Analyzing Partially Nested Designs with Irregular Nesting: A Cautionary Case Study

Abstract

Models with partially nested fixed effect structures arise when two‐way structures include a factor that can be partitioned according to a nested structure. In such cases, it is likely that the nesting will have an irregular structure with unequal numbers of nested factor levels among nesting factor levels. If a priori hypotheses correspond to the nested structure, these might be tested using the two‐way model and writing contrast statements. Alternatively, a more complex partially nested model might be used in an attempt to obtain the desired tests via model respecification. Comparing analyses based on the two‐way model and on the partially nested model established that the partially nested model correctly partitions sums of squares for the nested structure but that Type III non‐nested factor main effect hypotheses and sums of squares differed. Additionally non‐nested factor least squares means differed between the two models, and the partially nested model Type III non‐nested factor main effect hypothesis coefficients did not correspond to a comparison of the least squares means from either model. For the equal replications case, Type I hypotheses from the partially nested model produced the desired analysis but Type III hypotheses did not. For the unequal replications case, researchers might avoid writing contrast statements by running both models and selecting appropriate Type III tests and estimates from each analysis.