Skip to main content

Advertisement

SpringerLink
Log in

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

  • Regular Article
  • Published:
Advances in Data Analysis and Classification Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  • Aitken AC (1926) On Bernoulli’s numerical solution of algebraic equations. Proc R Soc Edinb 46:289–305

    Article  Google Scholar 

  • Basso RM, Lachos VH, Cabral CRB, Ghosh P (2010) Robust mixture modeling based on scale mixtures of skew-normal distributions. Comput Stat Data Anal 54:2926–2941

    Article  MathSciNet  Google Scholar 

  • Booth J, Casella G, Hobert J (2008) Clustering using objective functions and stochastic search. J R Stat Soc Ser B 70:119–139

    Article  MathSciNet  Google Scholar 

  • Celeux G, Martin O, Lavergne C (2005) Mixture of linear mixed models for clustering gene expression profiles from repeated microarray experiments. Stat Model 5:243–267

    Article  MathSciNet  Google Scholar 

  • Dang U, Punzo A, McNicholas P, Ingrassia S, Browne R (2017) Multivariate response and parsimony for Gaussian cluster-weighted models. J Classif 34:4–34

    Article  MathSciNet  Google Scholar 

  • Dayton C, Macready G (1988) Concomitant-variable latent-class models. J Am Stat Assoc 83:173–178

    Article  MathSciNet  Google Scholar 

  • De la Cruz-Mesía R, Quintana FA, Marshall G (2008) Model-based clustering for longitudinal data. Comput Stat Data Anal 52:1441–1457

    Article  MathSciNet  Google Scholar 

  • Dempster A, Laird N, Rubin D (1977) Maximum likelihood from incomplete data via the EM algorithm (with discussion). J R Stat Soc Ser B 9:1–38

    Article  Google Scholar 

  • Fraley C, Raftery AE (1998) How many clusters? Which clustering method? Answers via model-based cluster analysis. Comput J 41:578–588

    Article  Google Scholar 

  • Fraley C, Raftery A (2002) Model-based clustering, discriminant analysis, and density estimation. J Am Stat Assoc 97:611–612

    Article  MathSciNet  Google Scholar 

  • Frühwirth-Schnatter S (2006) Finite mixture and markov switching models. Springer, New York

    Google Scholar 

  • Gaffney S, Smyth P (2003) Curve clustering with random effects regression mixtures. In: Bishop C, Frey B (Eds.) Proceedings of the ninth international workshop on artificial intelligence and statistics. Akademiai Kiado, KeyWest, FL

  • Gershenfeld N (1997) Nonlinear inference and cluster-weighted modeling. Ann NY Acad Sci 808:18–24

    Article  Google Scholar 

  • Goldfeld S, Quandt R (1973) A Markov model for switching regression. J Econom 1:3–15

    Article  Google Scholar 

  • Grün B, Leisch F (2008) Flexmix version 2: finite mixtures with concomitant variables and varying and constant parameters. J Stat Softw 28:1–35

    Article  Google Scholar 

  • Hammer S, Squires K, Hughes M, Grimes J, Demeter L, Currier J, Eron J, Feinberg J, Balfour H, Deyton L, Chodakewitz J, Fischl M (1997) A controlled trial of two nucleoside analogues plus indinavir in persons with human immunodeficiency virus infection and CD4 cell counts of per cubic millimeter or less. N Engl J Med 337:725–733

    Article  Google Scholar 

  • Hartigan JA, Wong MA (1979) Algorithm AS 136: a \(K\)-means clustering algorithm. J R Stat Soc C 28:100–108

    Google Scholar 

  • Hennig C (2000) Identifiability of models for clusterwise linear regression. J Classif 17:273–296

    Article  MathSciNet  Google Scholar 

  • Huang Y, Dagne G (2011) A Bayesian approach to joint mixed effects models with a skew-normal distribution and measurement errors in covariates. Biometrics 67:260–269

