Publication date: 08/13/2020

## Construction of Hierarchical Terms

Hierarchical terms are constructed using a tree structure that is analogous to a Partition analysis. However, the criterion that is maximized is the sum of squares between groups (SSB).

For a nominal variable with k levels, the k levels are split into two groups of levels that have maximum SSB. Call these two groups of levels A1 and A2, where A1 has the smaller mean and A2 has the larger mean. The two groups of levels in A1 and A2 are used to define an indicator variable with values of 1 for the levels in A1 and -1 for the levels in A2. This variable is the first hierarchical term for the nominal variable.

For the levels within each of the initial two groups A1 and A2, the split into two groups of levels with the maximum SSB is identified. Suppose that the groups of levels with maximum SSB are among the levels in A1. Call the two groups B1 and B2, where B1 has the smaller mean and B2 has the larger mean. The two groups of levels in B1 and B2 are used to define a hierarchical variable with values of 1 for the levels in B1, -1 for the levels in B2, and 0 for the levels in A2. To construct the next variable, splits of the levels in B1, B2, and A2 are considered. The split that maximizes SSB defines the next hierarchical variable. The process continues until k-1 hierarchical terms are constructed.

For an ordinal variable, the groups of levels considered in splitting contain only levels that are contiguous in the ordering. This ensures that the constructed terms respect the level ordering.

#### Rules and Hierarchical Terms

When you use the Combine rule or the Restrict rule, a term cannot enter the model unless all the terms above it in the hierarchy have been entered. When you use the Whole Effects rule and enter a term for a categorical variable, all of its associated terms are entered. For an example, see Construction of Hierarchical Terms in Example.