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I have data on 18 participants responding to a 30-item instrument. I also have a paper in which the researchers performed a principal axis factor analysis on the same survey where they report the item loadings (4 factors).

Is it possible to convert the data that I have, and compute scores for each participant on the four factors based on the published loadings?

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  • $\begingroup$ Are the data sets the same? I don't think this would make sense if you are dealing with different data from possibly different populations. $\endgroup$ – Michael R. Chernick Jul 21 '12 at 14:17
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    $\begingroup$ A bundle of questions to you. 1) Is your task this one: Somebody's done FA on large sample and published loadings. You have 18 more cases and wish to compute factor scores for these. 2) By "scores" in the last paragraph you mean 18x30 data values, don't you? 3) Do you have the original large sample at hand? 4) Did they publish (a) means and st. deviations of their 30-item data; (b) factor score regression coefficient matrix? 4) What does PCA tag do here? - PCA is not quite identical to principal axis factoring. $\endgroup$ – ttnphns Jul 21 '12 at 14:22
  • $\begingroup$ P.S. Maybe they published their 30x30 correlation matrix? $\endgroup$ – ttnphns Jul 21 '12 at 14:32
  • $\begingroup$ @DanielStark - Should the first sentence of the second paragraph read "factor analysis on a larger sample who filled in the same survey"? I'm presuming so as otherwise I don't see how the question (or indeed that sentence) makes sense, but I didn't make the edit in case I'd misunderstood. $\endgroup$ – Peter Ellis Jul 21 '12 at 21:18
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Most questionnaires have instructions for how the scale scores are meant to be computed. To foster comparability across studies, it is generally wise to follow those instructions where available.

Unit weighted scale scores

  • The most common approach is to take the rotated factor loadings, and work out which item loads most on which factor. Then the factor score is just the mean or sum for an individual on the items that load most for that factor. There are many good reasons for adopting this approach. The mean creates a scale that matches the original response scale. It is easy to compare means across uses of the test. It is simple to compute. The meaning of the scale can be inferred from the content of the items. And there is a certain robustness in the unit-weighted approach.

Computing scale scores for published factor analyses

  • I don't think you can compute true factor saved scores from factor loadings. As @ttnphns mentions, you would probably need the factor score coefficients (see this example). If you were able to request the raw data from the people who published the previous study then you could then obtain the factor score coefficients.

  • If you could obtain the factor score coefficient matrix then you would still need to be careful. In particular you would need to check that you used the exact same response format. You would also want to be careful that there weren't other subtle differences between your samples.

Using factor loadings to compute factor saved scores

  • If all you have is the factor loading matrix, then you could use them as weights to formed a weighted composite of each factor. If all the items on the same response scale and have roughly the same SD, then you could apply the loadings as they are. Otherwise, you might want to convert the items to z-scores first. Note that this approach is a bit of a hack. It's not giving you true factor saved scores. It's just a composite variable which weights items in a factor score proportion to their loadings.
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