Interpretation of Interaction when the results in one of the two groups is not statistically significant Here is the question. Suppose we have a drug A, and we have conducted a RCT evaluating the risk of death in patients randomised to receive A or not. The results of a Cox-regression model shows HR: 0.60 (95%CI: 0.50-0.70) (please don't focus on the numbers, this is only for demonstrative purposes).
Then we want to explore whether there is an interaction between sex and the effect of A. Females were approximately 35% of the cohort.
We therefore perform a Cox-Regression including an interaction term:
outcome~(Sex*A)+A+Sex
and the results are the following:
                   coef exp(coef) p
A               -0.6733    0.5100 0.0032
sexFemale       -0.2543    0.7754 0.2834
A:sexFemale      0.2689    1.3085 0.4503

Manipulating the coefficients, we can have:
A*sex=Female: HR: 0.67 (95%CI: 0.40-1.20)
A*sex=Male: HR: 0.51 (95%CI: 0.43-0.82) [again, don't focus on the number]
interaction p: 0.450
Now, here's the question, related to the interpretation of the results presented: I think one can conclude that no statistically significant interaction was observed between males and females for the effect of A (with effect being broadly consistent?). However, how can one interpret the non-statistically significant results observed for females, while the effect remained statistically significant in males?
My guess is that one should not speculate onto this, given that this is mainly due to the reduced sample size in females group, but I'd like to hear more expert opinions.
 A: 
how can one interpret the non-statistically significant results observed for females, while the effect remained statistically significant in males?

Fundamentally, the tests on the interaction coefficient and on the estimated (combinations of) coefficients for the effects of intervention A on each sex separately are evaluating different things. Then you add on evaluating "significance" based on a fixed p-value cutoff and you have a recipe for confusion. I'll focus on the (combinations of) coefficients here, as those are the primary result of the regression and they have an asymptotic multivariate normal distribution.
The test on the individual coefficient for A, representing its association with outcome in males, is whether you can distinguish its value from 0 at p < 0.05. In this case, it can.
The test on the A:sexFemale interaction coefficient is effectively whether you can distinguish the effect of A between males and females at p < 0.05. In this case, you can't.
A plot of the probability distribution of coefficient estimates for males and females, based on the values that you provided, shows what's going on. There's a lot of overlap in the estimates for the 2 sexes, but only that for males is sufficiently far from the value of 0 to qualify as "significant."

The above is based on: the coefficient for A of -0.6733 with a standard error of 0.2284 estimated from the p-value, and the coefficient combination providing the effect of A on females (log of 0.67 = -0.4005) with a standard error of 0.28 estimated from the reported 95% CI.
Although this type of thing might sometimes result from differences in sample sizes, that's not required. Sample sizes primarily affect the widths of the distributions. The coefficient mean-value estimates also matter. If the two curves overlap like this but only one is far enough away from 0 to be "significant," then you will get the same phenomenon with equal sample sizes.
A: Interaction terms are tricky! But there is nothing out of the ordinary here.
Note: I'm assuming that the variables "A" and "female" are both coded 1/0 (where 0 means "didn't get the treatment" or "male") If not then you need to recode them that way or the interpretation  of the main effects  in the interanion model will be all messed up.
But assuming they are coded that way...here is what your results mean.
The coefficient for "A" tells you the estimated effect of getting the treatment (vs not getting it) FOR MALES (that is for people who's "sexFemale" value is ZERO)
The coefficient for "sexFemale" tells you the estimated effect of BEING FEMALE (vs male) for people who did NOT get the treatment (i.e. for people who's "A" value is ZERO).
The coefficient for the interaction term tells you whether the estimated effect of A is significantly different for females, compared to the effect for males (which is what the first coefficient showed you).
This means that [purely looking in terms of significance at the 95% level):
-MEN who got the treatment have a significantly lower hazard than men who did not (p<.01)
-Among the control group, being female had no impact on the hazard rate.
-The effect of the treatment for females is not significantly different from the effect estimated for males.
In short: The treatment seems to work, and it seems to work basically the same way for males and females. Also gender itself doesn't seem to impact the hazard of dying in this population.
