How to handle imbalanced covariate (Gender) in logistic regression classification model?

Question

I have classification model with OSA (Obstructive Sleep Apnea) status as dependent variable and a continuous biomarker as independent variable, adjusting for BMI and Age. I get the following result.

Call:
glm(formula = OSAclass ~ ANGPTL7 + BMI + AgeSurgery, family = binomial, 
    data = d)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-1.7803  -0.8112  -0.6614   1.0924   2.0289  

Coefficients:
              Estimate Std. Error z value Pr(>|z|)   
(Intercept) -6.0448112  1.9004997  -3.181  0.00147 **
ANGPTL7      0.0003061  0.0001445   2.119  0.03410 * 
BMI          0.0693494  0.0329811   2.103  0.03549 * 
AgeSurgery   0.0409950  0.0179479   2.284  0.02237 * 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 156.72  on 124  degrees of freedom
Residual deviance: 143.14  on 121  degrees of freedom
  (9 observations deleted due to missingness)
AIC: 151.14

Number of Fisher Scoring iterations: 4

But when I add Gender to the model. I lose the significance of ANGPTL7.

Coefficients:
              Estimate Std. Error z value Pr(>|z|)    
(Intercept) -7.2699222  2.0282067  -3.584 0.000338 ***
ANGPTL7      0.0002434  0.0001493   1.630 0.103020    
BMI          0.0903125  0.0351067   2.573 0.010096 *  
GenderMale   1.6716022  0.6147563   2.719 0.006545 ** 
AgeSurgery   0.0443312  0.0186087   2.382 0.017205 *  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

I tried the model with Gender == Male and Gender == Female separately. Still the ANGPTL7 is not significant.

Following is the proportion of Male and Female in the sample.

       Non-OSA OSA
Male         6  11
Female      85  33

Is the losing statistical significance of ANGPTL7 in the model because of adjusting with imbalanced covariate (Gender)? Can someone please help me how to understand these results?

Answer 1

You are using a probability model, not a classification model. Logistic regression does not classify Y; it uses an already classified Y to predict the probability of class membership.

Your sample size is just barely large enough to fit the intercept in the logistic model, i.e., to handle the case where there are zero covariates. So take all of your analyses with a grain of salt.

Study the association between ANGPTL7 and sex. If they are collinear, test them with a 2 d.f. "chunk" test (e.g., a likelihood ratio $\chi^2$ test) instead of testing them individually. Let them combine forces instead of compete.

The imbalanced distribution of sex is not an issue. This is really about joint distributions.

Losing tremendous power by using OSA as a binary variable and not using the underlying apnea index it was derived from is a very strange way to analyze data.

Answer 2

P-value is a rough estimate of variable significance. There could be committed variable bias in the first model. You need to do an F-test to check the significance of ANGPTL7. Maybe, it is still significant in the second model, but maybe it was never significant