coefficients and discourages the coefficients from having extreme values in Φ . The same value of λ for all φ$_{jj}$ ′ s implies that the same degree of sparsity is enforced across all pairs of the items. In practice it is possible to impose different levels of penalties on the respective φ$_{jj}$ ′ s—for example, when the prior knowledge about the item relationships is available due to the testlet design. Also note that no penalty is imposed on the diagonal elements in the above expression. This is because it has been assumed that φ$_{jj}$ = 0 for all j = 1 , . . . , J and that any nonzero value can be inferred from the DCM. Even if φ$_{jj}$ s were not assumed to be zero, the diagonal entries of the design matrix should not be penalized because they impart the effects of the DCM of concern. In this case constraints on the parameters are needed to ensure identifiability of the model. Simple calculus (Donoho & Johnstone., 1994) suggests that the solution to the problem (9) has a closed-form expression. A coordinate-wise update for φ$_{jj}$ ′ is obtained as where S ( z, λ ) is the soft-thresholding operator with the value and r$_{ijj}$ ′ = y ( t ) ij − y ( t ) ij, − j ′ is the partial residual for fitting φ$_{jj}$ ′ . Once the estimate Φ ( t +1) is obtained from the above procedure, B ( t +1) and π ( t +1) are obtained sequentially via the coordinate descent method. The B -matrix is updated with a maximizer of the function The optimization problem is solved for each parameter separately, holding all other parameters fixed at the latest values. It should be pointed out that the maximization in (11) is enforced only φ ( t +1) jj ′ ←− S ( 1 N N ∑ i =1 ∑ α f ( α | x$_{i}$ , θ ( t $^{)}$) ω ( t ) ij x$_{ij}$ ′ r$_{ijj}$ ′ , λ$^{)}$ 1 N N ∑ i =1 ∑ α f ( α | x$_{i}$ , θ ( t $^{)}$) ω ( t ) ij x 2 ij ′ , (10) S ( z, λ ) = sign( z )( | z | − λ )$_{+}$ =        z − λ if z > 0 and λ < | z | z + λ if z < 0 and λ < | z | 0 if λ ≥ | z | , N ∑ i =1 ∑ α f ( α | x$_{i}$ , θ ( t $^{)}$) J ∑ j =1 log f ( x$_{ij}$ | x$_{i,}$$_{−}$$_{j}$ , α , B , Φ ) . (11) 12