Would the group means of PC scores differ from the PC scores of group means?

Question

I have $2$ $n\times p$ matrices, where $n$ are the rows (samples), and $p$ the columns (measurements). Each matrix has samples and measurements from different groups. I call these the "raw" data. I've conducted a principal components analyses of the complete raw data, and computed the mean of each PC score by group. The latter I call the mean of the PC scores by group.

My question is whether the means of the PC scores by group (raw-data $\rightarrow$ PCA $\rightarrow$ mean PCs by group) would differ from the PC scores derived from a PCA conducted on the "raw" group means (raw data $\rightarrow$ mean by group $\rightarrow$ PCA)?

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Example analysis of simulated data

set.seed(123) 
a <- matrix(rnorm(900),ncol=3,byrow=F) 
a[1:100,] <- 4 + a[1:100,] 
a[101:200,] <- -4 + a[101:200,]
# compute PCA and extract PC scores
pc <- prcomp(a)$x 
    plot(pc[,1:2],col=rep(c("red","blue","green"),each=100))
    # compute PC means and plot
    m <-rbind(colMeans(pc[1:100,1:2]),colMeans(pc[101:200,1:2]),colMeans(pc[201:300,1‌​‌​:2]))
    points(m,col="black", pch=19,cex=1)
    # compute means of raw data by group
    b <- rbind(colMeans(a[1:100,]),colMeans(a[101:200,]),colMeans(a[201:300,]))
    # conduct PCA on "raw means" and plot 
    pc2 <- prcomp(b)$x
points(pc2[,1:2],col="black", pch=17,cex=1)

Answer 1

The answer to your question should logicaly be "Yes." The group means of the PC should differ from the PC of the means. This should happen for two reasons.

1. You're transforming your variables into PCs, which try to maximize the total interia. This depends on the spread of the data in the different variables.
2. Once you take the means you eliminate most of the inertia. The PCA will be much more "accurate" (not surprisingly almost 100% of the intertia is explained by the first component), but the composition of the PC will be different because there is no longer much intertia to explain.

You can think of this in terms of how a PCA operates. The PCA is computed using squared distances, maximized for the first PCA. If you remove all this variability, the estimations change.