Applied Bayesian Modelling - Free

330 STRUCTURAL EQUATION AND LATENT VARIABLE MODELSposterior variance ± suggesting that the mathematics self-concept is least aligned with aputative single self-concept. The l coefficients in the higher order model are very similarto those in Table 8.1.Continuing the data and indicator format as above, Byrne et al. (1989) compare twogroups (denoted high and low track) with a low track group as reference. Then the mthindicator for the ith subject in the gth group is modelled asY img ˆ k m l m c ijg u imgwhere the errors u are independently Normal, the reference group scores arec ij2 ˆ d ij j ˆ 1, 4; i ˆ 1, : : 248and the other group's scores are given byc ij1 ˆ d ij A j j ˆ 1, 4; i ˆ 1, : : 582with the d ij taken to be multivariate Normal. In line with factorial invariance, theloadings are not group specific and as above loadings l 1 , . . . l 4 are preset and so thecovariance matrix S for the d ij may be estimated. Hence, there are 15 intercepts (11 for themeasurement model, and 4 for the factor model) together with seven loadings to estimate.The differential intercepts A j in the high track group are the major focus of interest.A two chain run of 10 000 iterations (convergent from around 1000) accordinglyshows higher intercepts on the Academic, Linguistic and Mathematics Self Concepts inthe `High Track' students (see Table 8.2). This model adequately represents the datasince a posterior predictive check is 0.54, based on comparing error sum of squares ofTable 8.2 Self-concept: group comparisonsunder invariance, high track interceptsMean 2.5% 97.5%A 1 0.87 2.83 1.16A 2 10.18 8.77 11.58A 3 4.47 3.33 5.61A 4 7.48 5.24 9.73Correlations between factorsCorr[c1, c2] 0.38 0.31 0.46Corr[c1, c4] 0.30 0.22 0.37Corr[c1, c3] 0.24 0.17 0.31Corr[c2, c3] 0.61 0.54 0.68Corr[c2, c4] 0.62 0.56 0.67Corr[c3, c4] 0.02 0.10 0.05Factor loadingsl 5 0.52 0.48 0.56l 6 0.36 0.34 0.39l 7 0.52 0.47 0.56l 8 1.36 1.25 1.48l 9 0.67 0.60 0.74l 10 0.68 0.65 0.70l 11 0.44 0.42 0.46