It should be noted that the three hypothesis tests we have learned for testing the existence of a linear relationship — the ttest for \(H_{0} \colon \beta_{1}= 0\), the ANOVA Ftest for \(H_{0} \colon \beta_{1} = 0\), and the ttest 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 husband's age ("HAge") as the response and wife's age ("WAge") as the predictor, each test yields a Pvalue of 0.000... < 0.001 (Husband and Wife data):
Analysis of Variance
Source 
DF 
Adj SS 
Adj MS 
FValue 
PValue 
Regression 
1 
20577 
20577 
1242.51 
0.000 
Error 
168 
2782 
17 


Total 
169 
23359 



Model Summary
S 
Rsq 
Rsq(adj) 
Rsq(pred) 
4.06946 
88.09% 
88.02% 
87.84% 
Coefficients
Predictor 
Coef 
SE Coef 
TValue 
PValue 
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 
PValue 
0.000 
And similarly, if we treat wife's age ("WAge") as the response and husband's age ("HAge") as the predictor, each test yields of Pvalue of 0.000... < 0.001:
Analysis of Variance
Source 
DF 
Adj SS 
Adj MS 
FValue 
PValue 
Regression 
1 
19396 
19396 
1242.51 
0.000 
Error 
168 
2623 
16 


Total 
169 
22019 



Model Summary
S 
Rsq 
Rsq(adj) 
3.951 
88.1% 
88.0% 
Coefficients
Predictor 
Coef 
SE Coef 
TValue 
PValue 
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 
PValue 
0.000 
Technically, then, it doesn't matter what test you use to obtain the Pvalue. You will always get the same Pvalue. 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 ttest or Ftest 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 ttest for testing \(H_{0} \colon \rho = 0\). (Does it only make sense to look for an association between x and y?)