Statistics Homework -
Q1. Exploratory Factor Analysis
1. Below is a loading plot for two factors of a 10-item test. The two axes describe the unrotated factor solution for the two factors, with each axis reflecting the item loadings on the two factors. From this plot, (a) recreate the matrix of factor loadings, in tabular form (hint: a ruler should make this easy). (b) Explain whether a rotation is needed and if so, what kind? (hint: you may need to draw on a printout to figure this out) (c) Do you see any problems with any of the items (i.e., either with or without rotation)? (d) If all of the factors (i.e., not just the two extracted here) accounted for 100% of the variance in the set of 10 items, what kind of analysis was run? (e) For this kind of analysis (the one you identified in part d), if the first factor had an eigenvalue of 5 and the second factor had a value of 2, then what was the average percentage of variance accounted for by the remaining factors?
Q2. Confirmatory Factor Analysis
You have a 10-item test of leadership potential. Four items assess agency (i.e., being a good task-master), four items assess communion (i.e., being a good listener), and two items assess conscientiousness (i.e., being responsible). In the space below, (a) draw the overall confirmatory factor model as completely as possible. (b) Is this model likely to be identified? How do you know this? (And I mean specifically, with numbers to back up your knowledge) (c) Based on what you know from parts a and b, what problems might you anticipate with the measure? (d) After examining some initial data, the modification indices suggest that the residual of the fourth item of the agency factor is correlated with the residual of the second item of the conscientiousness factor. Draw this parameter in the model and explain whether it would be a good idea or not to include it in future models of the measure.
Q3. Measurement Equivalence
You are also interested in whether your test of leadership potential functions equivalently for males and females. (a) Recreate 2 drawings of the overall model from problem 5, though leave out the correlated measurement error (one for males and one for females, ideally side-by-side or top and bottom on the same sheet if you can fit it legibly). Using lowercase letters, indicate the constraints that are needed to examine metric equivalence. (b) Using numbers, indicate in the figure the constraints that are needed to examine measurement error equivalence. There is no need to include constraints for scalar equivalence. (c) If you find that a model estimating all parameters separately for males and females fits significantly better (i.e., according to the χ2) than the model in part a, what would you conclude? (d) What would you conclude if the model estimating all parameters separately for males and females fit worse than the model in part a?