Robustness, semiparametric estimation and goodness-of-fit of latent trait models
This thesis studies the one-factor latent trait model for binary data. In examines the sensitivity of the model when the assumptions about the model are violated, it investigates the information about the prior distribution when the model is estimated semi-parametrically and it also examines the goodness-of-fit of the model using Monte-Carlo simulations. Latent trait models are applied to data arising from psychometric tests, ability tests or attitude surveys. The data are often contaminated by guessing, cheating, unwillingness to give the true answer or gross errors. To study the sensitivity of the model when the data are contaminated we derive the Influence Function of the parameters and the posterior means, a tool developed in the frame of robust statistics theory. We study the behaviour of the Influence Function for changes in the data and also the behaviour of the parameters and the posterior means when the data are artificially contaminated. We further derive the Influence Function of the parameters and the posterior means for changes in the prior distribution and study empirically the behaviour of the model when the prior is a mixture of distributions. Semiparametric estimation involves estimation of the prior together with the item parameters. A new algorithm for fully semiparametric estimation of the model is given. The bootstrap is then used to study the information on the latent distribution than can be extracted from the data when the model is estimated semiparametrically. The use of the usual goodness-of-fit statistics has been hampered for latent trait models because of the sparseness of the tables. We propose the use of Monte-Carlo simulations to derive the empirical distribution of the goodness-of-fit statistics and also the examination of the residuals as they may pinpoint to the sources of bad fit.