## Bayesian covariance selection in generalized linear mixed models (2006)

Venue: | Biometrics |

Citations: | 9 - 3 self |

### BibTeX

@ARTICLE{Cai06bayesiancovariance,

author = {Bo Cai and David B. Dunson and Thank Beth Gladen},

title = {Bayesian covariance selection in generalized linear mixed models},

journal = {Biometrics},

year = {2006},

volume = {62},

pages = {446--457}

}

### OpenURL

### Abstract

SUMMARY. The generalized linear mixed model (GLMM), which extends the generalized linear model (GLM) to incorporate random effects characterizing heterogeneity among subjects, is widely used in analyzing correlated and longitudinal data. Although there is often interest in identify-ing the subset of predictors that have random effects, random effects selection can be challenging, particularly when outcome distributions are non-normal. This article proposes a fully Bayesian approach to the problem of simultaneous selection of fixed and random effects in GLMMs. Inte-grating out the random effects induces a covariance structure on the multivariate outcome data, and an important problem which we also consider is that of covariance selection. Our approach relies on variable selection-type mixture priors for the components in a special LDU decomposition of the random effects covariance. A stochastic search MCMC algorithm is developed, which relies on Gibbs sampling, with Taylor series expansions used to approximate intractable integrals. Simu-lated data examples are presented for different exponential family distributions, and the approach is applied to discrete survival data from a time-to-pregnancy study.

### Citations

433 |
Random-effects models for longitudinal data
- Laird, Ware
- 1982
(Show Context)
Citation Context ...utine implementation, the generalized linear mixed model (GLMM) has become very widely used in analyses of correlated and longitudinal data (McCulloch and Searle, 2001). Analogous to the mixed model (=-=Laird and Ware, 1982-=-) extension of the linear model, the GLMM extends the generalized linear model (GLM) to incorporate random effects characterizing heterogeneity among subjects or clusters. By integrating out the rando... |

372 |
Approximate inference in generalized linear mixed models
- Breslow, Clayton
- 1993
(Show Context)
Citation Context ..., Z)π(ζ|Σ)dζ. (9) Let l(β, φ, Σ; y) = log L(β, φ, Σ; y), suppressing the conditioning on X and Z as shorthand. To approximate (9), the classical way is Laplace’s approximation (Solomon and Cox, 1992; =-=Breslow and Clayton, 1993-=-; Lin, 1997; Chipman et al., 2003, among others). From (6), we note that Σ depends on the standard deviation of random effects which is proportional to λ. When λ = 0, the likelihood (9) reduces to ord... |

369 |
Variable selection via Gibbs sampling
- George, McCulloch
- 1993
(Show Context)
Citation Context ...dels, φ is σ 2 , and we follow common practice in choosing a gamma prior, G(c0, d0), for σ −2 . Posterior computation relies on a stochastic search variable selection (SSVS) Gibbs sampling algorithm (=-=George and McCulloch, 1993-=-) in which we iteratively sample from the full conditional distributions of each of the parameters. For β, λ and γ, these posteriors will have a mixture structure consisting of point mass at 0 and non... |

351 | Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments, in - Geweke - 1992 |

110 | Approximate Bayes factors and accounting for model uncertainty in generalised linear models - Raftery - 1996 |

93 | Maximum likelihood algorithms for generalized linear mixed models - McCulloch - 1997 |

88 | Gibbs sampling for Bayesian non-conjugate and hierarchical models by using auxiliary variables - Damien, Wakefield, et al. - 1999 |

78 |
Adaptive rejection Metropolis sampling within Gibbs sampling
- Gilks, Best, et al.
- 1995
(Show Context)
Citation Context ...ions. In calculating the point mass probabilities, we rely on the approximation described in Section 2.4. To sample from the non-conjugate distribution, we use adaptive rejection Metropolis sampling (=-=Gilks et al., 1995-=-). One can use an alternative non-rejection-based sampling algorithm via latent variables proposed by Damien, Wakefield and Walker (1999). Let δ1,k denote an indicator variable which is one if H0k hol... |

74 |
Generalized, linear, and mixed models
- McCulloch, Searle
- 2001
(Show Context)
Citation Context ...Introduction With improvements in computation permitting routine implementation, the generalized linear mixed model (GLMM) has become very widely used in analyses of correlated and longitudinal data (=-=McCulloch and Searle, 2001-=-). Analogous to the mixed model (Laird and Ware, 1982) extension of the linear model, the GLMM extends the generalized linear model (GLM) to incorporate random effects characterizing heterogeneity amo... |

70 | Variable Selection and Model Comparison in Regression,” Working Paper
- Geweke
- 1994
(Show Context)
Citation Context ... . . . , p. The prior 8 (7)sprobability of the vth predictor being excluded is then π2,v0 = Pr(βv = 0). Similar mixture priors have been widely used in the Bayesian variable selection literature (cf. =-=Geweke, 1996-=-). We also allow zero off-diagonal elements in the covariance matrix by choosing mixture priors with masses at 0 for the γ’s. We choose a zero-inflated normal density, ZI-N(γmk; π3,mk,0, µ3,mk,0, σ 2 ... |

47 | Nonconjugate Bayesian estimation of covariance matrices and its use in hierarchical models - DANIELS, KASS - 1999 |

45 | Efficient estimation of covariance selection models - Wong, Carter, et al. - 2003 |

42 | Accounting for model uncertainty in survival analysis improves predictive performance (with discussion - RAFTERY, MADIGAN, et al. - 1996 |

