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I have a question regarding the best practice for dealing with cross-loadings on factors after conducting an exploratory factor analysis using a promax rotation. Just to give a bit of background information, I am trying to determine the factor structure of a set of 20 questions that I created about spirituality (based on PCA which identified 4 components). After conducting the EFA with 184 participants, I noticed that quite a few of the questions are cross-loading onto two or three factors. I read online that you should eliminate questions whose cross-loadings are less than or equal to .2, but I am confused because doesn't an oblique rotation (promax) imply that there is inter-correlation? For example, if question 2 loads onto factor 1 (.234), factor 2 (.346), and factor 3 (.212), what should I do?

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  • $\begingroup$ What do you mean under "loadings" in case of oblique rotation? Do you mean pattern matrix entries (the regression coefficients of the factor model) or structure matrix entries (the correlation coefficients between the factors and the variables)? $\endgroup$ – ttnphns Aug 18 '12 at 9:41
  • $\begingroup$ @ttnphns the OP almost certainly means the regression coefficients, as they are commonly referred to as factor loadings within psychology $\endgroup$ – richiemorrisroe Aug 18 '12 at 10:25
  • $\begingroup$ richiemorrisroe is correct - under pattern matrix $\endgroup$ – Madeline Aug 18 '12 at 10:50
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Firstly, principal components and factor analysis are quite different methods. PCA is normally used more as a data reduction technique, while factor analysis is more concerned with finding a latent structure.

On the cross loadings, the oblique rotation allows the factors to be correlated, but typically one would not want items to load on multiple factors. In this case, I would probably examine the factor loadings using other oblique rotations such as oblimin to see if these cross-loadings still appear.

Cross loadings of below .3 are often ignored, but if you have multiple samples with the same cross-loadings, then this may be an indication that the item is indeed associated with more than one factor. Typically, these items are discarded, and I would probably do so unless you have a strong theoretical or practical rationale for retaining them.

Finally, it sounds like you have two samples. In this case, I would perform EFA on your first sample, and then use the second sample to validate your model. This will raise the probability that you are modelling something real, rather than noise.

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  • $\begingroup$ Thank you for your answer! I still a bit confused though. I have two samples technically (from two separate semesters). I analyzed the first sample with a PCA for data reduction purposes then used a EFA with the second sample (as far as I know it isn't advisable to use PCA and EFA on the same sample). I will try analyzing the data with the other oblique rotations. If the cross-loadings still appear, then what? And when you say cross-loadings below .3, would that mean for example that I say question 2 loads on factor one if it is .314 on factor 1, .213 on factor 2 and .254 on factor 3? $\endgroup$ – Madeline Aug 18 '12 at 10:46
  • $\begingroup$ @Madeline, it is more correct to say "factor loads on variable", not "variable loads on factor". Multi-loaded variables are quite normal in EFA. Although in factor-validation of a questionnaire they are usually avoided since you aim to parcel up items into scales non-crossingly. If you have many multi-loaded items and don't want to discard them from your instrument, try to extract more factors to see what happens then, also try other oblique rotations, as richiemorrisroe has adviced $\endgroup$ – ttnphns Aug 18 '12 at 11:11
  • $\begingroup$ I tried increasing the number of factors to 5 (with eigenvalues above 1.0), but when I tried this I ended up with a factor that only had two questions??? And when I tried other oblique rotations, the results were basically the same. Is there anything else I can do to keep the multi-loaded items? $\endgroup$ – Madeline Aug 18 '12 at 11:22
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    $\begingroup$ @Madeline, I think you ought not to feel knocked out. Remember that rotation is an arbitrary tool that only aids interpretation, it doesn't change relative positions of the items. What I think I'd do in your situation: either (a) base myself on the 5-factor solution, plus maybe discard those 2 questions that split away from the rest and redo then 4-factor model; or (b) conclude that some of the items are worded unhappily to be multi-meaning, and reword them to make them less so. $\endgroup$ – ttnphns Aug 18 '12 at 11:40
  • $\begingroup$ @ttnphns, Thanks for your encouragement! I discarded the two items (so 18 questions now), and would it be okay to follow richiemorrisroe's advice where I ignore cross-loadings under .3? So, for example, can I keep question 6 if it falls on factor 1 (.645) and factor 3 (.315)? $\endgroup$ – Madeline Aug 18 '12 at 12:00
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@Madeline, if, after various rotation and solutions, you find items consistently load (> 0.3) across major factors - you might consider the existence of a higher-order construct. For example, an item assessing general well-being might cross-load on more specific factors associated with clarity of thought, social engagement, and existential purpose. If you have several of these items (3 or more) you might use a hierarchical (higher order) factorial model, or simply pull them out and examine them using a separate run. Factor analysis is intended as a structural validation tool not an interpretive tool (which is best left for other validation procedures).

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