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.

在半监督学习设置下，使用案例-对照抽样方法采集到的带有结果和协变量的标记数据集以及仅包含协变量的未标记数据集，可以通过构建观测到的标记和未标记数据的似然函数，并通过迭代算法获得最大似然估计，从而识别出截距参数并同时提高斜率参数的估计效率。

基于病例对照研究的逻辑回归有效半监督推断