In clinical trials, a biomarker (S ) that is measured after randomization and is strongly associated with the true endpoint (T) can often provide information about T and hence the effect of a treatment (Z ) on T. A useful biomarker can be measured earlier than T and cost less than T. In this article, we consider the use of S as an auxiliary variable and examine the information recovery from using S for estimating the treatment effect on T, when S is completely observed and T is partially observed. In an ideal but often unrealistic setting, when S satisfies Prentice's definition for perfect surrogacy, there is the potential for substantial gain in precision by using data from S to estimate the treatment effect on T. When S is not close to a perfect surrogate, it can provide substantial information only under particular circumstances. We propose to use a targeted shrinkage regression approach that data-adaptively takes advantage of the potential efficiency gain yet avoids the need to make a strong surrogacy assumption. Simulations show that this approach strikes a balance between bias and efficiency gain. Compared with competing methods, it has better mean squared error properties and can achieve substantial efficiency gain, particularly in a common practical setting when S captures much but not all of the treatment effect and the sample size is relatively small. We apply the proposed method to a glaucoma data example.