The approach used in ANOVA using Regression and Unbalanced Factorial ANOVA can be extended to more than two factors. In this section we show how to use perform three factor ANOVA via regression using the Three Factor ANOVA Real Statistics data analysis tool.
Example 1: Perform the analysis for Example 1 of ANOVA with more than Two Factors using regression.
The input data from Example 1 of ANOVA with more than Two Factors is repeated in range AT3:BH11 of Figure 1.
Figure 1 – Three Factor ANOVA using regression
To perform the analysis, click on cell BJ1 (where the output will start), enter Ctrl-m and select the Three Factor ANOVA option from the menu that appears. The dialog box in Figure 2 will now appear.
Figure 2 – Dialog box for Three Factor Anova
Enter AT3:BH11 in the Input Range, click on Column headings included with data, select Std by Rows as the Input Format, select Regression as the Analysis Type and click on the OK button.
The input data is first converted to standard format by columns as shown on the right side of Figure 1 (reformatted to fit more easily in the figure). From this reformatted data the regression model is created by the Real Statistics software as described in ANOVA using Regression. The following dummy variables are employed:
t1 = 1 if Gender = Male and t1 = -1 if Gender = Female
t2 = 1 if Country = Italian and t2 = -1 if Country = Foreign
t3 = 1 if Position = Seated and t3 = -1 if Position = Prone
The variables in the model are then t1, t2, t3, t1*t2, t1*t3, t2*t3, t1*t2*t3 and y, where y represents the scores.
All this is done automatically by the software and is not displayed. The results are descriptive statistics and ANOVA analysis, which are exactly as displayed in Figure 3 of ANOVA with more than Two Factors.
Observation: If the input data had been in column format then the analysis would have proceeded exactly as described above except that no data conversion would have been necessary.
Example 2: Repeat the analysis in Example 1 of with the data in Figure 3 (unbalanced model).
Figure 3 – Unbalanced Three Factor ANOVA
To perform the analysis you repeat the steps used for Example 1. The output is displayed in Figure 4 (only the first 29 terms out of 89 in the conversion to column format are shown).
Figure 4 – Unbalanced Three Factor ANOVA
Real Statistics Function: The following array supplemental function is contained in the Real Statistics Resource Pack:
SSAnova3(R1) – returns a column array with SSA, SSB, SSC, SSAB, SSAC, SSBC, SSABC and SSW for a three factor ANOVA for the data in R1 using a regression model where the data in R1 is assumed to be in standard format by columns without column headings
Observation: The approach described in this section requires that all the interactions have at least one element in common. E.g. if one of the rows in Figure 3 contains no data, then the output from the analysis will be in error.
Observation: When the Regression option of the Three Factor ANOVA data analysis tool is chosen you are limited to 64 independent variables (i.e. the same limitation as the Linear Regression data analysis tool, as described in Multiple Regression Analysis). This means that if a = the number of levels for factor A, b = the number of levels for factor B and c = the number of levels for factor C, then abc can be at most 64.