Video 1. Introduction to Factorial Design of Experiment DOE and the Main Effect Calculation Explained Example.

In a Factorial Design of Experiment, all possible combinations of the levels of a factor can be studied against all possible levels of other factors. Therefore, the factorial design of experiments is also called the crossed factor design of experiments. Due to the crossed nature of the levels, the factorial design of experiments can also be called the completely randomized design (CRD) of experiments. Therefore, the proper name for the factorial design of experiments would be completely randomized factorial design of experiments.

In an easy to understand study of human comfort, two levels of the temperature factor (or independent variable), including 0^OF and 75^OF; and two levels of the humidity factor, including 0% and 35% were studied with all possible combinations (Figure 1). Therefore, the four (2X2) possible treatment combinations, and their associated responses from human subjects (experimental units) are provided in Table 1.

Table 1. Data Structure/Layout of a Factorial Design of Experiment

Coding Systems for the Factor Levels in the Factorial Design of Experiment

As the factorial design is primarily used for screening variables, only two levels are enough. Often, coding the levels as (1) low/high, (2) -/+ or (3) -1/+1 is more convenient and meaningful than the actual level of the factors, especially for the designs and analyses of the factorial experiments. These coding systems are particularly useful in developing the methods in factorial and fractional factorial design of experiments. Moreover, general formula and methods can only be developed utilizing the coding system. Coding systems are also useful in response surface methodology. Often, coded levels produce smooth, meaningful and easy to understand contour plots and response surfaces. Moreover, especially in complex designs, the coded levels such as the low- and high-level of a factor are easier to understand.

How to graphically represent the design?

An example graphical representation of a factorial design of experiment is provided in Figure 1 .

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 6 on a scale of 0 (least comfortable) to 10 (most comfortable) if the temperature increases from 0- to 75-degree Fahrenheit.

Similarly, the main effect of B is calculated by ignoring all other factors assuming we don’t have anything else other than the interested factor, which is B, the humidity factor.

Therefore, the main effect of the humidity factor can be calculated as B= (2+9)/2- (5+0)/2=5.5-2.5=3. The calculation can be seen in figure 3.

= 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.

Figure 2. Graphical representation of the main effect of the temperature (factor A).

Video 3. How to calculate Two Factors Interaction Effect in Any Design of Experiments DOE Explained Examples.

In contrast to the main effects (the independent effect of a factor), in real world, factors (variables) may interact between each other to affect the responses. For an example, the temperature and the humidity may interact with each other to affect the human comfort.

At the low humidity level (0%), the comfort increases by 5 (=5-0) if the temperature increases from 0- to 75-degree Fahrenheit. However, at the high humidity level (35%), the comfort increases by 7 (=9-2) if the temperature increases form 0- to 75-degree Fahrenheit (Figure 4). Therefore, at different levels of the humidity factor, the changes in comfort are not the same even if the temperature change is same (from 0 to 75 degree). The effect of temperature (factor A) is different across the level of the factor B (humidity). This phenomenon is called the Interaction Effect, which is expressed by AB.

The average difference or change in comfort can be calculated as AB= (7-5)/2= 2/2=1.

= the change in comfort level increases by 1 more at the high level as compared to the low level of humidity (factor B) if the temperature (factor A) increase from the low level (0-degree) to the high level (75-degree).

Similarly, the interaction effect can be calculated for the humidity factor across the level of temperature factor as follows.

At the low level of A, effect of B = 2-0 = 2; at the high level of A, the effect of B = 9-5 = 4 (Figure 5). Therefore, the average difference or change in comfort can be calculated as AB= (4-2)/2= 2/2=1.

= the change in comfort level increases by 1 more at the high level as compared to the low level of temperature (factor A) if the humidity (factor B) increase from the low level (0 %) to the high level (35%).

The interaction effect is same whether it is calculated across the level of factor A or factor B.

Figure 4. Interaction effects of the temperature (factor A) and the humidity (factor B).

Figure 6. Visualization of a strong interaction effect.

Equation 1

How to estimate the regression coefficients from the main and the interaction effects.

Using the -1/+1 coding system (Figure 8), the average comfort level increases by 6 ((9+5-2-0)/2=6) if the temperature is increased by two units (-1 to 0 and then from 0 to +1). Therefore, for one unit increase of the temperature, the comfort level is increased by 3. As the estimate for the coefficients, beta for an example is the increase of the response for every one unit increase of the factor level, the estimated regression coefficient is calculated as one-half of the respective estimated effect.

The regression coefficient is one-half of the calculated effect. The regression constant is calculated by averaging all four responses. Therefore, the regression equation can be written as in Equation.

Equation 2

Figure 9. Visualization of the contour plot for the effect of temperature and humidity on human comfort.