Exercise 3: This question deals with multiple parameter tests for the fertility study example. Review Module 1 Exercise 5, Module 2 Exercise 3 and their solutions, for the fertility study where mothers having premature births have different fertility to mothers having full term births. In those exercises we studied a likelihood function that was a function of two parameters, the fertility parameter for mothers with premature births (p2), and the relative fertility rate (γ). Start by writing down the log-likelihood function from part b of Module 1 Exercise 5.
Our goal is to test the hypothesis that the fertility rates are identical, that is:
1: s versu1:10≠=γγHH
a. For the sample described in Module 1 Exercise 4 [“In a particular study we observe n1=78 mothers having full-term births and n2=51 mothers having premature births. The mothers having full-term births had a total of pregnancy attempts, and the mothers having premature births had a total of pregnancy attempts.”], evaluate the MLEs of p2 and γ. These MLEs have been derived previously (Module 2 Exercise 3). Hence evaluate the maximum value achieved by the log-likelihood under the full model, Σ=207ixΣ=212iy).ˆ,ˆ(2γp
b. Write down the log-likelihood function under the assumption that the null hypothesis is true. Hence derive and evaluate the MLE of p2 under the assumption that the null hypothesis is true and evaluate the maximum value achieved by the log-likelihood under the null model.
c. Based on your results in parts a and b, write down the likelihood ratio test statistic for testing the above hypotheses, and carry out a likelihood ratio test. Evaluate and interpret the p-value for this test.
d. Using your results from part d of Module 2 Exercise 3, evaluate the approximate Wald test statistic for testing the above hypotheses. Carry out the test and calculate and interpret the p-value.
e. Recall your results from parts d and e in Module 2 Exercise 3. Explain why these are consistent with the tests conducted above.
I have attached the assignment and the two background questions. It is only question 3 in the PDF labelled module 4 that I require.