We show how to conduct Three Factor ANOVA using capabilities found in the Real Statistics Resource Pack.

**Real Statistics Data Analysis Tool**:** **The Real Statistics Resource Pack provides the **Three Factor ANOVA** data analysis tool. Two types of input formats are supported. The first format, which we will call **standard format by rows** is like the input in Example 1 of ANOVA with more than Two Factors.

We will start by looking at this format. We will look at the other format, **standard format by columns** shortly.

**Example 1**: Repeat the analysis conducted in Example 1 of ANOVA with more than Two Factors using the Real Statistics’s Three Factor ANOVA data analysis tool.

To use the tool for the analysis of Example 1, click on cell Q1 (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 1 will now appear.

**Figure 1 – Dialog box for Three Factor Anova**

Enter A4:O11 in the **Input Range**, click on **Column headings included with data**, select **Std by Rows **as the **Input Format** and select **Anova** as the **Analysis Type** and click on the **OK** button. The output is shown in Figure 2.

**Figure 2 – Three Factor ANOVA data analysis**

The left side of the figure contains a table of descriptive statistics. Cell T30 contains the total sample size, cell U30 contains the total mean and cell U31 contains *SS _{T}*. The right side of the figure contains the Anova table, which is similar to the analysis shown in Figure 3 of ANOVA with more than Two Factors..

**Observation**: As mentioned previously, two input formats are supported by the data analysis tool. In addition to the input format shown above, the data analysis tool also supports standard format by columns. This is similar to the stacked format used for Two Factor Anova.

**Example 2**: Perform an analysis of variance for the data in range A3:D38 of Figure 3.

**Figure 3 – Three Factor ANOVA on data in stacked format**

To do this, enter **Ctrl-m** and select the **Three Factor ANOVA **option** **from the menu that appears. When the dialog box in Figure 1 appears, enter A3:D38 in the **Input Range**, unclick **Column headings included with data**, select **Std by Columns **as the **Input Format**, select **ANOVA** as the **Analysis Type** and click on the **OK** button. The output is shown in Figure 3.

The output is similar to that obtained when the standard by rows input format is used and includes descriptive statistics tables and the ANOVA results. In addition, you have the option to convert the input data into the standard by rows format (in sorted order). To do this, also check **Convert data in input range to Rows format** in the dialog box shown in Figure 1. The result is displayed in Figure 4.

**Figure 4 – Converting from by column to by row format**

**Real Statistics Functions**: The Real Statistics Resource Pack contains the following supplemental array functions for converting between the two types of Three Factor Anova formats.

**Anova3Rows**(R1): takes the data in R1 which is in standard format by columns and outputs an array with the same data in standard format by rows.

**Anova3Cols**(R1): takes the data in R1 which is in standard format by rows and outputs an array with the same data in standard format by columns.

**Observation**: Referring to Figure 3 and 4, note that =Anova3Cols(A3:D38) generates the output in range T3:V14 and =Anova3Row(T3:V14) generates output similar to that in range A3:D38, but in sorted order.

**Observation**: In the all examples given the group samples have the same number of elements (**balanced model**). In Three Factor ANOVA using Regression we show how to use the **Regression** option of the **Three Factor Anova** data analysis tool to analyze unbalanced models.

Also when the **Regression** option is used for input data in standard format by rows the data is automatically converted to standard format by columns (see Three Factor ANOVA using Regression).

Dear Charles,

I have used you 2 way ANOVA follow-up for my data before and it was really useful. Now I have extended my experiments and I need to do post-hoc tests on my 3 way ANOVA (2 factors with 2 levels and one with 3). I was wondering if I could do the 2 way Anova follow up 7 times to account for all the intereactions or if I would need a specific 3 way ANOVA follow up to account for other factors I don’t know about? (my background is not stats…)

Thanks!

Dear Dr. Zaiontz,

Can I use 2 factors that are discrete and one is not discrete?

Boris,

Yes, but you still need to take the assumptions of normality and homogeneity of variances into account.

Charles

Dear Charles

I am having 3 factors at 3 levels with 3 replications with one Control. 28 treatments. How to analyse please guide me

This situation is described on the referenced webpage. What parts of it are you having trouble with?

Charles

I running a 3 factor (say time, temp and brand) with 2 levels for each factor test. I have 1 result for each test, for example time=10mins, temp=180degC and brand=1 the result is 23units. When I run a 3 factor ANOVA test the ‘within’ column is 0 (zero) and hence the F value cannot be calculated. What am I doing wrong?

If I have 2 results for each test then it works.

Nick,

This version of the test is with replication and so requires at least 2 values per interaction.

Charles

Hey Charles,

I’ve been running 3-way ANOVA’s on my data for a while now but only noticed something curious today. It may not even be an issue, but I thought I would bring it up in the event that it is.

When looking at the ANOVA table output for the 3-way ANOVA, the p-value generated for the A*C interaction uses the degrees of freedom (df) for A*B instead of its own df. I noticed this because I ran some numbers through Minitab and sometimes found that your program and Minitab would give different results. However, when I changed the A*C p-value formula to use the df of A*C the p-value matched the p-value given by Minitab perfectly.

I was wondering if there was any reason for using the df of A*B instead of A*C when calculating the p-value for A*C?

Thanks,

Aaron 🙂

Hi Aaron,

You are correct. There is a bug in the software, which I will correct shortly.

Thank you very much for finding this error. I apologize to you and everyone else for not identifying this error previously.

Charles

Hi Aaron,

I have now fixed the bug in the software. The corrected version is now available for download (Rel 4.4.3).

Thanks again for your help.

Charles

Is it possible to use this factorial design with 8 different factors?

The factorial design described on the referenced webpage is limited to 3 factors. The same approach will work for 8 factors, but the actual design will require considerable modification. In any case, I am not anyone would be able to analyze an Anova model with 8 factors; it would be too complicated.

Charles