If the fitted hazard ratio from a Cox model is approximately 1, does that mean the covariate is not significant?

Question

Does a hazard ratio nearly 1 indicate that the covariate's effect on survival time is 0? If so would the corresponding p-value routinely indicate statistical insignificance? In other words, can a covariate both have 0 effect on survival time and be statistically significant?

Answer 1

A hazard ratio is the increase or decrease in the probability of an event. So given that out null hypotheses is no change in the hazard ratio, than a true 1 should be not significant, because there is no increased probability of experiencing the event, per unit of time, with changing values of the covariate.

You asked (emphasis added):

hazard ratio nearly 1 indicate that the covariate's effect on survival time is 0? If so would the corresponding p-value routinely indicate statistical insignificance?

With not exactly 1 however, the answer changes completely. As with all significance tests, it heavily relies on sample size. Think of it this way - in the population, even the tiniest of deviations from 0 (or 1 here) are "significant". The larger our sample size, the more confident we can be in finding smaller findings.

Whatever the effect size, if it is 1.96 times larger than the underlying standard error, it will be significant (p<0.05).

Edit: it is important to note that most statistical packages will truncate results so a ratio of 1.0004 might appear as 1, but be significant.