Efficient semi-supervised inference for logistic regression under
  case-control studies
Zhuojun Quan Yuanyuan Lin Kani Chen Wen Yu
Semi-supervised learning has received increasingly attention in statistics
and machine learning. In semi-supervised learning settings, a labeled data set
with both outcomes and covariates and an unlabeled data set with covariates
only are collected. We consider an inference problem in semi-supervised
settings where the outcome in the labeled data is binary and the labeled data
is collected by case-control sampling. Case-control sampling is an effective
sampling scheme for alleviating imbalance structure in binary data. Under the
logistic model assumption, case-control data can still provide consistent
estimator for the slope parameter of the regression model. However, the
intercept parameter is not identifiable. Consequently, the marginal case
proportion cannot be estimated from case-control data. We find out that with
the availability of the unlabeled data, the intercept parameter can be
identified in semi-supervised learning setting. We construct the likelihood
function of the observed labeled and unlabeled data and obtain the maximum
likelihood estimator via an iterative algorithm. The proposed estimator is
shown to be consistent, asymptotically normal, and semiparametrically
efficient. Extensive simulation studies are conducted to show the finite sample
performance of the proposed method. The results imply that the unlabeled data
not only helps to identify the intercept but also improves the estimation
efficiency of the slope parameter. Meanwhile, the marginal case proportion can
be estimated accurately by the proposed method.