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[A question about what we optimize in FA, is FA a clustering of variables, and when/how we choose the number of factors]

I have read some tutorials and looked at some of the questions here, as well, still I am unsure about the questions I ask below. I would really appreciate if you could help me clarify them, if possible with concrete examples.

If I understand it correctly, the underlying model for factor analysis is a system of equations as follows:

My overall understanding

$X_j = a_1F_1 + ... + a_mF_m + E_j$ (E is some error variance term of the variable, not explained by the factors; see)

for any of the variables X, and the m factors, and the corresponding “loadings” (the "a"). Now we can use any of the optimization methods to estimate the parameters of the model, the a’s, such that the correlation (or covariance) matrix of the variables X can be approximated as closely as possible. The highest loading will then also attribute each variable to the corresponding factor (hence dimension reduction).Is this correct so far?

My questions now

(1) As we want to have clusters of Variables which are correlated with each other in one and the same factor, where exactly do we optimize for this cohesity? In other words, how do the loadings of the variables, especially their primary loadings, reflect the proximity of the variables X to each other (which is after all the basis of grouping them together to factors)?

(2) What are the Factors, numerically, when we try to optimize the model (ie the equations above for minimizing)? I assume that if we want to optimize for the loadings, we must already know the values for the Fs – or not? I am thinking for instance about regression, where we have the dependent and the independent variables of several datasamples, and we try to optimize the regression line.

(3) When we optimize the model above, are there other criteria then just reducing the distance between the estimated correlation matrix and the original correlation matrix? Especially I always read that the factors should each capture as much overall variance as possible – is this just a byproduct or an actual criteria of the optimization process?

(4) Determination of the Number of Factors: I know that after we do have some number of Factors, we can use some of various criteria to get rid of those who do not capture enough variance. But what determines how many factors we have before throwing any of them out? Is this initial number of factors determined by the dimensions of the correlation matrix we start with? So we can say how many factors are initially there even without having yet determined the loadings etc?

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(1) As we want to have clusters of Variables which are correlated with each other in one and the same factor, where exactly do we optimize for this cohesity?

Factor analysis is not cluster analysis, it is latent variable modeling. It could be seen as a "clustering" method for correlated variables, however, the different "clusters" may show up only after rotation (such as varimax) to a simple structure where different subsets of items appear loaded by different factors. Yet, reconstruction of correlations holds even without any rotation. (See: A point in "Contrast list PCA vs FA" here; the starting section here.)

In other words, how do the loadings of the variables ... reflect the proximity (i.e. their correlations) to each other...?

Reconstruction (= explaining) of correlations/covariances is the primary aim of FA. The reconstruction is formalized in fundamental factor theorem (see it in footnote 2 here). In short, correlation (or covariance) between two variables X and Y is restored, with (ideally small) residual, by those variables' m factor loadings: $r_{xy} \approx a_{x1}a_{y1}+a_{x2}a_{y2}+...+a_{xm}a_{ym}$.

(2) What are the Factors, numerically, when we try to optimize the model (i.e. the equations above for minimizing)? I assume that if we want to optimize for the loadings, we must already know the values for the Fs – or not? I am thinking for instance about regression, where we have the dependent and the independent variables of several datasamples, and we try to optimize the regression line.

FA model is like a regressional model, still it is not regression analysis. In regression, the aim is to minimize the error. In FA, the main task is to restore correlations. Maximize variance explained by common factors (= hence minimize error or "unique variance") is second, not as principal, aim. Different extraction methods may put different accents in such two-fold (even 2+ fold) task.

We do the FA optimization on the level of loadings, the parameters a, not factor values. We do everything basic by analyzing the correlation/covariance matrix, not casewise dataset. Optimization tries to minimize error of the factor theorem (i.e. restore the matrix), not the "error" of the factor model. So no, we don't need to know F's values to optimize what we want and we don't estimate those values at the FA factor extraction phase optimization.

