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Extending finite mixtures of nonlinear mixed-effects models with covariate-dependent mixing weights

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Abstract

Finite mixtures of nonlinear mixed-effects models have emerged as a prominent tool for modeling and clustering longitudinal data following nonlinear growth patterns with heterogeneous behavior. This paper proposes an extended finite mixtures of nonlinear mixed-effects model in which the mixing proportions are related to some explanatory covariates. A logistic function is incorporated to describe the relationship between the prior classification probabilities and the covariates of interest. For parameter estimation, we develop an analytically simple expectation conditional maximization algorithm coupled with the first-order Taylor approximation to linearize the model with pseudo data. The calculation of the standard errors of estimators via a general information-based method and the empirical Bayes estimation of random effects are also discussed. The methodology is illustrated through several simulation experiments and an application to the AIDS Clinical Trials Group Protocol 315 study.

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Acknowledgements

The authors gratefully acknowledge the Associate Editor and three anonymous referees for their insightful comments which helped to improve the quality of the paper. In addition, the authors are grateful to Mr. Yi-Cong Li for his skillful assistance of initial graphical outputs. This work was supported in part by the Ministry of Science and Technology of Taiwan under Grant Nos. 110-2118-M-006-006-MY3 and 109-2118-M-005-005-MY3.

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Authors and Affiliations

  1. Institute of Statistics, National Chung Hsing University, Taichung, 402, Taiwan

    Tsung-I Lin

  2. Department of Public Health, China Medical University, Taichung, 404, Taiwan

    Tsung-I Lin

  3. Department of Statistics and Institute of Data Science, National Cheng Kung University, Tainan, 701, Taiwan

    Wan-Lun Wang

  4. Department of Applied Mathematics, National Chung Hsing University, Taichung, 402, Taiwan

    Yu-Chen Yang

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  1. Wan-Lun Wang
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Appendix: Explicit expressions for the score vector and Hessian matrix

Appendix: Explicit expressions for the score vector and Hessian matrix

Taking the first partial derivative of (20) for the jth individual with respect to each entry of \({\varvec{\theta }}_i\) gives \({\varvec{s}}_{{\varvec{\theta }}_i}^{(j)}=({\varvec{s}}_{{\varvec{\beta }}_i}^{{(j)}^\top },s_{\sigma _i^2}^{(j)},{\varvec{s}}_{{\varvec{\alpha }}_i}^{{(j)}^\top })^\top \), where

$$\begin{aligned} {\varvec{s}}^{(j)}_{{\varvec{\beta }}_i}= & {} \frac{1}{\sigma _i^2}\tilde{{\varvec{X}}}_{ij}^{\top }\tilde{{\varvec{\varLambda }}}^{-1}_{ij}(\tilde{{\varvec{y}}}_{ij}-\tilde{{\varvec{X}}}_{ij}{\varvec{\beta }}_i),\\ s^{(j)}_{\sigma ^2_i}= & {} \left\{ -\frac{(s_j+q)}{2\sigma ^2_i}+\frac{1}{2\sigma ^4_i}(\tilde{{\varvec{y}}}_{ij}-\tilde{{\varvec{X}}}_{ij}{\varvec{\beta }}_i)^{\top }\tilde{{\varvec{\varLambda }}}^{-1}_{ij}(\tilde{{\varvec{y}}}_{ij}-\tilde{{\varvec{X}}}_{ij}{\varvec{\beta }}_i)\right\} , \end{aligned}$$

and

$$\begin{aligned} {[{\varvec{s}}^{(j)}_{{\varvec{\alpha }}_i}]}_l=\frac{1}{2}\left\{ \frac{1}{\sigma _i^2}\text{ tr }\Big [(\tilde{{\varvec{y}}}_{ij}-\tilde{{\varvec{X}}}_{ij}{\varvec{\beta }}_i)(\tilde{{\varvec{y}}}_{ij}-\tilde{{\varvec{X}}}_{ij}{\varvec{\beta }}_i)^{\top }\tilde{{\varvec{\varLambda }}}^{-1}_{ij}\dot{\tilde{{\varvec{\varLambda }}}}_{ijl}\tilde{{\varvec{\varLambda }}}^{-1}_{ij}\Big ]-\text{ tr }\big (\tilde{{\varvec{\varLambda }}}^{-1}_{ij}\dot{\tilde{{\varvec{\varLambda }}}}_{ijl}\big )\right\} , \end{aligned}$$

with \(\dot{\tilde{{\varvec{\varLambda }}}}_{ ijl}=\partial \tilde{{\varvec{\varLambda }}}_{ij}/\partial ({\varvec{\alpha }}_i)_l=\tilde{{\varvec{Z}}}_{ij}\frac{\partial {\varvec{D}}_i}{\partial ({\varvec{\alpha }}_i)_l}\tilde{{\varvec{Z}}}_{ij}^{\top }\) if \(({\varvec{\alpha }}_i)_l=\text{ vech }({\varvec{D}}_i)\), and \(\dot{\tilde{{\varvec{\varLambda }}}}_{ ijl}=\frac{\partial {\varvec{C}}_{ij}({\varvec{\phi }}_i,{\varvec{t}}_j)}{\partial ({\varvec{\alpha }}_i)_l}\) if \(({\varvec{\alpha }}_i)_l={\varvec{\phi }}_i\), for \(l=1,\ldots ,q^*\) where \(q^*=q(q+1)/2+1\). The expressions for the entries of \({\varvec{H}}_{{\varvec{\theta }}_i{\varvec{\theta }}_i}^{(j)}\) derived from the minus second partial derivative of (20) for the jth individual are

