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I know we can explain the reasons why an independent variable is insignificant in a linear regression and therefore has no effect on the independent variable in the model but do you think we can still interpret the coefficient of this independent variable or is interpreting the coefficient useless since we can't reject the null hypothesis and therefore it has no effect on the independent variable?

Thank you in advance.

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It is not true in general that an insignificant variable has no effect on the response. A variable can be insignificant because the sample size is too low or the random variation too large to find a clear significant effect even if an effect in fact exists, or because it is correlated with other variables and the data cannot know how much of the effect of the correlated variables belongs to what individual variable. Insignificance only means that the data don't provide evidence of an effect; it doesn't mean that such an effect cannot exist.

The coefficient of an insignificant variable can in principle still be interpreted if it is appropriately expressed that any interpretation is unreliable due to random variation and that there is no conclusive evidence that the variable has any effect at all. Chances are however that in most situations the interpretation of such a coefficient is not of much interest, as it comes with too much uncertainty.

What makes more sense is to interpret a confidence interval for the coefficient, as this also expresses the uncertainty.

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    $\begingroup$ Sample size calculations can also help to interpret insignificant coefficients. With a small sample size and low power, you wouldn't expect to see a significant result even if the coefficient is truly non-zero. But with a large sample size and sufficient power, an insignificant result is more interpretable - you failed to reject the null because the data didn't support it, rather than failing to reject the null because you didn't collect enough data in the first place. Seeing no difference carries little meaning if you haven't looked hard enough, but carries more meaning if you have. $\endgroup$ Jan 20 at 15:35

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