Algorithm 1.

Create a negative version −xj of each predictor xj and expand the predictor space to X̃ = (X, −X). Set the initial values at step s = 0 for the coefficients β(0) =(β1, ⋯, β2p) = 0. The initial estimates of the intercept αc and random effects ui are obtained from the null random coefficient model. Find the predictor xj, j = 1, ⋯, 2p with the largest negative gradient of the log-likelihood $-\frac{\partial \text{log}L(𝜶,𝜷,{\mathbf{u}}_{\mathbf{i}},{\mathbf{G}}_{\mathbf{i}};{\mathbf{x}}_{\mathbf{i}},{\mathbf{z}}_{\mathbf{i}})}{\partial {\beta }_j}$ evaluated at the current estimate β(s). Update the coefficient estimate of the selected predictor xj in step 2 with ${\beta }_j^{(s+1)}\leftarrow {\beta }_j^{(s)}+\epsilon$, where ε is a small positive amount; a rational choice is ε = 1 × 10−4. Repeat steps 2 and 3 many times until it meets either criteria: 1) the difference between two successive log-likelihood is smaller than a given value δ; 2) the number of features having a nonzero coefficient estimates is less than a specified value. Fit a sequence of random coefficient ordinal response models using features selected from GMIFS at steps immediately preceding the step where a new feature enters the active set. The parsimonious model is based on model fitting criteria, e.g. AIC, BIC.