Question

Human-to-mosquito transmission of P. falciparum malaria is mediated by sexual stage parasites called gametocytes. Mosquito feeding assay is a method to assess whether the gametocytes in a patient are transmissible to mosquitos. Through the detection of oocysts in the mosquito midgut by microscopy, researchers can confirm such transmissibility. A random sample of n subjects were assayed, and the outcomes are denoted Yı, ,Ý,, where Y? = 1 if oocysts were detected in the i-th subject and Y 0 otherwise In what follows, show all your work and derivations, not just the final answer (a) As a simple initial model, suppose that the probability of transmission is p- EY,1 is n. Derive the maximum likelihood estimator (MLE) of p and its large sample distribution (b) For a given α (0, 1), find either an exact or an approximate (1-α) confidence interval or p (c) Researchers hypothesize that transmissibility depends on gametocyte density in the pa- tient's blood. Let r, denote the gametocyte density in the i-th subject, 1 <i < n. Researchers propose the following model for transmissibility, where p, is the subject-specific transmission probability. Assuming Ao is known, show that the maximum likelihood estimator (MLE) of Pi satisfies the equation i-1 where y is the observed value of Y (d) To test the association between transmissibility and gametocyte density, the researchers propose the null hypothesis Ho B0 versus the alternative hypothesis H Find an asymptotic test with type-I error probability α B 0 (e) The study aims to estimate the gametocyte density threshold 0, defined to be the density with transmission probability equal to 0.95. One biostatistician suggested that, using the MLE in (c), a feasible estimator of θ is θ-k/a. Find the constant k such that θ is the MLE of θ. Further, find either an exact or an approximate 95% confidence interval for θ

Answer 1