Interpretation of measures is greatly simplified if the measure assesses a single dimension (underlying factor). For example, imagine that a measure of college aptitude were developed. Unbeknownst to the test developers the items reflect a major dimension of IQ and a secondary dimension of conscientiousness. These two dimensions have only a minimal correlation. Both dimensions may predict good performance in many classes. But the conscientiousness dimension may be a far better predictor of performance in a history course in which large amounts of material must be regularly learned. In contrast, IQ may be a far better predictor of performance in a calculus course. By separating the two dimensions, we can gain a far greater understanding of the influence of the two dimensions in performance in different college classes. Indeed, the interpretation of the body of research associated with several classic measures of personality has been difficult because of the existence of multiple dimensions underlying the personality scale (see Briggs & Cheek, 1986; Carver, 1989; Neuberg, Judice, & West, 1997 for discussions). Finch and West (1997) discussed testing of measures in cross-sectional studies that are hypothesized to have more complex, multidimensional structures.
In longitudinal research, these issues only become more difficult because dimensions within a scale may change at different rates. For example, Khoo, Butner, and lalongo (2004) found that a preventive intervention led to a linear decrease on a dimension of general aggression, but no change on a secondary dimension of indirect aggression toward property during the elementary school years. Such findings make it necessary to consider a more complex measurement structure in assessing longitudinal effects on the aggression scale.
The most commonly used method of assessing the dimensionality of measures in cross-sectional studies is confirmatory factor analysis (see Eid, Lis-chetzke, & Nussbeck, chap. 20, this volume; Hattie, 1985 for a review). In this approach, the researcher hypothesizes that a specific measurement model consisting of one or more latent factors underlies a set of items. The measurement model is then tested against data with two aspects of the results of the test being of special interest, (a) The procedure provides an overall %2 test (likelihood ratio test) of whether the hypothesized model fits the observed covariances between the items. If the value of the obtained x2 is not significant, then the hypothesized model fits the data. For large samples, the %2 test may reject even close-fitting models so that various fit indices such as the RMSEA and the CFI, which are less dependent on sample size, may be used to assess whether the model is adequate, (b) The strength of the relationship between the factor and each item (A = factor loading) is estimated. In some models, the As can be expressed in standardized form, in which case they represent the correlation between the latent factor and each item. Alternatively, one of the items may be treated as a reference variable (A = 1). The strength of each of the other loadings is interpreted relative to the reference variable, values of A >1 indicate a relatively larger change, and values of A < 1 indicate a relatively smaller change in the measured variable corresponding to a one-unit change in the latent factor (see Steiger, 2002).
Confirmatory factor analysis can also be used to estimate coefficient alpha. We noted earlier that coefficient alpha assumes that all measures are equally good measures of the underlying construct. This assumption means that the factor loadings of all the items on the factor are equal, known as the assumption of essential tau equivalence. Comparing the fit of a model in which the As are constrained to be equal, versus an alternative model in which the As are freely estimated, tests essential tau equivalence. If the fit of the two models does not differ, then the assumption of essential tau equivalence is reasonable. McDonald (1999) and Raykov (1997) provide procedures for estimating a both when the assumption of essential tau equivalence is and is not met. Later in this chapter we will extend the idea of testing of assumptions about measurement structure to longitudinal data. To the extent measures have the same structure at two (or more) time points, the results of analyses using the measures become more interpretable.
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