Investigating Husband and Wife Data
It should be noted that the three hypothesis tests we have learned for testing the existence of a linear relationship — the t-test for \(H_{0} \colon \beta_{1}= 0\), the ANOVA F-test for \(H_{0} \colon \beta_{1} = 0\), and the t-test for \(H_{0} \colon \rho = 0\) — will always yield the same results. For example, when evaluating whether or not a linear relationship exists between a husband's age and his wife's age if we treat the husband's age ("HAge") as the response and the wife's age ("WAge") as the predictor, each test yields a P-value of 0.000... < 0.001 (Husband and Wife data):
Analysis of Variance
Source |
DF |
Adj SS |
Adj MS |
F-Value |
P-Value |
Regression |
1 |
20577 |
20577 |
1242.51 |
0.000 |
Error |
168 |
2782 |
17 |
|
|
Total |
169 |
23359 |
|
|
|
Model Summary
S |
R-sq |
R-sq(adj) |
R-sq(pred) |
4.06946 |
88.09% |
88.02% |
87.84% |
Coefficients
Predictor |
Coef |
SE Coef |
T-Value |
P-Value |
Constant |
3.590 |
1.159 |
3.10 |
0.002 |
WAge |
0.96670 |
0.02742 |
35.25 |
0.000 |
Regression Equation
HAge = 3.59 + 0.967 WAge
*48 rows unused
Correlation: HAge, WAge
Pearson correlation |
0.939 |
P-Value |
0.000 |
And similarly, if we treat the wife's age ("WAge") as the response and the husband's age ("HAge") as the predictor, each test yields of P-value of 0.000... < 0.001:
Analysis of Variance
Source |
DF |
Adj SS |
Adj MS |
F-Value |
P-Value |
Regression |
1 |
19396 |
19396 |
1242.51 |
0.000 |
Error |
168 |
2623 |
16 |
|
|
Total |
169 |
22019 |
|
|
|
Model Summary
S |
R-sq |
R-sq(adj) |
3.951 |
88.1% |
88.0% |
Coefficients
Predictor |
Coef |
SE Coef |
T-Value |
P-Value |
Constant |
1.574 |
1.150 |
1.37 |
0.173 |
HAge |
0.91124 |
0.02585 |
35.25 |
0.000 |
Regression Equation
WAge = 1.57 + 0.911 HAge
*48 rows unused
Correlation: WAge, HAge
Pearson Correlation |
0.939 |
P-Value |
0.000 |
Technically, then, it doesn't matter what test you use to obtain the P-value. You will always get the same P-value. But, you should report the results of the test that make sense for your particular situation:
- If one of the variables can be clearly identified as the response, report that you conducted a t-test or F-test results for testing \(H_{0} \colon \beta_{1} =0\). (Does it make sense to use x to predict y?)
- If it is not obvious which variable is the response, report that you conducted a t-test for testing \(H_{0} \colon \rho = 0\). (Does it only make sense to look for an association between x and y?)