binomial class has fewer than 8 observations; dangerous ground
Warning in lognet(xd, is.sparse, ix, jx, y, weights, offset, alpha, nobs, : one
multinomial or binomial class has fewer than 8 observations; dangerous ground
Warning in lognet(xd, is.sparse, ix, jx, y, weights, offset, alpha, nobs, : one
multinomial or binomial class has fewer than 8 observations; dangerous ground
Warning in lognet(xd, is.sparse, ix, jx, y, weights, offset, alpha, nobs, : one
multinomial or binomial class has fewer than 8 observations; dangerous ground
Warning in lognet(xd, is.sparse, ix, jx, y, weights, offset, alpha, nobs, : one
multinomial or binomial class has fewer than 8 observations; dangerous ground

coef <- coef(fit, s = fit$lambda.min)
coef <- as.matrix(coef)
coef[which(coef != 0), 1]

[1] -1.455287

plot(fit)

A plot depicting binomial deviance against discrete values of lambda, with red points showing the average mean squared error and grey error bars showing the 1 standard error interval around them. The MSE decreases as log lambda increases, indicating a smaller prediction error. Two dashed lines indicate the lambda value corresponding to the lowest overall MSE value and the lambda value corresponding to the lambda value with MSE with one standard error of the minimum, with both being exactly on the right side of the plot, indicating an intercept-only model.

In this case, the results aren’t very interesting! We select an intercept-only model. However, as highlighted by the warnings above, we should not trust this result too much as the data was too small to obtain reliable results! We only included it here to provide the code that could be used to perform penalised regression for binary outcomes (i.e. penalised logistic regression).

tidymodels

A lot of the packages for fitting predictive models like regularised regression have different user interfaces. To do predictive modelling, it’s important to consider things like choosing a good performance metric and how to run normalisation. It’s also useful to compare different model “engines”.

To this end, the tidymodels R framework exists. We’re not doing a course on advanced topics in predictive modelling so we are not covering this framework in detail. However, the code below would be useful to perform repeated cross-validation. More information about tidymodels, including installation instructions, can be found here.

library("tidymodels")
all_data <- as.data.frame(cbind(age = age, methyl_mat))
split_data <- initial_split(all_data)

norm_recipe <- recipe(training(split_data)) %>%
    ## everything other than age is a predictor
    update_role(everything(), new_role = "predictor") %>%
    update_role(age, new_role = "outcome") %>%
    ## center and scale all the predictors
    step_center(all_predictors()) %>%
    step_scale(all_predictors()) 

## set the "engine" to be a linear model with tunable alpha and lambda
glmnet_model <- linear_reg(penalty = tune(), mixture = tune()) %>% 
    set_engine("glmnet")

## define a workflow, with normalisation recipe into glmnet engine
workflow <- workflow() %>%
    add_recipe(norm_recipe) %>%
    add_model(glmnet_model)

## 5-fold cross-validation repeated 5 times
folds <- vfold_cv(training(split_data), v = 5, repeats = 5)

## define a grid of lambda and alpha parameters to search
glmn_set <- parameters(
    penalty(range = c(-5, 1), trans = log10_trans()),
    mixture()
)
glmn_grid <- grid_regular(glmn_set)
ctrl <- control_grid(save_pred = TRUE, verbose = TRUE)

## use the metric "rmse" (root mean squared error) to grid search for the
## best model
results <- workflow %>%
    tune_grid(
        resamples = folds,
        metrics = metric_set(rmse),
        control = ctrl
    )
## select the best model based on RMSE
best_mod <- results %>% select_best("rmse")
best_mod
## finalise the workflow and fit it with all of the training data
final_workflow <- finalize_workflow(workflow, best_mod)
final_workflow
final_model <- final_workflow %>%
    fit(data = training(split_data))

## plot predicted age against true age for test data
plot(
    testing(split_data)$age,
    predict(final_model, new_data = testing(split_data))$.pred,
    xlab = "True age",
    ylab = "Predicted age",
    pch = 16,
    log = "xy"
)
abline(0:1, lty = "dashed")

Further reading

• An introduction to statistical learning.
• Elements of statistical learning.
• glmnet vignette.
• tidymodels.

Footnotes

Key Points

• Regularisation is a way to fit a model, get better estimates of effect sizes, and perform variable selection simultaneously.

• Test and training splits, or cross-validation, are a useful way to select models or hyperparameters.

• Regularisation can give us a more predictive set of variables, and by restricting the magnitude of coefficients, can give us a better (and more stable) estimate of our outcome.

• Regularisation is often very fast! Compared to other methods for variable selection, it is very efficient. This makes it easier to practice rigorous variable selection.

Principal component analysis

Overview

Teaching: 90 min
Exercises: 40 min

Questions

• What is principal component analysis (PCA) and when can it be used?

• How can we perform a PCA in R?

• How many principal components are needed to explain a significant amount of variation in the data?

• How to interpret the output of PCA using loadings and principal components?

Objectives

• Identify situations where PCA can be used to answer research questions using high-dimensional data.

• Perform a PCA on high-dimensional data.

• Select the appropriate number of principal components.

• Interpret the output of PCA.

Introduction

If a dataset contains many variables ($p$), it is likely that some of these variables will be highly correlated. Variables may even be so highly correlated that they represent the same overall effect. Such datasets are challenging to analyse for several reasons, with the main problem being how to reduce dimensionality in the dataset while retaining the important features.

In this episode we will explore principal component analysis (PCA) as a popular method of analysing high-dimensional data. PCA is a statistical method which allows large datasets of correlated variables to be summarised into smaller numbers of uncorrelated principal components that explain most of the variability in the original dataset. As an example, PCA might reduce several variables representing aspects of patient health (blood pressure, heart rate, respiratory rate) into a single feature capturing an overarching “patient health” effect. This is useful from an exploratory point of view, discovering how variables might be associated and combined. The resulting principal component could also be used as an effect in further analysis (e.g. linear regression).

Challenge 1

Descriptions of three datasets and research questions are given below. For which of these might PCA be considered a useful tool for analysing data so that the research questions may be addressed?

1. An epidemiologist has data collected from different patients admitted to hospital with infectious respiratory disease. They would like to determine whether length of stay in hospital differs in patients with different respiratory diseases.
2. A scientist has assayed gene expression levels in 1000 cancer patients and has data from probes targeting different genes in tumour samples from patients. She would like to create new variables representing relative abundance of different groups of genes to i) find out if genes form subgroups based on biological function and ii) use these new variables in a linear regression examining how gene expression varies with disease severity.
3. Both of the above.

Solution

In the first case, a regression model would be more suitable; perhaps a survival model. In the second example, PCA can help to identify modules of correlated features that explain a large amount of variation within the data.

Therefore the answer here is 2.

Principal component analysis

PCA transforms a dataset of continuous variables into a new set of uncorrelated variables called “principal components”. The first principal component derived explains the largest amount of the variability in the underlying dataset. The second principal component derived explains the second largest amount of variability in the dataset and so on. Once the dataset has been transformed into principal components, we can extract a subset of the principal components in order of the variance they explain (starting with the first principal component that by definition explains the most variability, and then the second), giving new variables that explain a lot of the variability in the original dataset. Thus, PCA helps us to produce a lower dimensional dataset while keeping most of the information in the original dataset.

To see what these new principal components may look like, the figure below shows the log-transformed prostate specific antigen (lpsa) versus cancer volume (lcavol) data from the prostate data set introduced in Episode 1. Principal components are a collection of new, artificial data points, one for each individual observation called scores. The red line on the plot represents the line passing through the scores (points) of the first principal component for each observation. The angle that the first principal component line passes through the data points at is set to the direction with the highest variability. The plotted first principal components can therefore be thought of reflecting the effect in the data that has the highest variability. The second principal component explains the next highest amount of variability in the data and is represented by the line perpendicular to the first (the green line). The second principal component can be thought of as capturing the overall effect in the data that has the second-highest variability.

Scatter plots of the prostate data with the first principal component in red and second principal component in green.

The animation below illustrates how principal components are calculated from data. You can imagine that the black line is a rod and each red dashed line is a spring. The energy of each spring is proportional to its squared length. The direction of the first principal component is the one that minimises the total energy of all of the springs. In the animation below, the springs pull the rod, finding the direction of the first principal component when they reach equilibrium. We then use the length of the springs from the rod as the first principal component. This is explained in more detail on this Q&A website.

Animation showing the iterative process by which principal components are found. Data points are shown in blue, distances from the line as red dashed lines and the principal component is shown by a black line.

Mathematical description of PCA

Mathematically, each principal component is a linear combination of the variables in the dataset. That is, the first principal component values or scores, $Z_1$, are a linear combination of variables in the dataset, $X_1…X_p$, given by

\[Z_1 = a_{11}X_1 + a_{21}X_2 +....+a_{p1}X_p,\]

where $a_{11}…a_{p1}$ represent principal component loadings.

In summary, the principal components’ values are called scores. The loadings can be thought of as the degree to which each original variable contributes to the principal component scores. In this episode, we will see how to perform PCA to summarise the information in high-dimensional datasets.

How do we perform a PCA?

To illustrate how to perform PCA initially, we revisit the low-dimensional prostate dataset. The data come from a study which examined the correlation between the level of prostate specific antigen and a number of clinical measures in 97 men who were about to receive a radical prostatectomy. The data have 97 rows and 9 columns.

The variables in the dataset (columns) are:

• lcavol (log-transformed cancer volume),
• lweight (log-transformed prostate weight),
• lbph (log-transformed amount of benign prostate enlargement),
• svi (seminal vesicle invasion),
• lcp (log-transformed capsular penetration; amount of spread of cancer in outer walls of prostate),
• gleason (Gleason score; grade of cancer cells),
• pgg45 (percentage Gleason scores 4 or 5),
• lpsa (log-transformed prostate specific antigen; level of PSA in blood).
• age (patient age in years).

We will perform PCA on the five continuous clinical variables in our dataset so that we can create fewer variables representing clinical markers of cancer progression.

First, we will examine the prostate dataset (originally part of the lasso2 package):

library("here")
prostate <- readRDS(here("data/prostate.rds"))

head(prostate)

      lcavol  lweight age      lbph svi       lcp gleason pgg45       lpsa
1 -0.5798185 2.769459  50 -1.386294   0 -1.386294       6     0 -0.4307829
2 -0.9942523 3.319626  58 -1.386294   0 -1.386294       6     0 -0.1625189
3 -0.5108256 2.691243  74 -1.386294   0 -1.386294       7    20 -0.1625189
4 -1.2039728 3.282789  58 -1.386294   0 -1.386294       6     0 -0.1625189
5  0.7514161 3.432373  62 -1.386294   0 -1.386294       6     0  0.3715636
6 -1.0498221 3.228826  50 -1.386294   0 -1.386294       6     0  0.7654678

Note that each row of the dataset represents a single patient.

