Taken variables that did not show association with a main variable to a multivariate analysis using the Cox proportional hazards model

Question

Apologies if this is a bit basic question but examples and other questions I have seen so far do not address my problem. Here it is:

I'm trying to understand multivariate analysis for survival data and I've got the following problem. I've got a small sample size study with a main variable for which I want to do a survival analysis. I did some fisher's/chi-square tests to see whether there were associations between my main variable and other variables present on the data. There were none.

Now, I did a survival analysis (Kaplan-Meier and log-rank test) using my main variable and I got significance. I fitted also a Cox proportional hazards model and I got significance. Someone suggested to adjust my model for one of the other variables and that resulted on a significant effect. However, I've got quite small sample size in two of the subgroups (N=4 for two of the subgroups) and given that there was no association between these two variables using the fisher's exact test, my opinion would be to not to take this second variable to the model since there was no association and I've got very small sample size.

Can I do that? Should I still include that second variable? If so, how can I interpret the results all together (with the no-association found)? Can anybody suggest any book/paper/blog/link where they address this issue?

Thank you so much

Edits following the comments:

Number of total events: 28

As I mentioned: Initially I just wanted to look one variable and I was suggested to adjust for another variable, so in total, my model would have two variables.

All patients are dead at the end of the analysis. No censored data. No recurrent events.

Answer 1

With only 28 events you are just barely able to evaluate 2 predictor variables in Cox regression. The usual rule of thumb is 10-20 events per predictor that you are evaluating. A categorical variable with multiple levels counts in this as one fewer than its number of levels.

Note that the small number of cases limits your ability to find a true relationship between your main variable of interest and any other predictor. So the finding of "no association" between your main variable and a second variable might simply represent a small sample size hiding a true relationship. On that basis it might make sense to adjust for a second variable by including it in your model. If that second variable is categorical with only 2 levels, or is continuous, then your 28 events should be enough for this.

If you want to adjust for more variables (you seem to have several) you could consider using a propensity score that combines several variables into a single predictor to use in the survival model. That's a 2-step process. First, calculate the propensity score for each case; that's a measure of how all the other variables are related to the values of your main variable of interest. For a 2-category main variable of interest, a logistic regression model is often used to calculate this propensity score. Then you can use the propensity score as a single predictor, rather than all the individual predictor variables, in your survival model to help adjust for them as you evaluate the relation to survival of your main variable of interest.