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Example 7-9: Spouse Data (Paired Hotelling's)
Section* *

Download the text file containing the data here: spouse.txt

#### Using SAS

The Spouse Data may be analyzed using the SAS program as shown below:

Download the SAS Program: spouse.sas

View the video below to see how to compute the Paired Hotelling's \(T^2\) using the SAS statistical software application.

The downloadable output is given by spouse.lst.

The first page of the output just gives a list of the raw data. You can see all of the data are numbers between 1 and 5. In fact if you look at them closely, you can see that they are mostly 4's and 5's, a few 3's. I don't think we see a 1 in there are all.

You can also see the columns for d1 through d4, and you should be able to confirm that they are indeed equal to the differences between the husbands and wives responses to the four questions.

The second page of the output gives the results of the iml procedure. First, it gives the hypothesized values of the population mean under the null hypothesis. In this case, it is just a column of zeros. The sample means of the differences are given in the next column. So the mean of the differences between husband and wives response to the first question is 0.0666667. This is also copied into the table below. The differences for the next three questions follow.

Following the sample mean vector is the sample variance-covariance matrix. The diagonal elements give the sample variances for each of the questions. So, for example, the sample variance for the first question is 0.8229885, which we have rounded off to 0.8230 and copied into the table below as well. The second diagonal element gives the sample variance for second question, and so on.

The results of the Hotelling's *T*-square statistic are given at the bottom of the output page.

#### Using Minitab

View the video below to see how to compute the Paired Hotelling's \(T^2\) using the Minitab statistical software application.

#### Analysis

The sample variance-covariance matrix from the output of the SAS program:

\(\mathbf{S} =\left(\begin{array}{rrrr}0.8230 & 0.0782 & -0.0138 & -0.0598 \\ 0.0782 & 0.8092 & -0.2138 & -0.1563 \\ -0.0138 & -0.2138 & 0.5621 & 0.5103 \\ -0.0598 & -0.1563 & 0.5103 & 0.6023 \end{array}\right)\)

Sample means and variances of the differences in responses between the husbands and wives.

Question |
Mean \((\bar {y})\) |
Variance \((s_{Y}^{2})\) |

1 | 0.0667 | 0.8230 |

2 | -0.1333 | 0.8092 |

3 | -0.3000 | 0.5621 |

4 | -0.1333 | 0.6023 |

Here we have \(T^{2}\) = 13.13 with a corresponding *F*-value of 2.94, with 4 and 26 degrees of freedom. The 4 corresponds with the number of questions asked of each couple. The 26 comes from subtracting the sample size of 30 couples minus the 4 questions. The *p*-value for the test is 0.039.

The results of our test are that we can reject the null hypothesis that the mean difference between husband and wife responses is equal to zero.

### Conclusion

Husbands do not respond to the questions in the same way as their wives (\(T^{2}\)= 13.13; *F* = 2.94; *d.f.* = 4, 26; *p* = 0.0394).

This indicates that husbands respond differently on at least one of the questions from their wives. It could be one question or more than one question.

The next step is to assess on which question do the husband and wives differ in their responses. This step of the analysis will involve the computation of confidence intervals.