The mixed effects model for repeated measures has been widely used for the analysis of longitudinal clinical data collected at a number of fixed time points. We propose a robust extension of the mixed effects model for repeated measures for skewed and heavy-tailed data on basis of the multivariate skew-t distribution, and it includes the multivariate normal, t, and skew-normal distributions as special cases. An efficient Markov chain Monte Carlo algorithm is developed using the monotone data augmentation and parameter expansion techniques. We employ the algorithm to perform controlled pattern imputations for sensitivity analyses of longitudinal clinical trials with nonignorable dropouts. The proposed methods are illustrated by real data analyses. Sample SAS programs for the analyses are provided in the online supplementary material.
SiddiquiOHungJHMO'NeillR. MMRM vs. LOCF: A comprehensive comparison based on simulation study and 25 NDA datasets. J Biopharmaceut Stat2009; 19: 227–246.
2.
MallinckrodtCHLanePWSchnellD, et al.Recommendations for the primary analysis of continuous endpoints in longitudinal clinical trials. Drug Inform J2008; 42: 303–319.
3.
ICH E9. Statistical principles for clinical trials: ICH harmonized tripartite guideline. Stat Med 1999; 18: 1905– 1942.
4.
CHMP. Guideline on adjustment for baseline covariates in clinical trials. London: CHAMP, 2015.
5.
LairdNMLangeNStramD. Maximum likelihood computations with repeated measures: application of the EM algorithm. J Am Stat Assoc1987; 82: 97–105.
6.
TangY. Closed-form REML estimators and sample size determination for mixed effects models for repeated measures under monotone missingness. Stat Med2017; 36: 2135–2147.
7.
LuKMehrotraDV. Specification of covariance structure in longitudinal data analysis for randomized clinical trials. Stat Med2010; 29: 474–488.
8.
GurkaMJEdwardsLJMullerKE. Avoiding bias in mixed model inference for fixed effects. Stat Med2011; 30: 2696–2707.
9.
Jacqmin-GaddaHSibillotSProustC, et al.Robustness of the linear mixed model to misspecified error distribution. Computat Stat Data Analys2007; 51: 5142–5154.
10.
GomezEVSchaaljeGBFellinghamGW. Performance of the Kenward-Roger method when the covariance structure is selected using AIC and BIC. Commun Stat-Simul Computat2005; 34: 377–392.
11.
LittleRYauL. Intent-to-treat analysis for longitudinal studies with drop-outs. Biometrics1996; 52: 1324–1333.
12.
CarpenterJRogerJKenwardM. Analysis of longitudinal trials with protocol deviation: a framework for relevant, accessible assumptions, and inference via multiple imputation. J Biopharmaceut Stat2013; 23: 1352–1371.
13.
MallinckrodtCRogerJChuang-steinC, et al.Missing data: turning guidance into action. Stat Biopharmaceut Res2013; 5: 369–382.
14.
TangY. An efficient multiple imputation algorithm for control-based and delta-adjusted pattern mixture models using SAS. Stat Biopharmaceut Res2017; 9: 116–125.
15.
RubinDB. Inference and missing data. Biometrika1976; 63: 581–92.
CHMP. EMA guideline on missing data in confirmatory clinical trials (EMA/CPMP/EWP/1776/99). London: CHAMP, 2010.
18.
National Research Council. Panel on handling missing data in clinical trials. Committee on National Statistics, Division of Behavioral and Social Sciences and Education. The prevention and treatment of missing data in clinical trials. Washington, DC: The National Academies Press, 2010.
19.
LittleRJA. Pattern-mixture models for multivariate incomplete data. J Am Stat Assoc1993; 88: 125–134.
20.
TangY. An efficient monotone data augmentation algorithm for multiple imputation in a class of pattern mixture models. J Biopharmaceut Stat2017; 27: 620–638.
TangY. Controlled pattern imputation for sensitivity analysis of longitudinal binary and ordinal outcomes with nonignorable dropout. Stat Med2018; 37: 1467–1481.
23.
