In the case-cohort studies conducted within the Atherosclerosis Risk in Communities (ARIC) study, it is of interest to assess and compare the effect of high-sensitivity C-reactive protein (hs-CRP) on the increased risks of incident coronary heart disease and incident ischemic stroke. Empirical cumulative hazards functions for different levels of hs-CRP reveal an additive structure for the risks for each disease outcome. Additionally, we are interested in estimating the difference in the risk for the different hs-CRP groups. Motivated by this, we consider fitting marginal additive hazards regression models for case-cohort studies with multiple disease outcomes. We consider a weighted estimating equations approach for the estimation of model parameters. The asymptotic properties of the proposed estimators are derived and their finite-sample properties are assessed via simulation studies. The proposed method is applied to analyze the ARIC Study.