# Efficient semi-supervised inference for logistic regression under case-control studies

arXiv: https://arxiv.org/abs/2402.15365 · HTML: https://mlsanity.com/paper/2402.15365/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 37 references that are in the map.
- First posted on arXiv 23 Feb 2024.

## Builds on (most cited first, 7 of 7)

- [Semi-supervised Inference: General Theory and Estimation of Means](https://mlsanity.com/paper/1606.07268/) (2016)
- [Efficient and Adaptive Linear Regression in Semi-Supervised Settings](https://mlsanity.com/paper/1701.04889/) (2017)
- [Local case-control sampling: Efficient subsampling in imbalanced data sets](https://mlsanity.com/paper/1306.3706/) (2013)
- [Semi-supervised Inference for Explained Variance in High-dimensional Linear Regression and Its Applications](https://mlsanity.com/paper/1806.06179/) (2018)
- [Semi-Supervised Approaches to Efficient Evaluation of Model Prediction Performance](https://mlsanity.com/paper/1711.05663/) (2017)
- [Combining case-control studies for identifiability and efficiency improvement in logistic regression](https://mlsanity.com/paper/2106.00939/) (2021)
- [Robust and Efficient Semi-Supervised Estimation of Average Treatment Effects with Application to Electronic Health Records Data](https://mlsanity.com/paper/1804.00195/) (2018)

## Built on by (most cited first, 1 of 1)

- [Deep non-parametric logistic model with case-control data and external summary information](https://mlsanity.com/paper/2409.01829/) (2024)

## Abstract (arXiv)

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.