A case-control study is efficient for investigating the association between outcomes and exposures. After conducting the primary outcome analysis, researchers can utilize the existing case-control study data to perform a secondary outcome analysis. Several methods have been proposed for analyzing secondary outcomes in case-control studies over the past few decades, but few of them have focused on the study design aspect. We propose optimal sampling strategies under a budget constraint for case-control studies with binary and Poisson secondary outcomes. We then extend our optimal sampling strategy by considering a confounder and derive the parameter of interest using doubly-weighted estimating equations. The term "optimal" refers to minimizing the variance of the estimator of the parameter of interest. We elucidate our proposed methods by developing the asymptotic variance of the estimator of the coefficient using weighted estimating equations and doubly-weighted estimating equations. Furthermore, we derive the optimal sampling ratio formulas through the Lagrange multiplier method based on certain monetary constraints. We verify our proposed methods through Monte Carlo simulation studies. Additionally, we apply our methods to empirical epidemiological studies that motivated the method development.
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