We will create a subset of the data including only the clinical variables we want to use in the PCA.

pros2 <- prostate[, c("lcavol", "lweight", "lbph", "lcp", "lpsa")]
head(pros2)

      lcavol  lweight      lbph       lcp       lpsa
1 -0.5798185 2.769459 -1.386294 -1.386294 -0.4307829
2 -0.9942523 3.319626 -1.386294 -1.386294 -0.1625189
3 -0.5108256 2.691243 -1.386294 -1.386294 -0.1625189
4 -1.2039728 3.282789 -1.386294 -1.386294 -0.1625189
5  0.7514161 3.432373 -1.386294 -1.386294  0.3715636
6 -1.0498221 3.228826 -1.386294 -1.386294  0.7654678

Do we need to scale the data?

PCA derives principal components based on the variance they explain in the data. Therefore, we may need to apply some pre-processing to scale variables in our dataset if we want to ensure that each variable is considered equally by the PCA. Scaling is essential if we want to avoid the PCA ignoring variables that may be important to our analysis just because they take low values and have low variance. We do not need to scale if we want variables with low variance to carry less weight in the PCA.

In this example, we want each variable to be treated equally by the PCA since variables with lower values may be just as informative as variables with higher values. Let’s therefore investigate the variables in our dataset to see if we need to scale our variables first:

apply(pros2, 2, var)

  lcavol  lweight     lbph      lcp     lpsa 
1.389157 0.246642 2.104840 1.955102 1.332476 

par(mfrow = c(1, 2))
hist(pros2$lweight)
hist(pros2$lbph)

Histograms of two variables from the prostate data set.

Note that variance is greatest for lbph and lowest for lweight. Since we want each of the variables to be treated equally in our PCA, but there are large differences in the variances of the variables, we need to scale each of the variables before including them in a PCA to ensure that differences in variances do not drive the calculation of principal components. In this example, we standardise all five variables to have a mean of 0 and a standard deviation of 1.

Challenge 2

1. Why might it be necessary to standardise variables before performing a PCA?
   Some ideas:
   • To make the results of the PCA interesting.
   • If you want to ensure that variables with different ranges of values contribute equally to analysis.
   • To allow the feature matrix to be calculated faster, especially in cases where there are a lot of input variables.
   • To allow both continuous and categorical variables to be included in the PCA.
   • All of the above.
2. Can you think of datasets where it might not be necessary to standardise variables?

Solution

1. Scaling the data isn’t guaranteed to make the results more interesting. It also won’t affect how quickly the output will be calculated, whether continuous and categorical variables are present or not.

   We use scaling to ensure that features with different ranges of values have equal weighting in the resulting principal components (point 2).

2. You may not want to standardise datasets which contain continuous variables all measured on the same scale (e.g. gene expression data or RNA sequencing data). In this case, variables with very little sample-to-sample variability may represent only random noise, and standardising the data would give these extra weight in the PCA.

Performing PCA

Throughout this episode, we will carry out PCA using the Bioconductor package PCAtools. This package provides several functions that are useful for exploring data via PCA and producing useful figures and analysis tools. The package is made for the somewhat unusual Bioconductor style of data tables (observations in columns, features in rows). When using Bioconductor data sets and PCAtools, it is thus not necessary to transpose the data. You can use the help files in the package documentation to find out about the pca() function (type help("pca") or ?pca in R). Although we focus on PCA implementation using this package, there are several other options for performing PCA, including a base R function, prcomp(). Equivalent functions and output labels for the common implementations are summarised below.

Let’s use the PCAtools package to perform PCA on the scaled prostate data. First, let’s load the PCAtools package.

library("PCAtools")

The input data (pros2) is in the form of a matrix and we need to take the transpose of this matrix and convert it to a data frame for use within the PCAtools package. We set the scale = TRUE argument to standardise the variables to have a mean 0 and standard deviation of 1 as above.

pros2t <- data.frame(t(pros2))  # create transposed data frame
colnames(pros2t)<-seq(1:dim(pros2t)[2]) # rename columns for use within PCAtools
pmetadata= data.frame("M" = rep(1, dim(pros2t)[2]), row.names = colnames(pros2t)) # create fake metadata for use with PCA tools

## implement PCA
pca.pros <- pca(pros2t, scale = TRUE, center = TRUE, metadata = pmetadata)
summary(pca.pros)

           Length Class      Mode     
rotated     5     data.frame list     
loadings    5     data.frame list     
variance    5     -none-     numeric  
sdev        5     -none-     numeric  
metadata    1     data.frame list     
xvars       5     -none-     character
yvars      97     -none-     character
components  5     -none-     character

How many principal components do we need?

We have calculated one principal component for each variable in the original dataset. How do we choose how many of these are necessary to represent the true variation in the data, without having extra components that are unnecessary?

Let’s look at the relative importance of (percentage of the variance explained by) the components:

prop.var <- round(pca.pros$variance, digits = 4)
prop.var

    PC1     PC2     PC3     PC4     PC5 
48.9767 27.3063 11.1094  8.0974  4.5101 

This returns the proportion of variance in the data explained by each of the principal components. In this example, the first principal component, PC1, explains approximately 49% of variance in the data, PC2 27% of variance, PC3 a further 11%, PC4 approximately 8% and PC5 around 5%.

Let’s visualise this. A plot of the amount of variance accounted for by each principal component is also called a scree plot. Often, scree plots show a characteristic pattern where initially, the variance drops rapidly with each additional principal component. But then there is an “elbow” after which the variance decreases more slowly. The total variance explained up to the elbow point is sometimes interpreted as structural variance that is relevant and should be retained versus noise which may be discarded after the elbow.

We can create a scree plot using the PCAtools function screeplot():

screeplot(pca.pros, axisLabSize = 5, titleLabSize = 8, 
          drawCumulativeSumLine = FALSE, drawCumulativeSumPoints = FALSE) +
    geom_line(aes(x = 1:length(pca.pros$components), y = as.numeric(pca.pros$variance))) 

Scree plot showing the percentage of the variance explained by the principal components calculated from the prostate data.

The scree plot shows that the first principal component explains most of the variance in the data (>50%) and each subsequent principal component explains less and less of the total variance. The first two principal components explain >70% of variance in the data.

Selecting a subset of principal components results in loss of information from the original dataset but selecting principal components to summarise the majority of the information in the original dataset is useful from a practical dimension reduction perspective. There are no clear guidelines on how many principal components should be included in PCA. We often look at the ‘elbow’ on the scree plot as an indicator that the addition of principal components does not drastically contribute to explaining the remaining variance. In this case, the ‘elbow’ of the scree plot appears to be at the third principal component. We may therefore conclude that three principal components are sufficient to explain the majority of the variability in the original dataset. We may also choose another criterion, such as an arbitrary cut-off for the proportion of the variance explained compared to other principal components. Essentially, the criterion used to select principal components should be determined based on what is deemed a sufficient level of information retention for a specific dataset and question.

Loadings and principal component scores

Most PCA functions will produce two main output matrices: the principal component scores and the loadings. The matrix of principal component scores has as many rows as there were observations in the input matrix. These scores are what is usually visualised or used for down-stream analyses. The matrix of loadings (also called rotation matrix) has as many rows as there are features in the original data. It contains information about how the (usually centered and scaled) original data relate to the principal component scores.

We can print the loadings for the PCAtools implementation using

pca.pros$loadings

              PC1         PC2         PC3         PC4         PC5
lcavol  0.5616465 -0.23664270  0.01486043  0.22708502 -0.75945046
lweight 0.2985223  0.60174151 -0.66320198 -0.32126853 -0.07577123
lbph    0.1681278  0.69638466  0.69313753  0.04517286 -0.06558369
lcp     0.4962203 -0.31092357  0.26309227 -0.72394666  0.25253840
lpsa    0.5665123 -0.01680231 -0.10141557  0.56487128  0.59111493

The principal component scores are obtained by carrying out matrix multiplication of the (usually centered and scaled) original data times the loadings. The following callout demonstrates this.

Computing a PCA “by hand”

The rotation matrix obtained in a PCA is identical to the eigenvectors of the covariance matrix of the data. Multiplying these with the (centered and scaled) data yields the principal component scores:

pros2.scaled <- scale(pros2) # centre and scale the prostate data
pros2.cov <- cov(pros2.scaled)   # generate covariance matrix
pros2.cov

           lcavol   lweight         lbph          lcp      lpsa
lcavol  1.0000000 0.1941283  0.027349703  0.675310484 0.7344603
lweight 0.1941283 1.0000000  0.434934636  0.100237795 0.3541204
lbph    0.0273497 0.4349346  1.000000000 -0.006999431 0.1798094
lcp     0.6753105 0.1002378 -0.006999431  1.000000000 0.5488132
lpsa    0.7344603 0.3541204  0.179809410  0.548813169 1.0000000

pros2.eigen <- eigen(pros2.cov) # preform eigen decomposition
pros2.eigen # The slot $vectors = rotation of the PCA

eigen() decomposition
$values
[1] 2.4488355 1.3653171 0.5554705 0.4048702 0.2255067

$vectors
           [,1]        [,2]        [,3]        [,4]        [,5]
[1,] -0.5616465  0.23664270  0.01486043  0.22708502  0.75945046
[2,] -0.2985223 -0.60174151 -0.66320198 -0.32126853  0.07577123
[3,] -0.1681278 -0.69638466  0.69313753  0.04517286  0.06558369
[4,] -0.4962203  0.31092357  0.26309227 -0.72394666 -0.25253840
[5,] -0.5665123  0.01680231 -0.10141557  0.56487128 -0.59111493

# generate principal component scores by by hand, using matrix multiplication
my.pros2.pcs <- pros2.scaled %*% pros2.eigen$vectors
# compare results
par(mfrow=c(1,2))
plot(pca.pros$rotated[,1:2], main="PCAtools")
abline(h=0, v=0, lty=2)
plot(my.pros2.pcs[,1:2], main="\"By hand\"", xlab="PC1", ylab="PC2")
abline(h=0, v=0, lty=2)

Scatter plots of the second versus first principal components calculated by prcomp() (left) and by hand (right).

par(mfrow=c(1,1))
# Note that the axis orientations may be swapped but the relative positions of the dots should be the same in both plots.

