William P. Gardiner

A STUDY OF TYPE II AND TYPE III POWER FOR TESTING HYPOTHESES FROM UNBALANCED FACTORIAL DESIGNS

Methods for analysing unbalanced factorial designs are well documented when there is at least one observation for all treatment combinations. The Type II and Type III methods, as they have become known, are the methods of choice for hypothesis testing purposes, but there is no consensus about which is more suitable. The aim of this paper is to assess how the deterioration of the balanced structure in a given design influences Type II and Type III power, both when negligible/insignificant interactions and no interactions exist. A simulation study was set up using 726 unbalanced designs which stem from a 2 × 3 factorial design with three replicates per cell. The sampling scheme was chosen so that the interaction effect was negligible and associated with low power. A separate study investigated a 20% random sample of the unbalanced designs identified above, but fixing the interaction effect to be zero. The results from the simulation study showed that, regardless of how many observations were lost from the balanced design, the median Type II power was greater and the inter-quartile range of Type II power wider than the corresponding values for Type III power. This is an important message for practitioners, namely that the Type II method is, on average, more powerful than the Type III method but is also more influenced by cell patterning than the Type III method. There was also some evidence to suggest that up to a certain point, which is particular to the factorial design set-up, as more observations are lost the Type II method will be increasingly more powerful than the Type III method.

A study of simple unbalanced factorial designs that use type ii and type iii sums of squares

Methods for analysing unbalanced factorial designs can be traced back to the work of Frank Yates in the 1930s . Yet, still today the question on how his methods of fitting constants (Type II) and weighted squares of means (Type III) behave when negligible or insignificant interactions exist, is still unanswered. In this paper, by means of a simulation study, Type II and Type III ANOVA results are examined for all unbalanced structures originating from a 2x3 balanced factorial design within homogeneous groups (design types) accounting for structure, number of observations lost and which cells contained the missing observations. The two level factor is further analysed to test the null hypothesis, for both Type II and Type III analyses, that the unbalanced structures within each design type provide comparable F values. These results are summarised and the conclusion shows that this work agrees with statements made by Yates Burdick and Herr and Shaw and Mitchell-Olds, but there are some results which require further investigation.