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For the special case of balanced one-way random effects ANOVA, it has been established that the generalized likelihood ratio test (LRT) and Wald’s test are largely equivalent in testing the variance component. We extend these results to explore the relationships between Wald’s F test, and the LRT for a much broader class of linear mixed models; the generalized split-plot models. In particular, we explore when the two tests are equivalent and prove that when they are not equivalent, Wald’s F test is more powerful, thus making the LRT test inadmissible. We show that inadmissibility arises in realistic situations with common number of degrees of freedom. Further, we derive the statistical distribution of the LRT under both the null and alternative hypotheses $$H_0$$ and $$H_1$$ where $$H_0$$ is the hypothesis that the between variance component is zero. Providing an exact distribution of the test statistic for the LRT in these models will help in calculating a more accurate p-value than the traditionally used p-value derived from the large sample chi-square mixture approximations.

For testing a linear hypothesis about fixed effects in a normal mixed linear model, a popular approach is to use a Wald test, in which the test statistic is assumed to have a null distribution that is approximately chi-squared. This approximation is questionable, however, for small samples. In 1997 Kenward and Roger constructed a test that addresses this problem. They altered the Wald test in three ways: (a) adjusting the test statistic, (b) approximating the null distribution by a scaled F distribution, and (c) modifying the formulas to achieve an exact F test in two special cases. Alterations (a) and (b) lead to formulas that are somewhat complicated but can be explained by using Taylor series approximations and a few convenient assumptions. The modified formulas used in alteration (c), however, are more mysterious. Restricting attention to models with linear variance–covariance structure, we provide details of a derivation that justifies these formulas. We show that similar but different derivations lead to different formulas that also produce exact F tests in the two special cases and are equally justifiable. A simulation study was done for testing the equality of treatment effects in block-design models. Tests based on the different derivations performed very similarly. Moreover, the simulations confirm that alteration (c) is worthwhile. The Kenward–Roger test showed greater accuracy in its p values than did the unmodified version of the test.

While giving subjective responses, a small fraction of respondents overstate or understate their true status. The extent of over (or under) evaluation and the proportions of such respondents differs from population to population. These biases alter the significance level and power of the commonly used tests. In this paper the authors determine the effects of these biases on the commonly used Z test, F test of Anova, and χ2 test for testing the equality of multinominal distributions.

In the linear model Y i = x i′ β + σe i, i=1,…,n, with unknown (β, σ), β∈{\open R}p, σ>0, and with i.i.d. errors e 1,…,e n having a continuous distribution F, we test for the goodness-of-fit hypothesis H 0:F(e)≡F 0(e/σ), for a specified symmetric distribution F 0, not necessarily normal. Even the finite sample null distribution of the proposed test criterion is independent of unknown (β,σ), and the asymptotic null distribution is normal, as well as the distribution under local (contiguous) alternatives. The proposed tests are consistent against a general class of (nonparametric) alternatives, including the case of F having heavier (or lighter) tails than F 0. A simulation study illustrates a good performance of the tests.

In this article, we establish a method of selecting the best regression equation on the basis of F-test. The basic idea is to select the most significant regression equation, which corresponds to the minimum P-value of F-test. We also present upper and lower bounds for the P-value together with approximations for these bounds which are simple to compute. Using this method, not only can we find out the best regression equation, we also obtain its significance probability. When the method is applied to the well-known data from Hald (1952) and Gorman & Toman (1966), the results are satisfactory.

It is well known that dropping variables in regression analysis decreases the variance of the least squares (LS) estimator of the remaining parameters. However, after elimination estimates of these parameters are biased, if the full model is correct. In his recent paper, Boscher (1991) showed that the LS-estimator in the special case of a mean shift model (cf. Cook and Weisberg, 1982) which assumes no “outliers” can be considered in the framework of a linear regression model where some variables are deleted. He derived conditions under which this estimator outperforms the LS-estimator of the full model in terms of the mean squared error (MSE)-matrix criterion. We demonstrate that this approach can be extended to the general set-up of dropping variables. Necessary and sufficient conditions for the MSE-matrix superiority of the LS-estimator in the reduced model over that in the full model are derived. We also provide a uniformly most powerful F-statistic for testing the MSE-improvement.

As a test statistic for testing goodness-of-fit of a linear regression model, we propose a ratio of quadratic forms measuring the distance between parametric and nonparametric fits, relative to the estimated error variance. The test statistic is a modification of the statistic suggested by Hardle and Mammen (1988). The asymptotic distribution under the hypothesis is established. The finite sample behaviour of the test is investigated in a Monte Carlo study, and is illustrated for two applications.

A rather general class of mixed linear models is treated in this paper. Fixed models are included as a special case. For certain linear hypotheses the analysis-of-variance-test (F-Test) is shown to be uniformly most powerfull among all tests being invariant and unbiased. Those are for instance the usual hypotheses of treatment effects and interactions in latin squares and (possibly unbalanced) split-plot-models, both with random blocks.

For testing the significance of the difference between the unknown means of two normal populations having different unknown variances, we have the known Fisher-Behren’s-Welch test statistic. Several expressions for the distribution of this test statistic are available, and this paper is yet another attempt in the same direction. However, the expression given here appears to be, perhaps, more elegant and straight-forward than the ones available at present.