One way to visualise how principal components relate to the original variables is by creating a biplot. Biplots usually show two principal components plotted against each other. Observations are sometimes labelled with numbers. The contribution of each original variable to the principal components displayed is then shown by arrows (generated from those two columns of the rotation matrix that correspond to the principal components shown). See help("PCAtools::biplot") for arguments and their meaning. For instance, lab or colBy may be useful. Note that there are several biplot implementations in different R libraries. It is thus a good idea to specify the desired package when calling biplot(). A biplot of the first two principal components can be generated as follows:

PCAtools::biplot(pca.pros)

Scatter plot of the first two principal components with observations numerically labelled.

This biplot shows the position of each patient on a 2-dimensional plot where loadings can be observed via the red arrows associated with each of the variables. The variables lpsa, lcavol and lcp are associated with positive values on PC1 while positive values on PC2 are associated with the variables lbph and lweight. The length of the arrows indicates how much each variable contributes to the calculation of each principal component.

The left and bottom axes show normalised principal component scores. The axes on the top and right of the plot are used to interpret the loadings, where loadings are scaled by the standard deviation of the principal components (pca.pros$sdev) times the square root the number of observations.

Advantages and disadvantages of PCA

Advantages:

• It is a relatively easy to use and popular method.
• Various software/packages are available to run a PCA.
• The calculations used in a PCA are simple to understand compared to other methods for dimension reduction.

Disadvantages:

• It assumes that variables in a dataset are correlated.
• It is sensitive to the scale at which input variables are measured. If input variables are measured at different scales, the variables with large variance relative to the scale of measurement will have greater impact on the principal components relative to variables with smaller variance. In many cases, this is not desirable.
• It is not robust against outliers, meaning that very large or small data points can have a large effect on the output of the PCA.
• PCA assumes a linear relationship between variables which is not always a realistic assumption.
• It can be difficult to interpret the meaning of the principal components, especially when including them in further analyses (e.g. inclusion in a linear regression).

Using PCA to analyse gene expression data

The dataset we will be analysing in this lesson includes two subsets of data:

• a matrix of gene expression data showing microarray results for different probes used to examine gene expression profiles in 91 different breast cancer patient samples.
• metadata associated with the gene expression results detailing information from patients from whom samples were taken.

In this section, we will work through Challenges and you will apply your own PCA to the high-dimensional gene expression data using what we have learnt so far.

We will first load the microarray breast cancer gene expression data and associated metadata, downloaded from the Gene Expression Omnibus.

library("SummarizedExperiment")
cancer <- readRDS(here("data/cancer_expression.rds"))
mat <- assay(cancer)
metadata <- colData(cancer)

head(mat)

head(metadata)

# Check that column names and row names match, if they do should return TRUE
all(colnames(mat) == rownames(metadata))

[1] TRUE

The mat variable contains a matrix of gene expression profiles for each sample. Rows represent gene expression measurements and columns represent samples. The metadata variable contains the metadata associated with the gene expression data including the name of the study from which data originate, the age of the patient from which the sample was taken, whether or not an oestrogen receptor was involved in their cancer and the grade and size of the cancer for each sample (represented by rows).

Microarray data are difficult to analyse for several reasons. Firstly, formulating a research question using microarray data can be difficult, especially if not much is known a priori about which genes code for particular phenotypes of interest. Secondly, they are typically high-dimensional and therefore are subject to the same difficulties associated with analysing high-dimensional data. Finally, exploratory analysis, which can be used to help formulate research questions and display relationships, is difficult using microarray data due to the number of potentially interesting response variables (i.e. expression data from probes targeting different genes).

If researchers hypothesise that groups of genes (e.g. biological pathways) may be associated with different phenotypic characteristics of cancers (e.g. histologic grade, tumour size), using statistical methods that reduce the number of columns in the microarray matrix to a smaller number of dimensions representing groups of genes would help visualise the data and address research questions regarding the effect different groups of genes have on disease progression.

Using PCAtools, we will perform PCA on the cancer gene expression data, plot the amount of variation in the data explained by each principal component and plot the most important principal components against each other as well as understanding what each principal component represents.

Challenge 3

Apply a PCA to the cancer gene expression data using the pca() function from PCAtools.

Let us assume we only care about the principal components accounting for the top 80% of the variance in the dataset. Use the removeVar argument in pca() to remove the principal components accounting for the bottom 20%.

As in the example using prostate data above, examine the first 5 rows and columns of rotated data and loadings from your PCA.

Solution

pc <- pca(mat, metadata = metadata) # implement a PCA
# We can check the scree plot to see that many principal components explain a very small amount of
# the total variance in the data

# Let's remove the principal components with lowest 20% of the variance
pc <- pca(mat, metadata = metadata, removeVar = 0.2)
# Explore other arguments provided in pca
pc$rotated[1:5, 1:5] # obtain the first 5 principal component scores for the first 5 observations

               PC1        PC2        PC3        PC4        PC5
GSM65752 -29.79105  43.866788   3.255903 -40.663138 15.3427597
GSM65753 -37.33911 -15.244788  -4.948201  -6.182795  9.4725870
GSM65755 -29.41462   7.846858 -22.880525 -16.149669 22.3821009
GSM65757 -33.35286   1.343573 -22.579568   2.200329 15.0082786
GSM65758 -40.51897  -8.491125   5.288498  14.007364  0.8739772

pc$loadings[1:5, 1:5] # obtain the first 5 principal component loadings for the first 5 features

                    PC1          PC2          PC3        PC4          PC5
206378_at -0.0024680993 -0.053253543 -0.004068209 0.04068635  0.015078376
205916_at -0.0051557973  0.001315022 -0.009836545 0.03992371  0.038552048
206799_at  0.0005684075 -0.050657061 -0.009515725 0.02610233  0.006208078
205242_at  0.0130742288  0.028876408  0.007655420 0.04449641 -0.001061205
206509_at  0.0019031245 -0.054698479 -0.004667356 0.01566468  0.001306807

which.max(pc$loadings[, 1]) # index of the maximal loading for the first principal component

[1] 49

# principal component loadings for the feature
# with the maximal loading on the first principal component:
pc$loadings[which.max(pc$loadings[, 1]), ] 

                   PC1          PC2         PC3          PC4          PC5
215281_x_at 0.03752947 -0.007369379 0.006243377 -0.008242589 -0.004783206
                   PC6          PC7         PC8         PC9          PC10
215281_x_at 0.01194012 -0.002822407 -0.01216792 0.001137451 -0.0009056616
                   PC11          PC12       PC13         PC14          PC15
215281_x_at -0.00196034 -0.0001676705 0.00699201 -0.002897995 -0.0005044658
                     PC16        PC17         PC18        PC19         PC20
215281_x_at -0.0004547916 0.002277035 -0.006199078 0.002708574 -0.006217326
                  PC21        PC22        PC23        PC24        PC25
215281_x_at 0.00516745 0.007625912 0.003434534 0.005460017 0.001477415
                   PC26         PC27          PC28         PC29       PC30
215281_x_at 0.002350428 0.0007183107 -0.0006195515 0.0006349803 0.00413627
                    PC31        PC32         PC33         PC34         PC35
215281_x_at 0.0001322301 0.003182956 -0.002123462 -0.001042769 -0.001729869
                    PC36        PC37        PC38          PC39        PC40
215281_x_at -0.006556369 0.005766949 0.002537993 -0.0002846248 -0.00018195
                     PC41        PC42         PC43          PC44         PC45
215281_x_at -0.0007970789 0.003888626 -0.008210075 -0.0009570174 0.0007998935
                     PC46         PC47        PC48        PC49         PC50
215281_x_at -0.0006931441 -0.005717836 0.005189649 0.002591188 0.0007810259
                   PC51        PC52         PC53         PC54        PC55
215281_x_at 0.006610815 0.005371134 -0.001704796 -0.002286475 0.001365417
                   PC56         PC57        PC58         PC59         PC60
215281_x_at 0.003529892 0.0003375981 0.009895923 -0.001564423 -0.006989092
                   PC61        PC62         PC63          PC64        PC65
215281_x_at 0.000971273 0.001345406 -0.003575415 -0.0005588113 0.006516669
                    PC66        PC67       PC68         PC69        PC70
215281_x_at -0.008770186 0.006699641 0.01284606 -0.005041574 0.007845653
                   PC71        PC72         PC73         PC74        PC75
215281_x_at 0.003964697 -0.01104367 -0.001506485 -0.001583824 0.003798343
                   PC76         PC77         PC78         PC79          PC80
215281_x_at 0.004817252 -0.001290033 -0.004402926 -0.003440367 -0.0001646198
                   PC81        PC82          PC83         PC84        PC85
215281_x_at 0.003923775 0.003179556 -0.0004388192 9.664648e-05 0.003501335
                   PC86        PC87          PC88         PC89         PC90
215281_x_at -0.00112973 0.006489667 -0.0005039785 -0.004296355 -0.002751513
                   PC91
215281_x_at -0.00383085

which.max(pc$loadings[, 2]) # index of the maximal loading for the second principal component

[1] 27

# principal component loadings for the feature
# with the maximal loading on the second principal component:
pc$loadings[which.max(pc$loadings[, 2]), ] 

