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This chapter presents basics of factor analysis modeling, focusing on its use and interpretations in scale and test development contexts. Two approaches to such analyses are discussed: exploratory factor analysis (EFA), in which one explores what factor structure the data represent; and confirmatory factor analysis (CFA), where one attempts to confirm a hypothesized factor structure. Text and illustrations demonstrate how one can use factor analysis results for deciding which items should be retained or deleted from a scale or test instrument. Model fit evaluation is discussed through the chi-square statistic, as well as some fit indices and information criteria. Uses of a CFA with covariates (MIMIC) and multiple-group CFA approaches for measurement invariance studies are also demonstrated. Lastly, basics of item response theory (IRT) modeling are introduced by demonstrating its parameter estimation and interpretations.

This chapter focuses on using multiple-group confirmatory factor analysis (CFA) to examine the appropriateness of CFA models across different groups and populations. Multiple-group CFA involves simultaneous CFAs in two or more groups, using separate variance-covariance matrices (or raw data) for each group. Measurement invariance is be tested by placing equality constraints on parameters in the groups. Two examples of multiple-group CFA from the social work literature are discussed, and then a detailed multiple-group CFA building on the Job Satisfaction Scale (JSS) example presented in the previous chapter is presented. This is one of the more complex uses of CFA, and this chapter briefly introduces this topic; other resources are provided at the end of the chapter for more information.

This chapter discusses multiple family psychoeducational group interventions and their rationale for use with the family members of persons who have schizophrenia. Psychoeducation offered in multiple family groups (MFGs) for schizophrenia can be helpful in decreasing stress to family caregivers of people with severe mental illness and preventing relapse and hospitalization. Common group topics include theories of schizophrenia, its course, the positive and adverse effects of medications, the relationship between stress and symptoms, available resources for persons with schizophrenia and their families, the nature of professional interventions, managing family emotional responses to the illness, and mobilizing strengths for improved family functioning. The group process, as exemplified through a case study, is intended to help families experience less burden and stress, reduce their sense of stigma, help them assist each other with parenting, and normalize their communications.

If the treatment (T) and control (C) groups are different in unobserved variables e as well as in observed variables x, and if e affects both the treatment and response, then the difference in outcome y cannot be attributed to the difference in the treatment d. The difference in x causing overt bias can be removed with one of the methods discussed in the preceding chapters, but the difference in e causing hidden bias is hard to deal with. In this and the following chapter, hidden bias due to the difference in e is dealt with. An econometrician’s first reaction to hidden bias (or endogeneity problem) is to ‘use an instrument’. But good instruments are hard to come by. Much easier, but less conclusive, is exploring ways to detect the presence of hidden bias; this is done in the name of ‘coherence’ (or consistency, if one does not mind the abuse of this term), whether the main scenario of the treatment effect is coherent with other auxiliary findings. This task can be done with multiple treatment groups, multiple responses, or multiple control groups, which are easier to find than instruments; checking coherence leads to an emphasis on study design rather than estimation techniques. The treatment-effect literature sheds new light on instrumental variables, i.e., the instrumental variable estimator is shown to be for the effect on those whose treatment selection is affected by the instrument.

This chapter describes Type A Weyl group multiple Dirichlet series. It begins by defining the basic shape of the class of Weyl group multiple Dirichlet series. To do so, the following parameters are introduced: Φ‎, a reduced root system; n, a positive integer; F, an algebraic number field containing the group μ‎₂ₙ of 2n-th roots of unity; S, a finite set of places of F containing all the archimedean places, all places ramified over a ℚ; and an r-tuple of nonzero S-integers. In the language of representation theory, the weight of the basis vector corresponding to the Gelfand-Tsetlin pattern can be read from differences of consecutive row sums in the pattern. The chapter considers in this case expressions of the weight of the pattern up to an affine linear transformation.

This chapter shows that Weyl group multiple Dirichlet series are expected to be Whittaker coefficients of metaplectic Eisenstein series. The fact that Whittaker coefficients of Eisenstein series reduce to the crystal description that was given in Chapter 2 is proved for Type A. On the adele group, the corresponding local computation reduces to the evaluation of a type of λ‎-adic integral. These were considered by McNamara, who reduced the integrals to sums over crystals by a very interesting method. A full treatment of this topic is outside the scope of this work, but it is introduced in this chapter by considering the case where n = 1. In this chapter, F is used to denote a nonarchimedean local field and F to denote a global field. The values of the Whittaker function are Schur polynomials multiplied by the normalization constant.

This chapter translates the definitions of the Weyl group multiple Dirichlet series into the language of crystal bases. It reinterprets the entries in these arrays and the accompanying boxing and circling rules in terms of the Kashiwara operators. Thus, what appeared as a pair of unmotivated functions on Gelfand-Tsetlin patterns in the previous chapter now takes on intrinsic representation theoretic meaning. The discussion is restricted to crystals of Cartan type Aᵣ. The Weyl vector, denoted by ρ‎, is considered as an element of the weight lattice, and the bijection between Gelfand-Tsetlin patterns and tableaux is described. The chapter also examines the λ‎-part of the multiple Dirichlet series in terms of crystal graphs.

This chapter reinterprets Statements A and B in a different context, and yet again directly proves that the reinterpreted Statement B implies the reinterpreted Statement A in Theorem 19.10. The p-parts of Weyl group multiple Dirichlet series, with their deformed Weyl denominators, may be expressed as partition functions of exactly solved models in statistical mechanics. The transition to ice-type models represents a subtle shift in emphasis from the crystal basis representation, and suggests the introduction of a new tool, the Yang-Baxter equation. This tool was developed to prove the commutativity of the row transfer matrix for the six-vertex and similar models. This is significant because Statement B can be formulated in terms of the commutativity of two row transfer matrices. This chapter presents an alternate proof of Statement B using the Yang-Baxter equation.

This chapter discusses three advanced structural equation modeling topics: how to conduct a power analysis for SEM; how to prevent and solve problems of underidentification; and how to conduct a multiple-group analysis.