Excel doesn’t provide tools for ANOVA with more than two factors. On this webpage we show how to construct such tools by extending the analysis provided in the previous sections. See Real Statistics Support for Three Factor ANOVA for how perform the same sort of analysis using the Real Statistics Three Factor ANOVA data analysis tool.
Alternatively, see ANOVA using Regression for how to perform ANOVA with any number of factors using regression (either using the standard Excel Regression data analysis tool or the Real Statistics Linear Regression data analysis tool).
We begin by extending the definitions from Two Factor ANOVA with Replication to three factors.
Definition 1: Using the terminology of Definition 1 of Two Factor ANOVA with Replication (where we use a and b instead of r and c), define
As before, we can also define the between groups terms. This time there are four types:
And similarly for SSBetAC, dfBetAC, SSBetBC, dfBetBC,. There is also the following version:
As usual, we can define the error terms:
Proof: The proof is similar to the proof of Property 1 of Two Factor ANOVA with Replication.
Similar properties hold for the between AC and BC terms. We also have the following properties about the between ABC term:
Property 3: If a sample is made as described in Definition 1 with the xijkl independently and normally distributed and with all (or ) equal, then
If all μi are equal and all are equal then
If all μj are equal and all are equal then
Also, under certain circumstances,
Proof: The result follows from Property 1 and Theorem 1 of F Distribution.
Here terms like (βγ)jk2 are not a product of a beta and gamma term squared. Instead such as term refers to a single parameter named (βγ)jk which is squared.
Observation: We use the following tests:
- All samples are drawn from normally distributed populations
- All populations have a common variance
- All samples are drawn independently from each other
- Within each sample, the observations are sampled randomly and independently of each other
We now show how to conduct the above tests in Excel. Excel doesn’t have a three factor ANOVA data analysis tool, and so we will need to carry out the analysis using Excel formulas.
Example 1: An Italian research psychologist decides to conduct an experiment to understand the ability of subjects to perform simple tasks when instructed in Italian. She creates 8 sample groups, each with 12 subjects. The three factors are: gender (male/female) of the subject, nationality of the subject (Italian/foreign) and whether the subject performs the task seated or lying down (Seated/Prone). Each participant is given a test to measure their ability to perform the required tasks, with the scores recorded in the table shown in Figure 16.19. The psychologist wants to know if there are any significant differences between the groups.
Figure 1 – Data for Example 1
We begin by constructing the tables shown in Figure 2.
Figure 2 – Counts and means for Example 1
Now using the information in Figure 1 and 2, we can construct the ANOVA analysis as shown in Figure 3.
Figure 3 – ANOVA for Example 1
We conclude there are significant differences between the nationalities, but no significant differences between the genders or positions. There are also significant interactions between gender and position as well as between all the factors.