                   PC1        PC2          PC3        PC4          PC5
211122_s_at 0.01649085 0.05090275 -0.003378728 0.05178144 -0.003742393
                    PC6         PC7          PC8        PC9        PC10
211122_s_at -0.00543753 -0.03522848 -0.006333521 0.01575401 0.004732546
                   PC11        PC12        PC13        PC14       PC15
211122_s_at 0.004687599 -0.01349892 0.005207937 -0.01731898 0.02323893
                   PC16       PC17        PC18       PC19        PC20
211122_s_at -0.02069509 0.01477432 0.005658529 0.02667751 -0.01333503
                    PC21        PC22       PC23         PC24        PC25
211122_s_at -0.003254036 0.003572342 0.01416779 -0.005511838 -0.02582847
                  PC26        PC27       PC28        PC29       PC30      PC31
211122_s_at 0.03405417 -0.01797345 0.01826328 0.005123959 0.01300763 0.0127127
                   PC32       PC33       PC34        PC35        PC36
211122_s_at 0.002477672 0.01933214 0.03017661 -0.01935071 -0.01960912
                   PC37        PC38        PC39        PC40       PC41
211122_s_at 0.004411188 -0.01263612 -0.02019279 -0.01441513 -0.0310399
                   PC42         PC43        PC44        PC45        PC46
211122_s_at -0.02540426 0.0007949801 -0.00200195 -0.01748543 0.006881834
                   PC47         PC48        PC49         PC50        PC51
211122_s_at 0.006690698 -0.004000732 -0.02747926 -0.006963189 -0.02232332
                     PC52        PC53        PC54        PC55       PC56
211122_s_at -0.0003089115 -0.01604491 0.005649511 -0.02629501 0.02332997
                   PC57        PC58        PC59        PC60         PC61
211122_s_at -0.01248022 -0.01563245 0.005369433 0.009445262 -0.005209349
                  PC62       PC63       PC64        PC65        PC66
211122_s_at 0.01787645 0.01629425 0.02457665 -0.02384242 0.002814479
                    PC67        PC68         PC69         PC70       PC71
211122_s_at 0.0004584731 0.007939733 -0.009554166 -0.003967123 0.01825668
                   PC72        PC73        PC74        PC75        PC76
211122_s_at -0.00580374 -0.02236727 0.001295688 -0.02264723 0.006855855
                   PC77         PC78       PC79         PC80        PC81
211122_s_at 0.004995447 -0.008404118 0.00442875 -0.001027912 0.006104406
                   PC82        PC83         PC84       PC85       PC86
211122_s_at -0.01988441 0.009667348 -0.008248781 0.01198369 0.01221713
                    PC87        PC88        PC89        PC90        PC91
211122_s_at -0.003864842 -0.02876816 -0.01771452 -0.02164973 0.004593411

The function pca() is used to perform PCA, and uses as input a matrix (mat) containing continuous numerical data in which rows are data variables and columns are samples, and metadata associated with the matrix in which rows represent samples and columns represent data variables. It has options to centre or scale the input data before a PCA is performed, although in this case gene expression data do not need to be transformed prior to PCA being carried out as variables are measured on a similar scale (values are comparable between rows). The output of the pca() function includes a lot of information such as loading values for each variable (loadings), principal component scores (rotated) and the amount of variance in the data explained by each principal component.

Rotated data shows principal component scores for each sample and each principal component. Loadings the contribution each variable makes to each principal component.

Scaling variables for PCA

When running pca() above, we kept the default setting, scale=FALSE. That means genes with higher variation in their expression levels should have higher loadings, which is what we are interested in. Whether or not to scale variables for PCA will depend on your data and research question.

Note that this is different from normalising gene expression data. Gene expression data have to be normalised before donwstream analyses can be carried out. This is to reduce technical and other potentially confounding factors. We assume that the expression data we use had been normalised previously.

Choosing how many components are important to explain the variance in the data

As in the example using the prostate dataset, we can use a scree plot to compare the proportion of variance in the data explained by each principal component. This allows us to understand how much information in the microarray dataset is lost by projecting the observations onto the first few principal components and whether these principal components represent a reasonable amount of the variation. The proportion of variance explained should sum to one.

Challenge 4

This time using the screeplot() function in PCAtools, create a scree plot to show proportion of variance explained by the first 20 principal component (hint: components = 1:20). Explain the output of the scree plot in terms of proportion of the variance in the data explained by each principal component.

Solution

pc <- pca(mat, metadata = metadata)
screeplot(pc, components = 1:20) +
    ylim(0, 80)

A scree plot of the gene expression data.

The first principal component explains around 33% of the variance in the microarray data, the first 4 principal components explain around 50%, and 20 principal components explain around 75%. Many principal components explain very little variation. A first ‘elbow’ appears to be around 4-5 principal components, indicating that this may be a suitable number of principal components. However, these principal components cumulatively explain only 51-55% of the variance in the dataset. Although the fact we are able to summarise most of the information in the complex dataset in 4-5 principal components may be a useful result, we may opt to retain more principal components (for example, 20) to capture more of the variability in the dataset depending on research question. A second ‘elbow’ around 12 principal components may provide a good middleground. Note that first principal component (PC1) explains more variation than other principal components (which is always the case in PCA). The scree plot shows that the first principal component only explains ~33% of the total variation in the microarray data and many principal components explain very little variation. The red line shows the cumulative percentage of explained variation with increasing principal components.

Note that in this case 18 principal components are needed to explain over 75% of variation in the data. This is not an unusual result for complex biological datasets including genetic information as clear relationships between groups are sometimes difficult to observe in the data. The scree plot shows that using a PCA we have reduced 91 predictors to 18 in order to explain a significant amount of variation in the data. See additional arguments in scree plot function for improving the appearance of the plot.

Investigating the principal components

Once the most important principal components have been identified using screeplot() and explored, these can be explored in more detail by plotting principal components against each other and highlighting points based on variables in the metadata. This will allow any potential clustering of points according to demographic or phenotypic variables to be seen.

We can use biplots to look for patterns in the output from the PCA. Note that there are two functions called biplot(), one in the package PCAtools and one in stats. Both functions produce biplots but their scales are different!

Challenge 5

Create a biplot of the first two principal components from your PCA using biplot() function in PCAtools. See help("PCAtools::biplot") for arguments and their meaning. For instance, lab or colby may be useful.

Examine whether the data appear to form clusters. Explain your results.

Solution

PCAtools::biplot(pc, lab = NULL, colby = 'Grade', legendPosition = 'top')

A biplot of PC2 against PC1 in the gene expression data, coloured by Grade.

The biplot shows the position of patient samples relative to PC1 and PC2 in a 2-dimensional plot. Note that two groups are apparent along the PC1 axis according to expressions of different genes while no separation can be seem along the PC2 axis. Labels of patient samples are automatically added in the biplot. Labels for each sample are added by default, but can be removed if there is too much overlap in names. Note that PCAtools does not scale biplot in the same way as biplot using the stats package.

Let’s consider this biplot in more detail, and also display the loadings:

Challenge 6

Use colby and lab arguments in biplot() to explore whether these two groups may cluster by patient age or by whether or not the sample expresses the oestrogen receptor gene (ER+ or ER-).

Note: You may see a warning about ggrepel. This happens when there are many labels but little space for plotting. This is not usually a serious problem - not all labels will be shown.

Solution

  PCAtools::biplot(pc,
    lab = paste0(pc$metadata$Age,'years'),
    colby = 'ER',
    legendPosition = 'right')

Warning: ggrepel: 36 unlabeled data points (too many overlaps). Consider
increasing max.overlaps

A biplot of PC2 against PC1 in the gene expression data, coloured by ER status with patient ages overlaid.

It appears that one cluster has more ER+ samples than the other group.

So far, we have only looked at a biplot of PC1 versus PC2 which only gives part of the picture. The pairplots() function in PCAtools can be used to create multiple biplots including different principal components.

pairsplot(pc)

Multiple biplots produced by pairsplot().

The plots show two apparent clusters involving the first principal component only. No other clusters are found involving other principal components. Each dot is coloured differently along a gradient of blues. This can potentially help identifying the same observation/individual in several panels. Here too, the argument colby allows you to set custom colours.

Finally, it can sometimes be of interest to compare how certain variables contribute to different principal components. This can be visualised with plotloadings() from the PCAtools package. The function checks the range of loadings for each principal component specified (default: first five principal components). It then selects the features in the top and bottom 5% of these ranges and displays their loadings. This behaviour can be adjusted with the rangeRetain argument, which has 0.1 as the default value (i.e. 5% on each end of the range). NB, if there are too many labels to be plotted, you will see a warning. This is not a serious problem.

plotloadings(pc, c("PC1"), rangeRetain = 0.1)

Plot of the features in the top and bottom 5 % of the loadings range for the first principal component.

plotloadings(pc, c("PC2"), rangeRetain = 0.1)

Plot of the features in the top and bottom 5 % of the loadings range for the second principal component.

plotloadings(pc, c("PC1", "PC2"), rangeRetain = 0.1)

Plot of the features in the top and bottom 5 % of the loadings range for the either the first or second principal components.

You can see how the third code line produces more dots, some of which do not have extreme loadings. This is because all loadings selected for any principal component are shown for all other principal components. For instance, it is plausible that features which have high loadings on PC1 may have lower ones on PC2.

Using PCA output in further analysis

The output of PCA can be used to interpret data or can be used in further analyses. For example, the PCA outputs new variables (principal components) which represent several variables in the original dataset. These new variables are useful for further exploring data, for example, comparing principal component scores between groups or including the new variables in linear regressions. Because the principal components are uncorrelated (and independent) they can be included together in a single linear regression.

Principal component regression

PCA is often used to reduce large numbers of correlated variables into fewer uncorrelated variables that can then be included in linear regression or other models. This technique is called principal component regression (PCR) and it allows researchers to examine the effect of several correlated explanatory variables on a single response variable in cases where a high degree of correlation initially prevents them from being included in the same model. Principal component regression (PCR) is just one example of how principal components can be used in further analysis of data. When carrying out PCR, the variable of interest (response/dependent variable) is regressed against the principal components calculated using PCA, rather than against each individual explanatory variable from the original dataset. As there as many principal components created from PCA as there are variables in the dataset, we must select which principal components to include in PCR. This can be done by examining the amount of variation in the data explained by each principal component (see above).

Further reading

• James, G., Witten, D., Hastie, T. & Tibshirani, R. (2013) An Introduction to Statistical Learning with Applications in R. Chapter 6.3 (Dimension Reduction Methods), Chapter 10 (Unsupervised Learning).
• Jolliffe, I.T. & Cadima, J. (2016) Principal component analysis: a review and recent developments. Phil. Trans. R. Soc A 374..
• Johnstone, I.M. & Titterington, D.M. (2009) Statistical challenges of high-dimensional data. Phil. Trans. R. Soc A 367.
• PCA: A Practical Guide to Principal Component Analysis, Analytics Vidhya.
• A One-Stop Shop for Principal Component Analysis, Towards Data Science.

Key Points

• A principal component analysis is a statistical approach used to reduce dimensionality in high-dimensional datasets (i.e. where $p$ is equal or greater than $n$).

• PCA may be used to create a low-dimensional set of features from a larger set of variables. Examples of when a PCA may be useful include reducing high-dimensional datasets to fewer variables for use in a linear regression and for identifying groups with similar features.

• PCA is a useful dimensionality reduction technique used in the analysis of complex biological datasets (e.g. high throughput data or genetics data).

