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

  1. Q-mode PCA is equivalent to R-mode PCA of the transposed data matrix!
  2. 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?

amoeba
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Tiago
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    A similar question about R and Q mode PCA http://stats.stackexchange.com/q/20101/3277. – ttnphns Nov 26 '15 at 11:44
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    Your second point is not abput "R" vs "Q" mode but is about the equivalences found between PCA and PCoA (Principal coordinates analysis aka Torgerson's MDS). – ttnphns Nov 26 '15 at 11:56
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    As DJonhson points out these terms (R, Q etc) are obsolete nowadays. PCA is based on SVD and as such it is a form of Biplot. So, R and Q modes are potentially symmetric to each other. If no centerings or scalings were done on the data table prior the svd or if those transformations were done symmetrically on its rows and columns then then the results of one mode analysis is identical on the results of the other mode analysis. – ttnphns Nov 26 '15 at 12:21
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    Instead of using labels (R, Q...) which few people know one should better report thoroughly, what he considers random (and what fixed) units in the analysis (to average over, in the analysis): rows, columns or both; what centering and/or scaling is being applied to (rows, columns, both) what specifically type of scaling; and so on. – ttnphns Nov 26 '15 at 12:25
  • @ttnphns about Comment 2. What's the difference between PCA and PCoA? I understood from Norman MacLeod's "Minding Your Rs and Qs - Part 1" (http://www.palass.org/modules.php?name=palaeo_math&page=10) that Principal Coordinate Analysis was simply another name for Q mode Principal Component Analysis. If not, are they completely different procedures? (ps. I read the guidelines about not citing internet sources as references, but I didn't find this information anywhere else, and the original text is too long too reproduce here). – Tiago Dec 02 '15 at 12:31
  • @ttnphns comment 3 and Answer: If (even being obsolete terms) Q and R mode PCA are potentialy symmetric to each other, that means that I could 'potentialy' perform an R-mode PCA on a matrix with less rows than columns. Why then there is a prompt "that's impossible!" answer for this question? – Tiago Dec 02 '15 at 12:45
  • You always can do PCA with such n>p data matrix, you don't have to refer to R or Q notions to do it. PCA is tightly related to PCoA, and you don't need R or Q again. I just would not recommend to use these labels altogether for terminologic Occam razor's reason. – ttnphns Dec 02 '15 at 13:30
  • @ttnphns. In the last comment I meant n<p . Sorry for repeating those terms, I was just too used to them to mean PCA with variables on the columns (R) and PCA with samples on the columns (Q). – Tiago Dec 02 '15 at 14:37
  • @ttnphns. The overall goal of my question was to understand why people are usually advised not to perform PCA when they have more variables than samples (n<p), because they will have a matrix with more columns than raws. However, in the more usual case, when they have more samples than variables, transposing the data matrix to work on the samples, and getting so a resulting matrix with more columns than rows, is acceptable. – Tiago Dec 02 '15 at 14:38
  • There is nothing wrong doing PCA on n<p (yes, I meant this, not n>p, in my comment). Programmically, however, if you do PCA via eigen-decomposition it will be slower for n<p than for n>p. Therefore it is often advised to transpose the data first. Then, the eigenvalues are the same but the eigenvectors are left (row), not right (clolumn) ones, and so right eigenvectors are to compute indirectly (which is simple). This theme was discussed on this site numerous times. – ttnphns Dec 02 '15 at 15:22
  • Maybe they are obsolete, but they are still in the documentation for princomp in R: "princomp only handles so-called R-mode PCA, that is feature extraction of variables. If a data matrix is supplied (possibly via a formula) it is required that there are at least as many units as variables. For Q-mode PCA use prcomp." – Peter Flom Jul 09 '19 at 12:45

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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:

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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).

user78229
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    I'd assert that those who extended the original notation of R and Q to other modes helped to kill it. Some decades ago I encountered the R and Q distinction often enough to retain a memory of what it implied. But schemes like this, unless you use them permanently, become useless. What was intended as an aid to understanding becomes a cryptic notation that has to be looked up each time. No disagreement with @DJohnson here, I think: he is just the messenger. (An explanation in terms of persons clearly is not at all general. PCA/FA are much used on quite different data.) – Nick Cox Nov 26 '15 at 12:32