Next we will return to the second question posed earlier in this lesson.

Question 2: Do the husbands and wives accurately perceive the responses of their spouses?

To understand this question, let us return to the four questions asked of each husband and wife pair. The questions were:

1. What is the level of passionate love you feel for your partner?
2. What is the level of passionate love your partner feels for you?
3. What is the level of companionate love you feel for your partner?
4. What is the level of companionate love your partner feels for you?

Notice that these questions are all paired. The odd numbered questions ask about how each person feel about their spouse, while the even numbered questions ask how each person thinks their spouse feels towards them. The question that we are investigating now asks about perception, so here we are trying to see if the husbands accurately perceive the responses of their wives and conversely to the wives accurately perceive the responses of their husbands.

In more detail, we may ask:

In this case we are asking if the wife accurately perceives the amount of passionate love her husband feels towards her.

Here, we are asking if the husband accurately perceives the amount of passionate love his wife feels towards him.

• Does the husband's answer to question 1 match the wife's answer to question 2.
• Secondly, does the wife's answer to question 1 match the husband's answer to question 2.
• Similarly, does the husband's answer to question 3 match the wife's answer to question 4, and
• Does the wife's answer to question 3 match the husband's answer to question 4.

To address the research question we need to define four new variables as follows:

• \(Z_{i1} = X_{1i1} - X_{2i2}\) - (for the \(i^{th}\) couple, the husbands response to question 1 minus the wives response to question 2.)
• \(Z_{i2} = X_{1i2} - X_{2i1}\) - (for the \(i^{th}\) couple, the husbands response to question 2 minus the wives response to question 1.)
• \(Z_{i3} = X_{1i3} - X_{2i4}\) - (for the \(i^{th}\) couple, the husbands response to question 3 minus the wives response to question 4.)
• \(Z_{i4} = X_{1i4} - X_{2i3}\) - (for the \(i^{th}\) couple, the husbands response to question 4 minus the wives response to question 3.)

These Z's can then be collected into a vector. We can then calculate the sample mean for that vector...

\(\mathbf{\bar{z}} = \dfrac{1}{n}\sum_{i=1}^{n}\mathbf{Z}_i\)

as well as the sample variance-covariance matrix...

\(\mathbf{S}_Z = \dfrac{1}{n-1}\sum_{i=1}^{n}\mathbf{(Z_i-\bar{z})(Z_i-\bar{z})'}\)

Here we will make the usual assumptions about the vector \(Z_{i}\) containing the responses for the \(i^{th}\) couple:

1. The \(Z_{i}\)'s have common mean vector \(\mu_{Z}\)
2. The \(Z_{i}\)'s have common variance-covariance matrix \(\Sigma_{Z}\)
3. Independence. The \(Z_{i}\)'s are independently sampled.
4. Multivariate Normality. The \(Z_{i}\)'s are multivariate normally distributed.

Question 2 is equivalent to testing the null hypothesis that the mean \(\mu_{Z}\) is equal to 0, against the alternative that \(\mu_{Z}\) is not equal to 0 as expressed below:

\(H_0\colon \boldsymbol{\mu}_Z = \mathbf{0}\) against \(H_a\colon \boldsymbol{\mu}_Z \ne \mathbf{0}\)

We may then carry out the Paired Hotelling's T-Square test using the usual formula with the sample mean vector z-bar replacing the mean vector y-bar from our previous example, and the sample variance-covariance matrix \(S_{Z}\) replacing the the sample variance-covariance matrix \(S_{Y}\) also from our previous example:

\(n\mathbf{\bar{z}'S^{-1}_Z\bar{z}}\)

We can then form the F-statistic as before:

\(F = \frac{n-p}{p(n-1)}T^2 \sim F_{p, n-p}\)

And, under \(H_{o} \colon \mu_{Z} = 0\) we will reject the null hypothesis \(H_{o}\) at level \(\alpha\) if this F-statistic exceeds the critical value from the F-table with p and n-p degrees of freedom evaluated at \(α\).

\(F > F_{p, n-p, \alpha}\)