Hypotheses about the mechanisms of action by which a treatment affects a clinical outcome may prompt consideration of an alternative outcome as a potential surrogate. In many cases, due to costs or other factors, the candidate for surrogacy will only be measured for patients randomized to a substudy within the main trial. In this situation, the substudy patients provide information about links between the true and surrogate outcomes and the treatment, and these links can be exploited, using methods for handling missing covariates, to allow available information for patients not in the substudy to be incorporated into the analysis. The increased precision with which the treatment effect can be estimated using these methods, compared with using substudy data alone, in turn allows more precise estimates of measures of surrogacy. This paper reviews and compares some methods for handling missing covariate data and applies the methodology to a large heart attack trial, in order to investigate the properties of four measures for assessing the extent to which early response to thrombolytic therapy, as measured by an improvement in coronary blood flow, can be regarded as a surrogate for 30-day survival following heart attack. Design issues for substudies intended to assess treatment mechanisms are also considered. In particular, we consider how the precision of surrogate measures varies with the size of the substudy relative to the main trial. The results suggest that for reasonable surrogates, substudies substantially smaller than the main study can extract most of the available information regarding surrogacy.