Dropping insignificant interactions I have two general questions regarding interactions in a Cox regression.

*

*I have a hypothesis that marker X has different effect in groups G. The interaction self and the marker are insignificant. When the interaction is dropped, the effect of a marker is significant. How legitimate is it in statistics in general to drop the interaction, just to show that the marker is overall a significant predictor of an outcome?


*Study participants are grouped according to other variables into groups (G). For example, when A or B is true, they are group 1, C or D they are group 2 and E, F group 3.
In the model I've left out the variables A-F. When the interaction with G is dropped, the marker overall is significant, however there is strong evidence that marker has different effect depending on A. Can one include both interactions in a model (with A and G)? Or is it advised to build a completely new model with A-F and without Group variable?

My sense is that you started out by trying to fit too complex a model for the data that you have. Frank Harrell has extensive online notes to accompany his Regression Modeling Strategies text, which provide valuable guidance for how to build regression models (including survival models) properly.
For a survival model you typically need about 15 events (not just participants) per coefficient you are estimating, to fit a model reasonably without overfitting. Furthermore, a large number of coefficients necessarily reduces the power to find any of them to be significant.
If your model has a single marker X, 3 groups in G, and the X-G interaction, you are fitting at least 5 coefficients.* Unless you had on the order of 75 events you probably should not have even attempted such a complex model (without penalization). Without the interaction you would be down to only 3 coefficients.
An alternate approach, if you don't care about the association of G with outcome except insofar as the associations of X with outcome might depend on the group, would be to set up a Cox model stratified by group and an interaction of X with the strata. Then you are down again to only 3 coefficients to evaluate your "hypothesis that marker X has different effect in groups G."
There's also a question about how you are evaluating whether "the effect of a marker is significant." In a model with an interaction, you can't evaluate that significance solely from the individual coefficient for X (as I often see attempted in questions on this site). That's only for the association of X with outcome in the reference category of G.
To evaluate the  association of X with outcome in the presence of interactions, you need to evaluate all of the coefficients involving it together, as in the "Type II" Wald tests provided by the Anova() function in the car package or the anova() function specific to objects in Harrell's rms package (not the basic anova() function in R).
Assuming that the model with interactions was significantly different from a null model and you can demonstrate (e.g., by bootstrapping) that the model wasn't overfit, you have identified some combination of predictors associated with outcome. Lack of significance of particular predictors means that you aren't yet able to tell exactly which specific combinations. What you might do in that case is to present the results as you found them for the two models sequentially, but to acknowledge that the p-values in the model without interactions, which you developed after seeing the initial results, aren't valid--as the answer from @Dave (+1) rightly points out. That will allow your work to serve as a guide to others (and you) for later exploration.

*In general, I'm skeptical of survival models that omit covariates that are known to be associated with outcome, so these counts of coefficients are ideally underestimates, as are the numbers of events needed. There's a big risk that what you "discover" about X or the groups is just that they are associated with some other well-known outcome-associated factor that you left out of the model.
