Likelihood-based estimation methods for models for concurrent continuous and discrete responses

Recently, continuous responses have attracted interest as covariates accompanying discrete responses. One may think of the response time needed to answer an item in a computerized adaptive testing situation. In the present report, a model is proposed to analyze such data. In this model, dichotomous responses are modeled by a multidimensional extension of the three-parameter logistic model, continuous responses are modeled by a multidimensional extension of a model proposed by Mellenbergh (1994), and the relation between the responses is modeled by a variant of the testlet model by Bradlow, Wainer, and Wang (1999). Amarginal maximum likelihood method is outlined and a simulation study is conducted to test the precision of this estimation procedure and the Type I error rate and power of likelihood-ratio tests pertaining to the covariance structure.