    Article  MathSciNet  Google Scholar 

  • Huang M, Yao W (2012) Mixture of regression models with varying mixing proportions: a semiparametric approach. J Am Stat Assoc 107:711–724

    Article  MathSciNet  Google Scholar 

  • Hubert L, Arabie P (1985) Comparing partitions. J Classif 2:193–218

    Article  Google Scholar 

  • Hunter D, Lange K (2004) A tutorial on MM algorithms. Am Stat 58:30–37

    Article  MathSciNet  Google Scholar 

  • Ingrassia S, Minotti S, Punzo A (2014) Model-based clustering via linear cluster-weighted models. Comput Stat Data Anal 71:159–182

    Article  MathSciNet  Google Scholar 

  • Ingrassia S, Punzo A (2016) Decision boundaries for mixtures of regressions. J Korean Stat Soc 45:295–306

    Article  MathSciNet  Google Scholar 

  • Ingrassia S, Punzo A (2020) Cluster validation for mixtures of regressions via the total sum of squares decomposition. J Classif 37:526–547

    Article  MathSciNet  Google Scholar 

  • Ingrassia S, Punzo A, Vittadini G, Minotti S (2015) The generalized linear mixed cluster-weighted model. J Classif 32:85–113

    Article  MathSciNet  Google Scholar 

  • Keribin C (2000) Consistent estimation of the order of mixture models. Sankhyõ A 62:49–66

    MathSciNet  Google Scholar 

  • Konrad S, Skinner S, Kazadi G, Gartner K, Lim H (2013) HIV disease progression to CD4 count\(<\)200 cells/\(\mu \)L and death in Saskatoon, Saskatchewan. Can J Infect Dis Med Microbiol 24:97–101

    Article  Google Scholar 

  • Krentz H, Auld M, Gill M (2004) The high cost of medical care for patients who present late (CD4\(<\)200 cells/\(\mu \)L) with HIV infection. HIV Med 5:93–98

    Article  Google Scholar 

  • Laird N, Ware J (1982) Random effects models for longitudinal data. Biometrics 38:963–974

    Article  Google Scholar 

  • Lamont A, Vermunt J, Horn MV (2016) Regression mixture models: Does modeling the covariance between independent variables and latent classes improve the results? Multivar Behav Res 51:35–52

    Article  Google Scholar 

  • Lavielle M, Aarons L (2016) What do we mean by identifiability in mixed effects models? J Pharmacokinet Pharmacodyn 43:111–122

    Article  Google Scholar 

  • Lederman M, Connick E, Landay A, Kuritzkes D, Spritzler J, Clair M, Kotzin B, Fox L, Chiozzi M, Leonard J, Rousseau F, Wade M, Roe J, Martinez A, Harold K (1998) Immunologic responses associated with 12 weeks of combination antiretroviral therapy consisting of zidovudine, lamivudine, and ritonavir: results of AIDS clinical trials group protocol 315. J Infect Dis 178:70–79

    Article  Google Scholar 

  • Lee W, Chen Y, Hsieh K (2003) Ultrasonic liver tissues classification by fractal feature vector based on M-band wavelet transform. IEEE Trans Med Imaging 22:382–392

    Article  Google Scholar 

  • Liang H, Wu H, Carroll R (2003) The relationship between virologic and immunologic responses in AIDS clinical research using mixed effects varying coefficient models with measurement error. Biostatistics 4:297–312

    Article  Google Scholar 

  • Lin TI, Lachos VH, Wang WL (2018) Multivariate longitudinal data analysis with censored and intermittent missing responses. Stat Med 37(19):2822–2835

    Article  MathSciNet  Google Scholar 

  • Lin TI, Wang WL (2013) Multivariate skew-normal at linear mixed models for multi-outcome longitudinal data. Stat Model 13:199–221

    Article  MathSciNet  Google Scholar 

  • Lin TI, Wang WL (2017) Multivariate-\(t\) nonlinear mixed models with application to censored multi-outcome AIDS studies. Biostatistics 18:666–681

