Hierarchical models with normal and conjugate random effects: a review (invited article)

• Geert Molenberghs ^[1] ; Geert Verbeke ^[1] ; Clarice G.B. Demétrio ^[2]
  1. [1] University of Hasselt

     University of Hasselt

     Arrondissement Hasselt, Bélgica

     
  2. [2] Universidade de São Paulo

     Universidade de São Paulo

     Brasil

     
• Localización: Sort: Statistics and Operations Research Transactions, ISSN 1696-2281, Vol. 41, Nº. 2, 2017, págs. 191-254
• Idioma: inglés
• Enlaces
  • Texto completo (pdf)
• Resumen

  • Molenberghs, Verbeke, and Demétrio (2007) and Molenberghs et al. (2010) proposed a general framework to model hierarchical data subject to within-unit correlation and/or overdispersion. The framework extends classical overdispersion models as well as generalized linear mixed models. Subsequent work has examined various aspects that lead to the formulation of several extensions. A unified treatment of the model framework and key extensions is provided. Particular extensions discussed are: explicit calculation of correlation and other moment-based functions, joint modelling of several hierarchical sequences, versions with direct marginally interpretable parameters, zero-inflation in the count case, and influence diagnostics. The basic models and several extensions are illustrated using a set of key examples, one per data type (count, binary, multinomial, ordinal, and time-to-event).

