Investigating Husband and Wife Data Section
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?)