A factorial repeated measures ANOVA (or two-way repeated measures ANOVA) is quite similar to a factorial ANOVA with the difference that there is dependency between groups in the data set like in a repeated measures ANOVA. This means that subjects have been measured repeatedly in time or in different circumstances or treatments.

The null hypothesis H0 in this test is that all means are equal.

As for the “simple” repeated measures ANOVA, you may fit a linear mixed effect model with the function lme() that is found in the package nlme.

Let’s take an example where 10 rats (5 rats from strain A and 5 rats from strain B) are weighed 4 times with intervals of 4 weeks (week8 to week20). Here is the code for the dataframe:

# response variable
rat.weight <- c(166,168,155,159,151,166,170,160,162,153,220,230,223,233,229,262,274,267,283,274,261,275,264,280,282,343,354,351,359,349,297,311,305,315,303,375,399,388,405,395)

# predictor variables
rat.strain <- as.factor(rep(c(rep("strainA",5),rep("strainB",5)),4))
time.point <- as.factor(c(rep("week08",10), rep("week12",10), rep("week16",10), rep("week20",10)))

# individual ID
rat.ID <- as.factor(rep(c("rat01","rat02","rat03","rat04","rat05","rat06","rat07","rat08","rat09","rat10"),4))

#dataframe
my.dataframe <- data.frame(rat.ID,rat.strain,time.point,rat.weight)

Lets visualize the data with a line plot:

Assumptions

The assumptions are:

• each individual is represented in the form of a measurement in each of the tested groups (there cannot be any missing value),
• normality of distribution (use the Shapiro-Wilk),
• sphericity, which means that equality of variance is verified when comparing any two groups (all possible pairs of groups) in the experimental design (commonly checked via Mauchly’s test),
• groups contain no outliers.

Running the test

Since lme() is part of the package nlme(), you need to install and load it. Here is how to do it:

install.packages("nlme")
library(nlme)

We first need to fit the linear mixed effect model with lme(). The syntax is as follows: lme(response ~ predictor1 * predictor2, random=~1|subjects, data = my.dataframe), where response is the response variable, predictor1 and predictor2 are the predictor variables or factors (which categorize the observations), subjects is the variable that defines the individuals, and dataframe the name of the dataframe that contains the data. In our example, the predictor variables are time.point and rat.strain, the response variable is rat.weight, and the subjects are described by the variable rat.ID. We will store the model in the object model, and we finally compute and display the table for the analysis using anova():

model <- lme(rat.weight ~ time.point * rat.strain, random=~1|rat.ID, data=my.dataframe)
anova(model)

##                       numDF denDF   F-value p-value
## (Intercept)               1    24 22147.089  <.0001
## time.point                3    24  1750.535  <.0001
## rat.strain                1     8   217.521  <.0001
## time.point:rat.strain     3    24    94.598  <.0001

The large F-value and the very small p-value on the lines rat.strain and time.point indicate that there is indeed a main effect of both rat.strain and time.point on rat.weight. Further down, the last line of the table (time.point:rat.strain) also shows a very low p-value and thus indicates an interaction between time.point and rat.strain.

But this does not tell us anything about the groups which means are significantly different.

Indeed, this test tells you about main effects and interactions, but does not tell you which groups are significantly different from others. For that, we will need post hoc tests. We can run multiple comparisons in linear mixed effect models [factorial design].

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