Using the linear mixed model to analyze nonnormal data distributions in longitudinal designs

Using a Monte Carlo simulation and the Kenward–Roger (KR) correction for degrees of freedom, in this article we analyzed the application of the linear mixed model (LMM) to a mixed repeated measures design. The LMM was first used to select the covariance structure with three types of data distribution: normal, exponential, and log-normal. This showed that, with homogeneous between-groups covariance and when the distribution was normal, the covariance structure with the best fit was the unstructured population matrix. However, with heterogeneous between-groups covariance and when the pairing between covariance matrices and group sizes was null, the best fit was shown by the between-subjects heterogeneous unstructured population matrix, which was the case for all of the distributions analyzed. By contrast, with positive or negative pairings, the within-subjects and between-subjects heterogeneous first-order autoregressive structure produced the best fit. In the second stage of the study, the robustness of the LMM was tested. This showed that the KR method provided adequate control of Type I error rates for the time effect with normally distributed data. However, as skewness increased—as occurs, for example, in the log-normal distribution—the robustness of KR was null, especially when the assumption of sphericity was violated. As regards the influence of kurtosis, the analysis showed that the degree of robustness increased in line with the amount of kurtosis.

[1]  J. Ware,et al.  Random-effects models for longitudinal data. , 1982, Biometrics.

[2]  Russell D. Wolfinger,et al.  Repeated Measures Analysis Using Mixed Models: Some Simulation Results , 1997 .

[3]  R. Horner,et al.  Age at onset of Alzheimer's disease: clue to the relative importance of etiologic factors? , 1987, American journal of epidemiology.

[4]  R C Littell,et al.  Mixed Models: Modelling Covariance Structure in the Analysis of Repeated Measures Data , 2005 .

[5]  Y. W. Wu,et al.  A comparison of traditional approaches to hierarchical linear modeling when analyzing longitudinal data. , 1999, Research in nursing & health.

[6]  Guillermo Vallejo,et al.  Modified Brown–Forsythe Procedure for Testing Interaction Effects in Split-Plot Designs , 2006, Multivariate behavioral research.

[7]  F. W. Preston.  PSEUDO-LOGNORMAL DISTRIBUTIONS' , 1981 .

[8]  Guillermo Vallejo,et al.  Consequences of Misspecifying the Error Covariance Structure in Linear Mixed Models for Longitudinal Data , 2008 .

[9]  F. Vaida,et al.  Conditional Akaike information for mixed-effects models , 2005 .

[10]  Rand R. Wilcox,et al.  A one-way random effects model for trimmed means , 1994 .

[11]  Ellián Tuero-Herrero,et al.  Selecting the best unbalanced repeated measures model , 2011, Behavior research methods.

[12]  Elisa Lee,et al.  Statistical Methods for Survival Data Analysis: Lee/Survival Data Analysis , 2003 .

[13]  Allen I. Fleishman.  A method for simulating non-normal distributions , 1978 .

[14]  Nan M. Laird,et al.  Using the General Linear Mixed Model to Analyse Unbalanced Repeated Measures and Longitudinal Data , 1997 .

[15]  J. Ware.  Linear Models for the Analysis of Longitudinal Studies , 1985 .

[16]  Gilbert W. Fellingham,et al.  Performance of the Kenward–Roger Method when the Covariance Structure is Selected Using AIC and BIC , 2005 .

[17]  Jeff Miller,et al.  Information processing models generating lognormally distributed reaction times , 1993 .

[18]  Paula Fernández,et al.  Procedimientos estadísticos alternativos para evaluar la robustez mediante diseños de medidas repetidas , 2006 .

[19]  Robert L. Winkler,et al.  Comment Bayesian Model Building and Forecasting , 1985 .

[20]  M. Kenward,et al.  Small sample inference for fixed effects from restricted maximum likelihood. , 1997, Biometrics.

[21]  Russell D. Wolfinger,et al.  A comparison of two approaches for selecting covariance structures in the analysis of repeated measurements , 1998 .

[22]  Jan de Leeuw,et al.  A Review of Two different Approaches for the Analysis of Growth Data Using Longitudinal Mixed Linear Models: Comparing Hierarchical Linear Regression (ML3), HLM) and Repeated Measures Designs with Structured Covariance Matrices (BMDP5V) , 1996 .

