In the design of experiments and analysis of variance, a main effect is the effect of an independent variable on a dependent variable averaged across the levels of any other independent variables.
The term is frequently used in the context of factorial designs and regression models to distinguish main effects from interaction effects.
Relative to a factorial design, under an analysis of variance, a main effect test will test the hypotheses expected such as H0, the null hypothesis.
Running a hypothesis for a main effect will test whether there is evidence of an effect of different treatments.
However, a main effect test is nonspecific and will not allow for a localization of specific mean pairwise comparisons (simple effects).
A main effect test will merely look at whether overall there is something about a particular factor that is making a difference.
The contrast of a factor between levels over all levels of other factors is the main effect.
The difference between the marginal means of all the levels of a factor is the main effect of the response variable on that factor .
[1] Main effects are the primary independent variables or factors tested in the experiment.
[2] Main effect is the specific effect of a factor or independent variable regardless of other parameters in the experiment.
[3] In design of experiment, it is referred to as a factor but in regression analysis it is referred to as the independent variable.
In factorial designs, thus two levels each of factor A and B in a factorial design, the main effects of two factors say A and B be can be calculated.
"ab" is the represents both factors at level 2.
Finally, 1 represents when both factors are set to level 1.
[2] Consider a two-way factorial design in which factor A has 3 levels and factor B has 2 levels with only 1 replicate.
[4] The main effect for factor A can be computed with 2 degrees of freedom.
This variation is summarized by the sum of squares denoted by the term SSA.
Likewise the variation from factor B can be computed as SSB with 1 degree of freedom.
The expected value for the mean of the responses in column i is
while the expected value for the mean of the responses in row j is
SSA and SSB are main-effects sums of squares.
The two remaining degrees of freedom can be used to describe the variation that comes from the interaction between the two factors and can be denoted as SSAB.
[4] A table can show the layout of this particular design with the main effects (where
factorial design (2 levels of two factors) testing the taste ranking of fried chicken at two fast food restaurants.
Let taste testers rank the chicken from 1 to 10 (best tasting), for factor X: "spiciness" and factor Y: "crispiness."
Suppose that five people (5 replicates) tasted all four kinds of chicken and gave a ranking of 1-10 for each.
The table of hypothetical results is given here: The "Main Effect" of X (spiciness) when we are at Y1 (not crunchy) is given as:
Likewise, the "Main Effect" of X at Y2 (crunchy) is given as:
, upon which we can take the simple average of these two to determine the overall main effect of the Factor X, which results as the above formula, written here as:
For the Chicken tasting experiment, we would have the resulting main effects: