The t-distribution allows the incorporation of outlier robustness into statistical models while retaining the elegance of likelihood-based inference. In this paper, we develop and implement a linear mixed model for the general design of the linear mixed model using the univariate t-distribution. This general design allows a considerably richer class of models to be fit than is possible with existing methods. Included in this class are semi-parametric regression and smoothing and spatial models.
Aitkin M
(1999)
A general maximum likelihood analysis of variance components in generalized linear models
. Biometrics, 55,
117–28
.
2.
Booth JG
and
Hobert JP
(1999)
Maximizing generalized linear mixed model likelihoods with an automated Monte Carlo EM algorithm
. Journal of the Royal Statistics Society, Series B, 61,
265–85
.
3.
Breusch TS
,
Robertson JC
and
Welsh AH
(1997)
The emperor’s new clothes: a critique of the multivariate t regression model
. Statistica Neerlandica, 51,
269–86
.
4.
Caffo BS
,
Jank W
and
Jones GL
(2005)
Ascent-based Monte Carlo expectation maximization
. Journal ofthe Royal Statistical Society, Series B: Statistical Methodology, 67,
235–51
.
5.
Crainiceanu C
,
Ruppert D
and
Wand MP
(2005)
Bayesian analysis for penalized spline regression using WinBUGS
. Journal of Statistical Software, 14.
6.
Dempster AP
,
Laird NM
and
Rubin DB
(1977)
Maximum likelihood from incomplete data via the EM algorithm
. Journal ofthe Royal Statistics Society, Series B, 39,
1–22
.
7.
Ghidey W
,
Lesaffre E
and
Eilers P
(2004)
Smooth random effects distribution in a linear mixed model
. Biometrics, 60,
945–53
.
8.
Hagobian TA
and
Braun B
(2006)
Interactions between energy surplus and short-term exercise on glucose and insulin responses in healthy people with induced, mild insulin insensitivity
. Metabolism, 55,
402–408
.
9.
Henderson CR
(1975)
Best linear unbiased estimation and prediction under a selection model
. Biometrics, 31,
423–47
.
10.
Jara A and Quintana F (2006) Unpublished MS. Linear mixed models with skew-elliptical distributions: a Bayesian approach.
11.
Kammann EE
and
Wand MP
(2003)
Geoadditive models
. Journal ofthe Royal Statistical Society,SeriesC, 52,
1–18
.
12.
Kleinman K
and
Ibrahim JG
(1998)
A semi-parametric Bayesian approach to the random effects model
. Biometrics, 54,
921–38
.
13.
Kotz S
and
Nadarajah S
(2004) Multivariate t-distributions and their applications.
Cambridge: Cambridge University Press
.
14.
Laird NL
and
Ware JH
(1982)
Random-effects models for longitudinal data
. Biometrics, 38,
963–74
.
15.
Lange KL
,
Little RJA
and
Taylor JMG
(1989)
Robust statistical modeling using the t-distribution
. Journal of the American Statistical Association, 84,
881–96
.
16.
Levine RA
and
Casella G
(2001)
Implementations of the Monte Carlo EM algorithm
. Journal ofComputational and Graphical Statistics, 10,
422–39
.
17.
Lin X
and
Zhang D
(1999)
Inference in generalized additive mixed models by using smoothing splines
. Journal ofthe Royal Statistical Society, Series B, 61,
381–400
.
18.
Liu C
,
Rubin DB
and
Wu YN
(1998)
Parameter expansion to accelerate EM: The PX-EX algorithm
. Biometrika, 85,
755–70
.
19.
Louis TA
(1982)
Finding the observed information matrix when using the EM algorithm
. Journal of the Royal Statistical Society, Series B, 44,
226–33
.
20.
McCulloch CE
and
Searle SR
(2001) Generalized, linear, and mixed models.
London: Chapman & Hall
.
21.
Meng XL
and
Rubin DB
(1993)
Maximum likelihood estimation via the ECM algorithm: a general framework
. Biometrika, 80,
267–78
.
22.
O’Connell MA
and
Wolfinger RD
(1997)
Spatial regression models, response surfaces, and process optimization
. Journal of Computational and Graphical Statistics, 6,
224–41
.
23.
Pinheiro JC
,
Liu C
and
Wu YN
(2001)
Efficient algorithms for robust estimation in linear mixed-effects models using the multivariate t-distribution
. Journal ofComputational and Graphical Statistics, 10,
249–76
.
24.
Robinson GK
(1991)
That BLUP is a good thing: the estimation of random effects
. Statistical Science, 6,
15–51
.
25.
Rosa GJM
,
Gianola D
and
Padovani CR
(2004)
Bayesian longitudinal data analysis with mixed models and thick-tailed distributions using MCMC
. Journal ofApplied Statistics, 31,
855–73
.
26.
Rosa GJM
,
Padovani CR
and
Gianola D
(2003)
Robust linear mixed models with normal/independent distributions and Bayesian MCMC implementation
. Biometrical Journal, 45,
573–90
.
27.
Ruppert D
,
Wand MP
and
Carroll RJ
(2003) Semiparametric regression modelling.
Cambridge: Cambridge University Press
.
28.
Smith M
and
Kohn R
(1996)
Nonparametric regression using Bayesian variable selection
. Journal of Econometrics, 75,
317–44
.
29.
Speed T
(1991)
Comment on paper by Robinson
. Statistical Science, 6,
42–44
.
30.
Stranden I
and
Gianola D
(1998)
Mixed effects linear models with t-distributions forquantitative genetic analysis: a Bayesian approach
. Genetics, Selection, Evolution, 31,
25–42
.
31.
Tao H
,
Palta M
,
Yandel B
and
Newton MA
(1999)
An estimation method for the semiparametric mixed effects model
. Biometrics, 55,
102–110
.
32.
van Dyk DA
(2000)
Nesting EM algorithms for computational efficiency
. Statistica Sinica, 10,
203–25
.
33.
Wahba G
(1978)
Improper priors, spline smoothing and the problem of guarding against model errors in regression
. Journal ofthe Royal Statistical Society, Series B, 40,
364–72
.
34.
Wei GCG
and
Tanner MA
(1990)
A Monte Carlo implementation of the EM algorithm and the poor man s data augmentation algorithm
. Journal ofthe American Statistical Association, 85,
699–704
.
35.
Welsh AH
and
Richardson AM
(1997) Approaches to the robust estimation of mixed models. In
Maddala GS
and
Rao CR
. (eds) Handbook ofStatistics, Volume 15,
New York: Elsevier Science B.V
.
36.
Wu CFJ
(1983)
On the convergence properties of the EM algorithm
. The Annals ofStatistics, 11,
95–103
.
37.
Zhang D
and
Davidian M
(2001)
Linear mixed models with flexible distributions of random effects for longitudinal data
. Biometrics, 57,
795–802
.
38.
Zhao Y
,
Staudenmayer J
,
Coull BA
and
Wand MP
(2006)
General design Bayesian generalized linear mixed models
. Statistical Science, 21,
35–51
.
