So, for example, using the Iris data and treating iris species as the predictor variable and sepal length, sepal width, petal length, and petal width as the dependent variables we get MANOVA output that looks like this:
set.seed(2) # Creating a matrix of the 4 dependent variables (DVs) Y <- as.matrix(iris[,c(1:4)]) # MANOVA looking at the effect of species on DVs summary(manova(Y ~ iris$Species)) # Df Pillai approx F num Df den Df Pr(>F) # iris$Species 2 1.1919 53.466 8 290 < 2.2e-16 ***
That seems to make sense. Species has a significant effect on our DVs. Now what if we add another predictor variable (a random one which we shouldn’t expect to have an effect on the DVs)?
# Creating a random dummy variable to be used as a predictor variable iris$random.dummy <- sample(x = c(0,1), size = 150, replace = TRUE) # MANOVA looking at the effect of species + our random dummy on DVs summary(manova(Y ~ iris$Species + iris$random.dummy)) # Df Pillai approx F num Df den Df Pr(>F) # iris$Species 2 1.19339 53.263 8 288 <2e-16 *** # iris$random.dummy 1 0.03784 1.406 4 143 0.2349
That also seems to make sense. Species is significant still, but our random dummy variable is not. Now what if we simply switch the order of those variables?
# Switching the order of our two predictor variables in the formula summary(manova(Y ~ iris$random.dummy + iris$Species)) # Df Pillai approx F num Df den Df Pr(>F) # iris$random.dummy 1 0.13031 5.357 4 143 0.0004764 *** # iris$Species 2 1.19526 53.470 8 288 < 2.2e-16 ***
Now, the Pillai’s trace and approximate F-values change and our random dummy variable has become significant.
So my questions are these.
Why do the results of a MANOVA change when the order of the predictor variables is changed?
What does this mean for those of us trying to use and interpret a MANOVA?