• Referencias bibliográficas
  • Abrams, S., Aerts, M., Molenberghs, G. and Hens, N. (2017). Parametric overdispersed frailty models for current status data. Biometrics, DOI:...
  • Aerts, M., Geys, H., Molenberghs, G. and Ryan, L. (2002). Topics in Modelling of Clustered Data. London: Chapman & Hall.
  • Agresti, A. (2002). Categorical Data Analysis (2nd ed.). New York: John Wiley & Sons.
  • Aitkin, M. (l999). A general maximum likelihood analysis of variance components in generalized linear models. Biometrics, 55, 117–128.
  • Alonso, A., Bigirumurame, T., Burzykowski, T., Buyse, M., Molenberghs, G., Muchene, L., Perualila, N.J., Shkedy, Z. and Van der Elst, W. (2017)....
  • Alfò, M. and Aitkin, M. (2000). Random coefficient models for binary longitudinal responses with attrition. Statistics and Computing, 10,...
  • Aregay, M., Shkedy, Z. and Molenberghs, G. (2013). A hierarchical Bayesian approach for the analysis of longitudinal count data with overdispersion:...
  • Aregay, M., Shkedy, Z. and Molenberghs, G. (2015). Comparison of additive and multiplicative Bayesian models for longitudinal count data with...
  • Ashford, J.R. and Sowden, R.R. (1970) Multivariate probit analysis. Biometrics, 26, 535–546.
  • Bennett, S. (1983). Log-logistic regression models for survival data. Applied Statistics, 32, 165–171.
  • Böhning, D. (2000) Computer-assisted Analysis of Mixtures and Applications. Meta-analysis, Disease Mapping and Others. London: Chapman &...
  • Booth, J.G., Casella, G., Friedl, H. and Hobert, J.P. (2003). Negative binomial loglinear mixed models. Statistical Modelling, 3, 179–181.
  • Borgermans, L., Goderis, G., Van Den Broeke, C., Verbeke, G., Carbonez, A., Ivanova, A., Mathieu, C., Aertgeerts, B., Heyrman, J. and Grol,...
  • Breslow, N. (1984). Extra-Poisson variation in log-linear models. Applied Statistics, 33, 38–44.
  • Breslow, N.E. and Clayton, D.G. (1993). Approximate inference in generalized linear mixed models. Journal of the American Statistical Association,...
  • Breslow, N.E. and Lin, X. (1995). Bias correction in generalized linear mixed models with a single component of dispersion. Biometrika, 82,...
  • Collett, D. (2003). Modelling Survival Data in Medical Research (2nd ed.). Boca Raton: CRC Press.
  • Cook, R.D. (1986). Assessment of local influence. Journal of the Royal Statistical Society, Series B, 48, 133–169.
  • Cox, D.R. and Hinkley, D.V. (1974). Theoretical Statistics. London: Chapman & Hall/CRC.
  • Dean, C.B. (1991). Estimating equations for mixed-Poisson models. In: Estimating Functions, V.P. Godambe (Ed.). Oxford: Oxford University...
  • De Backer, M., De Keyser, P., De Vroey, C. and Lesaffre, E. (1996). A 12-week treatment for dermatophyte toe onychomycosis: terbinafine 250...
  • Del Fava, E., Shkedy, Z., Aregay, M. and Molenberghs, G. (2014). Modelling multivariate, overdispersed binomial data with additive and multiplicative...
  • Duchateau, L. and Janssen, P. (2007). The Frailty Model. New York: Springer.
  • Efendi, A. and Molenberghs, G. (2013). A multilevel model for hierarchical, repeated, and overdispersed time-to-event outcomes and its estimation...
  • Efendi, A., Molenberghs, G. and Iddi, S. (2014). A Marginalized combined gamma frailty and normal random-effects model for repeated, overdispersed...
  • Engel, B. and Keen, A. (1994). A simple approach for the analysis of generalized linear mixed models. Statistica Neerlandica, 48, 1–22.
  • Faught, E., Wilder, B.J., Ramsay, R.E., Reife, R.A., Kramer, L.D., Pledger, G.W. and Karim, R.M. (1996). Topiramate placebo-controlled dose-ranging...
  • Gentle, J.E. (2003). Random Number Generation and Monte Carlo Methods. New York: Springer.
  • Ghebretinsae, A.H., Faes, C., Molenberghs, G., De Boeck, M. and Geys, H. (2013). A Bayesian generalized frailty model for comet assays. Journal...
  • Ghebretinsae, A., Faes, C., Molenberghs, G., Geys, H. and Van der Leede, B.-J. (2012). Joint modelling of hierarchically clustered and overdispersed...
  • Gibbons, R.D. and Hedeker, D. (1997). Random effects probit and logistic regression models for three-level data. Biometrics, 53, 1527–1537.
  • Greene, W. (1994). Accounting for Excess Zeros and Sample Selection in Poisson and Negative-Binomial Regression Models. Working Paper EC-94–10,...
  • Griswold, M.E. and Zeger, S.L. (2004). On Marginalized Multilevel Models and their Computation. John Hopkins University, Dept. of Biostatistics...
  • Guilkey, D.K. and Murphy, J.L. (1993). Estimation and testing in the random effects probit model. Journal of Econometrics, 59, 301–317.
  • Heagerty, P.J. (1999). Marginally specified logistic-normal models for longitudinal binary data Biometrics, 55, 688–698.
  • Heagerty, P.J. and Zeger, S.L. (2000). Marginalized multilevel models and likelihood inference, Statistical Science, 15, 1–26.
  • Hedeker, D. and Gibbons, R.D. (1994). A random-effects ordinal regression model for multilevel analysis. Biometrics, 51, 933–944.
