I have fitted a Cox PH model with three covariates, age (continuous), risk group (factor) and treatment (factor). The variable of interest is the treatment, which is statistically significant but the other two covariates have large p-values (>0.05). However, the p-values of the model for the Likelihood Ratio and Wald tests are <0.05. Based on plots, the assumption of the PH is met. So my question is, how do I decide whether I should include the non statitsically significant covariates in the model in this case?
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$\begingroup$ Which test did give you the p-values that are greater than 0.05? $\endgroup$– Márcio Augusto DinizCommented Dec 31, 2021 at 2:41
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$\begingroup$ Wald test. It is an output from R. $\endgroup$– Kaτ_τCommented Dec 31, 2021 at 6:13
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$\begingroup$ If the p-values that are greater than 0.05 are from Wald test, then your statement: "However, the p-values of the model for the Likelihood Ratio and Wald tests are <0.05." is confusing. Could you clarify it? $\endgroup$– Márcio Augusto DinizCommented Dec 31, 2021 at 8:18
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$\begingroup$ I have an outrput from R that shows that the coefficients of some of the covariates are not statistcially significant. The Likelihood Ratio Test and Wald Test for the model are statustically significant. $\endgroup$– Kaτ_τCommented Dec 31, 2021 at 8:55
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$\begingroup$ Can you copy and paste those outputs in your question? $\endgroup$– Márcio Augusto DinizCommented Dec 31, 2021 at 8:58
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1 Answer
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There is no reason to remove "insignificant" predictors from a model that shows significance overall. In fact, it is typically wrong to do so--especially with things like binary or Cox regression models.
The "insignificant" predictors of age and risk_group were presumably included to control for associations of those covariates with outcome. Although your data set might not be large enough to show "significant" associations with outcome, they presumably were expected (based on your understanding of the subject matter) to have some associations with outcome.
If you remove those "insignificant" predictors a couple of bad things can happen. First, you no longer are controlling for those predictors. So if the treatment groups have differences in age or risk_group, apparent treatment differences might in part represent those differences in covariate values rather than true treatment differences. That's a standard cause of omitted-variable bias in all regression models. Second, as you would have used the outcomes of the model to select the predictors, the p-values and confidence intervals for treatment after removing the "insignificant" predictors wouldn't be valid.
For models like yours with primary interest in estimating an effect, Frank Harrell says in Section 4.12.2 of Regression Modeling Strategies, 2nd edition:
There is even less gain from having a parsimonious model than when developing overall predictive models, as estimation is usually done at the time of analysis. Leaving insignificant predictors in the model increases the likelihood that the confidence interval for the effect of interest has the stated coverage.
Omitting "insignificant" predictors is even more of a problem for models like logistic or Cox regression, where omitting any predictor that is associated with outcome (even if not "significantly" so in your data set, or even if uncorrelated with the included predictors) can bias the magnitudes of coefficients for included predictors downward toward 0.
If you have enough data, you might be better off including even more variables potentially associated with outcome even if they aren't "significant" by the arbitrary cutoff at p = 0.05.
