Factor analysis has several rotation methods, such as varimax, quartimax, equamax, promax, oblimin, etc. I am unable to find any information that relates their names to their actual mathematical or statistical doings. Why is it called "equa-max" or "quarti-max"? In what way are the axes or matrices rotated so they have such name?

Unfortunately, most of them were invented in 1950s-1970s, so I cannot contact their authors.

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    $\begingroup$ I edited your question and its title so it is better formed. Feel free to edit it or revert the changes if you don't agree with them. The question is quite interesting so in my opinion it is important for it to be well-formed. $\endgroup$ – Tim Dec 5 '15 at 20:27
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    $\begingroup$ @Tim, I edited further, and in particular added the recently created [factor-rotation] tag which you perhaps have not encountered yet. $\endgroup$ – amoeba Dec 5 '15 at 21:41

This answer succeeds this general question on rotations in factor analysis (please read it) and briefly describes a number of specific methods.

Rotations are performed iteratively and on every pair of factors (columns of the loading matrix). This is needed because the task to optimize (maximize or minimize) the objective criterion simultaneously for all the factors would be mathematically difficult. However, in the end the final rotation matrix $\bf Q$ is assembled so that you can reproduce the rotation yourself with it, multiplying the extracted loadings $\bf A$ by it, $\bf AQ=S$, getting the rotated factor structure matrix $\bf S$. The objective criterion is some property of the elements (loadings) of resultant matrix $\bf S$.

Quartimax orthogonal rotation seeks to maximize the sum of all loadings raised to power 4 in $\bf S$. Hence its name ("quarti", four). It was shown that reaching this mathematical objective corresponds enough to satisfying the 3rd Thurstone's criterion of "simple structure" which sounds as: for every pair of factors there is several (ideally >= m) variables with loadings near zero for any one of the two and far from zero for the other factor. In other words, there will be many large and many small loadings; and points on the loading plot drawn for a pair of rotated factors would, ideally, lie close to one of the two axes. Quartimax thus minimizes the number of factors needed to explain a variable: it "simplifies" the rows of the loading matrix. But quartimax often produces the so called "general factor" (which most of the time is not desirable in FA of variables; it is more desirable, I believe, in the so called Q-mode FA of respondents).

Varimax orthogonal rotation tries to maximize variance of the squared loadings in each factor in $\bf S$. Hence its name (variance). As the result, each factor has only few variables with large loadings by the factor. Varimax directly "simplifies" columns of the loading matrix and by that it greatly facilitates the interpretability of factors. On the loading plot, points are spread wide along a factor axis and tend to polarize themselves into near-zero and far-from-zero. This property seems to satisfy a mixture of Thurstones's simple structure points to an extent. Varimax, however, is not safe from producing points lying far away from the axes, i.e. "complex" variables loaded high by more than one factor. Whether this is bad or ok depends of the field of the study. Varimax performs well mostly in combination with the so called Kaiser's normalization (equalizing communalities temporarily while rotating), it is advised to always use it with varimax (and recommended to use it with any other method, too). It is the most popular orthogonal rotation method, especially in psychometry and social sciences.

Equamax (rarely, Equimax) orthogonal rotation can be seen as a method sharpening some properties of varimax. It was invented in attempts to further improve it. Equalization refers to a special weighting which Saunders (1962) introduced into a working formula of the algorithm. Equamax self-adjusts for the number of the being rotated factors. It tends to distribute variables (highly loaded) more uniformly between factors than varimax does and thus further is less prone to giving "general" factors. On the other hand, equamax wasn't conceived to give up the quartimax's aim to simplify rows; equamax is rather a combination of varimax and quartimax than their in-between. However, equamax is claimed to be considerably less "reliable" or "stable" than varimax or quartimax: for some data it can give disastrously bad solutions while for other data it gives perfectly interpretable factors with simple structure. One more method, similar to equamax and even more ventured in quest of simple structure is called parsimax ("maximizing parsimony") (See Mulaik, 2010, for discussion).

I am sorry for stopping now and not reviewing the oblique methods - oblimin ("oblique" with "minimizing" a criterion) and promax (unrestricted procrustes rotation after varimax). The oblique methods would require probably longer paragraphs to describe them, but I didn't plan any long answer today. Both methods are mentioned in Footnote 5 of this answer. I may refer you to Mulaik, Foundations of factor analysis (2010); classic old Harman's book Modern factor analysis (1976); and whatever pops out in the internet when you search.

See also The difference between varimax and oblimin rotations in factor analysis; What does “varimax” mean in SPSS factor analysis?

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Rotation methods optimise heuristic fuctions with the aim of "simplifying" factor loadings. Simplicity can be defined in many different ways. The most commonly used ones come from Thurnstone [2]: sparsity, column simplicity and parsimony, row-simplicity (or complexity). Most rotation criteria address one or the other of both, their names are not really important.

Single criteria are included in families of criteria: the most comprehensive one is the Crawford-Ferguson one, which is equivalent to the Orthomax family for orthogonal rotations. These families provide a weighing of both simplicity requirements controlled by different parameters. By changing these, almost all known rotation criteria can be obtained. An excellent and accessible overview of rotation methods is the Browne paper.

[1] M. Browne, An overview of analytic rotation in exploratory factor analysis, Multivariate Behavioral Research 36 (2001), pp. 111–150.

[2] L. Thurstone, Multiple-factor analysis, The University of Chicago Press, 1947

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  • $\begingroup$ Thank you for your time that you gave to answer that question of mine dear Marco. However, the central point of my question was about "what the rotation names are meaning in relation to their mathematical or statistical manipulations they are doing in the data". Why their inventors or other people gave the name "Varimax" to that specific method of rotation as well to the other types of rotation methods ? What it was its meaning; @ttnphns answered that question quite well. If you think you can add more information on that topic feel free to do it! :) Thank you again for your time. $\endgroup$ – Elias Estatistics Sep 26 '18 at 8:18

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