  • Hinde, J. and Demétrio, C.G.B. (1998a). Overdispersion: models and estimation. Computational Statistics and Data Analysis, 27, 151–170.
  • Hinde, J. and Demétrio, C.G.B. (1998b). Overdispersion: Models and Estimation. São Paulo: XIII Sinape.
  • Iddi, S. and Molenberghs, G. (2012a). A combined overdispersed and marginalized multilevel model. Computational Statistics and Data Analysis,...
  • Iddi, S. and Molenberghs, G. (2012b). A joint marginalized multilevel model for continuous and binary longitudinal outcomes. Journal of Applied...
  • Iddi, S. and Molenberghs, G. (2013). A marginalized model for zero-inflated, overdispersed and correlated count data. Electronic Journal of...
  • Iddi, S., Molenberghs, G., Aregay, M. and Kalema, G. (2014). Empirical Bayes estimates for correlated hierarchical data with overdispersion....
  • Ivanova, A., Molenberghs, G. and Verbeke, G. (2014). A model for overdispersed hierarchical ordinal data. Statistical Modelling, 14, 399–415.
  • Ivanova, A., Molenberghs, G. and Verbeke, G. (2016). Mixed model approaches for joint modelling of different types of responses. Journal of...
  • Johnson, N.L., Kemp, A., Kotz, S. (2005). Univariate Discrete Distributions (3rd ed.). Hoboken: John Wiley & Sons.
  • Johnson, N.L. and Kotz, S. (1970). Distributions in Statistics, Continuous Univariate Distributions, Vol. 2. Boston: Houghton-Mifflin.
  • Kalema, G., Iddi, S. and Molenberghs, G. (2016). The combined model: a tool for simulating correlated counts with overdispersion. Communications...
  • Kalema, G. and Molenberghs, G. (2015). Generating correlated and/or overdispersed count data; a SAS implementation. Journal of Statistical...
  • Kassahun, W., Neyens, T., Faes, C., Molenberghs, G. and Verbeke, G. (2014a). A Zero-inflated overdispersed hierarchical Poisson model. Statistical...
  • Kassahun, W., Neyens, T., Molenberghs, G., Faes, C. and Verbeke, G. (2012). Modelling overdispersed longitudinal binary data from the Jimma...
  • Kassahun, W., Neyens, T., Molenberghs, G., Faes, C. and Verbeke, G. (2014b). Marginalized multilevel hurdle and zero-inflated models for overdispersed...
  • Kassahun, W., Neyens, T., Molenberghs, G., Faes C. and Verbeke, G. (2015). A joint model for hierarchical continuous and zero-inflated overdispersed...
  • Kenward, M.G. and Molenberghs, G. (2016). A taxonomy of mixing and outcome distributions based on conjugacy and bridging. Communications in...
  • Kleinman, J. (1973). Proportions with extraneous variance: single and independent samples. Journal of the American Statistical Association,...
  • Lambert, D. (1992). Zero-inflated Poisson regression, with an application to defects in manufacturing. Technometrics, 34, 1–14.
  • Lawless, J. (1987). Negative binomial and mixed Poisson regression. The Canadian Journal of Statistics, 15, 209–225.
  • Lee, Y. and Nelder, J.A. (1996). Hierarchical generalized linear models (with discussion). Journal of the Royal Statistical Society, Series...
  • Lee, Y. and Nelder, J.A. (2001a). Two ways of modelling overdispersion. Applied Statistics, 49, 591–598.
  • Lee, Y. and Nelder, J.A. (2001b). Hierarchical generalized linear models: a synthesis of generalized linear models, random-effect models and...
  • Lee, Y. and Nelder, J.A. (2003). Extended-REML estimators. Journal of Applied Statistics, 30, 845–856.
  • Lee, Y., Nelder, J.A. and Pawitan, Y. (2006). Generalized Linear Models with Random Effects: Unified Analysis via H-likelihood. Boca Raton:...
  • Lesaffre, E. and Molenberghs, G. (1991). Multivariate probit analysis: a neglected procedure in medical statistics. Statistics in Medicine,...
  • Lesaffre, E. and Verbeke, G. (1998). Local influence in linear mixed models. Biometrics, 54, 570–582.
  • Liang, K.Y. and McCullagh, P. (1993). Case studies in binary dispersion. Biometrics, 49, 623–630.
  • Liu, L. and Yu, Z. (2008) A likelihood reformulation method in non-normal random-effects models. Statistics in Medicine, 27, 3105–3124.
  • McCullagh, P. and Nelder, J.A. (1989). Generalized Linear Models. London: Chapman & Hall/CRC.
  • McCulloch, C.E. (1994). Maximum likelihood variance components estimation for binary data. Journal of the American Statistical Association,...
  • McLachlan, G. and Peel, D.A. (2000). Finite Mixture Models. New York: John Wiley & Sons.
  • Milanzi, E., Molenberghs, G., Alonso, A., Verbeke, G. and De Boeck, P. (2015). Reliability measures in item response theory: Manifest versus...
  • Molenberghs, G., Kenward, M.G., Verbeke, G., Efendi, A. and Iddi, S. (2013). On the connections between bridge distributions, marginalized...
  • Molenberghs, G. and Verbeke, G. (2005). Models for Discrete Longitudinal Data. New York: Springer.
  • Molenberghs, G. and Verbeke, G. (2007). Likelihood ratio, score, and Wald tests in a constrained parameter space. The American Statistician,...
  • Molenberghs, G. and Verbeke, G. (2011a). On the Weibull-Gamma frailty model, its infinite moments, and its connection to generalized log-logistic,...
