Model specification and SAS program for random coefficient multinomial logit model on health state among older Americans

The following random coefficient multinomial logit regression model is an extension of Equation (11.11). It is assumed that, in addition to the random effects on intercepts, the regression coefficients of time vary significantly over subjects, thereby specifying an extra random slope term. For analytic simplicity, the effects of time × time are assumed to remain fixed. Consequently, we have

$\begin{array}{l} P_{i j k} & = \Pr (Y_{i j} = k |X_{i j}) = {[1 + \sum_{l = 1}^{K} \exp (X_{i j}^{'} β_{l} + b_{0 i l} + b_{1 i l} T_{j} + ɛ_{i j l})]}^{- 1} \exp (X_{i j}^{'} β_{k} + b_{0 i k} + b_{1 i k} T_{j} + ɛ_{i j k}) \\ = {[1 + \sum_{l = 1}^{K} \exp (X_{i j}^{'} β_{l}) \exp (b_{0 i l} + b_{1 i l} T_{j} + ɛ_{i j l})]}^{- 1} \exp (X_{i j}^{'} β_{k}) \exp (b_{0 i k} + b_{1 i k} T_{j} + ɛ_{i j k}), \end{array}$ $\begin{array}{l} P_{i j k} & = \Pr (Y_{i j} = k |X_{i j}) = {[1 + \sum_{l = 1}^{K} \exp (X_{i j}^{'} β_{l} + b_{0 i l} + b_{1 i l} T_{j} + ɛ_{i j l})]}^{- 1} \exp (X_{i j}^{'} β_{k} + b_{0 i k} + b_{1 i k} T_{j} + ɛ_{i j k}) \\ = {[1 + \sum_{l = 1}^{K} \exp (X_{i j}^{'} β_{l}) \exp (b_{0 i l} + b_{1 i l} T_{j} + ɛ_{i j l})]}^{- 1} \exp (X_{i j}^{'} β_{k}) \exp (b_{0 i k} + b_{1 i k} T_{j} + ɛ_{i j k}), \end{array}$

(D.1)

where b_0ik is the between-subjects random effect on the intercept assumed to be distributed as

N (0, σ_{b_{0 k}}^{2})

$N (0, σ_{b_{0 k}}^{2})$

, and b_1ik is the between-subjects random effect on the regression coefficient of T assumed to be distributed as

N (0, σ_{b_{1 k}}^{2})

$N (0, σ_{b_{1 k}}^{2})$

. With the specification of a large number of parameters in the model, I only need to estimate the variances of the random effects, assuming all covariances to be zero.

Given the above specifications, SAS Program 11.2 is adapted for the creation of a random coefficient multinomial logit model by including two additional random terms. The following is the revised SAS program.

SAS Program D1:

With the AHEAD longitudinal data, SAS Program D1 derives the following output:

In the above output, the analytic results for the fixed effects are analogous to those from the corresponding random intercept multinomial logit model, and the two additional random terms are not statistically significant. Therefore, in this analysis, the application of the random coefficient approach is not necessary.

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Table of Contents for Appendix D: Model specification and SAS program for random coefficient multinomial logit model on health state among older Americans

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Model specification and SAS program for random coefficient multinomial logit model on health state among older Americans

Table of Contents for
Appendix D: Model specification and SAS program for random coefficient multinomial logit model on health state among older Americans