A semiparametric two-component "compound" mixture model and its application to estimating malaria attributable fractions.

Malaria remains a major epidemiologic problem in many developing countries. Malaria is defined as the presence of parasites and symptoms (usually fever) due to the parasites. In endemic areas, an individual may have symptoms attributable either to malaria or to other causes. From a clinical viewpoint, it is important to correctly diagnose an individual who has developed symptoms so that the appropriate treatments can be given. From an epidemiologic and economic viewpoint, it is important to determine the proportion of malaria-affected cases in individuals who have symptoms so that policies on intervention program can be developed. Once symptoms have developed in an individual, the diagnosis of malaria can be based on the analysis of the parasite levels in blood samples. However, even a blood test is not conclusive as in endemic areas many healthy individuals can have parasites in their blood slides. Therefore, data from this type of study can be viewed as coming from a mixture distribution, with the components corresponding to malaria and non-malaria cases. A unique feature in this type of data, however, is the fact that a proportion of the non-malaria cases have zero parasite levels. Therefore, one of the component distributions is itself a mixture distribution. In this article, we propose a semiparametric likelihood approach for estimating the proportion of clinical malaria using parasite-level data from a group of individuals with symptoms. Our approach assumes the density ratio for the parasite levels in clinical malaria and nonclinical malaria cases can be modeled using a logistic model. We use empirical likelihood to combine the zero and nonzero data. The maximum semiparametric likelihood estimate is more efficient than existing nonparametric estimates using only the frequencies of zero and nonzero data. On the other hand, it is more robust than a fully parametric maximum likelihood estimate that assumes a parametric model for the nonzero data. Simulation results show that the performance of the proposed method is satisfactory. The proposed method is used to analyze data from a malaria survey carried out in Tanzania.