TangY. A monotone data augmentation algorithm for multivariate nonnormal data: with applications to controlled imputations for longitudinal trials. Stat Med2019; 38: 1715–1733.
24.
AzzaliniA. A class of distributions which includes the normal ones. Scand J Stat1985; 12: 171–178.
25.
AzzaliniACapitanioA. Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t distribution. J Royal Stat Soc, Ser B2003; 65: 367–389.
26.
VerbekeGLesaffreE. The effect of misspecifying the random effects distribution in linear mixed models for longitudinal data. Computat Stat Data Analysis1997; 23: 541–556.
27.
PinheiroJCLiuCHWuYN. Efficient algorithms for robust estimation in linear mixed-effects models using a multivariate t-distribution. J Computat Graphics Stat2001; 10: 249–276.
28.
ZhangDDavidianM. Linear mixed models with flexible distributions of random effects for longitudinal data. Biometrics2001; 57: 795–802.
29.
ArnauJBendayanRBlancaMJ, et al.The effect of skewness and kurtosis on the robustness of linear mixed models. Behavior Res Meth2013; 45: 873–879.
30.
BoosDDHughes-OliverJM. How large does n have to be for z and t intervals?. Am Stat2000; 54: 121–128.
31.
von HippelPT. Should a normal imputation model be modified to impute skewed variables?. Sociol Meth Res2013; 42: 105–138.
32.
AzzaliniADalla ValleA. The multivariate skew-normal distribution. Biometrika1996; 83: 715–726.
33.
LiuC. Missing data imputation using the multivariate t distribution. J Multivariate Analys1995; 53: 139–158.
34.
JaraAQuintanaFMartinES. Linear mixed models with skew-elliptical distributions: a Bayesian approach. Computat Stat Data Analys2008; 52: 5033–5045.
35.
LachosVHDeyDKCanchoVG. Robust linear mixed models with skew-normal independent distributions from a Bayesian perspective. J Stat Plan Inference2009; 139: 4098–4110.
36.
ArnoldBCBeaverRJ. Skewed multivariate models related to hidden truncation and/or selective reporting. Test2002; 11: 7–54.
37.
Arellano-ValleRBBrancoMDGentonMG. A unified view on skewed distributions arising from selections. Canadian J Stat2006; 34: 581–601.
38.
LiseoBLoperfidoN. A note on reference priors for the scalar skew-normal distribution. J Stat Plan Inference2006; 136: 373–389.
39.
BayesCLBrancoMD. Bayesian inference for the skewness parameter of the scalar skew-normal distribution. Brazilian J Probabil Stat2007; 21: 141–163.
PewseyA. Problems of inference for Azzalini's skew-normal distribution. J Appl Stat2000; 27: 859–870.
42.
Arellano-ValleRAzzaliniA. The centred parametrization for the multivariate skew-normal distribution. J Multivariate Analys2008; 99: 1362–1382.
43.
LiseoBParisiA. Bayesian inference for the multivariate skew-normal model: a population Monte Carlo approach. Computat Stat Data Analys2013; 63: 125–138.
44.
FonsecaTCOFerreiraMARMigonHS. Objective Bayesian analysis for the Student-t regression model. Biometrika2008; 95: 325–333.
45.
LiuJSWuYN. Parameter expansion for data augmentation. J Am Stat Assoc1999; 94: 1264–1274.
46.
LiuJSSabattiC. Generalised Gibbs sampler and multigrid Monte Carlo for Bayesian computation. Biometrika2000; 87: 353–369.
47.
LeeSXMcLachlanGJ. On mixtures of skew normal and skew t-distributions. Adv Data Analysis Classification2013; 7: 241–266.
48.
Fruhwirth-SchnatterSPyneS. Bayesian inference for finite mixtures of univariate and multivariate skew-normal and skew-t distributions. Biostatistics2010; 11: 317–336.
GelmanA. Prior distributions for variance parameters in hierarchical models. Bayesian Analysis2006; 1: 515–534.
51.
TangY. An efficient monotone data augmentation algorithm for Bayesian analysis of incomplete longitudinal data. Stat Probabil Lett2015; 104: 146–52.
