Taking all observations from a sample into account, the objective function can be expressed as where x$_{i,}$$_{−}$$_{j}$ = ( x$_{i}$$_{1}$, . . . , x$_{i, j}$$_{−}$$_{1}$, x$_{i, j}$$_{+1}$, . . . , x$_{iJ}$ ). The optimization problem is then given by It can be shown that, after algebraic simplification, the Q -function has the form The posterior probability for an individual being classified into α given the response profile x$_{i}$ is obtained by 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 4 Computation Q ( θ | θ ( t $^{)}$) = N ∑ i =1 ∑ α f ( α | x$_{i}$ , θ ( t $^{)}$) J ∑ j =1 $^{[}$log f ( x$_{ij}$ | x$_{i,}$$_{−}$$_{j}$ , α , B , Φ ) + log f ( α | x$_{i,}$$_{−}$$_{j}$ , B , π$_{α}$ ) $^{]}$, ( ˆ B , ˆ Φ , ˆ π ) = arg max ( B , Φ , π ) Q ( θ | θ ( t $^{)}$) . (7) Q ( θ | θ ( t $^{)}$) = N ∑ i =1 ∑ α f ( α | x$_{i}$ , θ ( t $^{)}$) J ∑ j =1 [ log π$_{α}$ + x ⊤ i B α + 1 2 x ⊤ i Φx$_{i}$ − log$^{}$   ∑ ˜ α π$_{˜}$ $_{α}$exp$^{}$  ∑ j $^{′}$̸ = j x$_{ij}$ ′ β ⊤ j ′ ˜ α$^{}$   $_{}$1 + exp$^{}$ $_{}$β ⊤ j ˜ α + ∑ j $^{′}$̸ = j φ$_{jj}$ ′ x$_{ij}$ ′        ] . f ( α | x$_{i}$ , θ ( t $^{)}$) = f ( α | x$_{i}$ , B ( t $^{)}$, π ( t ) α ) = π ( t ) α exp $^{(}$x ⊤ i B ( t $^{)}$α ) ∑ ˜ α π ( t ) ˜ α exp $^{(}$x ⊤ i B ( t $^{)}$˜ α $_{)}$. (8) 10