• The first principal component represents the dimension along which there is maximum variation in the data. Subsequent principal components represent dimensions with progressively less variation.

• Scree plots and biplots may be used to show: 1. how much variation in the data is explained by each principal component and 2. how data points cluster according to principal component scores and which variables are associated with these scores.

Factor analysis

Overview

Teaching: 30 min
Exercises: 10 min

Questions

• What is factor analysis and when can it be used?

• What are communality and uniqueness in factor analysis?

• How to decide on the number of factors to use?

• How to interpret the output of factor analysis?

Objectives

• Perform a factor analysis on high-dimensional data.

• Select an appropriate number of factors.

• Interpret the output of factor analysis.

Introduction

Biologists often encounter high-dimensional datasets from which they wish to extract underlying features – they need to carry out dimensionality reduction. The last episode dealt with one method to achieve this, called principal component analysis (PCA), which expressed new dimension-reduced components as linear combinations of the original features in the dataset. Principal components can therefore be difficult to interpret. Here, we introduce a related but more interpretable method called factor analysis (FA), which constructs new components, called factors, that explicitly represent underlying (latent) constructs in our data. Like PCA, FA uses linear combinations, but uses latent constructs instead. FA is therefore often more interpretable and useful when we would like to extract meaning from our dimension-reduced set of variables.

There are two types of FA, called exploratory and confirmatory factor analysis (EFA and CFA). Both EFA and CFA aim to reproduce the observed relationships among a group of features with a smaller set of latent variables. EFA is used in a descriptive (exploratory) manner to uncover which measured variables are reasonable indicators of the various latent dimensions. In contrast, CFA is conducted in an a priori, hypothesis-testing manner that requires strong empirical or theoretical foundations. We will mainly focus on EFA here, which is used to group features into a specified number of latent factors.

Unlike with PCA, researchers using FA have to specify the number of latent variables (factors) at the point of running the analysis. Researchers may use exploratory data analysis methods (including PCA) to provide an initial estimate of how many factors adequately explain the variation observed in a dataset. In practice, a range of different values is usually tested.

Motivating example: student scores

One scenario for using FA would be whether student scores in different subjects can be summarised by certain subject categories. Take a look at the hypothetical dataset below. If we were to run and EFA on this, we might find that the scores can be summarised well by two factors, which we can then interpret. We have labelled these hypothetical factors “mathematical ability” and “writing ability”.

Student scores data across several subjects with hypothesised factors.

So, EFA is designed to identify a specified number of unobservable factors from observable features contained in the original dataset. This is slightly different from PCA, which does not do this directly. Just to recap, PCA creates as many principal components as there are features in the dataset, each component representing a different linear combination of features. The principal components are ordered by the amount of variance they account for.

Prostate cancer patient data

We revisit the [prostate]((https://carpentries-incubator.github.io/high-dimensional-stats-r/data/index.html) dataset of 97 men who have prostate cancer. Although not strictly a high-dimensional dataset, as with other episodes, we use this dataset to explore the method.

In this example, we use the clinical variables to identify factors representing various clinical variables from prostate cancer patients. Two principal components have already been identified as explaining a large proportion of variance in the data when these data were analysed in the PCA episode. We may expect a similar number of factors to exist in the data.

Let’s subset the data to just include the log-transformed clinical variables for the purposes of this episode:

library("here")
prostate <- readRDS(here("data/prostate.rds"))

nrow(prostate)

[1] 97

head(prostate)

      lcavol  lweight age      lbph svi       lcp gleason pgg45       lpsa
1 -0.5798185 2.769459  50 -1.386294   0 -1.386294       6     0 -0.4307829
2 -0.9942523 3.319626  58 -1.386294   0 -1.386294       6     0 -0.1625189
3 -0.5108256 2.691243  74 -1.386294   0 -1.386294       7    20 -0.1625189
4 -1.2039728 3.282789  58 -1.386294   0 -1.386294       6     0 -0.1625189
5  0.7514161 3.432373  62 -1.386294   0 -1.386294       6     0  0.3715636
6 -1.0498221 3.228826  50 -1.386294   0 -1.386294       6     0  0.7654678

#select five log-transformed clinical variables for further analysis
pros2 <- prostate[, c("lcavol", "lweight", "lbph", "lcp", "lpsa")]
head(pros2)

      lcavol  lweight      lbph       lcp       lpsa
1 -0.5798185 2.769459 -1.386294 -1.386294 -0.4307829
2 -0.9942523 3.319626 -1.386294 -1.386294 -0.1625189
3 -0.5108256 2.691243 -1.386294 -1.386294 -0.1625189
4 -1.2039728 3.282789 -1.386294 -1.386294 -0.1625189
5  0.7514161 3.432373 -1.386294 -1.386294  0.3715636
6 -1.0498221 3.228826 -1.386294 -1.386294  0.7654678

Performing exploratory factor analysis

EFA may be implemented in R using the factanal() function from the stats package (which is a built-in package in base R). This function fits a factor analysis by maximising the log-likelihood using a data matrix as input. The number of factors to be fitted in the analysis is specified by the user using the factors argument.

Challenge 1 (3 mins)

Use the factanal() function to identify the minimum number of factors necessary to explain most of the variation in the data

Solution

# Include one factor only
pros_fa <- factanal(pros2, factors = 1)
pros_fa

Call:
factanal(x = pros2, factors = 1)

Uniquenesses:
 lcavol lweight    lbph     lcp    lpsa 
  0.149   0.936   0.994   0.485   0.362 

Loadings:
        Factor1
lcavol  0.923  
lweight 0.253  
lbph           
lcp     0.718  
lpsa    0.799  

               Factor1
SS loadings      2.074
Proportion Var   0.415

Test of the hypothesis that 1 factor is sufficient.
The chi square statistic is 29.81 on 5 degrees of freedom.
The p-value is 1.61e-05 

# p-value <0.05 suggests that one factor is not sufficient 
# we reject the null hypothesis that one factor captures full
# dimensionality in the dataset

# Include two factors
pros_fa <- factanal(pros2, factors = 2)
pros_fa

Call:
factanal(x = pros2, factors = 2)

Uniquenesses:
 lcavol lweight    lbph     lcp    lpsa 
  0.121   0.422   0.656   0.478   0.317 

Loadings:
        Factor1 Factor2
lcavol   0.936         
lweight  0.165   0.742 
lbph             0.586 
lcp      0.722         
lpsa     0.768   0.307 

               Factor1 Factor2
SS loadings      2.015   0.992
Proportion Var   0.403   0.198
Cumulative Var   0.403   0.601

Test of the hypothesis that 2 factors are sufficient.
The chi square statistic is 0.02 on 1 degree of freedom.
The p-value is 0.878 

# p-value >0.05 suggests that two factors is sufficient 
# we cannot reject the null hypothesis that two factors captures
# full dimensionality in the dataset

#Include three factors
pros_fa <- factanal(pros2, factors = 3)

Error in factanal(pros2, factors = 3): 3 factors are too many for 5 variables

# Error shows that fitting three factors are not appropriate
# for only 5 variables (number of factors too high)

The output of factanal() shows the loadings for each of the input variables associated with each factor. The loadings are values between -1 and 1 which represent the relative contribution each input variable makes to the factors. Positive values show that these variables are positively related to the factors, while negative values show a negative relationship between variables and factors. Loading values are missing for some variables because R does not print loadings less than 0.1.

How many factors do we need?

There are numerous ways to select the “best” number of factors. One is to use the minimum number of features that does not leave a significant amount of variance unaccounted for. In practice, we repeat the factor analysis for different numbers of factors (by specifying different values in the factors argument). If we have an idea of how many factors there will be before analysis, we can start with that number. The final section of the analysis output then shows the results of a hypothesis test in which the null hypothesis is that the number of factors used in the model is sufficient to capture most of the variation in the dataset. If the p-value is less than our significance level (for example 0.05), we reject the null hypothesis that the number of factors is sufficient and we repeat the analysis with more factors. When the p-value is greater than our significance level, we do not reject the null hypothesis that the number of factors used captures variation in the data. We may therefore conclude that this number of factors is sufficient.

Like PCA, the fewer factors that can explain most of the variation in the dataset, the better. It is easier to explore and interpret results using a smaller number of factors which represent underlying features in the data.

Variance accounted for by factors - communality and uniqueness

The communality of a variable is the sum of its squared loadings. It represents the proportion of the variance in a variable that is accounted for by the FA model.

Uniqueness is the opposite of communality and represents the amount of variation in a variable that is not accounted for by the FA model. Uniqueness is calculated by subtracting the communality value from 1. If uniqueness is high for a given variable, that means this variable is not well explained/accounted for by the factors identified.

apply(pros_fa$loadings^2, 1, sum)  #communality

   lcavol   lweight      lbph       lcp      lpsa 
0.8793780 0.5782317 0.3438669 0.5223639 0.6832788 

1 - apply(pros_fa$loadings^2, 1, sum)  #uniqueness

   lcavol   lweight      lbph       lcp      lpsa 
0.1206220 0.4217683 0.6561331 0.4776361 0.3167212 

Visualising the contribution of each variable to the factors

Similar to a biplot as we produced in the PCA episode, we can “plot the loadings”. This shows how each original variable contributes to each of the factors we chose to visualise.

#First, carry out factor analysis using two factors
pros_fa <- factanal(pros2, factors = 2)

#plot loadings for each factor
plot(
  pros_fa$loadings[, 1], 
  pros_fa$loadings[, 2],
  xlab = "Factor 1", 
  ylab = "Factor 2", 
  ylim = c(-1, 1),
  xlim = c(-1, 1),
  main = "Factor analysis of prostate data"
)
abline(h = 0, v = 0)

#add column names to each point
text(
  pros_fa$loadings[, 1] - 0.08, 
  pros_fa$loadings[, 2] + 0.08,
  colnames(pros2),
  col = "blue"
)

Factor 2 loadings versus factor 1 loadings for each feature.

Challenge 2 (3 mins)

Use the output from your factor analysis and the plots above to interpret the results of your analysis.

What variables are most important in explaining each factor? Do you think this makes sense biologically? Consider or discuss in groups.

Solution

This plot suggests that the variables lweight and lbph are associated with high values on factor 2 (but lower values on factor 1) and the variables lcavol, lcp and lpsa are associated with high values on factor 1 (but lower values on factor 2). There appear to be two ‘clusters’ of variables which can be represented by the two factors.

