Medical, Pharma, Engineering, Science, Technology and Business

^{1}China University of Mining and Technology (Beijing), Beijing, China

^{2}State Key Laboratory of Coal Resource and Safe Mining (CUMT), Beijing, China

- *Corresponding Author:
- Li Z

China University of Mining and Technology (Beijing)

Beijing, China

**Tel:**0516-83592826

**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.

**Visit for more related articles at** Journal of Biometrics & Biostatistics

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 [3] 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 [4] 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 [10], varying coefficient mixed model-s VCMM [11]. Generalized linear mixed models GLMM [12], generalized additive mixed models GAMM [6], generalized varying coefficient mixed models GVCMM [13].

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. [7] 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 [8] 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 [12], Nguyen and Mentr [18], Li [19]. Lin and Zhang [6] considered GAMM and Zhang [13] 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 [20]. 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 [21] and Giampaoli and Singer [22] considered likelihood ratio tests (LRTs) according to Self and Liang [20] and Vu and Zhou [23] respectively; Crainiceanui and Ruppert [24] and Greven et al. [25] developed some algorithms for this nonstandard testing problem; Saville and Herring [26] applied Bayes factors in LMM. For other mixed models, Russo et al. [27] considered variance components testing in NLMM with elliptical distributions by score-type test SST [28]. Besides, Zhang and Lin [29] examined GLMM with normally distributed random effects by the adapted LRT based on the theory of Self and Liang [20] and SST based on Silvapulle and Silvapulle [28]. Sinha [30] 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. [31] and Li and Zhu [32] proposed the trace-based tests TDV KP and Tmtr for LMM respectively; Nobre et al. [33] developed a tractable U-test. Li and Zhu [10] proposed a difference-based test for the existence of random effects TmD in SMM. Li et al. [34] developed two distribution-free and easily tractable tests based on the quasi-likelihood for VCMM. Li et al. [35] studied ANOVA-type LMM and Li [36] 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 [34] and Chen et al. [37].

- 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.

Select your language of interest to view the total content in your interested language

- Adomian Decomposition Method
- Algebra
- Algebraic Geometry
- Algorithm
- Analytical Geometry
- Applied Mathematics
- Artificial Intelligence Studies
- Axioms
- Balance Law
- Behaviometrics
- Big Data Analytics
- Big data
- Binary and Non-normal Continuous Data
- Binomial Regression
- Bioinformatics Modeling
- Biometrics
- Biostatistics methods
- Biostatistics: Current Trends
- Clinical Trail
- Cloud Computation
- Combinatorics
- Complex Analysis
- Computational Model
- Computational Sciences
- Computer Science
- Computer-aided design (CAD)
- Convection Diffusion Equations
- Cross-Covariance and Cross-Correlation
- Data Mining Current Research
- Deformations Theory
- Differential Equations
- Differential Transform Method
- Findings on Machine Learning
- Fourier Analysis
- Fuzzy Boundary Value
- Fuzzy Environments
- Fuzzy Quasi-Metric Space
- Genetic Linkage
- Geometry
- Hamilton Mechanics
- Harmonic Analysis
- Homological Algebra
- Homotopical Algebra
- Hypothesis Testing
- Integrated Analysis
- Integration
- Large-scale Survey Data
- Latin Squares
- Lie Algebra
- Lie Superalgebra
- Lie Theory
- Lie Triple Systems
- Loop Algebra
- Mathematical Modeling
- Matrix
- Microarray Studies
- Mixed Initial-boundary Value
- Molecular Modelling
- Multivariate-Normal Model
- Neural Network
- Noether's theorem
- Non rigid Image Registration
- Nonlinear Differential Equations
- Number Theory
- Numerical Solutions
- Operad Theory
- Physical Mathematics
- Quantum Group
- Quantum Mechanics
- Quantum electrodynamics
- Quasi-Group
- Quasilinear Hyperbolic Systems
- Regressions
- Relativity
- Representation theory
- Riemannian Geometry
- Robotics Research
- Robust Method
- Semi Analytical-Solution
- Sensitivity Analysis
- Smooth Complexities
- Soft Computing
- Soft biometrics
- Spatial Gaussian Markov Random Fields
- Statistical Methods
- Studies on Computational Biology
- Super Algebras
- Symmetric Spaces
- Systems Biology
- Theoretical Physics
- Theory of Mathematical Modeling
- Three Dimensional Steady State
- Topologies
- Topology
- mirror symmetry
- vector bundle

- 6th International Conference on
**Biostatistics**and**Bioinformatics**

November 13-14, 2017, Atlanta, USA

- Total views:
**221** - [From(publication date):

June-2017 - Sep 22, 2017] - Breakdown by view type
- HTML page views :
**178** - PDF downloads :
**43**

Peer Reviewed Journals

International Conferences 2017-18