The Use of Score Tests for Inference on Variance Components

Whenever inference for variance components is required, the choice between one-sided and two-sided tests is crucial. This choice is usually driven by whether or not negative variance components are permitted. For two-sided tests, classical inferential procedures can be followed, based on likelihood ratios, score statistics, or Wald statistics. For one-sided tests, however, one-sided test statistics need to be developed, and their null distribution derived. While this has received considerable attention in the context of the likelihood ratio test, there appears to be much confusion about the related problem for the score test. The aim of this paper is to illustrate that classical (two-sided) score test statistics, frequently advocated in practice, cannot be used in this context, but that well-chosen one-sided counterparts could be used instead. The relation with likelihood ratio tests will be established, and all results are illustrated in an analysis of continuous longitudinal data using linear mixed models.

[1]  C B Dean,et al.  The use of mixture models for identifying high risks in disease mapping , 2001, Statistics in medicine.

[2]  C. Dean.  Testing for Overdispersion in Poisson and Binomial Regression Models , 1992 .

[3]  Mark Berman,et al.  Approximating Point Process Likelihoods with Glim , 1992 .

[4]  John A. Nelder,et al.  The interpretation of negative components of variance , 1954 .

[5]  J. Lawless,et al.  Tests for Detecting Overdispersion in Poisson Regression Models , 1989 .

[6]  R. Hines.  A comparison of tests for overdispersion in generalized linear models , 1997 .

[7]  David R. Cox,et al.  Some remarks on overdispersion , 1983 .

[8]  Xihong Lin.  Variance component testing in generalised linear models with random effects , 1997 .

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

[10]  T. Britton.  Tests to detect clustering of infected individuals within families. , 1997, Biometrics.

[11]  S. Paul,et al.  ANALYSIS OF PROPORTIONS IN THE PRESENCE OF OVER-/UNDER-DISPERSION , 1995 .

[12]  Daniel B. Hall,et al.  Order‐restricted score tests for homogeneity in generalised linear and nonlinear mixed models , 2001 .

[13]  Daniel F. Heitjan,et al.  Testing and Adjusting for Departures from Nominal Dispersion in Generalized Linear Models , 1993 .

[14]  D. Bates,et al.  Newton-Raphson and EM Algorithms for Linear Mixed-Effects Models for Repeated-Measures Data , 1988 .

[15]  Paramsothy Silvapulle,et al.  A Score Test against One-Sided Alternatives , 1995 .

[16]  S. le Cessie,et al.  Testing the fit of a regression model via score tests in random effects models. , 1995 .

[17]  Erik V. Nordheim,et al.  Hypothesis tests for normal means constrained by linear inequalities , 1986 .

[18]  L. de Ridder,et al.  Comparative effects of neonatal and prepubertal castration on craniofacial growth in rats. , 1998, Archives of oral biology.

[19]  K. Liang.  A locally most powerful test for homogeneity with many strata , 1987 .

[20]  A. Shapiro.  Towards a unified theory of inequality constrained testing in multivariate analysis , 1988 .

[21]  C B Dean,et al.  Detecting Interaction Between Random Region and Fixed Age Effects in Disease Mapping , 2001, Biometrics.

[22]  Wang-Shu. Lu,et al.  Score tests for overdispersion in poisson regression models , 1997 .

[23]  Geert Verbeke,et al.  The Effect of Drop‐Out on the Efficiency of Longitudinal Experiments , 1999 .

[24]  K. Liang,et al.  Asymptotic Properties of Maximum Likelihood Estimators and Likelihood Ratio Tests under Nonstandard Conditions , 1987 .

[25]  J. Miller,et al.  Asymptotic Properties of Maximum Likelihood Estimates in the Mixed Model of the Analysis of Variance , 1977 .

[26]  W. A. Thompson.  The Problem of Negative Estimates of Variance Components , 1962 .

[27]  D. Commenges,et al.  Tests of Homogeneity for Generalized Linear Models , 1995 .

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

[29]  D. Stram,et al.  Variance components testing in the longitudinal mixed effects model. , 1994, Biometrics.

[30]  R. Gueorguieva,et al.  A multivariate generalized linear mixed model for joint modelling of clustered outcomes in the exponential family , 2001 .

[31]  H. Chernoff.  On the Distribution of the Likelihood Ratio , 1954 .

[32]  Helen Brown,et al.  Applied Mixed Models in Medicine , 2000, Technometrics.

[33]  R. Gray.  Tests for Variation over Groups in Survival Data , 1995 .

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

[35]  P. M. E. Altham,et al.  Improving the Precision of Estimation by Fitting a Model , 1984 .