The correlation and association methods described so far are correlations between observed variables that are usually affected by measurement error. Latent variable models are statistical approaches designed to separate measurement error from "true" individual differences. Moreover, latent variable models allow the definition of latent variables that represent different sources of influence on the observed variables. The advantage here is that one can model complex structures that link latent trait-specific and latent method-specific variables to other latent variables. Concepts of criterion-related validity can, therefore, easily be linked to concepts of convergent and discriminant validity.

Latent variable models can be classified into four groups depending on whether the observed variables and the latent variables are categorical or metrical (Bartholomew, 1987). Models with categorical observed variables and metrical latent variables are models of item response theory (IRT) and models of factor analysis for categorical response variables. Models with categorical observed and categorical latent variables are models of latent class analysis. Models with metrical observed and metrical latent variables are models of factor analysis (for metrical observed variables) and, more generally, structural equation models (SEM), whereas models for metrical observed and categorical latent variables are latent profile models.

SEM and IRT are approaches for metrical latent variables. SEM is the methodological approach that has been most often applied to analyze multitrait-multimethod data. It offers a very flexible modeling framework for defining models for quite different purposes. SEM are very general models implying other approaches such as the composite direct model, covariance component models (Wothke, 1995), and models of analysis of variance as special cases. They allow easy extensions of existing models, for example, to consider multiple indicators of a trait-method unit. Eid et al. (chap. 22, this volume) provide an introduction to these models and present some models for analyzing MTMM data. Recent developments in IRT offer a similarly flexible modeling framework for categorical response variables. Rost and Walter (chap. 18, this volume) show how multicomponent IRT models can be applied to multimethod data structures.

SEM and multicomponent IRT models are very flexible methodological approaches. Several models for analyzing MTMM data have been developed in these frameworks. However, sometimes it might be necessary to adapt these models or to formulate new models for analyzing a research question. Therefore, the aim of the following chapters in this handbook is not to give a sufficient overview of all possible models that can be considered when conducting research, but to introduce the basic ideas of these approaches and to illustrate their advantages and limitations by referring to some important models and applications.

IRT models are models for categorical observed variables. SEM have been developed for metrical observed variables. However, there are also approaches for modeling dichotomous and ordinal variables with SEM. The development of new methods for estimating and testing SEM for ordinal variables (Muthen, 2002) makes it possible to analyze ordinal variables with structural equation modeling as well. In fact, Takane and de Leeuw (1987) have shown that SEM of ordinal variables are equivalent to special models of IRT. What are the differences between IRT models and SEM for ordinal variables with respect to the analysis of MTMM data?

SEM for ordinal variables are closely linked to the traditional way of structural equation modeling, which means they aim to explain a bivariate association structure (in this case the polychoric correlation matrix) by a set of latent variables. The great advantage of SEM for ordinal variables is that this association structure can be modeled by different latent variables representing trait- and method-specific influences as first- or higher-order factors. SEM for ordinal variables is variable-centered as trait- and method-specific influences are analyzed on the level of individual differences. The covariances between latent trait-method units are usually the starting point for SEM, and these covariances can be modeled in a very flexible way considering several latent variables.

The covariances of latent trait-method units are almost the end point of Rost and Walter's (chap. 18, this volume) presentation of multicomponent IRT models for multimethod data. The IRT models that they discuss are more restrictive with respect to the homogeneity of the items considered because differences in the discrimination parameters are not allowed (which are represented by different factor loadings in SEM). Moreover, these models are less variable centered because the modeling of the associations of the latent trait-method unit is not at the center of their focus. IRT models for multimethod data focus more strongly on a decomposition of item parameters and person parameters to detect general and item- and person-specific method influences. A strong advantage of IRT models is the many possibilities to decompose the item parameters (which is not the focus of SEM), the extension of these models to mixture distribution models to detect structurally different subgroups, and the estimation of individual person parameters (which is less intended by SEM). Moreover, the measurement theoretical basis of the multicomponent IRT models and their implications for the estimation of the model parameters is impressive. Hence, both types of models stress different kinds of multimethod influences, and an interesting domain of future psychometric research would be a closer integration of both traditions.

Latent class and latent profile analysis are approaches for categorical latent variables. There is good reason to assign latent class models to the family of IRT models, and therefore, Rost and Walter (chap. 18, this volume) also introduce latent class models and show how they can be combined with other IRT models to analyze MTMM data. Several other approaches have applied the latent class modeling framework for analyzing interrater agreement, and Nussbeck (chap. 17, this volume) refers to these approaches. Latent class models have been extended to log-linear models with latent variables (Hagenaars, 1993). Log-linear models with latent variables are comparable to SEM in their flexibility to model latent structures. Eid and Langeheine (1999, 2003) have shown how latent state-trait models (see Khoo et al., chap. 21, this volume) can be formulated for categorical latent variables using this framework. This type of model can also be adapted for MTMM research, but there are currently very few applications to MTMM data (Eid, Lischetzke, Nussbeck, & Geiser, 2004).

Latent profile models are latent class models for metrical observed variables. However, a systematic application of this approach to multimethod data is, to our knowledge, still missing. Rather, new and versatile computer programs such as Mplus (Muthén & Muthén, 2004) will certainly contribute to a broader application of these models for MTMM research.

In sum, latent variable approaches for multi-method data have typically been applied to situations with metrical latent variables (IRT, SEM), and these approaches will be described in more detail in the current handbook. However, modeling approaches for latent categorical variables (latent class analysis, latent profile analysis) offer manifold and versatile new ways of analyzing the convergent and discriminant validity of typological structures that are of great importance for different areas of psychology (e.g., clinical psychology). This will certainly be one of the major future domains of psychometric research concerning multimethod measurement.

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