Small sample inference for the fixed effects in the mixed linear model

Orly Manor*, David M. Zucker

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

35 Scopus citations


The small sample performance of several procedures for testing a given fixed effect in a mixed linear model is investigated. Using simulations, constructed on the basis of a study of growth of children with Gaucher's disease, standard normal-theory Wald tests for both ML and REML estimates, the likelihood ratio test (LRT), a modified LRT based on Bartlett correction, and a number of adjusted tests based on t and F distributions are evaluated. Methods used for determining the denominator degrees of freedom in the t and F tests include the residual degrees of freedom method, the between and within degrees of freedom, the containment method, the naive method and the Satterthwaite method. A test based on a sandwich-type estimator of the variance of the parameter estimate is evaluated as well and the effect of mis-specifying the random-effects distribution is considered. Results show that Type I error rates for the Wald-type test with chi-square approximation are substantially inflated, though less so with REML estimates than with ML estimates. The LRT based on ML estimates yielded Type I error rates similar to those observed for the Wald-type chi-square test with REML estimates. A substantial improvement in Type I error rates for testing on both the intercept and slope is provided by each of the three following modifications: the Satterthwaite and naive methods with REML-based estimates and the Bartlett-corrected LRT.

Original languageAmerican English
Pages (from-to)801-817
Number of pages17
JournalComputational Statistics and Data Analysis
Issue number4
StatePublished - 1 Jul 2004

Bibliographical note

Funding Information:
This research was supported by Grant 814/99 from the Israel Science Foundation.


  • Fixed effects
  • Longitudinal studies
  • Mixed linear model
  • Satterthwaite approximation
  • Wald-type test


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