Restricted maximum likelihood (REML) is now well established as a method for estimating the parameters of the general Gaussian linear model with a structured covariance matrix, in particular for mixed linear models. Conventionally, estimates of precision and inference for fixed effects are based on their asymptotic distribution, which is known to be inadequate for some small-sample problems. In this paper, we present a scaled Wald statistic, together with an F approximation to its sampling distribution, that is shown to perform well in a range of small sample settings. The statistic uses an adjusted estimator of the covariance matrix that has reduced small sample bias. This approach has the advantage that it reproduces both the statistics and F distributions in those settings where the latter is exact, namely for Hotelling T2 type statistics and for analysis of variance F-ratios. The performance of the modified statistics is assessed through simulation studies of four different REML analyses and the methods are illustrated using three examples.
Biometrics is a scientific journal emphasizing the role of statistics and mathematics in the biological sciences. Its object is to promote and extend the use of mathematical and statistical methods in pure and applied biological sciences by describing developments in these methods and their applications in a form readily assimilable by experimental scientists. JSTOR provides a digital archive of the print version of Biometrics. The electronic version of Biometrics is available at http://www.blackwell-synergy.com/servlet/useragent?func=showIssues&code;=biom. Authorized users may be able to access the full text articles at this site.
The International Biometric Society is an international society for the advancement of biological science through the development of quantitative theories and the application, development and dissemination of effective mathematical and statistical techniques. The Society welcomes as members biologists, mathematicians, statisticians, and others interested in applying similar techniques.
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