Albanese, Maria Teresinha
(1990)
Latent variable models for binary response data.
PhD thesis, London School of Economics and Political Science.
Abstract
Most of the results in this thesis are obtained for the logit/probit model for binary response data given by Bartholomew (1980), which is sometimes called the two-parameter logistic model. In most the cases the results also hold for other common binary response models. By profiling and an approximation, we investigate the behaviour of the likelihood function, to see if it is suitable for ML estimation. Particular attention is given to the shape of the likelihood around the maximum point in order to see whether the information matrix will give a good guide to the variability of the estimates. The adequacy of the asymptotic variance-covariance matrix is inwestigated through jackknife and bootstrap techniques. We obtain the marginal ML estimators for the Rasch model and compare them with those obtained from conditional ML estimation. We also test the fit of the Rasch model against a logit/probit model with a likelihood ratio test, and investigate the behaviour of the likelihood function for the Rasch model and its bootstrap estimates together with approximate methods. For both fixed and decreasing sample size, we investigate the stability of the discrimination parameter estimates ai, 1 when the number of items is reduced. We study the conditions which give rise to large discrimination parameter estimates. This leads to a method for the generation of a (p+1)th item with any fixed ap+1,1 and ap+1,0. In practice it is importante to measure the latent variable and this is usually done by using the posterior mean or the component scores. We give some theoretical and applied results for the relation between the linearity of the plot of the posterior mean latent variable values, the component scores and the normality of those posterior distributions.
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