Applied Bayesian Modelling - Free

CONFIRMATORY FACTOR ANALYSIS WITH A SINGLE GROUP 331actual and samples of replicate data. As in the single group analysis, the ASC factor ishighly correlated over 0.5 with both LSC and MSC, but the latter two are effectivelyindependent.Another model allows c ij1 and c ij2 to be drawn from different multivariate Normaldensities, and for the indicators to have different loadings l mg and intercepts k mg withingroups (Model F in Program 8.1). There is an improved predictive criterion comparedto the previous `invariance' model (about 429 000 vs. 438 000).Example 8.2 Mental ability indicators: robust CFA Yuan and Bentler (1998) illustratethe implications of non-Normality, including possible outliers that distort inferences,using mental ability test data analysed by Joreskog (1970) and first presented byHolzinger and Swineford (1939). There are in full 26 test items defined for 145 childrenin the seventh and eighth grades of two schools. Yuan and Bentler focus on 9 of the 26tests, namely1. Visual perception2. Cubes3. Lozenges4. Paragraph comprehension5. Sentence comprehension6. Word meanings7. Addition8. Counting dots9. Straight curved capitalsThey postulate three factors, the first spatial ability taken to explain X 1 X 3 ; the second,verbal ability, explaining X 4 X 6 ; and the third, speed in tasks, designed to explainX 7 X 9 .For these data, Yuan and Bentler use a number of robust frequentist techniquesand densities such as M±estimation and Multivariate t density models, which placelow weights on exceptional data points. They find an improved fit for these methodsover Normal theory based maximum likelihood. Here a scale mixture form ofthe Student t with degrees of freedom n, with precisions for case i and variable mdefined byf im ˆ F m w imwhere the F m ˆ 1=s 2 m is an overall precision for variable m, and the weights w im aredrawn from Gamma (n m =2, n m =2) densities. The degrees of freedom parameters aredrawn from an exponential prior with parameters Z m , and are constrained to be above1. The Z m parameters are drawn from a uniform (0.01, 1) prior corresponding approximatelyto means 1 and 100 for n m .The means m im for indicators X 1 to X 9 are given bym im ˆ k ml m, C[m] C i, C[m]where {C ˆ 1, 1, 1, 2, 2, 2, 3, 3, 3} is the factor index for variable m. The scale of theconstructs is defined by fixing l 11 ˆ l 42 ˆ l 73 ˆ 1, so that the full covariance matrixbetween factors is estimated. The loadings under the alternative method when theirvariance is set at 1 (and the covariance matrix becomes a correlation matrix) are derivedin parallel, namely