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I have a question about how to interpret a regression analysis I did following a factor analysis.

I did principal axis factoring (direct oblimin)

I got a 3 factor solution. The three factors were (1) feelings (2) kinship and (3) interactions. (I'm looking at father-child relationships)

Regarding factor loadings, (1) feelings had all positive factor loadings, (2) kinship had all negative factor loadings and (3) interactions had all negative loadings.

I did a multiple regression analysis using the factor scores - saved as regression scores - as outcome variables. I'm not bringing them together to form a total scale, so I don't think I need to do a MANOVA instead, or adjust the p-value.

For (1) feelings, all of my predictor variables have positive beta weights

For (2) kinship, one of my predictors (participation in healthcare) has a negative beta weight

For (3) interactions, again, one of my predictors (participation in healthcare) has a negative beta weight

I'm confused about whether the predictors with negative beta weights actually imply that less participation in healthcare means a higher score on kinship and on interactions. Or whether, because those two outcome variables have all negative factor loadings, should the negative beta be interpreted as a positive beta? (i.e. do the negatives cancel each other out?).

If the latter is the case, do the predictors with a positive beta for outcome variables (2) and (3) actually have negative betas?

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    $\begingroup$ As far as I know, factor loadings (and hence scores) are only defined up to a multiplicative constant, so the signs are not meaningful $\endgroup$
    – richiemorrisroe
    Jun 5, 2012 at 13:48
  • $\begingroup$ It sounds like you are using the same variables as explanatory variables for your regression as you used in the factor analysis? If so, the approach is probably not very helpful - see stats.stackexchange.com/questions/32319/…. If not, then what variables are you using for explanatory variables? $\endgroup$
    – Peter Ellis
    Aug 4, 2012 at 20:35
  • $\begingroup$ You can take a look here: (pdf) for a discussion on using factors scores in regression (see in particular Table 1, 3rd column, for the case of independent variable observed, dependent variable factor scores) $\endgroup$
    – Federico Tedeschi
    Jun 30, 2018 at 20:35

2 Answers 2

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Apparently your rotation produced meaningless results, as sometimes happens in exploratory factor analysis. If you want to have straightforward interpretation of your factors, then you would want to either flip the sign manually, or somehow make it work in your software that the signs of the loadings are positive. Your interpretation is kinda correct, in the sense that an increase in healthcare participation decreases the kinship factor score, which, however, in turn means that the variables contributing to kinship factor increase.

Note that you will likely be better off with a latent variable MIMIC model in which you have predictors, a latent variable, and its indicators; in any latent variable modeling software, you can specify one of the loadings to be 1 (the scaling indicator), and then other loadings will follow suit if the indicators are positively correlated with one another. Modeling the measurement part and the causal part jointly will produce more efficient estimates; you will fall into the confirmatory-after-exploratory-factor-analysis fallacy though.

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I don't know if your factor analysis produced meaningless results, you have to look at the results and see if they are meaningless. You have adapted some names for the factors, presumably based on those scores and your knowledge of the variables that went into them.

The regressions can be interpreted like any other regression results. But you need to be careful that your factors are in the direction you think they are. That is, e.g., is a higher score on interactions indicative of better or worse interactions?

It can be useful, as the others have said, to flip the sign of factors to make them more interpretable. In our heads, higher scores go naturally with more of something. But sometimes the results from a FA show that higher scores go with less of something.

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