Small Sample Inference for Fixed Effects from Restricted Maximum Likelihood
1997/09/01 by Michael G. Kenward, James H. Roger, James Roger · 4,568 citations
Agricultural and Biological Sciences · Mathematics · #Advanced Statistical Methods and Models #Computer science #Covariance #Covariance matrix #Estimation theory #Estimator #Genetics and Plant Breeding #Inference #Mathematics #Restricted maximum likelihood #Sample size determination #Sampling distribution #Statistic #Statistical Methods and Bayesian Inference #Statistical hypothesis testing #Statistical inference #Statistics #Wald test
paper · doi:10.2307/2533558
published in Biometrics 53(3), 983 (Oxford University Press)
openalex publication_date 1997/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Abstract
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.
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