Completely Randomized Design with More than One Treatment Factor

Last updated on 2025-05-30 | Edit this page

Overview

Questions

How is a CRD with more than one treatment factor designed and analyzed?

Objectives

When experiments are structured with two or more factors, these factors can be qualitative or quantitative. With two or more factors we face the same design issues. Which factors to choose? Which levels for each factor? A full factorial experiment includes all levels of all factors, which can become unwieldy when there are multiple levels for each factor. One option to manage an unwieldy design is to use only a fraction of the factor levels in a fractional factorial design. In this lesson we consider a full factorial design containing all levels of all factors.

A study aims to determine how dosage of a hypoglycemic drug and duration of daily exercise affect blood glucose levels in diabetic mice. The study has two quantitative factors with four levels each.

Drug dosage represents the amount of a new antidiabetic drug administered daily. The levels for this factor are in mg per kg body weight. Control mice receive no drug. The second factor, exercise duration, represents the number of minutes the mice run on a running wheel each day. Control mice do not have a running wheel to run on. A full factorial design is used, with each combination of drug dosage and exercise duration applied to a group of mice. For example, one group receives 5 mg/kg of the drug and exercises 15 minutes per day, another group receives 5 mg/kg and exercises 30 minutes per day, and so on.

There are 4 levels for each factor, leading to 16 treatment combinations (4 drug doses × 4 exercise durations). Each combination is replicated with a group of 5 mice, making the design balanced and allowing analysis of interactions. Fasting blood glucose level (mg/dL) was measured at the start and after 4 weeks of treatment. Load the data and summarize by mean change in glucose levels (Delta).

R

drugExercise <- read_csv("data/drugExercise.csv")
drugExercise$DrugDose <- as_factor(drugExercise$DrugDose)
drugExercise$Exercise <- as_factor(drugExercise$Exercise)

drugExercise %>%
   group_by(DrugDose, Exercise) %>%
  summarise(ChangeGlucose = mean(Delta))

OUTPUT

# A tibble: 9 × 3
# Groups:   DrugDose [3]
  DrugDose Exercise ChangeGlucose
  <fct>    <fct>            <dbl>
1 0        0              -0.0197
2 0        30             -0.542
3 0        60             -2.79
4 10       0              -4.75
5 10       30             -2.65
6 10       60             -1.47
7 20       0             -10.0
8 20       30             -5.40
9 20       60             -0.334

A heatmap is a good way to visualize the table of mean glucose changes. It shows the greatest changes with a drug dose of 20 mg/kg for 3 of the 4 exercise groups. The 5 mg/kg drug dosage group also shows a large change, but only when combined with 60 minutes of exercise per day.

Boxplots show the same pattern for 5 mg/kg drug dosage group combined with 60 minutes of exercise per day. They also show an increase in mean glucose with increasing exercise for the 20 mg/kg drug dosage group.

R

ggplot(drugExercise, aes(x = DrugDose, y = Delta, fill = Exercise)) +
  geom_boxplot() +
  labs(title = "Change in Glucose by Drug and Exercise",
       y = "Δ Glucose (mg/dL)", x = "Drug Dosage (mg/kg)")

Boxplots with exercise on the x-axis are not as easy to interpret since patterns for combinations of exercise and drug dose aren’t so apparent. Greater variability for some groups is apparent however. The length of the boxplots for the 0 mg/kg and 5 mg/kg drug dose groups indicates high within-group variability. The 20 mg/kg boxplots are more compact, indicating lesser variability within this group.

R

ggplot(drugExercise, aes(x = Exercise, y = Delta, fill = DrugDose)) +
  geom_boxplot() +
  labs(title = "Change in Glucose by Exercise and Drug",
       y = "Δ Glucose (mg/dL)", x = "Exercise duration (min/day)") +
  scale_fill_brewer(palette = "PuOr") # use a different color palette

Interaction between factors

We could analyze these data as if it were simply a completely randomized design with 16 treatments (4 drug doses and 4 exercise durations). The ANOVA would have 15 degrees of freedom for treatments and the F-test would tell us whether the variation among average changes in glucose levels for the 16 treatments was real or random. However, the factorial treatment structure lets us separate out the variability in glucose level changes among drug doses averaged over exercise durations. The ANOVA table would provide a sum of squares based on 3 degrees of freedom for the difference between the 4 treatment means (\(\bar{y}_i\)) and the pooled (overall) mean (\(\bar{y}\)).

Sum of squares for 16 treatments \(= n\sum(\bar{y}_i - \bar{y})^2\).

The sum of squares would capture the variability among the 4 drug dose levels. The variation among the 4 exercise levels would be captured similarly, with 3 degrees of freedom. That leaves 15 - 6 = 9 degrees of freedom left over. What variability do these remaining 9 degrees of freedom contain? The answer is interaction - the interaction between drug doses and exercise durations. Mean changes in glucose for each of the 16 treatments is given in the table below.

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We can visualize interactions for all combinations of drug dose and exercise duration with an interaction plot that shows mean change in glucose levels.

