2.8 - Equivalent linear relationship tests

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 husband's age ("HAge") as the response and 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 wife's age ("WAge") as the response and 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?)