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. .
- Verbeke G, Molenberghs G (2000) Linear mixed models for longitudinal data. Springer-Verlag, New York.
- Demidenko E (2004) Mixed models: Theory and Applications. Wiley, New York.
- Fisher RA (1918) The correlation between relatives on the supposition of Mendelian inheritance. Transactions of the Royal Society of Edinburgh. 52: 399-433.
- Diggle P, Heagerty P, Liang K, Zeger S (2002) Analysis of longitudinal data England. Oxford Univesity Press.
- Green PJ (1987) Penalized Likelihood for General Semi-parametric Regression Models. International Statistical Review 55: 245-260.
- Lin XH, Zhang DW (1999) Inference in generalized additive mixed models by using smoothing splines. J Roy Statist Soc B 61: 381-400.
- Henderson CR, Kempthorne O, Searle SR, Krosigk CM(1959) The Estimation of Environmental and Genetic Trends from Records Subject to Culling. Biometrics 15: 192-218.
- Laird N, Ware JH (1982) Random-effects models for longitudinal data. Biometrics 38: 963-974.
- Vonesh EF, Carter RL (1992) Mixed-effects nonlinear regression for unbalanced repeated measures. Biometrics 48: 1-17.
- Li ZX, Zhu LX (2010) On variance components in semi parametric mixed models for longitudinal data. Scand J Stat 37: 442-457.
- Li ZX, Wang YD, Wu P, Xu WL, Zhu LX (2012) Tests for variance components in varying coeffcient mixed models. Statistica Sinica 22: 123-148.
- Lin XH (1997) Variance component testing in generalized linear models with random effects. Biometrika 84: 309-326.
- Zhang DW(2004) Generalized linear mixed models with varying coeffcients for longitudinal data. Biometrics 60: 8-15.
- Laird N, Lange N, Stram D (1987) Maximum likelihood computation with repeated measures: Application of the EM algorithm. J Amer Statis Assoc 82: 97-105.
- Lindstrom MJ, Bates D (1988) Newtown-Raphson and EM algorithms for linear mixed-effects models for repeated-measures data. J Amer Statis Assoc 83: 1014-1022.
- Li ZX (2011) Estimation in linear mixed models for longitudinal data under linear restricted conditions. J Statist Plan Infer 141: 869-876.
- Li ZX (2013) Two kinds of variance/covariance estimates in linear mixed models. Metrika 76: 303-324.
- Nguyen TT, Mentr F (2014) Evaluation of the Fisher information matrix in non-linear mixed effect models using adaptive Gaussian quadrature. Computational Statistics and Data Analysis 80: 57-69.
- Li ZX (2017) Profile maximal likelihood estimation for nonlinear mixed models with longitudinal data. Communication in Statistics-Theory and Methods 46: 4449-4463.
- Self SG, Liang KY (1987) Asymptotic properties of maximum likelihood esti-mators and likelihood ratio tests under nonstandard conditions. J Amer Statis Assoc 82: 605-610.
- Stram DO, Lee JW (1994) Variance components testing in the longitudinal mixed effects model. Biometrics 50: 1171-1177.
- Giampaoli V, Singer JM (2009) Likelihood ratio tests for variance components in linear mixed models. J Statist Plan Infer 139: 1435-1448.
- Vu HTV, Zhou S (1997) Generalization of likelihood ratio tests under nonstandard conditions. Ann Statist 25: 897-916.
- Crainiceanui CM, Ruppert D (2004) Likelihood ratio tests in linear mixed models with one variance component. J Roy Statist Soc B 66: 165-185.
- Greven S, Crainiceanu CM, Kuchenhoff H, Peters A (2008) Restricted likelihood ratio testing for zero variance components in linear mixed models. Journal of Computational and Graphical Statistics 17: 870-891.
- Saville BR, Herring AH (2009) Testing random effects in the linear mixed model using approximate bayes factors. Biometrics 65: 369-376.
- Russo CM, Aoki R, Paula GA (2012) Assessment of variance components in nonlinear mixed-effect elliptical models. TEST 21: 519-545.
- Silvapulle MJ, Silvapulle P (1995) A score test against one-sided alternatives. J Amer Statis Assoc90: 342-349.
- Zhang DW, Lin XH (2008) Variance components testing in generalized linear mixed models for longitudinal/clustered data and other related topics. In: Random Effect and Latent variable Model Selection. Springer, Berlin.
- Sinha SJ (2009) Bootstrap tests for variance components in generalized linear mixed models. The Canadian Journal of Statistics 37: 219-234.
- Drikvandi R, Verbeke G, Khodadadi A, Nia VP (2013) Testing multiple variance components in linear mixed-effects models. Biostatistics 14: 144-159.
- Li ZX, Zhu LX (2013) A new test for random effects in linear mixed models with longitudinal data. Journal of Statistical Planning and Inference 143: 82-95.
- Nobre JS, Singer JM, Sen PK (2013) U-tests for variance components in linear mixed models. TEST 22: 580-605.
- Fan Y, Li R (2012) Variable selection in linear mixed effects models. The Annals of Statistics 40: 2043-2068.
- Li ZX, Chen F, Zhu LX (2014) Variance components testing in ANOVA-type mixed models. Scand J Statist 41: 482-496.
- Li ZX (2017) Inference of nonlinear mixed models for clustered data under moment conditions.
- Chen F, Li ZX, Shi L, Zhu LX (2015) Inference for mixed models of ANOVA type with high-dimensional data. J Mult Ana 133: 382-401.