Bayesian nonparametric modeling: alternative item response theory models
Bayesian item response theory modeling is a complex issue as it requires the estimation of many parameters (at least one parameter per respondent and one per item). The problem is especially intricate when Bayesian nonparametric item response theory models (BNIRMs) are used, as the number of parameters scale really quickly. Also, to guarantee the identifiability of the model, restrictions regarding the distribution of the true scores or the item response function (IRF) are used. The aim of the present study is to develop BNIRMs derived from optimal scoring, a new nonparametric psychometric approach that similar to Mokken Scale Analysis uses sum scores as initial guesses for estimating the IRFs. We propose four approaches for estimating the IRFs: the first two use basis expansion (Legendre and B-splines); the third one uses a single hidden layer neural network; and the last one is a new proposed way (developed in the present study) of doing of piecewise regression, which we call Rademacher basis. The priors for the regression coefficients of the bases follow a normal distribution with mean 0 and standard deviation equals to 1 for L2-regularization. For the priors of the latent true scores, we propose what we call a Kolmogorov-Smirnov prior, which uses the empirical cumulative distribution of the sum scores as an initial estimate for the distribution function. We provide Maximum a Posteriori estimation with Genetic Algorithm, as well as MCMC estimation with a Hit-and-Run algorithm. Comparisons between performances and future studies are discussed.