The clinical pulmonary infection score (CPIS) and bronchoalveolar lavage (BAL) are important diagnostic variables of pneumonia for forcefully ventilated patients who are susceptible to nosocomial infection. Because of its invasive nature, BAL is performed for patients only if the CPIS is greater than a certain threshold value. Thus, CPIS and BAL are closely related, yet BAL values are substantially missing. In a randomized clinical trial, the control and oral treatment groups were compared based on the outcomes from these procedures. Because of the relevance of both outcomes with respect to evaluating the efficacy of treatments, we propose and examine a nonparametric test based on these outcomes, which employs the empirical likelihood methodology. While efficient parametric methods are available when data are observed incompletely, performing appropriate goodness-of-fit tests to justify the parametric assumptions is difficult. Our motivation is to provide an approach based on no particular distributional assumption, which enables us to use all observed bivariate data, whether completed or not in an approximate likelihood manner. A broad Monte Carlo study evaluates the asymptotic properties and efficiency of the proposed method based on various sample sizes and underlying distributions. The proposed technique is applied to a data set from a pneumonia study demonstrating its practical worth.