content_image/322911.jpg content_image/322912.jpg content_image/322913.jpg content_image/322914.jpg
Taking all observations from a sample into account, the objective function can be expressed as 

<img src='content_image/322911.jpg'>

where x$_{i,}$$_{−}$$_{j}$ = ( x$_{i}$$_{1}$, . . . , x$_{i, j}$$_{−}$$_{1}$, x$_{i, j}$$_{+1}$, . . . , x$_{iJ}$ ). The optimization problem is then given by 

<img src='content_image/322912.jpg'>

It can be shown that, after algebraic simplification, the Q -function has the form 

<img src='content_image/322913.jpg'>

The posterior probability for an individual being classified into α given the response profile x$_{i}$ is obtained by 

<img src='content_image/322914.jpg'>

## 4 Computation 

The present study employs coordinate descent algorithm to solve the optimization problem (7). The algorithm minimizes the multivariate function − Q ( θ | θ ( t $^{)}$) with respect to each coordinate direction at a time and solves a sequence of univariate optimization subproblems in a loop. The procedure is simple to implement and computationally advantageous when the gradient of a func- tion is expensive to evaluate. When it comes to estimating the design matrix, the log pseudo- likelihood becomes a concave function, and thus, standard convex programming can be used to estimate Φ . This study employs the Newton’s algorithm to minimize the quadratic approximation of log f ( x$_{ij}$ | x$_{i,}$$_{−}$$_{j}$ , α , B , Φ ), similarly to solving the lasso problem (e.g., Friedman, Hastie, H¨ ofling, & Tibshirani, 2007; Friedman, Hastie, & Tibshirani, 2010). 

Let P ( t ) ij ( α ) denote the conditional probability of answering item j correctly for examinee i with 

