The Wald test of using REML fixed effects So, we now wish to test that a linear function of the fixed effects is some fixed value. Specifically, we test for a matrix of order and a vector of length Then following r×p r. immediately from the result for the distribution of we can say that is the Wald statistic. Note that the REML estimates of the variance parameters are used in this expression. The scaled Wald statistic therefore is F=W/r and this has an asymptotic 2 distribution with degrees of freedom. However, for the reasons just pointed out, the P values will be r of the true P values, so if the P value of the scaled Wald statistic is calculated using this asymptotic distribution then care needs to be taken with the interpretation in many cases. In 1997 Kenward and Roger (in Small Sample Inference for Fixed Effects from Restricted Biometrics, 53, 983–997) developed a method of improving the P Maximum Likelihood, values by scaling by a further factor: F* = F. They developed the equations necessary to  evaluate  as well as the denominator df (the numerator df = They showed by simulation r). that the new P values were much more reliable. In fact, two important properties of this approach can be stated: For an orthogonal design such as in ANOVA with no missing values, the P values of the scaled Wald statistic are that is, they reproduce the ANOVA F P values. exact, When = 1 (that is, testing the equality of two treatment the P values of the scaled r Wald statistic are the same as the Satterthwaite P values from an unpaired t test with unequal treatment variances.
The Wald test of fixed effects using REML So, we now wish to test that a linear function of the fixed effects is some fixed value. Specifically, we test for a matrix of order r×p and a vector of length r. Then following immediately from the result for the distribution of we can say that is the Wald statistic. Note that the REML estimates of the variance parameters are used in this expression. The scaled Wald statistic therefore is F=W/r and this has an asymptotic 2 distribution with r degrees of freedom. However, for the reasons just pointed out, the P values will be of the true P values, so if the P value of the scaled Wald statistic is calculated using this asymptotic distribution then care needs to be taken with the interpretation in many cases. In 1997 Kenward and Roger (in Small Sample Inference for Fixed Effects from Restricted Maximum Likelihood, Biometrics, 53, 983–997) developed a method of improving the P values by scaling by a further factor: F* =  F. They developed the equations necessary to evaluate  as well as the denominator df (the numerator df = r). They showed by simulation that the new P values were much more reliable. In fact, two important properties of this approach can be stated: For an orthogonal design such as in ANOVA with no missing values, the P values of the scaled Wald statistic are exact, that is, they reproduce the ANOVA F P values. When r = 1 (that is, testing the equality of two treatment the P values of the scaled Wald statistic are the same as the Satterthwaite P values from an unpaired t test with unequal treatment variances.
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