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 feels 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 toward her.

Here, we are asking if the husband accurately perceives the amount of passionate love his wife feels toward 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?
• 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 husband's response to question 1 minus the wife's response to question 2.)
• \(Z_{i2} = X_{1i2} - X_{2i1}\) - (for the \(i^{th}\) couple, the husband's response to question 2 minus the wife's response to question 1.)
• \(Z_{i3} = X_{1i3} - X_{2i4}\) - (for the \(i^{th}\) couple, the husband's response to question 3 minus the wife's response to question 4.)
• \(Z_{i4} = X_{1i4} - X_{2i3}\) - (for the \(i^{th}\) couple, the husband's response to question 4 minus the wife's 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 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 \(α\).

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

options ls=78;
title "Paired Hotelling's T-Square: Matching Perceptions";

 /* The differences 'd1' through 'd4' are defined
  * and added to the data set.
  */

data spouse;
  infile "D:\Statistics\STAT 505\data\spouse.csv" firstobs=2 delimiter=','
  input h1 h2 h3 h4 w1 w2 w3 w4;
  d1=h1-w2;
  d2=h2-w1;
  d3=h3-w4;
  d4=h4-w3;
  run;

proc print data=spouse;
  run;

 /* The iml code below defines and executes the 'hotel' module 
  * for calculating the one-sample Hotelling T2 test statistic. 
  * The calculations are based on the differences, which is why 
  * there is a single null vector consisting of only 0s.
  * The commands between 'start' and 'finish' define the 
  * calculations of the module for an input vector 'x'.
  * The 'use' statement makes the 'spouse' data set available, from 
  * which the difference variables are taken. The difference variables 
  * are then read into the vector 'x' before the 'hotel' module is called.
  */

proc iml;
  start hotel;
    mu0={0, 0, 0, 0};
    one=j(nrow(x),1,1);
    ident=i(nrow(x));
    ybar=x`*one/nrow(x);
    s=x`*(ident-one*one`/nrow(x))*x/(nrow(x)-1.0);
    print mu0 ybar;
    print s;
    t2=nrow(x)*(ybar-mu0)`*inv(s)*(ybar-mu0);
    f=(nrow(x)-ncol(x))*t2/ncol(x)/(nrow(x)-1);
    df1=ncol(x);
    df2=nrow(x)-ncol(x);
    p=1-probf(f,df1,df2);
    print t2 f df1 df2 p;
  finish;
  use spouse;
  read all var{d1 d2 d3 d4} into x;
  run hotel;