In IRT models—as opposed to CTT—the influence of the items and the respondents are explicitly modelled by distinct sets of parameters. In these models, an assumed continuous latent variable θ reflects the trait and every item is identified by thresholds β where a response in one category becomes more likely than a response in an adjacent category. It is usually assumed that the latent variables θj are drawn from a normal distribution, that is, θj are independently and identically distributed N(μ, σ2), though this assumption is not always necessary to identify the model parameters. The probability of the presence of the symptom i in individual j, p(Yij = 1), is a function of the difference between the individual’s trait score θj and the parameter βi, with βi indicating the location on the scale where the presence of a symptom becomes more probable than its absence. In the case of multiple symptoms, we have 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ p(Y_{{ij}} = 1) = \Phi (\theta _{j} - \beta _{i} ), $$\end{document}with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb}