R

# Interaction plot
interaction_plot <- drugExercise %>%
  group_by(DrugDose, Exercise) %>%
  summarise(MeanChange = mean(Delta), .groups = "drop")

ggplot(interaction_plot, aes(x = as.numeric(as.character(DrugDose)),
                             y = MeanChange,
                             color = Exercise, group = Exercise)) +
  geom_line() +
  geom_point() +
  labs(title = "Interaction Plot",
       x = "Drug Dosage (mg/kg)",
       y = "Mean Δ Glucose (mg/dL)")

The interaction plot shows wide variation in mean glucose changes among the groups at a drug dose of 20 mg/kg. As we saw earlier with the boxplots, mean glucose increased with increasing exercise. For the 0 and 15 minute/day exercise groups, increasing drug dosage led to decreasing mean glucose levels. For the 30 and 60 minute/day exercise groups, increasing drug dosage did not decrease mean glucose levels appreciably, with one exception. At 5 mg/kg dosage, the 60 minute/day exercise group saw a strong decrease in mean blood glucose.

If we plot exercise on the x-axis, the same patterns show up differently.

R

ggplot(interaction_plot, aes(x = as.numeric(as.character(Exercise)),
                             y = MeanChange,
                             color = DrugDose, group = DrugDose)) +
  geom_line() +
  geom_point() +
  labs(title = "Interaction Plot",
       x = "Exercise (min/day)",
       y = "Mean Δ Glucose (mg/dL)") +
  scale_color_brewer(palette = "PuOr") # use a different color palette

This second interaction plot shows wide variation in mean glucose changes within the 0 min/day exercise group, showing that an increase in drug dosage decreased mean glucose. For the 60 min/day exercise group, mean glucose change was nearly equal with the exception of the 5 mg/kg drug dosage group. At 5 mg/kg dosage, the 60 minute/day exercise group saw a strong decrease in mean blood glucose. At a drug dose of 20 mg/kg, increasing exercise led to increased mean glucose.

If lines were parallel we could assume no interaction between drug and exercise. Since they are not parallel we should assume interaction between exercise and drug dose. 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

# DrugDose*Exercise is the interaction
# main effects (DrugDose and Exercise separately) are included
anova(lm(Delta ~ DrugDose*Exercise, 
         data = drugExercise))

OUTPUT

Analysis of Variance Table

Response: Delta
                  Df  Sum Sq Mean Sq F value    Pr(>F)
DrugDose           2 128.142  64.071  81.048 4.674e-14 ***
Exercise           2  87.515  43.757  55.352 1.041e-11 ***
DrugDose:Exercise  4 195.265  48.816  61.752 1.271e-15 ***
Residuals         36  28.459   0.791
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

We can read the ANOVA table from the bottom up, starting with the interaction (DrugDose:Exercise). The F value for the interaction is 61.75 on 4 and 36 degrees of freedom for the interaction and error (Residuals) respectively. The p-value (Pr(>F)) is very low and as such the interaction between exercise and drug dose is significant, backing up what we see in the interaction plots.

If we move up a row in the table to Exercise, the F test compares the mean changes across drug dose groups. The F value for exercise is 55.35 on 2 and 36 degrees of freedom for exercise and residuals respectively. The p-value is high at 0 and so exercise is not significant. Finally, we move up to the row containing DrugDose to find an F value of 81.05 and a very low p-value again. Drug dose averaged over exercise is significant.

A summary of the linear model reiterates the observations we see in the plots and ANOVA.

R

summary(lm(Delta ~ DrugDose*Exercise, 
           data = drugExercise))

OUTPUT


Call:
lm(formula = Delta ~ DrugDose * Exercise, data = drugExercise)

Residuals:
     Min       1Q   Median       3Q      Max
-2.70282 -0.32734  0.07042  0.44838  1.76615

Coefficients:
                      Estimate Std. Error t value Pr(>|t|)
(Intercept)           -0.01975    0.39762  -0.050   0.9607
DrugDose10            -4.72607    0.56233  -8.405 5.20e-10 ***
DrugDose20            -9.97549    0.56233 -17.740  < 2e-16 ***
Exercise30            -0.52262    0.56233  -0.929   0.3589
Exercise60            -2.77026    0.56233  -4.926 1.88e-05 ***
DrugDose10:Exercise30  2.62200    0.79525   3.297   0.0022 **
DrugDose20:Exercise30  5.11979    0.79525   6.438 1.81e-07 ***
DrugDose10:Exercise60  6.04992    0.79525   7.608 5.33e-09 ***
DrugDose20:Exercise60 12.43128    0.79525  15.632  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.8891 on 36 degrees of freedom
Multiple R-squared:  0.9352,	Adjusted R-squared:  0.9208
F-statistic: 64.98 on 8 and 36 DF,  p-value: < 2.2e-16

DrugDose20 is significant, as are the interactions between 20 mg/kg dosage and 30 and 60 min/day exercise groups.

The partitioning of treatments sums of squares into main effect (average) and interaction sums of squares is a result of the crossed factorial structure (orthogonality) of the two factors. The complete combinations of these two factors provides clean partitioning between main effects and interactions. This is not to say that designs that don’t have full combinations of factors can’t be analyzed to estimate main effects and interactions. They can be using generalized linear models.
The development of efficient and informative multifactor designs that cleanly separate main effects from interactions is one of the most important contributions of statistical experimental design.

Key Points

Completely randomized designs can be structured with two or more factors.
Random assignment of treatments to experimental units in a single homogeneous group is the same.
Factorial structure of the experiment requires different analyses, primarily ANOVA.