What are the main effects in a factorial design?

Video 2. Introduction to Factorial Design of Experiment DOE and the Main Effect Calculation Explained Example.

The average effect of the factor A (called the main effect of A) can be calculated from the average responses at the high level of A minus the average responses at the low level of A (Figure 2). When the main effect of A is calculated, all other factors are ignored assuming that we don’t have anything else other than the interested factor, which is A, the temperature factor.

Therefore, the main effect of the temperature factor can be calculated as A = (9+5)/2 - (2+0)/2 = 7-1 = 6. The calculation can be seen in figure 2.

= the average comfort increases by 3 on a scale of 0 (least comfortable) to 10 (most comfortable) if the relative humidity increases from 0 to 35 percent.

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. In other words, it is a test examining differences amongst the levels of a single factor (averaging over the other factor and/or factors). Main effects are essentially the overall effect of a factor.

Definition[edit]

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 . Main effects are the primary independent variables or factors tested in the experiment. Main effect is the specific effect of a factor or independent variable regardless of other parameters in the experiment. In design of experiment, it is referred to as a factor but in regression analysis it is referred to as the independent variable.

Estimating Main Effects[edit]

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

A=12n[ab+a−b−1]{\displaystyle A={1 \over 2n}[ab+a-b-1]}

The main effect of B is given by

B=12n[ab+b−a−1]{\displaystyle B={1 \over 2n}[ab+b-a-1]}

Where n is total number of replicates. We use factor level 1 to denote the low level, and level 2 to denote the high level. The letter "a" represent the factor combination of level 2 of A and level 1 of B and "b" represents the factor combination of level 1 of A and level 2 of B. "ab" is the represents both factors at level 2. Finally, 1 represents when both factors are set to level 1.

Once again examining simple effects provides a means of breaking down the interaction and therefore it is only necessary to conduct these analyses when an interaction is present. When there is no interaction then the main effects will tell the complete and accurate story. To summarize, rather than averaging across the levels of the other independent variable, as is done in a main effects analysis, simple effects analyses are used to examine the effects of each independent variable at each level of the other independent variable(s). So a researcher using a 2×2 design with four conditions would need to look at 2 main effects and 4 simple effects. A researcher using a 2×3 design with six conditions would need to look at 2 main effects and 5 simple effects, while a researcher using a 3×3 design with nine conditions would need to look at 2 main effects and 6 simple effects. As you can see, while the number of main effects depends simply on the number of independent variables included (one main effect can be explored for each independent variable), the number of simple effects analyses depends on the number of levels of the independent variables (because a separate analysis of each independent variable  is conducted at each level of the other independent variable).

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