The grouping of weight and enlargement (lweight and lbph) makes sense biologically, as we would expect prostate enlargement to be associated with greater weight. The groupings of lcavol, lcp, and lpsa also make sense biologically, as larger cancer volume may be expected to be associated with greater cancer spread and therefore higher PSA in the blood.

Advantages and disadvantages of Factor Analysis

There are several advantages and disadvantages of using FA as a dimensionality reduction method.

Advantages:

• FA is a useful way of combining different groups of data into known representative factors, thus reducing dimensionality in a dataset.
• FA can take into account researchers’ expert knowledge when choosing the number of factors to use, and can be used to identify latent or hidden variables which may not be apparent from using other analysis methods.
• It is easy to implement with many software tools available to carry out FA.
• Confirmatory FA can be used to test hypotheses.

Disadvantages:

• Justifying the choice of number of factors to use may be difficult if little is known about the structure of the data before analysis is carried out.
• Sometimes, it can be difficult to interpret what factors mean after analysis has been completed.
• Like PCA, standard methods of carrying out FA assume that input variables are continuous, although extensions to FA allow ordinal and binary variables to be included (after transforming the input matrix).

Further reading

• Gundogdu et al. (2019) Comparison of performances of Principal Component Analysis (PCA) and Factor Analysis (FA) methods on the identification of cancerous and healthy colon tissues. International Journal of Mass Spectrometry 445:116204.
• Kustra et al. (2006) A factor analysis model for functional genomics. BMC Bioinformatics 7: doi:10.1186/1471-2105-7-21.
• Yong, A.G. & Pearce, S. (2013) A beginner’s guide to factor analysis: focusing on exploratory factor analysis. Tutorials in Quantitative Methods for Psychology 9(2):79-94.
• Confirmatory factor analysis can be carried out with the package Lavaan.
• A more sophisticated implementation of EFA is available in the packages EFA.dimensions and psych.

Key Points

• Factor analysis is a method used for reducing dimensionality in a dataset by reducing variation contained in multiple variables into a smaller number of uncorrelated factors.

• PCA can be used to identify the number of factors to initially use in factor analysis.

• The factanal() function in R can be used to fit a factor analysis, where the number of factors is specified by the user.

• Factor analysis can take into account expert knowledge when deciding on the number of factors to use, but a disadvantage is that the output requires careful interpretation.

K-means

Overview

Teaching: 60 min
Exercises: 20 min

Questions

• How do we detect real clusters in high-dimensional data?

• How does K-means work and when should it be used?

• How can we perform K-means in R?

• How can we appraise a clustering and test cluster robustness?

Objectives

• Understand what clusters look like in in high-dimensional data.

• Understand and perform K-means clustering in R.

• Assess clustering performance using silhouette scores.

• Assess cluster robustness using bootstrapping.

Introduction

As we saw in previous episodes, visualising high-dimensional data with a large amount of features is difficult and can limit our understanding of the data and associated processes. In some cases, a known grouping causes this heterogeneity (sex, treatment groups, etc). In other cases, heterogeneity may arise from the presence of unknown subgroups in the data. While PCA can be used to reduce the dimension of the dataset into a smaller set of uncorrelated variables and factor analysis can be used to identify underlying factors, clustering is a set of techniques that allow us to discover unknown groupings.

Cluster analysis involves finding groups of observations that are more similar to each other (according to some feature) than they are to observations in other groups and are thus likely to represent the same source of heterogeneity. Once groups (or clusters) of observations have been identified using cluster analysis, further analyses or interpretation can be carried out on the groups, for example, using metadata to further explore groups.

Cluster analysis is commonly used to discover unknown groupings in fields such as bioinformatics, genomics, and image processing, in which large datasets that include many features are often produced.

There are various ways to look for clusters of observations in a dataset using different clustering algorithms. One way of clustering data is to minimise distance between observations within a cluster and maximise distance between proposed clusters. Using this process, we can also iteratively update clusters so that we become more confident about the shape and size of the clusters.

What is K-means clustering?

K-means clustering groups data points into a user-defined number of distinct, non-overlapping clusters. To create clusters of ‘similar’ data points, K-means clustering creates clusters that minimise the within-cluster variation and thus the amount that data points within a cluster differ from each other. The distance between data points within a cluster is used as a measure of within-cluster variation.

To carry out K-means clustering, we first pick $k$ initial points as centres or “centroids” of our clusters. There are a few ways to choose these initial “centroids” and this is discussed below. Once we have picked intitial points, we then follow these two steps until appropriate clusters have been formed:

1. Assign each data point to the cluster with the closest centroid
2. Update centroid positions as the average of the points in that cluster.

We can see this process in action in this animation:

Animation showing the iterative process of K-means clustering on data y versus x.

While K-means has some advantages over other clustering methods (easy to implement and to understand), it does have some disadvantages, particularly difficulties in identifying initial clusters which observations belong to and the need for the user to specify the number of clusters that the data should be partitioned into.

Initialisation

The algorithm used in K-means clustering finds a local rather than a global optimum, so that results of clustering are dependent on the initial cluster that each observation is randomly assigned to. This initial configuration can have a significant effect on the final configuration of the clusters, so dealing with this limitation is an important part of K-means clustering. Some strategies to deal with this problem are:

• Choose $K$ points at random from the data as the cluster centroids.
• Randomly split the data into $K$ groups, and then average these groups.
• Use the K-means++ algorithm to choose initial values.

These each have advantages and disadvantages. In general, it’s good to be aware of this limitation of K-means clustering and that this limitation can be addressed by choosing a good initialisation method, initialising clusters manually, or running the algorithm from multiple different starting points.

Believing in clusters

When using clustering, it’s important to realise that data may seem to group together even when these groups are created randomly. It’s especially important to remember this when making plots that add extra visual aids to distinguish clusters. For example, if we cluster data from a single 2D normal distribution and draw ellipses around the points, these clusters suddenly become almost visually convincing. This is a somewhat extreme example, since there is genuinely no heterogeneity in the data, but it does reflect what can happen if you allow yourself to read too much into faint signals.

Let’s explore this further using an example. We create two columns of data (‘x’ and ‘y’) and partition these data into three groups (‘a’, ‘b’, ‘c’) according to data values. We then plot these data and their allocated clusters and put ellipses around the clusters using the stat_ellipse() function in ggplot.

Example of artificial clusters fitted to data points.

The randomly created data used here appear to form three clusters when we plot the data. Putting ellipses around the clusters can further convince us that the clusters are ‘real’. But how do we tell if clusters identified visually are ‘real’?

Initialisation

The algorithm used in K-means clustering finds a local rather than a global optimum, so that results of clustering are dependent on the initial cluster that each observation is randomly assigned to. This initial configuration can have a significant effect on the final configuration of the clusters, so dealing with this limitation is an important part of K-means clustering. Some strategies to deal with this problem are:

• Choose $k$ points at random from the data as the cluster centroids.
• Randomly split the data into $k$ groups, and then average these groups.
• Use the K-means++ algorithm to choose initial values.

These each have advantages and disadvantages. In general, it’s good to be aware of this limitation of K-means clustering and that this limitation can be addressed by choosing a good initialisation method, initialising clusters manually, or running the algorithm from multiple different starting points.

K-means clustering applied to the single-cell RNA sequencing data

Let’s carry out K-means clustering in R using some real high-dimensional data. We’re going to work with single-cell RNA sequencing data, scRNAseq, in these clustering challenges, which is often very high-dimensional. Commonly, experiments profile the expression level of 10,000+ genes in thousands of cells. Even after filtering the data to remove low quality observations, the dataset we’re using in this episode contains measurements for over 9,000 genes in over 3,000 cells.

One way to get a handle on a dataset of this size is to use something we covered earlier in the course - dimensionality reduction. Dimensionality reduction allows us to visualise this incredibly complex data in a small number of dimensions. In this case, we’ll be using principal component analysis (PCA) to compress the data by identifying the major axes of variation in the data, before running our clustering algorithms on this lower-dimensional data to explore the similarity of features.

The scater package has some easy-to-use tools to calculate a PCA for SummarizedExperiment objects. Let’s load the scRNAseq data and calculate some principal components.

library("SingleCellExperiment")
library("scater")

scrnaseq <- readRDS(here::here("data/scrnaseq.rds"))
scrnaseq <- runPCA(scrnaseq, ncomponents = 15)
pcs <- reducedDim(scrnaseq)[, 1:2]

The first two principal components capture almost 50% of the variation within the data. For now, we’ll work with just these two principal components, since we can visualise those easily, and they’re a quantitative representation of the underlying data, representing the two largest axes of variation.

We can now run K-means clustering on the first and second principal components of the scRNAseq data using the kmeans() function.

set.seed(42)
cluster <- kmeans(pcs, centers = 4)
scrnaseq$kmeans <- as.character(cluster$cluster)
plotReducedDim(scrnaseq, "PCA", colour_by = "kmeans")

Scatter plot of principal component 2 versus principal component 1 with points colour coded according to the cluster to which they belong.

We can see that this produces a sensible-looking partition of the data. However, is it totally clear whether there might be more or fewer clusters here?

Challenge 1

Perform clustering to group the data into $k=5$ clusters, and plot it using plotReducedDim(). Save this with a variable name that’s different to what we just used, because we’ll use this again later.

Solution

set.seed(42)
cluster5 <- kmeans(pcs, centers = 5)
scrnaseq$kmeans5 <- as.character(cluster5$cluster)
plotReducedDim(scrnaseq, "PCA", colour_by = "kmeans5")

Scatter plot of principal component 2 against principal component 1 in the scRNAseq data, coloured by clusters produced by k-means clustering.

K-medoids (PAM)

One problem with K-means is that using the mean to define cluster centroids means that clusters can be very sensitive to outlying observations. K-medoids, also known as “partitioning around medoids (PAM)” is similar to K-means, but uses the median rather than the mean as the method for defining cluster centroids. Using the median rather than the mean reduces sensitivity of clusters to outliers in the data. K-medioids has had popular application in genomics, for example the well-known PAM50 gene set in breast cancer, which has seen some prognostic applications. The following example shows how cluster centroids differ when created using medians rather than means.

x <- rnorm(20)
y <- rnorm(20)
x[10] <- x[10] + 10
plot(x, y, pch = 16)
points(mean(x), mean(y), pch = 16, col = "firebrick")
points(median(x), median(y), pch = 16, col = "dodgerblue")

Scatter plot of random data y versus x with the (mean(x), mean(y)) co-ordinate point shown in red and the (median(x), median(y)) co-ordinate point shown in blue.

