# Cox regression different results with different combinations of variables

I need help understanding how to use cox hazzard model to calculate risk of death in data from a cohort study.

I have a set of data on physical activity and I want to do a cox regression on mortality to calculate the risk of death from active time.

I have months from t1 as time, death as status and a selection of numeric data on physical activity.

One set with minutes of low intensity activity

One set with minutes of moderate to high intensity activity

One set with minutes of sedentary time

I also a set of dichotomous data on wether or not the participants fulfilled the recommendations for daily physical.

If I run all of these together as predictor variables, I get different Exp(b) values compared to running only a selection.

Should I run each variable in separate tests and put them together in a table?

Certainly you get different results from different models, that's to be expected.

Which variables you should include is a question of model building and variable selection, which have been discussed here many times. There are various schools of thought. There are automated methods such as backwards and forwards, but these are not good and are disdained by most of the people on this list. You can use better automated methods such as LASSO.

My own view is that you should include variables not based strictly on their statistical significance, AIC values or whatever but on what makes sense for your situation.

If you include all three types of active (high, low, sedentary) then you will be looking at the relationship between each of these and mortality after controlling for the others. This answers questions such as "for given amounts of high and low intensity activity, what is the relationship between mortality and amount of sedentary activity"?

You might even want to look at interactions among these variables, since the relationship between (say) sedentary activity and mortality might be different at different levels of high intensity activity.

It's a complex process. You might want to hire a consultant to help.

Any regression model can suffer from omitted-variable bias if a predictor related to outcome is not included in the model. As noted for example on this page, the coefficients for the included predictors can then be biased away from their true values. This occurs in linear regression only when the omitted predictors are correlated with the included predictors, but in logistic or Cox regressions it occurs regardless of such correlations among the predictors.

Thus your differences in Cox regression coefficients between a combined, multiple-regression, model and the single-predictor models is to be expected. The inherent omitted-variable bias in Cox regressions means that the coefficients from the single-predictor models are unlikely to represent the predictors' true relationships with outcome.

For a Cox model the best strategy is to include as many outcome-related predictors in a multiple-regression model as is reasonable without overfitting, given the scale of your study. To avoid overfitting, a usual rule of thumb is that 10-20 events (deaths in your study) are needed for each predictor that you evaluate in the model.

As @Peter Flom notes in his answer, in your study you might need to examine interactions among your predictors; if interaction terms are related to outcome, omitting them also risks omitted-variable bias in coefficient estimates. (In terms of avoiding overfitting, each interaction term adds an effective predictor with an associated requirement for more events.) So I agree strongly with his recommendation to work with a statistical consultant. If the study is important enough to undertake in the first place, it should be important enough to analyze the results in the way most likely to provide a clean answer to your underlying questions.