Can a coefficient technically operate as a test statistic?

#1
This is somewhat pedantic, but the debate I've had is whether a regression coefficient can technically operate as a test statistic. It's not being claimed that this is the normal setup or a better way to do anything, just a theoretical question.

My claim is that it can. Normally, you divide the coefficient by the standard error and compare to a t distribution: b/se ~ t

Instead, you can just compare the coefficient to a scaled t distribution: b ~ t(se)

Thus, for your test, you compare the coefficient to a calculable distribution under the null and it should be exactly equivalent to the normal setup. Therefore, the coefficient *can* operate as a test statistic. Correct logic?
 

hlsmith

Not a robit
#2
Coefficient is used in the calculation. Instead of using test statistic just use the coefficient with its respective confidence interval
 
#3
Coefficient is used in the calculation. Instead of using test statistic just use the coefficient with its respective confidence interval
Yes, that's substantively identical to what I'm describing. The question is when you take your estimated coefficient, then compare its size to a t distribution scaled by the standard error, is the coefficient technically operating as a test statistic? I say yes since a test statistic is any statistic that you test by comparing it to a distribution under the null hypothesis.
 

hlsmith

Not a robit
#4
Not sure what you are looking for beyond the above information. Presenting an estimate with confidence intervals based on the same information used in null hypothesis test will convey the same information, but actually more information in my opinion. Since a p-value won't tell the direction or magnitude of the estimate and are that palatable to the person in the field hoping to use the study results.
 
#5
Not sure what you are looking for beyond the above information. Presenting an estimate with confidence intervals based on the same information used in null hypothesis test will convey the same information, but actually more information in my opinion. Since a p-value won't tell the direction or magnitude of the estimate and are that palatable to the person in the field hoping to use the study results.
My question is simply whether it can make sense to describe a coefficient as a test statistic if you set up a test as I've described, namely comparing the coefficient to a t distribution scaled by the standard error. Not about presentation or anything, just a technical question about what can count as a "test statistic."