    Google Scholar 

  • Lin TI, Wang WL (2020) Multivariate-\(t\) linear mixed models with censored responses, intermittent missing values and heavy tails. Stat Meth Med Res 29(5):1288–1304

    Article  MathSciNet  Google Scholar 

  • Lindstrom MJ, Bates DM (1990) Nonlinear mixed effects models for repeated measures data. Biometrics 46:673–687

    Article  MathSciNet  Google Scholar 

  • Louis T (1982) Finding the observed information matrix when using the EM algorithm. J R Stat Soc Ser B 44:226–233

    Article  MathSciNet  Google Scholar 

  • Matos LA, Lachos VH, Lin TI, Castro LM (2019) Heavy-tailed longitudinal regression models for censored data: a robust parametric approach. Test 28:844–878

    Article  MathSciNet  Google Scholar 

  • McLachlan GJ, Krishnan T (2008) The EM algorithm and extensions, 2nd edn. John Wiley & Sons, New York

    Book  Google Scholar 

  • McLachlan G, Peel D (2000) Finite mixture models. Wiley, New York

    Book  Google Scholar 

  • Meng X, Rubin D (1993) Maximum likelihood estimation via the ECM algorithm: a general framework. Biometrika 80:267–278

    Article  MathSciNet  Google Scholar 

  • Muñoz A, Carey V, Schouten JP, Segal M, Rosner B (1992) A parametric family of correlation structures for the analysis of longitudinal data. Biometrics 48:733–742

    Article  Google Scholar 

  • Ng S, McLachlan G (2014) Mixture models for clustering multilevel growth trajectories. Comput Stat Data Anal 71:43–51

    Article  MathSciNet  Google Scholar 

  • Ng S, Mclachlan G, Wang K, Jones L, Ng S (2006) A mixture model with random-effects components for clustering correlated gene-expression profiles. Bioinformatrics 22:1745–1752

    Article  Google Scholar 

  • Perelson A, Essunger P, Cao Y, Vesanen M, Hurley A, Saksela K, Markowitz M, Ho D (1997) Decay characteristics of HIV-1-infected compartments during combination therapy. Nature 387:188–191

    Article  Google Scholar 

  • Pfeifer C (2004) Classification of longitudinal profiles based on semi-parametric regression with mixed effects. Stat Med 4:314–323

    MathSciNet  Google Scholar 

  • Pinheiro J, Bates D, Debroy S, Sarkar D, R Core Team (2016) nlme: linear and nonlinear mixed effects models. R package version 3.1-128. http://CRAN.R-project.org/package=nlme. Accessed from 8 Sep 2016

  • Punzo A (2014) Flexible mixture modeling with the polynomial Gaussian cluster-weighted model. Stat Model 14:257–291

    Article  MathSciNet  Google Scholar 

  • Quandt R (1972) A new approach to estimating switching regressions. J Am Stat Assoc 67:306–310

    Article  Google Scholar 

  • Quandt R, Ramsey J (1978) Estimating mixtures of normal distributions and switching regressions. J Am Stat Assoc 73:730–738

    Article  MathSciNet  Google Scholar 

  • R Core Team (2019) R: a Language and Environment for Statistical Computing. http://www.r-project.org/

  • Redner RA, Walker HF (1984) Mixture densities, maximum likelihood and the EM algorithm. SIAM Rev 26:195–239

    Article  MathSciNet  Google Scholar 

  • Regis M, Brini A, Nooraee N, Haakma R, van den Heuvel ER (2019) The \(t\) linear mixed model: model formulation, identifiability and estimation. Commun Stat Simul Comput. https://doi.org/10.1080/03610918.2019.1694153

    Article  Google Scholar 

  • Robin X, Turck N, Hainard A, Tiberti N, Lisacek F, Sanchez J, Müller M (2011) pROC: an open-source package for R and S+ to analyze and compare ROC curves. BMC Bioinform 12(1):77