  • Molenberghs, G. and Verbeke, G. (2011b). A note on a hierarchical interpretation for negative variance components. Statistical Modelling,...
  • Molenberghs, G., Verbeke, G. and Demétrio, C. (2007). An extended random-effects approach to modelling repeated, overdispersed count data....
  • Molenberghs, G., Verbeke, G., Demétrio, C.G.B. and Vieira, A. (2010). A family of generalized linear models for repeated measures with normal...
  • Molenberghs, G., Verbeke, G., Efendi, A., Braekers, R. and Demétrio, C.G.B. (2015). A combined gamma frailty and normal random-effects model...
  • Molenberghs, G., Verbeke, G., Iddi, S. and Demétrio, C.G.B. (2012). A combined beta and normal randomeffects model for repeated, overdispersed...
  • Moore, D.F. and Tsiatis, A.A. (1991). Robust estimation of the variance in moment methods for extrabinomial and extra-Poisson variation. Biometrics,...
  • Mullahy, J. (1986). Specification and testing of some modified count data models. Journal of Econometrics, 33, 341–65.
  • Nelder, J.A. and Wedderburn, R.W.M. (1972). Generalized linear models. Journal of the Royal Statistical Society, Series A, 135, 370–384.
  • Nelson, K.P., Lipsitz, S.R., Fitzmaurice, G.M., Ibrahim, J., Parzen, M. and Strawderman, R. (2006). Use of the probability integral transformation...
  • Neyens, T., Faes, C., and Molenberghs, G. (2012). A generalized Poisson-gamma model for spatially overdispersed data. Spatial and Spatio-temporal...
  • Njeru Njagi, E., Molenberghs, G., Rizopoulos, D., Verbeke, G., Kenward, M.G., Dendale, P. and Willekens, K. (2016). A flexible joint-modelling...
  • Oliveira, I.R.C., Molenberghs, G., Demétrio, C.G.B., Giolo, S. and Dias, C.T.S. (2016). Quantifying intraclass correlations for nonnegative...
  • Oliveira, I.R.C., Molenberghs, G., Verbeke, G., Demétrio, C.G.B. and Dias, C.T.S. (2017). Negative variance components for non-negative...
  • Ouwens, M.J.N.M., Tan, F.E.S. and Berger, M.P.F. (2001). Local influence to detect influential data structures for generalized linear mixed...
  • Pryseley, A., Tchonlafi, C., Verbeke, G. and Molenberghs, G. (2011). Estimating negative variance components from Gaussian and non-Gaussian...
  • Rakhmawati, T., Molenberghs, G., Verbeke, G. and Faes, C. (2016a). Local influence diagnostics for hierarchical count data models with overdispersion...
  • Rakhmawati, T., Molenberghs, G., Verbeke, G. and Faes, C. (2016b). Local influence diagnostics for incomplete overdispersed longitudinal counts....
  • Rakhmawati, T., Molenberghs, G., Verbeke, G. and Faes, C. (2017). Local influence diagnostics for generalized linear mixed models with overdispersion....
  • Renard, D., Molenberghs, G. and Geys, H. (2004). A pairwise likelihood approach to estimation in multilevel probit models. Computational Statistics...
  • Ridout, M., Demétrio, C.G.B. and Hinde, J. (1998). Models for count data with many zeros. In: International Biometric Conference XIX, Cape...
  • Rinne, H. (2009). The Weibull Distribution. A Handbook. Boca Raton: CRC/Chapman & Hall.
  • Rizzato, F.B., Leandro, R.A., Demétrio, C.G.B. and Molenberghs, G. (2016). A Bayesian approach to analyse overdispersed longitudinal count...
  • Roberts, D.T. (1992). Prevalence of dermatophyte onychomycosis in the United Kingdom: Results of an omnibus survey. British Journal of Dermatology,...
  • Schall, R. (1991). Estimation in generalized linear models with random effects. Biometrika, 78, 719–729.
  • Shepard, T.H., Mackler, B. and Finch, C.A. (1980). Reproductive studies in the iron-deficient rat. Teratology, 22, 329–334.
  • Shoukri, M.M., Mian, I.U.M. and Tracy, D.S. (1988). Sampling properties of estimators of the log-logistic distribution with application to...
  • Skellam, J.G. (1948). A probability distribution derived from the binomial distribution by regarding the probability of success as variable...
  • Skrondal, A. and Rabe-Hesketh, S. (2004). Generalized Latent Variable Modelling. London: Chapman & Hall/CRC.
  • Thall, P.F. and Vail, S.C. (1990). Some covariance models for longitudinal count data with overdispersion. Biometrics, 46, 657–671.
  • Vangeneugden, T., Molenberghs, G., Laenen, A., Alonso, A. and Geys, H. (2008). Generalizability in non-Gaussian longitudinal clinical trial...
  • Vangeneugden, T., Molenberghs, G., Laenen, A., Geys, H., Beunckens, C. and Sotto, C. (2010). Marginal correlation in longitudinal binary data...
  • Vangeneugden, T., Molenberghs, G., Verbeke, G. and Demétrio, C. (2011). Marginal correlation from an extended random-effects model for repeated...
  • Vangeneugden, T., Molenberghs, G., Verbeke, G. and Demétrio, C.G.B. (2014). Marginal correlation from logitand probit-beta-normal models...
  • Verbeke, G. andMolenberghs, G. (2000). Linear MixedModels for Longitudinal Data. NewYork: Springer.
  • Wolfinger, R. and O’Connell, M. (1993). Generalized linear mixed models: a pseudo-likelihood approach. Journal of Statistical Computation...
  • Zeger, S.L., Liang, K.-Y. and Albert, P.S. (1988). Models for longitudinal data: a generalized estimating equation approach. Biometrics, 44,...