PAM can be carried out using pam() from the cluster package.

Cluster separation

When performing clustering, it is important for us to be able to measure how well our clusters are separated. One measure to test this is silhouette width, which measures how similar a data point is to points in the same cluster compared to other clusters. The silhouette width is computed for every observation and can range from -1 to 1. A high silhouette width means an observation is closer to other observations within the same cluster. For each cluster, the silhouette widths can then be averaged or an overall average can be taken.

More detail on silhouette widths

In more detail, each observation’s silhouette width is computed as follows:

1. Compute the average distance between the focal observation and all other observations in the same cluster.
2. For each of the other clusters, compute the average distance between focal observation and all observations in the other cluster. Keep the smallest of these average distances.
3. Subtract (1.)-(2.) then divivde by whichever is smaller (1.) or (2).

Ideally, we would have only large positive silhouette widths, indicating that each data point is much more similar to points within its cluster than it is to the points in any other cluster. However, this is rarely the case. Often, clusters are very fuzzy, and even if we are relatively sure about the existence of discrete groupings in the data, observations on the boundaries can be difficult to confidently place in either cluster.

Here we use the silhouette() function from the cluster package to calculate the silhouette width of our K-means clustering using a distance matrix of distances between points in the clusters.

library("cluster")
dist_mat <- dist(pcs)
sil <- silhouette(cluster$cluster, dist = dist_mat)
plot(sil, border = NA)

Silhouette plot for each point according to cluster.

Let’s plot the silhouette score on the original dimensions used to cluster the data. Here, we’re mapping cluster membership to point shape, and silhouette width to colour.

pc <- as.data.frame(pcs)
colnames(pc) <- c("x", "y")
pc$sil <- sil[, "sil_width"]
pc$clust <- factor(cluster$cluster)
mean(sil[, "sil_width"])

[1] 0.7065662

ggplot(pc) +
    aes(x, y, shape = clust, colour = sil) +
    geom_point() +
    scale_colour_gradient2(
        low = "dodgerblue", high = "firebrick"
    ) +
    scale_shape_manual(
        values = setNames(1:4, 1:4)
    )

Scatter plot of y versus x coloured according to silhouette width and point characters grouped according to cluster membership.

This plot shows that silhouette values for individual observations tends to be very high in the centre of clusters, but becomes quite low towards the edges. This makes sense, as points that are “between” two clusters may be more similar to points in another cluster than they are to the points in the cluster one they belong to.

Challenge 2

Calculate the silhouette width for the k of 5 clustering we did earlier. Are 5 clusters appropriate? Why/why not?

Can you identify where the differences lie?

Solution

sil5 <- silhouette(cluster5$cluster, dist = dist_mat)
scrnaseq$kmeans5 <- as.character(cluster5$cluster)
plotReducedDim(scrnaseq, "PCA", colour_by = "kmeans5")

Scatter plot of principal component 2 against principal component 1 in the scRNAseq data, coloured by clusters produced by k-means clustering.

mean(sil5[, "sil_width"])

[1] 0.5849979

The average silhouette width is lower when k=5.

plot(sil5, border = NA)

Silhouette plot for each point according to cluster.

This seems to be because some observations in clusters 3 and 5 seem to be more similar to other clusters than the one they have been assigned to. This may indicate that k is too high.

Gap statistic

Another measure of how good our clustering is is the “gap statistic”. This compares the observed squared distance between observations in a cluster and the centre of the cluster to an “expected” squared distances. The expected distances are calculated by randomly distributing cells within the range of the original data. Larger values represent lower squared distances within clusters, and thus better clustering. We can see how this is calculated in the following example.

library("cluster")
gaps <- clusGap(pcs, kmeans, K.max = 20, iter.max = 20)
best_k <- maxSE(gaps$Tab[, "gap"], gaps$Tab[, "SE.sim"])
best_k
plot(gaps$Tab[,"gap"], xlab = "Number of clusters", ylab = "Gap statistic")
abline(v = best_k, col = "red")

Cluster robustness: bootstrapping

When we cluster data, we want to be sure that the clusters we identify are not a result of the exact properties of the input data. That is, we want to ensure that the clusters identified do not change substantially if the observed data change slightly. This makes it more likely that the clusters can be reproduced.

To assess this, we can bootstrap. We first resample the data with replacement to reproduce a ‘new’ data set. We can then calculate new clusters for this data set and compare these to those on the original data set, thus helping us to see how the clusters may change for small changes in the data. This is maybe easier to see with an example. First, we define some data:

data <- 1:5

Then, we can take a sample from this data without replacement:

sample(data, 5)

[1] 4 1 3 5 2

This sample is a subset of the original data, and points are only present once. This is the case every time even if we do it many times:

## Each column is a sample
replicate(10, sample(data, 5))

     [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,]    5    2    5    2    3    1    3    5    5     3
[2,]    4    5    4    5    4    4    1    3    1     2
[3,]    2    1    1    3    2    5    2    2    3     4
[4,]    1    4    2    1    5    3    5    1    2     5
[5,]    3    3    3    4    1    2    4    4    4     1

However, if we sample with replacement, then sometimes individual data points are present more than once.

replicate(10, sample(data, 5, replace = TRUE))

     [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,]    3    1    2    2    1    3    3    2    4     2
[2,]    1    3    2    4    2    5    2    1    2     5
[3,]    5    5    4    4    2    2    1    1    1     3
[4,]    1    1    4    2    1    4    4    5    5     4
[5,]    3    1    2    1    4    2    5    3    3     2

Bootstrapping

Bootstrapping is a powerful and common statistical technique.

We would like to know about the sampling distribution of a statistic, but we don’t have any knowledge of its behaviour under the null hypothesis.

For example, we might want to understand the uncertainty around an estimate of the mean of our data. To do this, we could resample the data with replacement and calculate the mean of each average.

boots <- replicate(1000, mean(sample(data, 5, replace = TRUE)))
hist(boots,
    breaks = "FD",
    main = "1,000 bootstrap samples",
    xlab = "Mean of sample"
 )

Histogram of mean of bootstapped samples.

In this case, the example is simple, but it’s possible to devise more complex statistical tests using this kind of approach.

The bootstrap, along with permutation testing, can be a very flexible and general solution to many statistical problems.

In applying the bootstrap to clustering, we want to see two things:

1. Will observations within a cluster consistently cluster together in different bootstrap replicates?
2. Will observations frequently swap between clusters?

In the plot below, the diagonal of the plot shows how often the clusters are reproduced in boostrap replicates. High scores on the diagonal mean that the clusters are consistently reproduced in each boostrap replicate. Similarly, the off-diagonal elements represent how often observations swap between clusters in bootstrap replicates. High scores indicate that observations rarely swap between clusters.

library("pheatmap")
library("bluster")
library("viridis")

km_fun <- function(x) {
    kmeans(x, centers = 4)$cluster
}
ratios <- bootstrapStability(pcs, FUN = km_fun, clusters = cluster$cluster)
pheatmap(ratios,
    cluster_rows = FALSE, cluster_cols = FALSE,
    col = viridis(10),
    breaks = seq(0, 1, length.out = 10)
)

Grid of empirical cluster swapping behaviour estimated by the bootstrap samples.

Yellow boxes indicate values slightly greater than 1, which may be observed. These are “good” (despite missing in the colour bar).

Challenge 3

Repeat the bootstrapping process with k=5. Do the results appear better or worse? Can you identify where the differences occur on the plotReducedDim()?

Solution

km_fun5 <- function(x) {
    kmeans(x, centers = 5)$cluster
}
set.seed(42)
ratios5 <- bootstrapStability(pcs, FUN = km_fun5, clusters = cluster5$cluster)
pheatmap(ratios5,
    cluster_rows = FALSE, cluster_cols = FALSE,
    col = viridis(10),
    breaks = seq(0, 1, length.out = 10)
)

Grid of empirical cluster swapping behaviour estimated by the bootstrap samples.

When k=5, we can see that the values on the diagonal of the matrix are smaller, indicating that the clusters aren’t exactly reproducible in the bootstrap samples.

Similarly, the off-diagonal elements are considerably lower for some elements. This indicates that observations are “swapping” between these clusters in bootstrap replicates.

Consensus clustering

One useful and generic method of clustering is consensus clustering. This method can use k-means or other clustering methods.

The idea behind this is to bootstrap the data repeatedly, and cluster it each time, perhaps using different numbers of clusters. If a pair of data points always end up in the same cluster, it’s likely that they really belong to the same underlying cluster.

This is really computationally demanding but has been shown to perform very well in some situations. It also allows you to visualise how cluster membership changes over different values of k.

Speed

It’s worth noting that a lot of the methods we’ve discussed here are very computationally demanding. When clustering data, we may have to compare points to each other many times. This becomes more and more difficult when we have many observations and many features. This is especially problematic when we want to do things like bootstrapping that requires us to cluster the data over and over.

As a result, there are a lot of approximate methods for finding clusters in the data. For example, the mbkmeans package includes an algorithm for clustering extremely large data. The idea behind this algorithm is that if the clusters we find are robust, we don’t need to look at all of the data every time. This is very helpful because it reduces the amount of data that needs to be held in memory at once, but also because it minimises the computational cost.

Similarly, approximate nearest neighbour methods like Annoy can be used to identify what the $k$ closest points are in the data, and this can be used in some clustering methods (for example, graph-based clustering).

Generally, these methods sacrifice a bit of accuracy for a big gain in speed.

Further reading

• Wu, J. (2012) Cluster analysis and K-means clustering: An Introduction. In: Advances in K-means Clustering. Springer Berlin, Heidelberg..
• Modern statistics for modern biology, Susan Holmes and Wolfgang Huber (Chapter 5).
• Understanding K-means clustering in machine learning, Towards Data Science.

Key Points

• K-means is an intuitive algorithm for clustering data.

• K-means has various advantages but can be computationally intensive.

• Apparent clusters in high-dimensional data should always be treated with some scepticism.

• Silhouette width and bootstrapping can be used to assess how well our clustering algorithm has worked.

Hierarchical clustering

Overview

Teaching: 70 min
Exercises: 20 min

Questions

• What is hierarchical clustering and how does it differ from other clustering methods?

• How do we carry out hierarchical clustering in R?

• What distance matrix and linkage methods should we use?

• How can we validate identified clusters?

Objectives

• Understand when to use hierarchical clustering on high-dimensional data.

