Sometimes in factorial experiments with three or more factors involving a random or mixed model, we determine that there is no exact test statistic for certain effects in the model. Satterthwaite (1946) proposed a test procedure which uses the **linear combinations** of the original mean squares to form the F-ratio. These linear combinations of the original mean squares are sometimes called “synthetic” mean squares. Details on how to build the test statistic and adjustments made to degrees of freedom based on Satterthwaite procedure can be found in Section 13.6.

Minitab will analyze these experiments and derive “synthetic” mean squares, although their “synthetic” mean squares are not always the best choice. Approximate tests based on large samples (which use modified versions of the Central Limit Theorem) are also available. Unfortunately, this is another case in which it is not clear that there is a best method.