Consider, for the moment, a balanced fixed two way layout of the analysis of variance assuming interactions and replications. Suppose a hypothesis of interest is that the row effects are all equal. A test statistic for such a hypothesis is the ratio of the mean square for rows (MSR) divided by the mean square for error (MSE). Another test procedure frequently used in practice is as follows: Test whether the interaction effects are zero. If it is decided that these effects are not zero, then test for the equality of row effects by MSR/MSE. If it is decided that the interaction effects are zero, then test for the equality of row effects by MSR divided by the pooled mean square error. That is, the pooled mean square error consists of the sum of squares for error plus the sum of squares for interaction divided by the sum of degrees of freedom for error and interaction. This latter type of procedure is called a "sometimes pooling" procedure. For a more general description and discussion of such procedures see Bozivich, Bancroft, and Hartley (1956). In this note we consider the general linear hypothesis model. We prove, in this general framework, that the "sometimes pooling" procedure is an admissible test procedure. The proof follows from a well known invariance result and a theorem of Matthes and Truax (1967). The "sometimes pooling" procedure can be viewed as a test procedure which depends on the outcome of a preliminary test. It is interesting to note that estimation procedures which depend on a preliminary test were found to be inadmissible for the squared error loss function. (See Cohen (1965).) In the next section we state the model and prove the admissibility result.
Arthur Cohen. "A Note on the Admissibility of Pooling in the Analysis of Variance." Ann. Math. Statist. 39 (5) 1744 - 1746, October, 1968. https://doi.org/10.1214/aoms/1177698157