    Article  Google Scholar 

  • Sarbo W, Cron W (1988) A maximum likelihood methodology for clusterwise linear regression. J Classif 5:249–282

    Article  MathSciNet  Google Scholar 

  • Schwarz G (1978) Estimating the dimension of a model. Ann Stat 6:461–464

    Article  MathSciNet  Google Scholar 

  • Stephens M (2000) Dealing with label switching in mixture models. J R Stat Soc Ser B 62:795–809

    Article  MathSciNet  Google Scholar 

  • Subedi S, Punzo A, Ingrassia S, McNicholas P (2015) Cluster-weighted \(t\)-factor analyzers for robust model-based clustering and dimension reduction. Stat Methods Appl 24:623–649

    Article  MathSciNet  Google Scholar 

  • Trabzuni D, Thomson P, The United KingdomBrain Expression Consortium (UKBEC) (2014) Analysis of gene expression data using a linear mixed model/finite mixture model approach: application to regional differences in the human brain. Bioinformatics 30:1555–1561

    Article  Google Scholar 

  • Verbeke G, Lesaffre E (1996) A linear mixed-effects model with heterogeneity in the random-effects population. J Am Stat Assoc 91:217–221

    Article  Google Scholar 

  • Wang W (2013) Identifiability of linear mixed effects models. Electron J Stat 7:244–263

    Article  MathSciNet  Google Scholar 

  • Wang W (2016) Identifiability of covariance parameters in linear mixed effects models. Linear Algebra Appl 506:603–613

    Article  MathSciNet  Google Scholar 

  • Wang WL (2019) Mixture of multivariate \(t\) nonlinear mixed models for multiple longitudinal data with heterogeneity and missing values. TEST 28:196–222

    Article  MathSciNet  Google Scholar 

  • Wang WL (2020) Bayesian analysis of multivariate linear mixed models with censored and intermittent missing responses. Stat Med 39(19):2518–2535

    Article  MathSciNet  Google Scholar 

  • Wang WL, Lin TI (2014) Multivariate t nonlinear mixed-effects models for multi-outcome longitudinal data with missing values. Stat Med 33:3029–3046

    Article  MathSciNet  Google Scholar 

  • Wang WL, Lin TI, Lachos VH (2018) Extending multivariate-\(t\) linear mixed models for multiple longitudinal data with censored responses and heavy tails. Stat Meth Med Res 27:48–64

    Article  MathSciNet  Google Scholar 

  • Wedel M (2002) Concomitant variables in finite mixture models. Stat Neerl 56:362–375

    Article  MathSciNet  Google Scholar 

  • Wedel M, Kamakura W (2000) Market segmentation: conceptual and methodological foundations, 2nd edn. Kluwer Academic Publishers, Boston

    Book  Google Scholar 

  • Wolfinger R, Lin X (1997) Two Taylor-series approximation methods for nonlinear mixed models. Comput Stat Data Anal 25:465–490

    Article  Google Scholar 

  • Yang YC, Lin TI, Luis MC, Wang WL (2020) Extending finite mixtures of \(t\) linear mixed-effects models with concomitant covariates. Comput Stat Data Anal 148:106961

    Article  MathSciNet  Google Scholar 

  • Yau K, Lee A, Ng S (2002) Finite mixture regression model with random effects: application to neonatal hospital length of stay. Comput Stat Data Anal 41:359–366

    Article  MathSciNet  Google Scholar 

  • Zeller CB, Cabral CRB, Lachos VH, Benites L (2019) Finite mixture of regression models for censored data based on scale mixtures of normal distributions. Adv Data Anal Classif 13:89–116

    Article  MathSciNet  Google Scholar 

Download references

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.

Author information

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

Authors
  1. Wan-Lun Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu-Chen Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tsung-I Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tsung-I Lin.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11634-022-00502-w

Keywords

Mathematics Subject Classification

Search

Navigation