34 | Bayesian treed models - Chipman, George, et al. - 2002 |

26 | Bayesian analysis of covariance matrices and dynamic models for longitudinal data - Daniels, Pourahmadi - 2002 |

20 | Random effects selection in linear mixed models - Chen, Dunson - 2003 |

19 | Bayesian tests and model diagnostics in conditionally independent hierarchical models - Albert - 1997 |

17 | Estimation in generalized mixed models - McGilchrist - 1994 |

16 | How many iterations of the Gibbs sampler? Bayesian Statistics 4 - Raftery, Lewis - 1992 |

13 |
Variance component testing in generalized linear models with random eects
- Lin
- 1997
(Show Context)
Citation Context ...undability varies among women). To assess whether one or more random effects should be included in the model, several authors have proposed frequentist score tests (Commenges and Jacqmin-Gadda, 1997; =-=Lin, 1997-=-; Hall and Praestgaard, 2001). In the Bayesian literature, Albert and Chib (1997) proposed an approach for testing whether a random intercept should be included, Sinharay and Stern (2001) developed a ... |

12 | Bayesian correlation estimation - Liechty, Mller |

10 | Bayesian inference on order-constrained parameters in generalized linear models - DUNSON, NEELON - 2003 |

10 | Predictive variable selection in generalized linear models - Meyer, Laud - 2002 |

9 |
Order-restricted score tests for homogeneity in generalised linear and nonlinear mixed models
- Hall, Præstgaard
- 2001
(Show Context)
Citation Context ...varies among women). To assess whether one or more random effects should be included in the model, several authors have proposed frequentist score tests (Commenges and Jacqmin-Gadda, 1997; Lin, 1997; =-=Hall and Praestgaard, 2001-=-). In the Bayesian literature, Albert and Chib (1997) proposed an approach for testing whether a random intercept should be included, Sinharay and Stern (2001) developed a more general approach for ca... |

8 | Prior elicitation for model selection and estimation in generalized linear mixed models - Chen, Ibrahim, et al. - 2002 |

8 |
Generalized score test of homogeneity based on correlated random effects models
- Commenges, Jacqmin-Gadda
- 1997
(Show Context)
Citation Context ..., that the effect of smoking on fecundability varies among women). To assess whether one or more random effects should be included in the model, several authors have proposed frequentist score tests (=-=Commenges and Jacqmin-Gadda, 1997-=-; Lin, 1997; Hall and Praestgaard, 2001). In the Bayesian literature, Albert and Chib (1997) proposed an approach for testing whether a random intercept should be included, Sinharay and Stern (2001) d... |

8 |
The mixed or multilevel model for behavior genetic analysis. Behav Genet
- Guo, Wang
- 2002
(Show Context)
Citation Context ...ce. This is certainly the case not only in fecundability studies, as we have illustrated, but also in general epidemiologic studies. In fact, in certain cases, such as family studies of heritability (=-=Guo and Wang, 2002-=-), inferences on the covariance structure are of primary interest. Since current methods for such inferences are limited 18sto criterion-based methods and simple score and likelihood ratio tests, our ... |

7 | Sampling schemes for Bayesian variables selection in generalized linear models - Nott, Leonte - 2004 |

6 |
Nonlinear component of variance models
- Solomon, Cox
- 1992
(Show Context)
Citation Context ...Z) = ℜ q π(y|β, φ, ζ, X, Z)π(ζ|Σ)dζ. (9) Let l(β, φ, Σ; y) = log L(β, φ, Σ; y), suppressing the conditioning on X and Z as shorthand. To approximate (9), the classical way is Laplace’s approximation (=-=Solomon and Cox, 1992-=-; Breslow and Clayton, 1993; Lin, 1997; Chipman et al., 2003, among others). From (6), we note that Σ depends on the standard deviation of random effects which is proportional to λ. When λ = 0, the li... |

5 | Bayes Factors for Variance Component Testing in Generalized Linear Mixed Models,” Doctoral dissertation - Sinharay - 2001 |

3 |
Reduced fertility among women employed as dental assistants exposed to high levels of nitrous oxide
- Rowland, Baird, et al.
- 1992
(Show Context)
Citation Context ...ncluded in the fixed and random effects components of the model, and when covariance structure modeling is the focus. As motivation, we consider data from an epidemiologic study of time to pregnancy (=-=Rowland et al., 1992-=-). In this study, dental assistants completed a demographic and exposure history questionnaire, while also providing information on the number of menstrual cycles during which the woman was having non... |

3 | Pitfalls inherent in retrospective time-to-event studies : the example of time to pregnancy - Weinberg, Baird, et al. - 1993 |

2 | Estimation in Generalized Linear Mixed Models with Random Effects - Schall - 1991 |

1 |
Bayesian Treed Generalized Linear Models. Bayesian Statistics 7
- Chipman, George, et al.
- 2003
(Show Context)
Citation Context ...log L(β, φ, Σ; y), suppressing the conditioning on X and Z as shorthand. To approximate (9), the classical way is Laplace’s approximation (Solomon and Cox, 1992; Breslow and Clayton, 1993; Lin, 1997; =-=Chipman et al., 2003-=-, among others). From (6), we note that Σ depends on the standard deviation of random effects which is proportional to λ. When λ = 0, the likelihood (9) reduces to ordinary GLM likelihood with Σ = 0. ... |

1 | Modelling the Random Effects Covariance - Daniels, Zhao - 2003 |

1 | Bayesian Variable Selection and Link Determination for Generalised Linear Models - Ntzoufras, Dellaportas, et al. - 2003 |