I have followed the King and Zeng paper and understand the consistency of the prior correction after oversampling in logistic regression. But I am trying to understand why the beta coefficients of the independent variables are not affected by the exercise? If I correctly, there is a proof of this in Minski's 1977 paper. But I cannot find that proof on the internet. Could anyone explain this to me in little detail?


Let 2 binary random variables $X$ and $Y$ e.g. exposure and disease status. Here is the logistic model:
$logit(\pi(X=x)) = \alpha + \beta x $ where $\pi(X=x)=P(Y=1|X=x)$.
Then, $\alpha = logit(\pi(X=0)) = \log (\frac{\pi(X=0)}{1-\pi(X=0)})$ so $\alpha$ is simply the log of odds for the unexposed, $X=0$. But why is the estimator of $\alpha$ biased when the diseased are oversampled?

This sampling scheme is called case-control. In case-control, the disease status $Y$ is fixed because one generally fixes the number of diseased and non-diseased and NOT the exposure status. Consequently, $X|Y \sim Binom$ so we can only estimate the probability of exposure given the disease status, which is not what we want. We want the probability of disease given exposure $\pi(X=x) = P(Y=1|X=x)$.

But in case control, $\pi(X=x)$ is not estimable. To see this, in case-control, we only have $P(Y=1|X=x, \text{Case Sampled}=1)$ and the marginal probability is
$P(Y=1|X=x) = P(Y=1|X=x, \text{Case Sampled}=0) P(\text{Case Sampled}=0) + P(Y=1|X=x, \text{Case Sampled}=1) P(\text{Case Sampled}=1).$
(Note: $P(\text{Case Sampled}|X=x)=P(\text{Case Sampled})$ because the sampling is not based on exposure status)

So, we've learnt that, unless you know the sampling probability, the estimator for $\pi(X=x) = P(Y=1|X=x)$ will be biased because $\pi(X=x) \neq \pi(X=x, Sampled=1)$, therefore, $\alpha = logit(\pi(X=0))$ is biased and this is general, not only in rare disease condition.

Moving on to $\beta$, you can see from the logistic model that
$\beta = logit(\pi(X=1)) - logit(\pi(X=0)) = \log(\frac{\pi(X=1)/(1-\pi(X=1))}{\pi(X=0)/(1-\pi(X=0))}) = \Omega$
which is the log odds ratio between exposed and unexposed. This has some nice properties, among which the most important one is its "unbiasedness" under different sampling. To see this, with simple probability exercise, we can show
$\Omega=\log(\frac{P(Y=1|X=1) P(Y=0|X=0)}{P(Y=1|X=0) P(Y=0|X=1)}) =\log(\frac{P(X=1|Y=1) P(X=0|Y=0)}{P(X=1|Y=0) P(X=0|Y=1)})$
We saw earlier that $X|Y \sim Bin$ in case-control so there is no problem estimating $P(X|Y)$, therefore, $\hat{\beta}$ is unbiased.

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