$$\begin{aligned} {\varvec{H}}^{(j)}_{{\varvec{\beta }}_i{\varvec{\beta }}_i}= & {} \frac{1}{\sigma _i^2}\tilde{{\varvec{X}}}^{\top }_{ij}\tilde{{\varvec{\varLambda }}}^{-1}_{ij}\tilde{{\varvec{X}}}_{ij},\\ {\varvec{H}}^{(j)}_{{\varvec{\beta }}_i\sigma ^2_i}= & {} \frac{1}{\sigma _i^4}\tilde{{\varvec{X}}}^{\top }_{ij}\tilde{{\varvec{\varLambda }}}^{-1}_{ij}(\tilde{{\varvec{y}}}_{ij}-\tilde{{\varvec{X}}}_{ij}{\varvec{\beta }}_i),\\ {\varvec{H}}^{(j)}_{{\varvec{\beta }}_i{\varvec{\alpha }}_i}= & {} \frac{1}{\sigma _i^2}\tilde{{\varvec{X}}}^{\top }_{ij}\tilde{{\varvec{\varLambda }}}^{-1}_{ij}\dot{\tilde{{\varvec{\varLambda }}}}_{ijl}\tilde{{\varvec{\varLambda }}}^{-1}_{ij}(\tilde{{\varvec{y}}}_{ij}-\tilde{{\varvec{X}}}_{ij}{\varvec{\beta }}_i),\\ H^{(j)}_{\sigma ^2_i\sigma ^2_i}= & {} \frac{1}{\sigma _i^6}(\tilde{{\varvec{y}}}_{ij}-\tilde{{\varvec{X}}}_{ij}{\varvec{\beta }}_i)^{\top }\tilde{{\varvec{\varLambda }}}^{-1}_{ij}(\tilde{{\varvec{y}}}_{ij}-\tilde{{\varvec{X}}}_{ij}{\varvec{\beta }}_i)-\frac{(s_j+q)}{2\sigma ^4_i},\\ {[{\varvec{H}}^{(j)}_{\sigma ^2_i{\varvec{\alpha }}_i}]}_l= & {} \frac{1}{2\sigma ^4_i}(\tilde{{\varvec{y}}}_{ij}-\tilde{{\varvec{X}}}_{ij}{\varvec{\beta }}_i)\tilde{{\varvec{\varLambda }}}^{-1}_{ij}\dot{\tilde{{\varvec{\varLambda }}}}_{ijl}\tilde{{\varvec{\varLambda }}}^{-1}_{ij}(\tilde{{\varvec{y}}}_{ij}-\tilde{{\varvec{X}}}_{ij}{\varvec{\beta }}_i), \end{aligned}$$

and

$$\begin{aligned} {[{\varvec{H}}^{(j)}_{{\varvec{\alpha }}_i{\varvec{\alpha }}_i}]}_{lr}= & {} -\frac{1}{2}\Big \{\text{ tr }(\tilde{{\varvec{\varLambda }}}^{-1}_{ij}\dot{\tilde{{\varvec{\varLambda }}}}_{ijr}\tilde{{\varvec{\varLambda }}}^{-1}_{ij}\dot{\tilde{{\varvec{\varLambda }}}}_{ijl}) -\frac{1}{\sigma _i^2}\text{ tr }\Big [(\tilde{{\varvec{y}}}_{ij}-\tilde{{\varvec{X}}}_{ij}{\varvec{\beta }}_i)(\tilde{{\varvec{y}}}_{ij}-\tilde{{\varvec{X}}}_{ij}{\varvec{\beta }}_i)^{\top }\\&\times (\tilde{{\varvec{\varLambda }}}^{-1}_{ij}\dot{\tilde{{\varvec{\varLambda }}}}_{ijr}\tilde{{\varvec{\varLambda }}}^{-1}_{ij}\dot{\tilde{{\varvec{\varLambda }}}}_{ijl}\tilde{{\varvec{\varLambda }}}^{-1}_{ij} +\tilde{{\varvec{\varLambda }}}^{-1}_{ij}\dot{\tilde{{\varvec{\varLambda }}}}_{ijl}\tilde{{\varvec{\varLambda }}}^{-1}_{ij}\dot{\tilde{{\varvec{\varLambda }}}}_{ijr}\tilde{{\varvec{\varLambda }}}^{-1}_{ij})\Big ]\Big \}, \end{aligned}$$

for \(l,r=1,\ldots ,q^*\).

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Wang, WL., Yang, YC. & Lin, TI. Extending finite mixtures of nonlinear mixed-effects models with covariate-dependent mixing weights. Adv Data Anal Classif 18, 271–307 (2024). https://doi.org/10.1007/s11634-022-00502-w

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