Random Intercept Item Factor Analysis

The common factor model assumes a linear relation between the observed variables and a set of underlying latent traits. It also assumes that the linear coefficients, intercepts and slopes (factor loadings), linking the observed variables to the latent traits are fixed coefficients (i.e., common for all subjects). When the observed variables are subjects' direct responses to stimuli, such as their responses to the items of a questionnaire, the assumption of common linear coefficients may be too restrictive. This may occur, for instance if respondents to questionnaire items consistently use the response scale idiosyncratically. To account for this phenomenon we partially relax the fixed coefficients assumption by letting the intercepts in the factor model change across subjects while keeping the factor loadings fixed. We show that, under suitable assumptions on this random component of the intercept, the covariance structure implied by a model with p factors and random intercepts is equivalent to the covariance structure model implied by a model with p + 1 factors model where one of the factors have common loadings