In the first example (Figure 27.1), the manifest variables were the individual symptom ratings. More typically, however, these procedures use summary scores as manifest variables (e.g., the summary score for the nine ADHD-IN, nine ADHD-HI, and eight ODD symptoms for the four sources), the focus of our second example.
Figure 27.2 shows the model for the summary scores. This model involves three latent trait factors (ADHD-IN, ADHD-HI, and ODD) and four latent
TIME 1 TIME 2
source factors (mothers, fathers, teachers, and aides) at Time 1 with the same factors at Time 2 (3 months later). The model involves 24 manifest variables, 12 at Time 1 and 12 at Time 2. Let us also assume, given the 3-month test-retest interval, that the instructions ask the sources to rate the children's behavior for the past month. This example allows us to discuss the convergent and discriminant validity of the manifest variables at Time 1 and Time 2, the discriminant validity of the latent traits and sources within each assessment, and the convergent and discriminant validity of the latent traits and sources across time.
There are six outcomes required for the ADHD-IN, ADHD-HI, and ODD measures (manifest variables) to have strong convergent and discriminant validity. Because four of these six outcomes are conceptually the same as for the example with the individual symptom ratings, the description is briefer here. First, each manifest variable must have a substantial loading on the appropriate trait factor. Such outcomes would provide support for the convergent validity of the ADHD-IN, ADHD-HI, and ODD measures. Second, each manifest variable is required to have a much stronger loading on its trait factor than on its source factor. These outcomes would provide support for the convergent and discriminant validity of each measure. Third, the ADHD-IN, ADHD-HI, and ODD latent trait factors must show convergent and discriminant validity across time (i.e., the correlation between the same trait across time is substantial, whereas the correlation between different traits across time is much smaller). Fourth, the ADHD-IN, ADHD-HI, and ODD latent trait factors are required to show discriminant validity within each assessment period (i.e., the correlations among the ADHD-IN, ADHD-HI, and ODD traits are small at Time 1 and at
Time 2). Fifth, the mother, father, teacher, and aide latent source factors must show convergent and discriminant validity across time, and sixth, the latent source factors are required to demonstrate discriminant validity at each assessment period. Of the six outcomes, outcomes 1 and 2 are again prerequisite conditions for the interpretation of the results relevant to outcomes 3 to 6.
If these six outcomes occurred, the results would indicate that the ADHD-IN, ADHD-HI, and ODD measures had strong convergent and discriminant validity (for this specific rating scale for these specific sources over this specific time interval, the validity thus being conditional on these parameters). Such a result (assuming good content validity) would indicate that the use of the scale by the four sources provided good measures of the three constructs.
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