Multivariate Statistics (Psych 716), Spring 2009
Homework #3 – Factor Analysis and more Factor Analysis
For the following questions, please provide information about specific results relevant to your interpretations. That is, be clear about which tables, plots, specific values, etc are the basis of your interpretations. You may provide values in the text of your answer, copy and paste specific pieces of output form SPSS, or print the key results (I don’t need everything!!) and clearly refer to facets of the output (it wouldn’t hurt to even mark on the output, so that I know exactly what you might be referring to). Your goal should be to convince me that you have a firm grasp on the issues in factor analysis, that you know what the results mean and how to interpret them.
Download the data set “Ch12BData - EFA.sav” from the class web site.
1 Begin by running a Principal Axis Factor analysis (PFA), with default settings for the “Eigenvalue greater than 1” option Don’t worry about rotation for now.
· Considering at least two rules-of-thumb that we discussed, what do the eigenvalues and scree plot suggest about the number of factors underyling the variables? Explain your answer.
2. Based on your evaluation of the eigenvalues and scree plot, run the appropriate analysis (ie, extracting however many factors you think appropriate), using a promax rotation. What do the results indicate about the factor structure?
· Does this solution seem to achieve simple structure (ie, a “clean” structure)? Explain – what seems more and less clean about the structure?
· How would you characterize the factors, psychologically speaking? Of course, this is a subjective issue, so be sure to offer clear statistical and logical support for your interpretation.
· To what degree are the factors correlated?
3. In class, we mentioned that principal components analysis usually provides results that are fairly similar to a factor analysis proper, say Principal Axis Factor analysis (PFA). Prove this to yourself.
· Run a PCA, with default settings for thee”Eigenvalue greater than 1” option, and with a varimax rotation.
· Run a PFA, with the same settings
· Compare some of the key results that you get from the two analyses (factor loadings, in particular). To what degree do the two analyses provide different results in terms of: the nature of the factors?
4. In class, we brushed off the “Communalities” output, and indeed, you don’t really need to examine these values in order to get the basic information from a factor analysis. Nevertheless, it might be useful to have a general idea about what these values represent. So, let’s do it!
a) Run a regression analysis predicting the first item from eight other items. Interpret the R2 value from this equation (don’t worry about whether it’s significant, just think about the R2 value itself).
b) Do the same thing with the second item, predicting it from the other eight items.
c) Run a principal axis factor analysis of the data and examine the “Initial Communality” values of the first two variables. So, in what sense does the term “communality” accurately describe these values? (Note, that this interpretation applies only for the communality estimate used for principal factor analysis, there are other types of estimates for other methods of conducting a factor analysis; however, this is a pretty common approach and conveys the general idea).
OVER
5. Analyze the data set “Factor Analysis data set HW 3.sav” on the class web site. These are real data taken from a personality questionnaire. Responses to the items were made on a 5-point Likert scale, so that larger values (ie, 5) indicate greater agreement with the item. The big goal – understand the factor structure underlying the set of items. This information would then help us understand how best to create subscales (if any) from these items.
· Conduct a factor analysis and describe your interpretation of the most plausible factor structure.
· Be sure to address the key questions in factor analysis, as discussed in class.
· Again, please be sure to provide information about specific results relevant to your interpretation.
· Also, for the purposes of our homework assignment, briefly discuss any alternative factor structures that you considered – what the structures were (eg, 17-factors, 2-factors, etc), why you considered them, and why you rejected them in favor of the one you finally settled on. Hint – looking at the results, there are at least three factor structures that I’d consider on the basis of various “rules-of-thumb” that we’ve discussed.
For these analyses, please note that the items with (R) in the label have been reverse-scored. So for example, item 2's label is "Tends to find fault with others (R)" - this means that the original item was phrased "Tends to find fault with others" where endorsements (ie, responses of 4 or 5 on the 5-point scale) indicated agreement with the item. However, the responses to this item have been reversed (5 changed to 1, 4 changed to 2, etc); so in the current data a 4 or 5 indicate the opposite of the apparent item content - they indicate tendency to not find fault with others.