[23]  Jan Fagerberg,et al.  Model-based prediction of phase III overall survival in colorectal cancer on the basis of phase II tumor dynamics. , 2009, Journal of clinical oncology : official journal of the American Society of Clinical Oncology.

[24]  R. Lomax,et al.  The Effect of Varying Degrees of Nonnormality in Structural Equation Modeling , 2005 .

[25]  Rand R. Wilcox,et al.  Analysing repeated measures or randomized block designs using trimmed means , 1993 .

[26]  Lisa M. Lix,et al.  Testing Repeated Measures Hypotheses When Covariance Matrices are Heterogeneous , 1993 .

[27]  Elisa T. Lee,et al.  Statistical Methods for Survival Data Analysis , 1994, IEEE Transactions on Reliability.

[28]  H. Akaike.  A new look at the statistical model identification , 1974 .

[29]  Willem J. van der Linden,et al.  A lognormal model for response times on test items , 2006 .

[30]  Russell D. Wolfinger,et al.  The Analysis of Repeated Measurements with Mixed-Model Adjusted F Tests , 2004 .

[31]  W. Stahel,et al.  Log-normal Distributions across the Sciences: Keys and Clues , 2001 .

[32]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[33]  Taesung Park,et al.  Covariance models for nested repeated measures data: analysis of ovarian steroid secretion data. , 2002, Statistics in medicine.

[34]  Roser Bono,et al.  Analyzing Small Samples of Repeated Measures Data with the Mixed-Model Adjusted F Test , 2009, Commun. Stat. Simul. Comput..

[35]  Roser Bono,et al.  General Linear Mixed Model for Analysing Longitudinal Data in Developmental Research , 2010, Perceptual and motor skills.

[36]  F. Uckun,et al.  Meta analysis of advanced cancer survival data using lognormal parametric fitting: a statistical method to identify effective treatment protocols. , 2007, Current pharmaceutical design.

[37]  F. E. Satterthwaite.  Synthesis of variance , 1941 .

[38]  R. Blair,et al.  A more realistic look at the robustness and Type II error properties of the t test to departures from population normality. , 1992 .

[39]  Hu Minghua,et al.  Estimation of air traffic longitudinal conflict probability based on the reaction time of controllers. , 2010 .

[40]  John M Ferron,et al.  Estimating individual treatment effects from multiple-baseline data: A Monte Carlo study of multilevel-modeling approaches , 2010, Behavior research methods.

[41]  N. Laird,et al.  Using the general linear mixed model to analyse unbalanced repeated measures and longitudinal data. , 1997, Statistics in medicine.

[42]  James Algina,et al.  An improved general approximation test for the main effect in a split‐plot design , 1995 .

[43]  G. B. Schaalje,et al.  Adequacy of approximations to distributions of test statistics in complex mixed linear models , 2002 .

[44]  Danielle D. Brown,et al.  Attention skills and looking to television in children from low income families , 2010 .

[45]  G. Molenberghs,et al.  Linear Mixed Models for Longitudinal Data , 2001 .

[46]  T. Micceri.  The unicorn, the normal curve, and other improbable creatures. , 1989 .

[47]  Pablo Livacic-Rojas,et al.  Comparison of Two Procedures for Analyzing Small Sets of Repeated Measures Data , 2005, Multivariate behavioral research.

[48]  H. Keselman,et al.  A comparison of recent approaches to the analysis of repeated measurements , 1999 .

[49]  Rand R. Wilcox,et al.  Testing Repeated Measures Hypotheses When Covariance Matrices are Heterogeneous: Revisiting the Robustness of the Welch-James Test Again , 2000 .

[50]  John Ferron,et al.  Effects of Misspecifying the First-Level Error Structure in Two-Level Models of Change , 2002, Multivariate behavioral research.

[51]  Rien van der Leeden,et al.  Multilevel Analysis of Repeated Measures Data , 1998 .

[52]  James Algina,et al.  Type I Error Rates of the Kenward-Roger Adjusted Degree of Freedom F-test for a Split-Plot Design with Missing Values , 2007 .