Main Effect

The term main effect typically is used with reference to FACTORIAL DESIGNS in ANALYSIS OF VARIANCE. Consider a factorial design in which the levels of one factor, A, are crossed with the levels of another factor, B, as follows in Table 1.

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Table 1
			Factor B
		Level B 1	Level B 2	Level B 3	Marginal Mean
Factor A	Level A1	MA1B1	MA1B2	MA1B3	MA1
	Level A2	MA2B1	MA2B2	MA2B3	MA2
	Marginal mean	MB1	MB2	MB2

The entries in each cell are means and the marginal entries are means for a given level of a factor collapsing across the levels of the other factor. For example, MA1 is the mean for Level 1 of Factor A collapsing across the levels of Factor B.

Informally, the term main effect refers to properties of the marginal means for a factor. Researchers use the term in different ways. Some will state that “there is a main effect of Factor A,” which implies that the marginal means for the factor are not all equal to one another in a POPULATION, a conclusion that is based on the results of a formal statistical test. Others simply refer to the “means comprising the main effect of Factor A,” drawing attention to the marginal means but without any implication as to their equality or the results of a statistical test that is applied to them.

There are different methods for calculating marginal means. One method uses unweighted means, whereby the marginal mean is the average of each cell mean that is being collapsed across. For example, MA1 is defined as (MA1B1 + MA1B2 + MA1B3)/3. A second method is based on weighted means, whereby each of the individual cell means that factor into the computation of the marginal mean is weighted in proportion to the SAMPLE SIZE on which it is based. Means based on larger sample sizes have more weight in determining the marginal mean.

The term main effect also is used in linear regression models. A “main effect” model is one that does not include any INTERACTION terms. For example, a “main effect” MULTIPLE REGRESSION model is

whereas an “interaction model” is

The presence of the product term in the second equation makes this an interaction model. In regression contexts, a main effect refers to any continuous PREDICTOR in a REGRESSION model that is not part of a product term or an interaction term. A main effect also refers to any DUMMY VARIABLE or set of dummy variables that is not part of a product term or interaction term. In the following model of all continuous predictors,

The coefficient for Q might be referred to as representing a “main effect” of Q, but the coefficients for X, Z, and XZ would not be so characterized.

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