Bayesian Analysis

Parameter Interpretation in Skewed Logistic Regression with Random Intercept

Cristiano C. Santos, Rosangela H. Loschi, and Reinaldo B. Arellano-Valle

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This paper aims at providing the prior and posterior interpretations for the parameters in the logistic regression model with random or cluster-level intercept when univariate and multivariate classes of skew normal distributions are assumed to model the random effects behavior. We obtain the prior distributions for the odds ratio and their medians under skew normality for the random effects. Original results related to linear combinations of skew-normal distributions are obtained as a by-product and, in the univariate case, a new class of log-skew-normal distribution is introduced. Robust results are obtained whenever a class of multivariate skew-normal distribution is assumed. We also evaluate the effect of the misspecification of the random effects distributions in the odds ratio estimation. We consider both simulated and the Teratogenic activity experiment datasets. The latter was previously analysed in the literature. We concluded that the misspecification of the random effects distribution yields poor odds ratios estimates and that the median odds ratio is not necessarily the best measure of heterogeneity among the clusters as suggested in the literature.

Article information

Bayesian Anal., Volume 8, Number 2 (2013), 381-410.

First available in Project Euclid: 24 May 2013

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Zentralblatt MATH identifier

Cluster mixed models random odds ratio skew normal distribution


Santos, Cristiano C.; Loschi, Rosangela H.; Arellano-Valle, Reinaldo B. Parameter Interpretation in Skewed Logistic Regression with Random Intercept. Bayesian Anal. 8 (2013), no. 2, 381--410. doi:10.1214/13-BA813.

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  • Alonso, A., Litière, S. and Molemberghs, G. (2008). A family of tests to detect misspecifications in the random-effects structure of generalized linear mixed models. Computational Statistics and Data Analysis, 52, 4474–4486.
  • Arellano-Valle, R.B. and Azzalini, A. (2006). On the unification of families of skew-normal distributions. Scandinavian Journal of Statistics, 33, 561–574.
  • Azzalini, A. (1985). A class of distributions which includes the normal ones. Scandinavian Journal of Statistics 12, 171–178.
  • Azzalini, A. and Dalla Valle, A. (1996). The multivariate skew-normal distribution. Biometrika, 83, 715–726.
  • Breslow, N. E. and Clayton, D. G. (1993). Approximate inference in generalized linear mixed models. Journal of the American Statistical Association, 88, 9–25.
  • Dalla-Valle, A., (2004). The skew-normal distribution. In Skew-Elliptical Distributions and Their Applications: A Journey Beyond Normality, M. G. Genton (ed.), Chapman & Hall / CRC, Boca Raton, FL, pp. 3–24.
  • Diggle, P. J., Heagerty, P., Liang, K. Y. and Zeger, S. L. (2002). Analysis of Longitudinal Data, New York: Oxford University Press.
  • Fitzmaurice, G. M., Laird, N. and Ware, J. H. (2004). Applied Longitudinal Analysis, New York: John Wiley.
  • Gibbons, R. D., Hedeker, D., Charles, S. C. and Frisch, P. (1994). A random-effects probit model for predicting medical malpractice claims. Journal of the American Statistical Association 89, 760–767.
  • Gilks, W. R., Best, N. G. and Tan, K. K. C. (1995). Adaptive rejection Metropolis sampling within Gibbs sampling, Applied Statatistics 44: 455–472.
  • Gilks, W. R. and Wild, P. (1992). Adaptive rejection sampling for Gibbs sampling, Applied Statatistics 41: 337–348.
  • Kuhn, E. and Lavielle, M.(2005). Maximum likelihood estimation in nonlinear mixed effects models. Computational Statistics and Data Analysis, 49, 1020–1038.
  • Larsen, K. and Merlo, J. (2005). Appropriate assessment of neighborhood effects on individual health: integrating random and fixed effects in multilevel logistic regression. American Journal of Epidemiology, 161, 81–88.
  • Larsen, K., Petersen, J. H., Budtz-Jørgensen, E. and Endahl, L. (2000). Interpreting parameters in the logistic regression model with random effects. Biometrics, 56, 909–914.
  • Litière, S., Alonso, A. and Molemberghs, G. (2008). The impact of a misspecified random-effect distribution on the estimation and the performance of inferential procedures in the generalized linear mixed models. Statistics in Medicine, 27, 3125–3144.
  • Liu, J. and Dey, D. K.(2008). Skew random effects in multilevel binomial models: an alternative to nonparametric approach. Statistical Modelling, 8(3), 221–241.
  • Liu, L. and Yu, Z. (2008). A likelihood reformulation method in non-normal random effects models. Statistics in Medicine 27, 3105–3124.
  • Marchenko, Y.V. and Genton, M. G. (2010). Multivariate log-skew-elliptical distributions with applications to precipitation data, Environmetrics, 21, 318–340.
  • Martins-Filho, S., Loschi, R.H. and Colosimo, E.A. (2010). Bayesian interpretation for the parameters of the mixed logistic model: A seed germination application. Unplublished manuscript.
  • McCulloch, C. and Searle, S. R. (2001). Generalized, Linear, and Mixed Models, Wiley: New York.
  • Nelson, K. P., Lipsitz, S. R., Fitzmaurice, G. M., Ibrahim, J., Parzen, M. and Strawderman, R. (2006). Use of the probability integral transformation to fit nonlinear mixed-effects models with nonnormal random effects. Journal of Computational and Graphical Statistics 15, 39–57.
  • Paulino, C. D., Silva, G. and Achcar, J. A. (2005). Bayesian analysis of correlated misclassified binary data. Computational Statistics and Data Analysis, 49, 1120–1131.
  • Schall, R. (1991). Estimation in generalized linear models with random effects. Biometrika 78, 719–727.
  • Souza, A. D. P. and Migon, H. S. (2010). Bayesian outlier analysis in binary regression. Journal of Applied Statistics 37, 1355–1368.
  • Ten Have, T. R. and Localio, A. R. (1999). Empirical Bayes estimation of random effects parameters in mixed effects logistic regression models. Biometrics, 55, 1022–1029.
  • Zeger, S. L. and Karim, M. (1991). Generalized linear mixed models with random effects: A Gibbs sampling approach. Journal of the American Statistical Association, 86, 79–86.