Q-mode vs. R-mode PCA I have some doubts on Q-mode and R-mode principal component analysis (PCA). I've read from different sources that:


*

*Q-mode PCA is equivalent to R-mode PCA of the transposed data matrix!

*Q-mode PCA (with squared Euclidean distance) is equivalent to R-mode PCA (of the covariance matrix)!


It seems to me that these two are not equivalent statements. Can someone clarify that? 
Q-mode(SEuclid) = R-mode(covar) is the only instance where (proper) Q-mode and R-mode PCAs give the same results?
If I perform an R-mode PCA on the transposed data matrix, wouldn't I work on a $n<p$ matrix? Would it be ok to perform a PCA in that case? If yes, what's the difference from performing R-mode PCA on a normal $n<p$ data matrix? If no, do I need more variables than observations for running Q-mode PCA?
 A: Discussions of Q, R, S, T, P and O as alternative modes of factor analyses are pretty rare these days. It's as if this typology has passed out of the literature. But if one can speak in terms of a "data cube" where each face of the cube is a different "mode" of information -- the unit of analysis, the description of the components and the approach to computing the association index -- then the typology unfolds from there. In other words, the alternatives can be related to the mode loaded or under analysis. Here's a table of how they can be interpreted:

Source: Dillon and Goldstein, Multivariate Statistics, p. 43
R-mode FAs are the most common type and, more commonly, are what most people refer to when speaking of FA or PCA. It's worth noting that, to your point, Q- and R-mode factor analyses flip modes of the data cube but they are agnostic wrt covariance vs correlation matrix inputs. Wrt Q-mode FAs, D&G write:

Q-mode FA has been employed in psychology and in other behavior
  sciences as a method for clustering persons. In Q-type analysis we
  interchange rows and columns of the basic data matrix so that the
  elements relate to the covariances or correlations between the
  individuals.

D&G go on to cite several problems with Q-mode FA that can complicate the assignments to clusters and the number of clusters it can create since the dimension of the matrix is limited to the min(of n,p).
