Transfer logistic regression odds ratio (based on stratified sample) to the population odds ratio

Question

In order to build prognostic model for predicting breast cancer risk, the sample of 321 women has been formed. Each woman answered the questionnaire, and the answers were used as predictors. The outcome was whether the women has breast cancer or not. We have collected 132 patients with breast cancer, and also 189 women without breast cancer answered the same questionnaire. So the incidence of breast cancer in the sample is about 0.41. But in population the incidence is about 0.0005 per year. Consequently, not only breast cancer, but each harmful factor has greater incidence in the sample than that in population. Now, I've obtained OR for smoking about 35.

Does it means, that in population OR will also be 35? If not, is there any way to transfer logistic regression odds ratio to the population odds ratio?

I can calculate relative risk by $RR\approx \frac{OR}{1-BR+BR \times OR}$. What should I use as a basis risk? If it is the risk that nonsmokers from population will have cancer, which is tiny, then $RR\approx OR$. If it is exp(intercept) of the model, then it is also close to 35. And if it is incidence among nonsmokers from sample, then $RR=2.7$. Is any of these options right?

Answer 1

• Yes. In logistic regression, the coefficient estimates for non-intercept terms $\beta_1$...$\beta_n$ should be unbiased even when oversampling events. The intercept, however, is not. There's a few methods for recovering an unbiased intercept coefficient in King and Zheng (2001) [an accesible PDF is a google away] if you need that.

• You should be using the risk from the population at large. As mentioned, the intercept is not unbiased.
  Furthermore, even when not oversampled, the intercept is usually not an estimate of the base risk (it's the estimate when all $X=0$, so only in the case where you have all continuous predictors and have centered them).