Quantile regression for longitudinal data based on latent Markov subject-specific parameters. (English) Zbl 1322.62206

Summary: We propose a latent Markov quantile regression model for longitudinal data with non-informative drop-out. The observations, conditionally on covariates, are modeled through an asymmetric Laplace distribution. Random effects are assumed to be time-varying and to follow a first order latent Markov chain. This latter assumption is easily interpretable and allows exact inference through an ad hoc EMtype algorithm based on appropriate recursions. Finally, we illustrate the model on a benchmark data set.


62M05 Markov processes: estimation; hidden Markov models
62G30 Order statistics; empirical distribution functions
62J02 General nonlinear regression


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[1] Akaike, H.: Information theory as an extension of the maximum likelihood principle. In: Petrov, B.N., Csaki, F. (eds.) Second International Symposium on Information Theory, pp. 267–281. Akademiai Kiado, Budapest (1973) · Zbl 0283.62006
[2] Andrews, D.W.K., Buchinsky, M.: A three-step method for choosing the number of bootstrap repetitions. Econometrica 68, 23–52 (2000) · Zbl 1056.62516
[3] Bartolucci, F.: Likelihood inference for a class of latent Markov models under linear hypotheses on the transition probabilities. J. R. Stat. Soc. Ser. B 68, 155–178 (2006) · Zbl 1100.62078
[4] Bartolucci, F., Farcomeni, A.: A multivariate extension of the dynamic logit model for longitudinal data based on a latent Markov heterogeneity structure. J. Am. Stat. Assoc. 104, 816–831 (2009) · Zbl 1388.62158
[5] Bartolucci, F., Farcomeni, A., Pennoni, F.: An overview of latent Markov models for longitudinal categorical data (2010). arXiv:1003.2804 · Zbl 1341.62002
[6] Boucheron, S., Gassiat, E.: An information-theoretic perspective on order estimation. In: Rydén, T., Cappé, O., Moulines, E. (eds.) Inference in Hidden Markov Models, pp. 565–602. Springer, Berlin (2007)
[7] Buchinsky, M.: Estimating the asymptotic covariance matrix for quantile regression models: a Monte Carlo study. J. Econom. 68, 303–338 (1995) · Zbl 0825.62437
[8] Burnham, K.P., Anderson, D.R.: Model Selection and Multi-Model Inference: A Practical Information-Theoretic Approach. Springer, New York (2002) · Zbl 1005.62007
[9] Chernozhukov, V.: Extremal quantile regression. Ann. Stat. 33, 806–839 (2005) · Zbl 1068.62063
[10] Dardanoni, V., Forcina, A.: A unified approach to likelihood inference on stochastic orderings in a nonparametric context. J. Am. Stat. Assoc. 93, 1112–1123 (1998) · Zbl 1063.62547
[11] Davis, S.: Semi-parametric and non-parametric methods for the analysis of repeated measurements with applications to clinical trials. Stat. Med. 10, 1959–1980 (1991)
[12] Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood from incomplete data via the EM algorithm (with discussion). J. R. Stat. Soc. Ser. B 39, 1–38 (1977) · Zbl 0364.62022
[13] Geraci, M., Bottai, M.: Quantile regression for longitudinal data using the asymmetric Laplace distribution. Biostatistics 8, 140–154 (2007) · Zbl 1170.62380
[14] Hsiao, C.: Analysis of Panel Data. Cambridge University Press, New York (2005) · Zbl 1320.62003
[15] Jung, S.-H.: Quasi-likelihood for median regression models. J. Am. Stat. Assoc. 91, 251–257 (1996) · Zbl 0871.62060
[16] Karlsson, A.: Nonlinear quantile regression estimation of longitudinal data. Commun. Stat., Simul. Comput. 37, 114–131 (2008) · Zbl 1139.62021
[17] Koenker, R.: Quantile regression for longitudinal data. J. Multivar. Anal. 91, 74–89 (2004) · Zbl 1051.62059
[18] Koenker, R.: Quantile Regression. Cambridge University Press, New York (2005) · Zbl 1111.62037
[19] Koenker, R., Bassett, G.: Regression quantiles. Econometrica 46, 33–50 (1978) · Zbl 0373.62038
[20] Koenker, R., d’Orey, V.: Computing regression quantiles. Appl. Stat. 36, 383–393 (1987)
[21] Koenker, R., d’Orey, V.: A remark on computing regression quantiles. Appl. Stat. 43, 410–414 (1993)
[22] Koenker, R., Machado, J.: Goodness of fit and related inference processes for quantile regression. J. Am. Stat. Assoc. 94, 1296–1309 (1999) · Zbl 0998.62041
[23] Lipsitz, S.R., Fitzmaurice, G.M., Molenberghs, G., Zhao, L.P.: Quantile regression methods for longitudinal data with drop-outs: application to CD4 cell counts of patients infected with the human immunodeficiency virus. J. R. Stat. Soc. Ser. C 46, 463–476 (1997) · Zbl 0908.62114
[24] Liu, Y., Bottai, M.: Mixed-effects models for conditional quantiles with longitudinal data. Int. J. Biostat. 5 (2009)
[25] MacDonald, I.L., Zucchini, W.: Hidden Markov and other Models for Discrete-Valued Time Series. Chapman and Hall, London (1997) · Zbl 0868.60036
[26] R Development Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna (2009)
[27] Schwarz, G.: Estimating the dimension of a model. Ann. Stat. 6, 461–464 (1978) · Zbl 0379.62005
[28] Shapiro, A.: Towards a unified theory of inequality constrained testing in multivariate analysis. Int. Stat. Rev. 56, 49–62 (1988) · Zbl 0661.62042
[29] Silvapulle, M.J., Sen, P.K.: Constrained Statistical Inference. Wiley, New York (2004)
[30] Vermunt, J.K., Langeheine, R., Böckenholt, U.: Discrete-time discrete-state latent Markov models with time-constant and time-varying covariates. J. Educ. Behav. Stat. 24, 179–207 (1999)
[31] Wang, H.J., Zhu, Z., Zhou, J.: Quantile regression in partially linear varying coefficient models. Ann. Stat. 37, 3841–3866 (2009) · Zbl 1191.62077
[32] Yu, K., Lu, Z., Stander, J.: Quantile regression: applications and current research areas. J. R. Stat. Soc. Ser. D 52, 331–350 (2003)
[33] Yu, K., Moyeed, R.A.: Bayesian quantile regression. Stat. Probab. Lett. 54, 437–447 (2001) · Zbl 0983.62017
[34] Yuan, Y., Yin, G.: Bayesian quantile regression for longitudinal studies with nonignorable missing data. Biometrics (2010). Available online · Zbl 1187.62183
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