MANOVA can be used instead of a Two Factor Repeated Measures ANOVA, especially when the sphericity assumption doesn’t hold. We illustrate the approach by repeating Example 1 of Two Factor Repeated Measures ANOVA.
Example 1 A new drug is tested on a random sample of insomniacs: 7 young people (20-40 yrs), 7 middle aged people (40-60 yrs) and 7 older people (60+ yrs). The number of minutes each person sleeps per night is recorded for 5 successive nights in Figure 1. Determine whether the drug is effective.
Figure 1 – Data for Repeated Measures Analysis
As described in Multivate One Factor Repeated Measures, we first calculate the differences between the data in the adjacent columns, as shown on the left side of Figure 2.
Figure 2 – Repeated Measures Analysis using MANOVA
Next we use the Real Statistics MANOVA data analysis tool by pressing Ctrl-m, double clicking on Analysis of Variance (or Multivariate Analyses) and selecting MANOVA from the list of options on the dialog box that appears. Finally we fill in the dialog box that now appears as shown in Figure 3 and press the OK button.
Figure 3 – MANOVA dialog box
The result is shown on the right side of Figure 2. All the tests show there is no significant difference between the mean hours of sleep for the three age groups. Interestingly enough this is the opposite conclusion from the univariate test.
Observation: It is important that none of the labels in range A6:A26 (and then H6:H26) be blank. This is acceptable for the univariate test since we had to inform the data analysis tool that there are 7 rows per group. For MANOVA the rows can appear in any order and the labels are mandatory.
Using Repeated Measures Option to MANOVA
We can also use the Repeated Measures option to the Real Statistics MANOVA data analysis tool to carry out this test directly. To perform the analysis for Example 1, as before, we press Ctrl-m, double click on Analysis of Variance and select MANOVA from the list of options on the dialog box that appears. This time when the dialog box shown in Figure 3 appears, we insert A5:F26 in the Input Range field and select Repeated Measures as the Analysis Type.
The output is similar to that shown in Figure 2.
Observation: Note that we can also select any of the various options shown in Figure 3, although in the current release of the software any Contrasts which rely on the fact that the columns are a repeated measures factor have to be performed using the Repeated Measures ANOVA data analysis tools using the input data in range H5:L26.
Observation: The approach described above using the Repeated Measures option to the MANOVA data analysis tool works provided that the input range has at least three columns of data. With only two columns of data, we use the approach described in Example 2.
Example 2: Perform the analysis from Example 1 using the data shown on the left side of Figure 4 (i.e. range A5:C26).
Figure 4 – Data for Repeated Measures Analysis with Two Dependent Variables
We begin as before by creating a table of differences, as shown on the right side of Figure 4 (i.e. range E5:F26). This can be done manually (e.g. by inserting the formula =C6-B6 in cell F6, highlighting range F6:F26 and pressing Ctrl-D) or by using the MANOVA data analysis tool and selecting Repeated Measures as before. This time the MANOVA data analysis will contain error cells, and so we will only use the range of differences (E5:F26).
To conduct the analysis we use the Single Factor ANOVA data analysis tool inserting E6:F26 in the Input Range field and selecting Standard (stacked) format as the Input Format. The output is shown in Figure 5.
Figure 5 – Analysis using Single Factor ANOVA