A reminder of the fallibility of the wald statistic

Computer programs often produce a parameter estimate <(theta)over cap> and estimated variance (var) over cap(<(theta)over cap>). Thus it is easy to compute a Wald statistic (<(theta)over cap> - theta(0)){(var) over cap(<(theta)over cap>)}(-1/2) to test the null hypothesis theta = theta(0). Hauck and Donner and Vaeth have identified situations in which the Wald statistic has poor power. We consider another example that is not in the classes discussed by those authors. We present data from a balanced one-way random effects analysis of variance (ANOVA) that illustrate the poor power of the Wald statistic compared to the usual F test, In this example the parameter of interest is the variance of the random effect. The power of the Wald test depends on the parameterization used, however, and a whole family of Wald statistics with p values ranging from 0 to 1 can be generated with power transformations of the random effect parameter.