Twin Research and Human Genetics

Articles

A Note on the Asymptotic Distribution of Likelihood Ratio Tests to Test Variance Components

Peter M. Visschera1 c1

a1 Queensland Institute of Medical Research, Brisbane,Australia; Institute of Evolutionary Biology, University of Edinburgh, United Kingdom. peter.visscher@qimr.edu.au

Abstract

When using maximum likelihood methods to estimate genetic and environmental components of (co)variance, it is common to test hypotheses using likelihood ratio tests, since such tests have desirable asymptotic properties. In particular, the standard likelihood ratio test statistic is assumed asymptotically to follow a χ2 distribution with degrees of freedom equal to the number of parameters tested. Using the relationship between least squares and maximum likelihood estimators for balanced designs, it is shown why the asymptotic distribution of the likelihood ratio test for variance components does not follow a χ2 distribution with degrees of freedom equal to the number of parameters tested when the null hypothesis is true. Instead, the distribution of the likelihood ratio test is a mixture of χ2 distributions with different degrees of freedom. Implications for testing variance components in twin designs and for quantitative trait loci mapping are discussed. The appropriate distribution of the likelihood ratio test statistic should be used in hypothesis testing and model selection.

(Received May 31 2006)

(Accepted June 06 2006)

Correspondence:

c1 Address for correspondence: Peter Visscher, Genetic Epidemiology, Queensland Institute of Medical Research, PO Royal Brisbane Hospital, QLD 4029, Australia.

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