Longitudinal confirmatory factor analysis: Common, time‐specific, item‐specific, and residual‐error components of variance

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Abstract
Following Cronbach (1970) and others, it is useful to decompose test score variation into common factor, time‐specific, item‐specific, and residual components. In the traditional approach to factor analysis, only two sources of variance can be estimated: common factor variance and a uniqueness term that confounds specific sources of variation and residual error. When the same items are measured on different occasions, however, it is possible to separate specific variance and residual error. Two approaches, the first‐order approach described by Raffalovich and Bohrnstedt (1987) and a second‐order approach based on Jöreskog and Sörbom (1989; Jöreskog, 1974) are considered initially. The two approaches, although based on different rationales, both suffer a similar weakness in that two of the four sources of variance are confounded. In the Raffalovich and Bohrnstedt approach, time‐specific variance is confounded with common factor variance that generalizes across items and time. In the second‐order approach based on Jöreskog and Sörbom, time‐specific variance is confounded with residual error. Here we demonstrate that by combining features from both approaches we can eliminate these weaknesses and estimate all four of Cronbach's sources of variance, and that this combined approach is easily generalized to a wide variety of applications.