Btw, in regression we technically estimate its parameters b from correlations, too, without assistance from casewise data.

In regression (its simple forms), the parameters relate to the data directly because no latent feature modeling is involed. In FA, we may estimate "true" factor loadings very adequately sometimes, but the values of those factors remain black box for us; it is known as "factor values indeterminacy problem": computed factor scores are never exact true values (read, read) unlike in PCA (read answer and comments).

(3) When we optimize the model above, are there other criteria then just reducing the distance between the estimated correlation matrix and the original correlation matrix? Especially I always read that the factors should each capture as much overall variance as possible – is this just a byproduct or an actual criteria of the optimization process?

This point was answered in my previous section at its beginning. It is a by-product. But often very desirable. In optimizations, we often can settle more than one objective function at once. (In this answer, I've tried to show visually on the 2nd picture how you can perform a factor extraction, and spoke of two objectives of "pulling the arrow by finger": one was to orthogonalize common factor to unique factors - and by that virtue to approximate correlation, and the other was to maximize the factor's variance. And here's a numerical example showing FA and PCA extractions step-by-step.)

(4) Determination of the Number of Factors: I know that after we do have [have extracted] some number of Factors, we can use ... criteria to get rid of those who do not capture enough variance. But what determines how many factors we have before [the extraction]? Is this initial number of factors determined by the dimensions of the correlation matrix we start with? So we can say how many factors are initially there even without having yet determined the loadings?

FA is fitting by modelling. This means we first select the number of factors m to extract - then extract them (and see the fit quality). You are not correct: we do not throw away some factors after extraction. If we don't like this or that factor (poor loadings, poor interpretation, etc) we usually reconsider m or revise variables to include in the analysis. Or simply ignore the factor, saying "this factor appears important for the overall fit but we have difficulty to describe/appreciate it". We don't "retain" factors in FA - unlike we do components in PCA; we might, however, in rare cases decide to rotate/interpret only a subset of the extracted factors.

To cite myself: After rotation, you may not say "this factor is more important than that" because they were rotated vis-a-vis each other (to be honest, in FA, unlike PCA, you may hardly say it even after the extraction because factors are modelled as already "important").

How do we determine m to extract? For this, please read some text on FA (including mayby this site's Q/A) - because the question is traditional/sacramental. Please find to read on 1) Kaiser's Eigenvalue >threshold criterion, 2) Cattell's Scree-plot elbow rule, 3) Very simple structure (VSS) and Minimum average partial (MAP) criterions, 4) Humphrey-Ilgen Parallel analysis, 5) post-extraction fit measures, such as size/pattern of correlation residuals, Chi-square goodness-of-fit, loadings st. errors, etc, 6) old good Percentage of variance explained, 7) Interpretability (sure!). And a number of other approaches.

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  • $\begingroup$ hey, I can't put into words how helpful your answer is..thank you really that^1000 much! However I have go through everything slowly and read up more, and might come back with questions (pretty surely) - will you see them here? Also, I have already one question: I know the criteria like Scree plot analysis, Kaiser's rule etc., but as far as I understand, these are criteria to decide how many of the factors to retain - hence in the optimization process with which to start with, m must be already given - my question is how? (sorry if I am misunderstanding you here) $\endgroup$ – Pegah Oct 22 '16 at 13:03
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    $\begingroup$ Please have a look at this recent Q and comments under it: stats.stackexchange.com/q/241032/3277. Scree-plot / Kaiser's rules are traditionally based on eigenvalues of PCA done before FA, not on FA's extracted factors. So, how many of the factors to retain - no; how many factors to decide to extract. In FA, you don't "retain" factors at all, we model them m factors, we know m (try out this or that m) prior extraction. $\endgroup$ – ttnphns Oct 22 '16 at 13:16
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    $\begingroup$ P.S. Please see also my reply to Gottfried comment here: stats.stackexchange.com/a/50758/3277. $\endgroup$ – ttnphns Oct 22 '16 at 13:25

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