Applied Bayesian Modelling - Free

356 STRUCTURAL EQUATION AND LATENT VARIABLE MODELSwith v* ˆ 0:619. This is non-ignorable, because c 2 is defined both by observed andmissing data at phase 2.Accordingly, values of (X j , j ˆ 1, 3) {Y j1 , j ˆ 1, 3} and {Y j2 , j ˆ 1, 3} are generatedand sampled data at wave 2 then removed according to the missingness model. One maythen compare (a) the estimates of the parameters {L, b, var(w j ), var(u m )} using theoriginal data with no imputed non-response (b) the parameters obtained when adoptinga missingness model based only on a MAR mechanism, and (c) the parameters obtainedadopting a missingness model based on the latent factors, for example as in Equation(8.15). Under (b), logit models for R i or R* i depend upon the fully observed observationsat wave 1, and under (c) such models depending on the constructs c 1 , c 2 and j.Using the 9 9 correlation matrix provided by Muthen et al., a full data set may begenerated and missingness then imputed according to Equation (8.13), (8.14) or (8.15).We adopt the option in Equation (8.14), where missingness is related to all indicators,whether subject to non-response at wave 2 or not, and take v* ˆ 0:27. To generate thedata, it is necessary to sample all the {X ji , Y jti } and then `remove' the sampled data formissing cases where R ij * is under the threshold. The form of the missingness models(8.13)±(8.15) means there is either complete non-response at wave 2 or completeresponse at unit level on all three indices. So individual item response indices R ij maybe replaced by a single unit response index, R i ˆ 0 for all missing observations at wave2, and R i ˆ 1 otherwise. There are 169 of the 600 observations with missingness atwave 3, a rate of 28%.Under the response mechanism in Equation (8.14), attrition is greater for lower statuspersons and more alienated persons: so missingess might be expected to be greater forsubjects with higher scores on c 1 and c 2 . In Model B in Program 8.11, the responsemodel relates p i ˆ Pr(R i ˆ 1) to the factor scores, namely,logit(p i ) ˆ v 0 v 1 j i v 2 c 1i v 3 c 2i (8:16)We then obtain the expected negative impacts on response of alienation (c 1 and c 2 ) and apositive impact ofstatus,j(see Table 8.13 obtained from iterations 500±5000 of a two chainrun). The impact of c 2 is as might be expected, less precisely estimated than that of c 1 Othermodels relating p i to (say) just j and c 2 might be tried. The coefficients of the structuralmodel (8.12) are close to the parameters obtained from the fully observed sample, thoughthe negatively signed impact b 21 of social status j on alienation c 2 at time 2 is enhanced.Instead one might assume a model with no information to predict missingess, i.e.logit(p i ) ˆ v 0 (8:17)(This is pi.1[ ] in Model B in Program 8.11.) In the present case, and with the particularsample of data from the covariance matrix of Muthen et al., this produces very similarestimates of structural and measurement coefficients to the non-ignorable model.Both models in turn provide similar estimates of the parameters to those based on thefully observed data set of 600 9 variables (Model C in Program 8.11). Model (8.16)allowing for non-ignorable missingness provides an estimate for l 23 closer to the fulldata parameter, but b 21 is better estimated under the MCAR model (8.17). So for thisparticular sampled data set, there is no benefit in using a missingness model linked tovalues on the latent constructs. However, to draw firm conclusions about the benefits ofignorable vs. non-ignorable missingess it would be necessary to repeat this analysis witha large number of replicate data sets.