52.
SimpsonDRueHRieblerA, et al.Penalising model component complexity: a principled, practical approach to constructing priors. Stat Sci2017; 32: 1–28.
53.
DanielsMJKassRE. Nonconjugate bayesian estimation of covariance matrices and its use in hierarchical models. JASA1999; 94: 1254–1263.
54.
PermuttT. Sensitivity analysis for missing data in regulatory submission. Stat Med2016; 35: 876–879.
55.
LuKLiDKochGG. Comparison between two controlled multiple imputation methods for sensitivity analyses of time-to-event data with possibly informative censoring. Stat Biopharmaceut Res2015; 7: 199–213.
56.
TangY. On the multiple imputation variance estimator for control-based and delta-adjusted pattern mixture models. Biometrics2017; 73: 1379–1387.
57.
TangY. Controlled pattern imputation for sensitivity analysis of longitudinal binary and ordinal outcomes with nonignorable dropout. Stat Med2018; 37: 1467–1481.
58.
TangY. Algorithms for imputing partially observed recurrent events with applications to multiple imputation in pattern mixture models. J Biopharmaceut Stat2018; 28: 518–533.
59.
BarnardJRubinDB. Small-sample degrees of freedom with multiple imputation. Biometrika1999; 86: 948–955.
60.
SpiegelhalterDBestNCarlinB, et al.Bayesian measures of model complexity and fit. J Royal Stat Soc, Ser B2002; 64: 583–639.
61.
GhideyWLesaffreEEilersP. Smooth random effects distribution in a linear mixed model. Biometrics2004; 60: 945–953.
62.
CabralCRBLachosVHMadrugaMR. Bayesian analysis of skew-normal independent linear mixed models with heterogeneity in the random-effects population. J Stat Plan Inference2012; 142: 181–200.
63.
LachosVHGhoshPArellano-ValleRB. Likelihood based inference for skew normal independent linear mixed models. Statistica Sinica2010; 20: 303–322.
64.
ChibSGreenbergE. Understanding the metropolis-hastings algorithm. Am Stat1995; 49: 327–335.
65.
SennS. Testing for baseline balance in clinical trials. Stat Med1994; 13: 1715–1726.
66.
LuK. On efficiency of constrained longitudinal data analysis versus longitudinal analysis of covariance. Biometrics2010; 66: 891–896.
67.
LuK. An analytic method for the placebo-based pattern-mixture model. Stat Med2014; 33: 1134–1145.
68.
KleinmanKPIbrahimJG. A semiparametric bayesian approach to the random effects model. Biometrics1998; 54: 921–938.
69.
VasconcellosKLPDa SilvaSG. Corrected estimates for student t regression models with unknown degrees of freedom. J Stat Computat Simulat2005; 52: 409–423.
70.
LangeKLLittleRJATaylorJMG. Robust statistical modeling using the t-distribution. J Am Stat Assoc1989; 84: 881–896.
71.
LukacsPMBurnhamKPAndersonDR. Model selection bias and Freedman's paradox. Ann Inst Stat Math2009; 62: 117–125.
72.
MundryRNunnCL. Stepwise model fitting and statistical inference: turning noise into signal pollution. Am Naturalist2009; 173: 119–123.
73.
WangJLuoS. Bayesian multivariate augmented beta rectangular regression models for patient-reported outcomes and survival data. Stat Meth Med Res2017; 26: 117–125.
74.
GlassGVPeckhamPDSandersJR. Consequences of failure to meet assumptions underlying the fixed effects analyses of variance and covariance. Rev Educ Res1972; 42: 237–288.
75.
MehrotraDVLiXLiuJ, et al.Analysis of longitudinal clinical trials with missing data using multiple imputation in conjunction with robust regression. Biometrics2012; 68: 1250–1259.
76.
KotzSNadarajahS. Multivariate t-distributions and their applications, Cambridge: Cambridge University Press, 2004.
Supplementary Material
Please find the following supplemental material available below.
For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.
For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.
