Main effect

A factor averaged over all other levels of the effects of other factors is termed as main effect (also known as marginal effect). 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. The main effect of A is given by

${\displaystyle A={1 \over 2n}[ab+a-b-1]}$

The main effect of B is given by

${\displaystyle B={1 \over 2n}[ab+b-a-1]}$

Where n is total number of replicates. The letter "a" represent the factor combination of level 1 of A and level 2 of B and "b" represents the factor combination of A level 2 of A and level 1 of B. "ab" is the represents both factors at 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. There are 6 treatments with 5 degrees of freedom. in this example, we have two null hypotheses. The first for Factor A is: ${\displaystyle H_{0}:\alpha _{1}=\alpha _{2}=\alpha _{3}=0}$ and the second for Factor B is: ${\displaystyle H_{0}:\beta _{1}=\beta _{2}=0}$.[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 SS_A. Likewise the variation from factor B can be computed as SS_B with 1 degree of freedom. The expected value for the mean of the responses in column i is ${\displaystyle \mu +\beta _{j}}$while the expected value for the mean of the responses in row j is ${\displaystyle \mu +\alpha _{i}}$where i corresponds to the level of factor in factor A and j corresponds to the level of factor in factor B. ${\displaystyle \alpha _{i}}$and ${\displaystyle \beta _{j}}$are main effects. SS_A and SS_B 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 SS_AB.[4] A table can show the layout of this particular design with the main effects (where ${\displaystyle x_{ij}}$ is the observation of the ith level of factor B and the jth level of factor A):

3x2 Factorial Experiment

Factor/Levels	${\displaystyle \alpha _{1}}$	${\displaystyle \alpha _{2}}$	${\displaystyle \alpha _{3}}$
${\displaystyle \beta _{1}}$	${\displaystyle x_{11}}$	${\displaystyle x_{12}}$	${\displaystyle x_{13}}$
${\displaystyle \beta _{2}}$	${\displaystyle x_{21}}$	${\displaystyle x_{22}}$	${\displaystyle x_{23}}$

Take a ${\displaystyle 2^{2}}$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." Level X1 is for "not spicy" chicken and X2 is for "spicy" chicken. Level Y1 is for "not crispy" and level Y2 is for "crispy" chicken. Suppose that five people (5 replicates) tasted all four kinds of chicken and gave a ranking of 1-10 for each. The hypotheses of interest would be: Factor X is: ${\displaystyle H_{0}:X_{1}=X_{2}=0}$ and for Factor Y is: ${\displaystyle H_{0}:Y_{1}=Y_{2}=0}$. The table of hypothetical results is given here:

The "Main Effect" of X (spiciness) when we are at Y1 (not crunchy) is given as:

${\displaystyle {\frac {[X_{2}Y_{1}]-[X_{1}Y_{1}]}{n}}}$ where n is the number of replicates. Likewise, the "Main Effect" of X at Y2 (crunchy) is given as:

${\displaystyle {\frac {[X_{2}Y_{2}]-[X_{1}Y_{2}]}{n}}}$, 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:

${\displaystyle A=X={1 \over 2n}[ab+a-b-1]}$ = ${\displaystyle {\frac {[X_{2}Y_{2}]+[X_{2}Y_{1}]-[X_{1}Y_{2}]-[X_{1}Y_{1}]}{2n}}}$

Likewise, for Y, the overall main effect will be:[5]

${\displaystyle B=Y={1 \over 2n}[ab+b-a-1]}$= ${\displaystyle {\frac {[X_{2}Y_{2}]+[X_{1}Y_{2}]-[X_{2}Y_{1}]-[X_{1}Y_{1}]}{2n}}}$

For the Chicken tasting experiment, we would have the resulting main effects:

${\displaystyle X:{\frac {[25]-[21]+[41]-[23]}{2*5}}=2.2}$

${\displaystyle Y:{\frac {[41]-[25]+[23]-[21]}{2*5}}=1.8}$

• McBurney, D.M., White, T.L. (2004). Research Methods. CA: Wadsworth Learning.
• Mook, Douglas G. (2001). Psychological Research: The Ideas Behind the Methods. NY: W. W. Norton & Company.