A Review on Mixed Models
1China University of Mining and Technology (Beijing), Beijing, China
2State Key Laboratory of Coal Resource and Safe Mining (CUMT), Beijing, China
- *Corresponding Author:
- Li Z
China University of Mining and Technology (Beijing)
E-mail: [email protected]
Received Date: May 19, 2017; Accepted Date: May 27, 2017; Published Date: May 31, 2017
Citation: Li Z (2017) A Review on Mixed Models. J Biom Biostat 8: 350. doi:10.4172/2155-6180.1000350
Copyright: © 2017 Li Z. This is an open-access article distributed under the
terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author and
source are credited.
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Mixed model/mixed modeling [1,2] is an important area/tool in
statistics. It includes fixed effects and random effects. In fact, random
effects (mixed) models were introduced by Fisher  where the
correlations of trait values between relatives were studied. One-way
analysis of variance (ANOVA) model and two way ANOVA are two
ordinary and widely-used mixed models. Now two kinds of mixed
models are mainly mentioned in literatures. One is to model clustered
data/repeated data/longitudinal data  where the response may
be divided into independent sub-vectors and the covariance matrix of random effects is very general, the other is similar to the two-way
ANOVA model and some is to act as a representation tool for the
nonparametric function [5,6] where the response may not be divided
into independent sub-vectors and the covariance matrix of random
effects is usually with special structures.
For different mean structures, mixed models usually include the
following: linear mixed models [7,8]. Nonlinear mixed models NLMM
[9,10]. Semi parametric mixed models SMM , varying coefficient
mixed model-s VCMM . Generalized linear mixed models GLMM
, generalized additive mixed models GAMM , generalized
varying coefficient mixed models GVCMM .
Statistical inference (estimation/prediction and hypothesis testing)
of mixed models is the main topic in this area.
As for the estimation/prediction of mixed models, Henderson et
al.  is the earliest literature to the best of my knowledge where LMM
is considered with the fixed effects estimated and the random effects
predicted. Laird and Ware  developed the EM algorithm to estimate
the fixed effect and the covariance matrix of random effects in the
framework of LMM for longitudinal data with normality assumptions.
There are many other literatures about estimation of LMM, for instance
[14-17]. Besides, NLMM is also estimated by many authors such as Lin
, Nguyen and Mentr , Li . Lin and Zhang  considered
GAMM and Zhang  studied GVCMM.
Most literatures about mixed models focus on the hypothesis testing
especially for the existence of random effects or their sub-vectors. The
testing problem is equivalent to testing whether the corresponding
(co)variances of random effects are zero or not since the mean of
random effects is zero. Since the true values are on the boundary
of the parametric space, it is a nonstandard testing problem and no
Wilks phenomenon holds . It is of interest and challenge. There
are two kinds of literatures: one is under parametric distributional
assumptions and Monte Carlo (MC) method is usually used, the other
is distribution-free and some are tractable in the sense that the critical
values do not resort to MC method.
Under the normality distributions about random effects and
random errors, LMM is studied by many authors. For instance, Stram
and Lee  and Giampaoli and Singer  considered likelihood
ratio tests (LRTs) according to Self and Liang  and Vu and Zhou
 respectively; Crainiceanui and Ruppert  and Greven et al.
 developed some algorithms for this nonstandard testing problem;
Saville and Herring  applied Bayes factors in LMM. For other
mixed models, Russo et al.  considered variance components
testing in NLMM with elliptical distributions by score-type test SST . Besides, Zhang and Lin  examined GLMM with normally
distributed random effects by the adapted LRT based on the theory of
Self and Liang  and SST based on Silvapulle and Silvapulle .
Sinha  also considered the existence of random effects in GLMM
where the responses are in the exponential family by a one-sided score
test based on SST.
For the distribution-free tests for random effects, some main
publications are as follows: Drikvandi et al.  and Li and Zhu
 proposed the trace-based tests TDV KP and Tmtr for LMM
respectively; Nobre et al.  developed a tractable U-test. Li and
Zhu  proposed a difference-based test for the existence of random
effects TmD in SMM. Li et al.  developed two distribution-free and
easily tractable tests based on the quasi-likelihood for VCMM. Li et al.
 studied ANOVA-type LMM and Li  investigated the existence
of any sub-vector of random effects in NLMM.
Moreover, the topics in mixed models include variable selection and high dimensional problems, which is of interest, too. For examples,
Fan and Li  and Chen et al. .
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- Demidenko E (2004) Mixed models: Theory and Applications. Wiley, New York.
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