# How to use variables derived from factor analysis as predictors in logistic regression?

### Context

I have a survey that asks 11 questions about self-efficacy. Each question has 3 response options (disagree, agree, strongly agree). Nine questions ask about self-esteem. I have used a factor analysis of the 11 self-efficacy items and extracted two factors.

$x_1$ to $x_{11}$ denote the 11 self-efficacy questions in the survey, and $f_1$ ($x_1$ to $x_6$) , $f_2$ ($x_7$ to $x_{11}$) denote the two factors I got from the factor analysis. $y$ is a Dependent variable.

Then I created two new variables:

   f1=mean(x1 to x6);
f2=mean(x7-x11).


So the logistic regression would looks like this:

   y=a+bf1+cf2+....


### My question:

• Can i use these two factors as predictor variables in my multivariate logistic regression model?
• Should I calculate the mean of each items in each factor and use this mean as a continuous variable in my logistic regression model?
• Is this an appropriate use of factor analysis?
• A few questions: Is $y$ univariate? Then you have a multiple logistic regression, not a multivariate logistic regression. How are you doing the factor analysis? While there are methods applicable to ordinal variables that are analogous to factor analysis for continuous manifest variables, it isn't clear to me what you're using. Finally, what would you consider to be an inappropriate use of factor analysis? If you look on it as a dimension reduction technique (turning 11 variables into 2) then there's nothing inherently wrong with it. Whether it's a good idea or not is another question.
– JMS
Feb 22 '11 at 5:07

If I understand you correctly, you are using FA to extract two subscales from your 11-item questionnaire. They are supposed to reflect some specific dimensions of self-efficacy (for example, self-regulatory vs. self-assertive efficacy).

Then, you are free to use individual mean (or sum) scores computed on the two subscales as predictors in a regression model. In others words, instead of considering 11 item scores, you are now working with 2 subscores, computed as described above for each individual. The only assumption that is made is that those scores reflect one's location on an "hypothetical construct" or latent variable, defined as a continuous scale.

As @JMS said, there are other issues that you might further clarify, especially which kind of FA was done. A subtle issue is that measurement error will not be accounted for by a standard regression approach. An alternative is to use Structural Equation Models or any latent variables model (e.g. those coming from the IRT literature), but here the regression approach should provide a good approximation. The analysis of ordinal variables (Likert-type item) has been discussed elsewhere on this site.

However, in current practice, your approach is what is commonly found when validating a questionnaire or constructing scoring rules: We use weighted or unweighted combination of item scores (hence, they are treated as numeric variables) to report individual location on the latent trait(s) under consideration.

### Using factor scores as predictors

Yes, you can use variables derived from a factor analysis as predictors in subsequent analyses.

Other options include running some form of structural equation model where you posit a latent variable with the items or bundles of items as observed variables.

### Mean as scale score

Yes, in your case, the mean would be a typical option for computing a scale score. If you have any reversed items, you have to deal with this.

You could also use factor saved scores instead of taking the mean. Although when all items load reasonably well on each factor and all items are on the same scale and all items are positively worded, there is rarely much difference between the mean and factor saved scores.

You could also look at methods that acknowledge the ordinal nature of the scale and therefore do not treat the scale options as equally distant.

• (+1) Good that you mentioned the use of Factor scores directly (and their correspondence with raw scores under certain conditions).
– chl
Feb 22 '11 at 13:23

Everything have be said by chl and Jeromy for the theorical part... If you don't have use sum/mean of variables you identify with FA you can use scores of FA.

Regarding the syntax you use you're probably using SAS. So to do a correct use of factor analysis you must use the score of observations and not the mean of variables.

You find below the code to obtain score for 2 factors with an FA. Scores you'll have to use will be call Factor1, Factor2, ... by SAS.

This is a 2 steps... 1) First FA then 2) call the proc score to compute Scores.

proc factor
data = Data
method = ml
rotate = promax
outstat = FAstats
n=3
heywood residuals msa score
;
var x:;
run;

proc score data=Data score=FAstats out=MyScores;
var x:;
run;


The variables to use are Factor1, Factor2, ... in MyScores datasets.

Continuous latent variables with discrete (polytomous in your case) manifest variables is part of item response analysis. Package 'ltm' in R covers a variety of such models. I refer you to this paper, which deals with exactly same problem.

• (+1) I have (re)read your paper, which looks pretty interesting, although I found amazing the use of a Rasch model in clusters of genes. Did you compare your results with a sparse PLS-DA approach?
– chl
Feb 23 '11 at 14:03
• @chl Not yet; working on that. Feb 23 '11 at 16:48