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In many contemporary applications, large amounts of unlabeled data arereadily available while labeled examples are limited. There has beensubstantial interest in semi-supervised learning (SSL) which aims to leverageunlabeled data to improve estimation or prediction. However, current SSLliterature focuses primarily on settings where labeled data is selectedrandomly from the population of interest. Non-random sampling, while posingadditional analytical challenges, is highly applicable to many real worldproblems. Moreover, no SSL methods currently exist for estimating theprediction performance of a fitted model under non-random sampling. In thispaper, we propose a two-step SSL procedure for evaluating a prediction rulederived from a working binary regression model based on the Brier score andoverall misclassification rate under stratified sampling. In step I, we imputethe missing labels via weighted regression with nonlinear basis functions toaccount for nonrandom sampling and to improve efficiency. In step II, weaugment the initial imputations to ensure the consistency of the resultingestimators regardless of the specification of the prediction model or theimputation model. The final estimator is then obtained with the augmentedimputations. We provide asymptotic theory and numerical studies illustratingthat our proposals outperform their supervised counterparts in terms ofefficiency gain. Our methods are motivated by electronic health records (EHR)research and validated with a real data analysis of an EHR-based study ofdiabetic neuropathy.

Efficient Estimation and Evaluation of Prediction Rules in Semi-Supervised Settings under Stratified Sampling