Randomized Complete Block Designs

Overview

Questions

• What is randomized complete block design?

Objectives

• A randomized complete block design randomizes treatments to experimental units within the block.
• Blocking increases the precision of treatment comparisons.

Blocking of experimental units, as presented in an earlier episode on Experimental Design Principles, can be critical for successful and valuable experiments. Blocking increases precision in treatment comparisons relative to an unblocked experiment with the same experimental units. A randomized block design can be thought of as a set of separate completely randomized designs for comparing the same treatments. Each member of the set is a block, and within this block, each of the treatments is randomly assigned. We can assess treatment differences within each block and determine whether treatment differences are consistent from block to block. In other words, we can determine whether there is an interaction between block and treatment.

Design issues

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The first issue to consider in this case is whether or not to block the experiment. Blocks serve to control natural variation among experimental units, and randomization within blocks accounts for “nuisance” variables or traits that are likely associated with the response. Shelf height and resulting differences in illumination is one example of a nuisance variable. Other variables like age and sex are characteristics of experimental units that can influence the treatment response. Blocking by sex and other characteristics is a best practice in experimental design.

In an earlier episode on Completely Randomized Designs, we presented the Generation 100 Study of 3 exercise treatments on men and women from 70 to 77 years of age. In the actual study, participants were stratified (blocked) by sex and cohabitation status (living with someone vs. living alone) to form 4 blocks. The three treatment levels (control, moderate- and high-intensity exercise) were randomly assigned within each block. As such, we would analyze the experiment as designed in blocks. Let’s revisit these data with blocking by sex in mind. Read in the data again if needed.

R

heart_rate <- read_csv("data/simulated_heart_rates.csv")

View the heart rate data separated by sex.

R

heart_rate %>% ggplot(aes(exercise_group, heart_rate)) + 
  geom_boxplot() + 
  facet_grid(rows = vars(sex))

Each panel in the plot above is one block, one for females, one for males. What patterns do you see? Does there appear to be a difference in heart rates between sexes? between exercise groups? between sexes and exercise groups?

Let’s extract the means and standard deviations for exercise groups by sex.

R

g100meansSD <- heart_rate %>%
  group_by(sex, exercise_group) %>%
  summarise(meanChange = round(mean(heart_rate), 3),
            stDev = sd(heart_rate))
g100meansSD

OUTPUT

# A tibble: 6 × 4
# Groups:   sex [2]
  sex   exercise_group     meanChange stDev
  <chr> <chr>                   <dbl> <dbl>
1 F     control                  62.5  5.27
2 F     high intensity           63.0  5.58
3 F     moderate intensity       73.0  4.92
4 M     control                  66.1  5.36
5 M     high intensity           60.4  5.30
6 M     moderate intensity       70.0  5.02

Use these summary statistics in an interaction plot to determine if there is an interaction between exercise (treatment) and sex (block).

R

ggplot(g100meansSD, aes(x=exercise_group, y=meanChange, group=sex, color=sex)) + 
    geom_line() +
    geom_point() +
    geom_errorbar(aes(ymin=meanChange-stDev, ymax=meanChange+stDev), width=.2,
                  position=position_dodge(0.05), alpha=.5) +
  labs(y = "Heart rate",
       title = "Mean change in heart rate by exercise group and sex") 

It appears that there is an interaction between exercise and sex given that the lines cross over one another. The effect of exercise is different depending on sex. The F-test from an ANOVA will tell us whether this apparent interaction is real or random, specifically whether it is more pronounced than would be expected due to random variation.

R

anova(lm(heart_rate ~ exercise_group*sex, data = heart_rate))

OUTPUT

Analysis of Variance Table

Response: heart_rate
                     Df Sum Sq Mean Sq  F value  Pr(>F)
exercise_group        2  26960 13480.1 489.6147 < 2e-16 ***
sex                   1    176   175.6   6.3779 0.01165 *
exercise_group:sex    2   3587  1793.7  65.1510 < 2e-16 ***
Residuals          1560  42950    27.5
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Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

As before, we read the ANOVA table from the bottom up starting with the interaction exercise_group:sex. Since the interaction is significant, you should not compare sexes across exercise groups because the exercise effects are not the same across sexes. They are different for each sex.

The ANOVA table looks similar to the previous example involving drug dose and exercise in mice. There is an important distinction between the two ANOVA tables, however. In the drug dose and exercise ANOVA table, the interaction was between two treatments - drug dose and exercise. In this ANOVA table, the interaction is between block and treatment. The block, sex, is not a treatment. It’s a characteristic of the experimental units. In the drug dose experiment, the experimental units (mice) are homogeneous and the treatments were randomized to the experimental units once only in a completely randomized design. In this case, the experimental units are heterogeneous and a separate randomization of treatments was applied to each block of experimental units. This is a randomized block design. The purpose of the blocks is to remove a source of variation, (e.g. sex), from the comparison of treatments.

Sizing a randomized block experiment

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True replication

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Balanced incomplete block designs

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Key Points

• Replication, randomization and blocking determine the validity and usefulness of an experiment.