Nested Factor Design

In nested factor design, two or more factors are not completely crossed; that is, the design does not include each possible combination of the levels of the factors. Rather, one or more factors are nested within the levels of another factor. For example, in a design in which a factor (factor B) has four levels and is nested within the two levels of a second factor (factor A), levels 1 and 2 of factor B would only occur in combination with level 1 of factor A and levels 3 and 4 of factor B would only be combined with level 2 of factor A. In other words, in a nested factor design, there are cells that are empty. In the described design, for example, no observations are made for the combination of level 1 of factor A and level 3 of factor B. When a factor B is nested under a factor A, this is denoted as B(A). In more complex designs, a factor can also be nested under combinations of other factors. A common example in which factors are nested is within treatments, for example, the evaluation of psychological treatments when therapists or treatment centers provide one treatment to more than one participant. Because each therapist or treatment center provides only one treatment, the provider or treatment center factor is nested under only one level of the treatment factor. Nested factor designs are also common in educational research in which classrooms of students are nested within classroom interventions. For example, researchers commonly assign whole classrooms to different levels of a classroom-intervention factor. Thus, each classroom, or cluster, is assigned to only one level of the intervention factor and is said to be nested under this factor. Ignoring a nested factor in the evaluation of a design can lead to consequences that are detrimental to the validity of statistical decisions. The main reason for this is that the observations within the levels of a nested factor are likely to not be independent of each other but related. The magnitude of this relationship can be expressed by a so-called intraclass correlation coefficient ρI.

The focus of this entry is on the most common nested design: the two-level nested design. This entry discusses whether nested factors are random or fixed effects and the implications of nested designs on statistical power. In addition, the criteria to determine which model to use and the consequences of ignoring nested factors are also examined.

Two-Level Nested Factor Design

The most common nested design involves two factors with a factor B nested within the levels of a second factor A. The linear structural model for this design can be given as follows:

where Yijk is the observation for the ith subject (i = 1, 2, …, n) in the jth level of factor A (j = 1, 2, …, p) and the kth level of the nested factor B (k = 1, 2, …, q), μ is the grand mean, αj is the effect for the jth treatment, βk(j) is the effect of the kth provider nested under the jth treatment, and εijk is the error of the observation (within cell variance). Note that because factors A and B are not completely crossed, the model does not include an interaction term because it cannot be estimated separately from the error term. More generally speaking, because nested factor designs have not as many cells as [Page 891]completely crossed designs, one cannot perform all tests for main effects and interactions.

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