• Perform hierarchical clustering on high-dimensional data and evaluate dendrograms.

• Understand and explore different distance matrix and linkage methods.

• Use the Dunn index to validate clustering methods.

Why use hierarchical clustering on high-dimensional data?

When analysing high-dimensional data in the life sciences, it is often useful to identify groups of similar data points to understand more about the relationships within the dataset. General clustering algorithms group similar data points (or observations) into groups (or clusters). This results in a set of clusters, where each cluster is distinct, and the data points within each cluster have similar characteristics. The clustering algorithm works by iteratively grouping data points so that different clusters may exist at different stages of the algorithm’s progression.

Here, we describe hierarchical clustering. Unlike K-means clustering, hierarchical clustering does not require the number of clusters $k$ to be specified by the user before the analysis is carried out.

Hierarchical clustering also provides an attractive dendrogram, a tree-like diagram showing the degree of similarity between clusters. The dendrogram is a key feature of hierarchical clustering. This tree-shaped graph allows the similarity between data points in a dataset to be visualised and the arrangement of clusters produced by the analysis to be illustrated. Dendrograms are created using a distance (or dissimilarity) that quantify how different are pairs of observations, and a clustering algorithm to fuse groups of similar data points together.

In this episode we will explore hierarchical clustering for identifying clusters in high-dimensional data. We will use agglomerative hierarchical clustering (see box) in this episode.

Agglomerative and Divisive hierarchical clustering

There are two main methods of carrying out hierarchical clustering: agglomerative clustering and divisive clustering. The former is a ‘bottom-up’ approach to clustering whereby the clustering approach begins with each data point (or observation) being regarded as being in its own separate cluster. Pairs of data points are merged as we move up the tree. Divisive clustering is a ‘top-down’ approach in which all data points start in a single cluster and an algorithm is used to split groups of data points from this main group.

The agglomerative hierarchical clustering algorithm

To start with, we measure distance (or dissimilarity) between pairs of observations. Initially, and at the bottom of the dendrogram, each observation is considered to be in its own individual cluster. We start the clustering procedure by fusing the two observations that are most similar according to a distance matrix. Next, the next-most similar observations are fused so that the total number of clusters is number of observations minus 2 (see panel below). Groups of observations may then be merged into a larger cluster (see next panel below, green box). This process continues until all the observations are included in a single cluster.

Example data showing two clusters of observation pairs.

Example data showing fusing of one observation into larger cluster.

A motivating example

To motivate this lesson, let’s first look at an example where hierarchical clustering is really useful, and then we can understand how to apply it in more detail. To do this, we’ll return to the large methylation dataset we worked with in the regression lessons. Let’s load and visually inspect the data:

library("minfi")
library("here")
library("ComplexHeatmap")

methyl <- readRDS(here("data/methylation.rds"))

# transpose this Bioconductor dataset to show features in columns
methyl_mat <- t(assay(methyl))

Looking at a heatmap of these data, we may spot some patterns – many columns appear to have a similar methylation levels across all rows. However, they are all quite jumbled at the moment, so it’s hard to tell how many line up exactly.

Heatmap of methylation data.

Looking at a heatmap of these data, we may spot some patterns – looking at the vertical stripes, many columns appear to have a similar methylation levels across all rows. However, they are all quite jumbled at the moment, so it’s hard to tell how many line up exactly. In addition, it is challenging to tell how many groups containing similar methylation levels we may have or what the similarities and differences are between groups.

We can order these data to make the patterns more clear using hierarchical clustering. To do this, we can change the arguments we pass to Heatmap() from the ComplexHeatmap package. Heatmap() groups features based on dissimilarity (here, Euclidean distance) and orders rows and columns to show clustering of features and observations.

Heatmap(methyl_mat,
  name = "Methylation level",
  cluster_rows = TRUE, cluster_columns = TRUE,
  row_dend_width = unit(0.2, "npc"),
  column_dend_height = unit(0.2, "npc"),
  show_row_names = FALSE, show_column_names = FALSE,
  row_title="Individuals", column_title = "Methylation sites"
)

Heatmap of methylation data clustered by methylation sites and individuals.

We can see that clustering the features (CpG sites) results in an overall gradient of high to low methylation levels from left to right. Maybe more interesting is the fact that the rows (corresponding to individuals) are now grouped according to methylation patterns. For example, 12 samples seem to have lower methylation levels for a small subset of CpG sites in the middle, relative to all the other samples. It’s not clear without investigating further what the cause of this is – it could be a batch effect, or a known grouping (e.g., old vs young samples). However, clustering like this can be a useful part of exploratory analysis of data to build hypotheses.

Hierarchical clustering

Now, let’s cover the inner workings of hierarchical clustering in more detail. Hierarchical clustering is a type of clustering that also allows us to estimate the number of clusters. There are two things to consider before carrying out clustering:

• how to define dissimilarity between observations using a distance matrix, and
• how to define dissimilarity between clusters and when to fuse separate clusters.

Defining the dissimilarity between observations: creating the distance matrix

Agglomerative hierarchical clustering is performed in two steps: calculating the distance matrix (containing distances between pairs of observations) and iteratively grouping observations into clusters using this matrix.

There are different ways to specify a distance matrix for clustering:

• Specify distance as a pre-defined option using the method argument in dist(). Methods include euclidean (default), maximum and manhattan.
• Create a self-defined function which calculates distance from a matrix or from two vectors. The function should only contain one argument.

Of pre-defined methods of calculating the distance matrix, Euclidean is one of the most commonly used. This method calculates the shortest straight-line distances between pairs of observations.

Another option is to use a correlation matrix as the input matrix to the clustering algorithm. The type of distance matrix used in hierarchical clustering can have a big effect on the resulting tree. The decision of which distance matrix to use before carrying out hierarchical clustering depends on the type of data and question to be addressed.

Defining the dissimilarity between clusters: Linkage methods

The second step in performing hierarchical clustering after defining the distance matrix (or another function defining similarity between data points) is determining how to fuse different clusters.

Linkage is used to define dissimilarity between groups of observations (or clusters) and is used to create the hierarchical structure in the dendrogram. Different linkage methods of creating a dendrogram are discussed below.

hclust() supports various linkage methods (e.g complete, single, ward D, ward D2, average, median) and these are also supported within the Heatmap() function. The method used to perform hierarchical clustering in Heatmap() can be specified by the arguments clustering_method_rows and clustering_method_columns. Each linkage method uses a slightly different algorithm to calculate how clusters are fused together and therefore different clustering decisions are made depending on the linkage method used.

Complete linkage (the default in hclust()) works by computing all pairwise dissimilarities between data points in different clusters. For each pair of two clusters, it sets their dissimilarity to the maximum dissimilarity value observed between any of these clusters’ constituent points. The two clusters with smallest dissimilarity value are then fused.

Computing a dendrogram

Dendrograms are useful tools that plot the grouping of points and clusters into bigger clusters. We can create and plot dendrograms in R using hclust() which takes a distance matrix as input and creates a tree using hierarchical clustering. Here we create some example data to carry out hierarchical clustering.

Let’s generate 20 data points in 2D space. Each point is generated to belong to one of three classes/groups. Suppose we did not know which class data points belonged to and we want to identify these via cluster analysis. Let’s first generate and plot our data:

#First, create some example data with two variables x1 and x2
set.seed(450)
example_data <- data.frame(
    x1 = rnorm(20, 8, 4.5),
    x2 = rnorm(20, 6, 3.4)
)

#plot the data and name data points by row numbers
plot(example_data$x1, example_data$x2, type = "n")
text(
    example_data$x1,
    example_data$x2,
    labels = rownames(example_data),
    cex = 0.7
)

Scatter plot of randomly-generated data x2 versus x1.

## calculate distance matrix using euclidean distance
dist_m <- dist(example_data, method = "euclidean")

Hierarchical clustering carried out on the data can be used to produce a dendrogram showing how the data is partitioned into clusters. But how do we interpret this dendrogram? Let’s explore this using our example data in a Challenge.

Challenge 1

Use hclust() to implement hierarchical clustering using the distance matrix dist_m and the complete linkage method and plot the results as a dendrogram using plot(). Why is hierarchical clustering and the resulting dendrogram useful for performing clustering this case?

Solution:

clust <- hclust(dist_m, method = "complete")
plot(clust)

A dendrogram of the randomly-generated data.

Hierarchical clustering is particularly useful (compared to K-means) when we do not know the number of clusters before we perform clustering. It is useful in this case since we have assumed we do not already know what a suitable number of clusters may be.

A dendrogram, such as the one generated in Challenge 1, shows similarities/differences in distances between data points. Each vertical line at the bottom of the dendrogram (‘leaf’) represents one of the 20 data points. These leaves fuse into fewer vertical lines (‘branches’) as the height increases. Observations that are similar fuse into the same branches. The height at which any two data points fuse indicates how different these two points are. Points that fuse at the top of the tree are very different from each other compared with two points that fuse at the bottom of the tree, which are quite similar. You can see this by comparing the position of similar/dissimilar points according to the scatterplot with their position on the tree.

Identifying the number of clusters

To identify the number of clusters, we can make a horizontal cut through the dendrogram at a user-defined height. The sets of observations beneath this cut and single points representing clusters above can be thought of as distinct clusters. Equivalently, we can count the vertical lines we encounter crossing the horizontal cut and the number of single points above the cut. For example, a cut at height 10 produces 3 clusters for the dendrogram in Challenge 1, while a cut at height 4 produces 8 clusters.

Dendrogram visualisation

We can first visualise cluster membership by highlight branches in dendrograms. In this example, we calculate a distance matrix between samples in the methyl_mat dataset. We then draw boxes round clusters obtained with cutree().

## create a distance matrix using euclidean distance
distmat <- dist(methyl_mat)
## hierarchical clustering using complete method
clust <- hclust(distmat)
## plot resulting dendrogram
plot(clust)

## draw border around three clusters
rect.hclust(clust, k = 3, border = 2:6) #border argument specifies the colours
## draw border around two clusters
rect.hclust(clust, k = 2, border = 2:6)

Dendrogram with boxes around clusters.

We can also colour clusters downstream of a specified cut using color_branches() from the dendextend package.

## cut tree at height = 4
cut <- cutree(clust, h = 50)

library("dendextend")
avg_dend_obj <- as.dendrogram(clust)      
## colour branches of dendrogram depending on clusters
plot(color_branches(avg_dend_obj, h = 50))