Modeling Change in Skewed Variables Using Mixed Beta Regression Models

The general linear model relies on the assumptions of independent and normally distributed residuals with constant variance. If these assumptions are not met, beta regression models represent a viable alternative to model variables that show skewness and heteroscedasticity. In this study, the beta regression model is extended to include random effects. Such a mixed beta regression model can account for the data dependency present in longitudinal data by allowing for participant-specific effects. The mixed beta regression model is illustrated using longitudinal data on complex choice reaction times. Results show that a model with fixed age and missingness effects as well as random intercept and age slope effects fitted the data best. Importantly, it fitted the data much better than a linear mixed model did. Alternatives to the mixed beta regression model and potential applications are described in the discussion.

[1]  S. MacDonald,et al.  Variability in reaction time performance of younger and older adults. , 2002, The journals of gerontology. Series B, Psychological sciences and social sciences.

[2]  J. Cerella.  Information processing rates in the elderly. , 1985, Psychological bulletin.

[3]  Michael Smithson,et al.  A better lemon squeezer? Maximum-likelihood regression with beta-distributed dependent variables. , 2006, Psychological methods.

[4]  B. McCullough,et al.  Regression analysis of variates observed on (0, 1): percentages, proportions and fractions , 2003 .

[5]  G. Molenberghs,et al.  Models for Discrete Longitudinal Data , 2005 .

[6]  Ana Ivelisse Avilés,et al.  Linear Mixed Models for Longitudinal Data , 2001, Technometrics.

[7]  T. Salthouse.  The processing-speed theory of adult age differences in cognition. , 1996, Psychological review.

[8]  Philip Paolino,et al.  Maximum Likelihood Estimation of Models with Beta-Distributed Dependent Variables , 2001, Political Analysis.

[9]  P. Rast,et al.  Verbal Learning Changes in Older Adults Across 18 Months , 2009, Neuropsychology, development, and cognition. Section B, Aging, neuropsychology and cognition.

[10]  P. McCullagh,et al.  Generalized Linear Models, 2nd Edn. , 1990 .

[11]  P. McCullagh,et al.  Generalized Linear Models , 1992 .

[12]  Arjun K. Gupta,et al.  Handbook of beta distribution and its applications , 2004 .

[13]  N. L. Johnson,et al.  Continuous Univariate Distributions. , 1995 .

[14]  S. Lewis,et al.  Regression analysis , 2007, Practical Neurology.

[15]  Fitting non‐linear structural relationships using SAS procedure NLMIXED , 2002 .

[16]  David R. Cox.  The analysis of binary data , 1970 .

[17]  S. Ferrari,et al.  Beta Regression for Modelling Rates and Proportions , 2004 .

[18]  D. Hultsch,et al.  Memory Change in the Aged , 1998 .

[19]  Richard J. Cleary.  Handbook of Beta Distribution and Its Applications , 2006 .

[20]  N. Draper,et al.  Applied Regression Analysis , 1966 .

[21]  Mike Martin,et al.  Can longitudinal changes in processing speed explain longitudinal age changes in fluid intelligence? , 2002, Psychology and aging.

[22]  James K. Lindsey,et al.  Applying Generalized Linear Models , 2000 .

[23]  D. Zimprich.  Cross-sectionally and Longitudinally Balanced Effects of Processing Speed on Intellectual Abilities , 2002, Experimental aging research.

[24]  H. Bozdogan.  Model selection and Akaike's Information Criterion (AIC): The general theory and its analytical extensions , 1987 .

[25]  Ole E. Barndorff-Nielsen,et al.  Some parametric models on the simplex , 1991 .

[26]  Donald Hedeker,et al.  Application of random-efiects pattern-mixture models for miss-ing data in longitudinal studies , 1997 .

[27]  Gordon K. Smyth,et al.  Generalized linear models with varying dispersion , 1989 .

[28]  Francisco Cribari-Neto,et al.  Influence diagnostics in beta regression , 2008, Comput. Stat. Data Anal..