Nested Designs and Latin Squares - Simple Between-Groups Nested Designs

In some studies it is not feasible to implement a complete factorial design, i.e., a design where each level of each factor co-occurs with each level of all others (the term factorial design was first introduced by Fisher, 1935a). For example, suppose you want to evaluate the effects of 4 different fertilizers (between-groups factor 1 with 4 levels) on the growth of corn. Imagine that the researcher used two fertilizers on each one of two different fields (between-groups factor 2 with 2 levels). The resulting design would be a 4 (Fertilizer) by 2 (Field) design; however, because only two levels of the first factor occur within each of the two levels of the second factor, the design is actually a 4 (nested within factor 2) by 2 design.

In general, designs are nested (the term was first used by Ganguli, 1941) when only some levels of a factor occur within the levels of another factor. In a sense, the levels of one factor are "placed" within the levels of the other factor like "eggs into a nest," hence the name "nested" design.

To return to the example, the data for this design can be entered as follows:

Fertilizer Field Growth
 (dependent variable)
1 1 24
1 1 34
2 1 25
2 1 28
3 2 45
3 2 42
4 2 33
4 2 31

Let us assume that the researcher took two samples of corn from each field and measured their size (on some arbitrary scale). Each row in the data set represents the data for one of those samples. The first variable (column) contains code numbers that uniquely identify what fertilizer was used for the respective corn sample; therefore, this is an independent or grouping variable. The second variable (columns) contains codes that uniquely identify the field from which the respective corn sample was taken; thus, this is the second independent or grouping variable.

Note: levels 1 and 2 of the first between-groups factor (Fertilizer) only appear within the first level (i.e., within level 1) of the second factor (Field), whereas levels 3 and 4 of the first factor only appear within level 2 of the second factor. Therefore, the first factor (Fertilizer) is nested within the second factor (Field).
Specifying the design
Note that nested designs must be analyzed via the General Linear Models (GLM) module. In order to specify this design, select Nested design ANOVA as the Type of analysis and Quick specs dialog as the Specification method in the GLM Startup Panel Quick tab and then click the OK button on the GLM (Startup Panel). Next click the Variables button and select Variable 3 in the Dependent variable list field and Variables 1 and 2 in the Categorical predictors. When specifying the code values (via the Factor codes button) that were used to indicate the four levels of the first between-groups factor, enter 1-4, i.e., specify the four values that were actually used in the data file.

Next, click the Between effects button to display the Nesting for Between-Group Factors dialog. For each factor, specify (1) whether or not the factor is nested, (2) in which other factors the respective factor is nested.

Testing main effects and interactions
Because of nesting, interactions of the nested factor with the factor(s) in which it is nested cannot be evaluated. Thus, in the present example you cannot test the hypothesis that the type of fertilizer and the particular field interact in their effect on the size of corn: It is simply impossible to determine, based on the current study, to what size the corn would have grown on field 1 (i.e., level 1 of the Field factor) had the third and fourth fertilizer (i.e., level 3 and level 4 of the Fertilizer factor) been used.
Very large designs, random effects, unbalanced nesting
Note that there are other modules of Statistica that can also analyze these types of designs. In particular the Variance Components and Mixed Model ANOVA/ANCOVA module is very efficient for analyzing designs with unbalanced nesting (when the nested factors have different numbers of levels within the levels of the factors in which they are nested), very large nested designs (e.g., with more than 200 levels overall), or hierarchically nested designs (with or without random factors).