Is a low significant coefficient stronger evidence for a low true coefficient compared with a higher insignificant coefficient? I had a discussion with a colleague about a research project he is involved in.
We're living in a regression framework, where the colleague aims to infer about the causal effect of an independent variable X on a dependent variable Y. A model is assumed:
$Y_i = \beta_0  + \beta_1X_i + ... +\epsilon_i$ 
where the typical assumptions are made about the distributions and the error etc.
Now, the colleague's 'personal' research hypothesis is that the true coefficient $\beta_1$ should be very small or even 0. Meaning: X has little or no effect on Y.
To learn about his hypothesis, he estimates the model parameters (coefficients) by using least squares, obtaining an estimate $\hat\beta_1$.
After obtaining the estimate a hypothesis test is conducted:
$H_0: \beta_1 = 0$ and the p-value of the test statistic is observed.
This is the standard test which is displayed after conducting simple ols, by using ordinary programs like R or Stata.
Case 1: 
Now, my colleague argues that observing a very low estimate for the coefficient e.g  $\hat\beta_1$ = 0.1 in combination with a very low p-value e. g. p=0.001 (highly significant) would be strong evidence for his personal hypothesis of the true $\beta_1$ being very small or 0.
Case 2:
He also says, that the Case 1 example values would be stronger evidence in favor of his hypothesis compared with an example where the coefficient estimate would be some higher value e.g. $\hat\beta_1$ = 0.6 and the p-value would be e.g. p=0.95 (highly insignificant).
To break the argument down in one sentence:
The claim is the case 1 values offer stronger evidence in favor of the 'personal' hypothesis that the true $\beta_1$ is very small or 0 compared with the values of case 2; meaning, X has little or no effect on Y.
Now I understand the argument behind the point but I don't really agree on this because in case 1 we reject the hypothesis that the true coefficient is 0, whereas in case 2 we cannot reject the hypothesis that it is actually 0.
Naturally, the question is:
Which case of values should a researcher desire who wants to obtain evidence in favor of their personal hypothesis of the true coefficient being very small or 0? (The personal hypothesis is that X has little or no effect on Y.)
Note: On purpose, I leave the expression 'very small or 0' as vague as it is. I also accept answers which demand that it cannot be answered without making this statement more explicit. Hoping for a discussion where the statement is as vague as it is though, because I often observe this in applied empirical discussions.
 A: Really your investigator needs to reframe the problem. As stated, if they wish to show (via significance testing) there is little or no effect, they must assume the contrary. Null hypothesis significance testing controls the type one error. The type two error cannot be controlled (in a two tailed test) because the "truth" can be arbitrarily close to the null. 
However a well designed study with high power and high precision is often treated as a confirmatory study, where either null or significant findings are treated as evidence for or against an effect.
To show that findings are consistent with a range of zero to small (non clinically significant) effects, you must conduct a noninferiority test which assumes a margin of effect. Choosing the margin is discussed elsewhere in the literature. The margin value is not a statistical problem, it is chosen based on the scientific understanding of the question. 
A: Sounds to me like the hypothesis is not $H_0: \beta_1=0$ but rather that you have $H_{0c}:|\beta_1|\leq c$ for some suitably chosen "small value" $c $. This value would likely be based on the scale of $X_1$.
You then have two situations.
   - $|\hat {\beta_1}|\leq c $...you cannot reject the null (p-value is $1$ because the mle is in the null hypothesis space)
   - $|\hat {\beta_1}|> c $...you might reject the null if the gap is big enough (as the data fits into the alternative hypothesis space)
The test could be done using likelihood ratio.
This justifies your friend's view, provided $c $ is larger than $0.1$ and smaller than $0.5$.
