Using the data in Table 16.1, the Project Talent test score data — What kinds preliminary screening would you need run evaluate whether assumptions for

Using the data in Table 16.1, the Project Talent test score data:

a. What kinds of preliminary data screening would you need to run to evaluate whether assumptions for factor analysis are reasonably well met? Run these analyses.

b. Using PAF as the method of extraction, do a factor analysis of this set of test scores: English, reading, mechanic, abstract, and math; the default criterion to retain factors with eigenvalues >1; varimax rotation; and the option to sort the factor loadings by size; also request that factor scores be computed using the regression method, and saved. Report and discuss your results. Do your results suggest more than one kind of mental ability? Do your results convince you that mental ability is “one-dimensional” and that there is no need to have theories about separate math and verbal dimensions of ability?

c. Compute unit-weighted scores to summarize scores on the variables that have high loadings on your first factor in the preceding analysis. Do this in two different ways: using raw scores on the measured variables and using z scores. (Recall that you can save the z scores for quantitative variables by checking a box to save standardized scores in the descriptive statistics procedure.) Run a Pearson correlation to assess how closely these three scores agree (the saved factor score from the factor analysis you ran in 2b, the sum of the raw scores, and the sum of the z scores). In future analyses, do you think it will make much difference which of these three different scores you use as a summary variable (the saved factor score, sum of raw scores, or sum of z scores)?

SOLUTION

a. For each variable, examine a histogram or other graph to assess whether scores are reasonably normally distributed, and to identify any univariate outliers. For all possible pairs of variables, examine a scatter plot to make sure that any relations between variables are linear, and to identify any bivariate outliers. After making sure that the assumptions for Pearson correlation are reasonably well satisfied, you would look at the matrix of correlations; if almost all of the correlations are very small, for example, r < .10 in absolute value, it probably would not be very useful to conduct a factor analysis. Other tests of “factorability” of a matrix (such as the KMO and Bartlett test) essentially test whether there are nonzero correlations among at least some of the measures.

b. SPSS output appears below.

Results

To assess how many different types or dimensions of mental ability were measured by a set of mental ability tests given to high school seniors in the Project Talent study, a Principal Axis Factor analysis was performed on scores on tests of math, reading, English, abstract reasoning, and mechanical reasoning. The default criterion to retain only factors with eigenvalues > 1 was used. Because only one factor had an eigenvalue greater than one, only one factor was retained; therefore, although Varimax rotation was requested, it could not be performed. In the initial solution, the first factor explained 58.68% of the variance in the scores on the mental ability tests; after the number of factors was reduced to one and the factor loadings were re-estimated, the final solution explained 49.14% of the variance. The (unrotated) loadings on this single factor were positive for all five of the mental ability tests; the mechanical reasoning test had the lowest loading (.526) on the mental ability factor, and the lowest communality or proportion of explained variance (.277). The results of this analysis suggest that the tests that were included in this study can be reasonably well understood as measures of one single kind of mental ability; among these tests, the one that was least strongly related to this single mental ability factor was mechanical reasoning. However, given the limited number of ability tests included in this study, it would not be reasonable to conclude that mental ability can be understood using a one dimensional model. There could be other mental ability tests that represent different types of ability than the five tests that were examined in this analysis.

Summary Table:

Loadings on One Factor for Five Mental Ability Tests; Results from a Principal Axis Factor Analysis.