Question

1. Let nj denote the number of eggs and xi the number of fertilised eggs for the ith woman. Let 1tj denote the fertilisation rate for the ith woman.

1. Explain why a binomial distribution may be valid to model the data.

2. write down the expression for the log likelihood of the observed data, assuming a binomial distribution with different fertilisation rates for each woman. Identify the logit function in your expression.

Answer 1

1) Given n_j is total number of eggs and x_i the number of fertilized eggs with constant fertilization rate t_j for each woman, we can see the number of fertilized eggs(x_i) as number of successes in n_j number of attempts with a probability parameter as t_j. Hence binomial distribution may be used to model data.

2) Log likelihood =   , where represents n_j choose x_i.

_{This can be simplified to . The last two terms are logit function in the expression.}