Facet design is part of a more general approach called facet theory, which uses the facet structure to generate hypotheses about similarities between items. Facet theory relies almost exclusively on producing low-dimensional geometric representations of the data, which are then interpreted in terms of the properties of the defining facets (cf. Borg & Shye, 1995). Other approaches include confirmatory factor analysis (cf. Mellenbergh, Kelderman, Stijlen, & Zondag, 1979). A problem with both types of approaches is that the analysis focuses on similarities between items and attempts to relate characteristics of the facet design to these similarities. However, as Borg (1994) explained, the relationship between characteristics of the facet design and the geometric ordering of the ensuing items is weak at best. This can be illustrated with the idea of a confirmatory factor analysis of the reasons for losing weight design. We have a source facet with four levels and a motivation facet with seven levels. Do we predict 4 + 7 = 11 factors, or do we predict 4 X 7 = 28 factors? Or should we assume that the facet design merely ensures the content validity of a one-dimensional instrument?
A classical reliability analysis of a (simulated) data set for 50 respondents responding to the 28 items generated by Gough's (1985) facet design produces a reliability coefficient (alpha) of 0.93. This is very high, but not unusual with facet data, because facet designs tend to produce items that are very similar in content and wording. A factor analysis (principal factors, eigenvalue >1, promax rotation) produces seven factors: four factors that are mostly based on the source facet, and three subsequent factors that are not readily interpretable.
Multilevel modeling of facet data takes a different viewpoint. The responses on the common response range are viewed as observations of what occurs when a specific person encounters a specific item. The goal of the multilevel analysis is to determine which item and person characteristics (as defined by the facet design) predict the outcome of this encounter. If all respondents respond to all items, a facet design produces cross-classified data, which can be handled by standard analysis methods such as ANOVA. However, a large-facet design generates too many items to include them all in a single instrument. Older research (cf. Borg & Shye, 1995) typically solved this problem by taking a subsample from all possible items. However, modern computer-assisted data collection methods make it easy to present a different sample of questions to each respondent. In this case, the facet design produces multilevel data, with items nested within respondents, with the response as the outcome variable and person and item characteristics as predictors. The item characteristics are predictors at the lowest (item) level, and the person characteristics are predictors at the person level.
A multilevel analysis of the reasons for weight-loss design requires that both the categorical source and motives facets be expressed as dummy variables. A multilevel analysis involving only these item characteristics shows that only the effects of the source dummies vary significantly across respondents; the effects of the motive dummies have no random variation at the respondent level. For the final model it is convenient to include all four source dummy variables in the regression equation so we can model the (co)variances of all regression coefficients of the source facet. Therefore, the intercept is no longer part of the equation. The seven motive elements are still represented by the usual set of 7 - 1 = 6 dummy variables. The final model is expressed in Equation (5):
where Sx to S4 are dummy variables that indicate the four elements of the source facet, and Mx to M6 are dummy variables that represent the six elements of the motivation facet. The variances cx2( to ct24 of the person-level residual error terms up u2, u3, and u4 are significant (using a likelihood-ratio test; see Hox, 2002), which indicates that there is significant slope variation across persons for the source's (1) own experience, (2) husband, (3) doctor, and (4) media. The variances of the regression slopes for the motivation dummies are not in the model because they were not significant, which means that there is no individual variation in the impact of the motivation facet.
This multilevel analysis produces several interesting estimates. Table 19.1 presents the regression coefficients and the variances for this model. The regression slopes for the item characteristics express overall differences between the item means related to the item content. The (significant) variances of the regression slopes for the predictors belonging to the source facet express differences between respondents in their sensitivity to item content coming from specific sources. The software HLM (Rauden-bush, Bryk, Cheong, & Congdon, 2000) calculates reliability estimates for the random slopes (when using other software these must be hand calculated using formulas presented in Raudenbush & Bryk, 2002). The reliability estimates for the slope variation of si, s2, s3, and s4 are 0.84, 0.83, 0.84, and 0.87, respectively. This means that variations in sensitivity to reasons originating from different sources can be measured with sufficient precision.
The slopes of doctor and media and of self and husband correlate strongly (0.93 and 0.89, respectively) , but the other slopes are relatively independent (correlations lower than 0.61). If we need to use these measurements in a different context, we can estimate residuals or posterior means for the slopes. These are estimates of the slopes for the individual respondents. This is especially convenient if we want to use the slope estimates as predic
Multilevel Analysis of Reasons for Attending Weight-Reduction Classes
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