# Technical Notes - Determining the Default Design

In Variance Estimation and Precision, default designs are determined (inferred) from the data according to the following set of rules. This is done to ensure that all effects in the default design will be estimable in most (practically all) cases.

1. The program will begin the search for inclusion of effects into the default design from the highest-order interaction as specified by the user. For example, if the default value of 8 is accepted and 8 or more factors are specified for the between-group design, then the program will begin the search by determining whether a factorial 8-way interaction effect or nested effect involving 8 factors can be constructed from the data (codes) found in the data file for the respective variables.

2. An interaction (factorial, crossed) effect (e.g., 8-way, 7-way, .., 2-way interaction) will be included into the default design if all cells in the respective interaction effect are observed in the data. Put another way, if there are any missing cells found in the respective interaction effect, then that effect will not be included in the default design (as a factorial effect).

3. A nested effect Effect1(Effect2) [Effect1 nested in Effect2, where Effect1 and Effect2 can be any main effect or interaction effect, or nested effect (symbolically encoded as an interaction effect, e.g.; A(B(C)) would be coded as A(B*C))] will be included in the default design, if the following conditions are met:

1. All unique combinations of codes (values or factor levels) in Effect1 will only occur once with each unique combinations of codes (values or factor levels) in Effect2 (this condition defines nesting, including unbalanced nesting).

2. If Effect1 occurs more than once in the design following these rules, then any lower-order duplicate effect (with Effect1) will be eliminated if the Effect2 portion of this nested effect is (symbolically) contained in the higher-order nested effect. For example, if there is an effect A(B*C), then lower-order nested effects A(B) and A(C) will be omitted from the default design.

3. Since Effect1 can be any effect, even an incomplete crossed effect (interaction with missing cells), there can be effects which contain identical factors. For example, there may be two effects A*B(C) and A(B*C). In those cases, the simpler lower-order effect (e.g., A(B*C) ) will take precedence over the higher order effect (e.g., A*B(C)), which will be removed from the final design.

1.  As a consequence of these rules, the default design will be constructed of:

1. Complete factorial effects, i.e., main effects and interactions, where all cells in the effects are observed;

2. Nested effects Effect1(Effect2), where both Effect1 and Effect2 can be a complete factorial effects or nested effects.

3. In hierarchically nested designs, Effect1 will not appear in combination with any subset of Effect2; so for example, note that in a design A, B(C), C(A*B), effect C(A) and C(B) are omitted.

Note that you can fit a customized design to the data using the options on the Define Custom Design dialog, which is accessible by clicking the Customize design button on the Define/Review Model dialog.