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I have a logistic regression model, where my outcome variable is a disease with yes and no outcome. We know that age is highly correlated with this disease. I have the data for a protein of interest and want to see whether this protein P has a relation with the disease. When I use protein alone as a predictor in the logistic model (in R), I get a significant p value for it. But when I add age in the model, my protein does not remain significant. I also looked at the correlation of my protein with age, and that is not correlated (0.08 correlation coefficient). So is it safe to omit age from the regression model?

DISEASE ~ PROTEIN   # significant ( p value : 0.00003)
DISEASE ~ PROTEIN + AGE 
    # Age is significant but protein no longer is (p value = 0.2)
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    $\begingroup$ Compute a likelihood ratio test to compare the two models. $\endgroup$
    – utobi
    Commented Mar 6 at 20:38
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    $\begingroup$ You might also want to look at DISEASE ~ AGE and how well the three models perform. Even better if you hold out a subset of your data, decide the three models and their coefficients on the rest of the data and then see how they perform on the data you initially held out. $\endgroup$
    – Henry
    Commented Mar 6 at 21:52
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    $\begingroup$ So you mean DISEASE ~ AGE DISEASE ~ PROTEIN DISEASE ~ PROTEIN + AGE and then look for lrt test? $\endgroup$
    – user3138373
    Commented Mar 6 at 22:10

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You should have given more information, such as sample size, and range of age variable, and shown some plots ... but: If the tange of age is large, effect may not be linear, so maybe spline it. Leaving out a variable, like age, which you know is important, cannot be sound, so always include it.

So then fit two models, with and without protein, and calculate a likelihood ratio test. Standard output by standard software often use the Wald test, which for logistic regression only is approximate, and can be a bad approximation for usual sample sizes. Illustrating with R code, assuming age is splined:

library(splines)
mod0 <- glm(DISEASE ~ ns(AGE, df=4), family=binomial, data=your_data)
mod1 <- glm(DISEASE ~ ns(AGE, df=4) + PROTEIN, family=binomial, 
                         data=your_data)
anova(mod0, mod1, test="LRT")
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