Analogy between two approaches to separately identify specific factors in factor analysis

by Wang, Jing.

John Tisak, Advisor An observed variable may be decomposed into three components: one or several common factors, a specific factor, and a measurement error. Unfortunately, the specific factor is usually combined with the measurement error and treated as the unique factor in a factor analytic model (FAM) (Harman, 1976; Muliak, 1972). The confounding of parameters leads to several potential problems, such as underestimated reliability, biased estimation of the intercepts, slopes, and measurement error variances. There are two approaches to separately identify the specific factors, the first- and secondorder FAM approach and the stationary longitudinal FAM approach, both by including several replicates for each variable from a classical FAM. The purpose of the dissertation is to fully discuss the analogy between the two approaches on separately identifying the specific factors and highlight the role played by the specific factors. First, the first- and second-order FAM approach is extended by including the mean structure and explicitly including the specific factor means and variances. Second, a special condition, i.e., equality of specific factor variances and covariances for the same variable, in the stationery longitudinal FAM approach is presented. To show the analogy between the two approaches on the model identification, all observed and latent variables are permutated to be variable ordered for the longitudinal model. After the permutations, we can easily see that the special condition allows us to build a second-order model on the specific factors, which shows the similarity between the two approaches. However, the first approach requires equal specific factor means for the replicates, while the second approach allows the specific factor means to change over time. From one illustration in a longitudinal FAM, the presence of the specific factors, exhibits themselves both in the mean iii structure, and in the variance/covariances structure. If the specific factors were omitted, the desired stationarity of the factor structure, i.e., equal intercepts and slopes, would be biased. Furthermore, neglecting the specific factors would lead to negatively biased reliabilities. iv