A method suggested by Wald for finding critical regions of almost constant size and various modifications are considered. Under reasonable conditions the $s$th step of this method gives a critical region of size $\alpha + R_s(\theta)$ where $\theta$ is the unknown value of the nuisance parameter, $R_s(\theta) = O(N^{-s/2})$ and $N$ is the sample size. The first step of this method gives the region which is obtained by assuming that an estimate $\hat \theta$ of the nuisance parameter is actually equal to $\theta$.