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Although this limitation is somewhat circumvented by sampling approaches to items (ie, in generalizability theory and similar approaches), in a practical sense most tests cannot be built as random samples from an item domain Second, the estimates for item properties and reliability are populationspeci c; that is, these indices are meaningful only in reference to a particular group of individuals In reference to another group of individuals, the indices lose meaning The true and error score model is used to justify estimates of population statistics that require variance estimates Third, item properties are not directly linked to behavior That is, knowing a person s score refers to an overall level relative to a group of persons Nothing is known about which items the person has likely passed or likely failed.

Thus, using item dif culty and discrimination to select items is justi ed by their impact on various population statistics, such as variances and reliabilities However, more modern techniques developed since the 1960s have changed measurement methods in several ways Although item response theory (IRT) methods enhance some CTT methods, perhaps more important is the impact of IRT for a new basis of score meaning, item selection, and design, as well as new methods of testing ITEM RESPONSE THEORY Item response theory methods have two distinct traditions In the United States, IRT is usually traced to Lord and Novick s (1968) classic book on measurement Preceding this volume was Lord s (1953) monograph on test theory models and Birnbaum s (see Lord & Novick, 1968, for.

and this separates the xed-effects ( rst four terms) from the random components (last three terms). This model is drawn as a path diagram in Figure 18.9. This diagram is the same as Figure 18.7, except here we have included the X as a predictor of the levels and slope components. This diagram gives the basic idea of external variable models, and other more complex alternatives are considered in later sections. In this simple latent growth model, as in more complex models to follow, we can always add other predictors X for the intercepts and the slopes because these models are simply latent growth models with extension variables (e.g., McArdle & Epstein, 1987). This kind of model (Equation 18.13 or 18.14) can also be seen as having two levels a rst-level equation for the observed scores, and a second-level equation for the intercepts and slopes. For these reasons, such models have been termed randomcoef cients or multilevel models, slopes as outcomes, or mixed-effects models (Bryk & Raudenbush, 1987, 1992; Littell et al., 1996). Variations on these models can be compared for goodness-of- t indices, and we can examine changes in the model variance explained at both the rst and second levels (see Snyders & Boskers, 1995). In any terminology, the between-group differences in the withingroup changes can be represented by the parameters in the model of Figure 18.9.

(See Section 7.3 for the rules.)

1

where 2 is the trait-level variance For SLAT trait levels, the variance is 823 Applying Equation 1714 with measurement variance (37242 = 1387), the composite reliability is 8315 As for CTT, lengthening the test will improve reliability However, improving the suitability of the items for the persons will also improve reliability Advanced IRT Models The models just presented have been useful for unidimensional tests with dichotomous item-response variables However, many measurement problems in psychology are more complex and require complex models to characterize the data (see Embretson & Reise, 2000, for a review) Although space does not permit a full review of these models, the reader may be interested in the following models: (a) those with restrictions on parameters, which are used to relate item dif culty to substantive sources (Fischer, 1973); (b) those for rating scales (eg.