Question

1. A toxicologist was interested in the effects of an insecticide on the mortality of a certain type of beetle. Six groups of beetles were exposed to various concentrations (C) of the insecticide. At the end of the experiment dead beetles in each group were counted.

concentration	10.8	11.6	12.1	12.6	13.1	13.5
death	15	24	26	24	29	29
group size	50	49	50	50	50	49

Consider the logistic regression model

logit [P(death at concentration C)] = β0 + β1 ×C

1. Write down the contribution to the likelihood function from the first group (that is, the group with death=15, size=50 and C = 10.8).
2. Re-write the above for the log-likelihood function.
3. Write down the contribution to the likelihood function from the last group (that is, the group with death=29, size=49 and C = 13.5).
4. Re-write the above for the log-likelihood function.
5. Use proc fcmp to write a SAS program that computes the log-likelihood function for the above application. In other words, your function accepts two numbers as values of β0 and β1, and outputs the log-likelihood function evaluated at the prescribed input values. Use the testing values β0 = 2 and β1 = 3 to make sure that you function works properly (it should give you a value of −5795.3).
6. Use SAS proc logistic to find the ML estimates and the corresponding maximized log-likelihood function value. Use your own function developed in part (5) to verify the SAS results.
7. By adding or subtracting 10% from the ML estimates, you can create four pairs of (β0,β1). Compute the log-likelihood function for all these 4 pairs of parameter values. Can you find a larger log-likelihood function value than that evaluated at the ML estimates? Why?

2) Revisit the toxicology example: six groups of beetles were exposed to various concentrations

(C) of the insecticide. At the end of the experiment dead beetles in each group were counted.

concentration	10.8	11.6	12.1	12.6	13.1	13.5
death	15	24	26	24	29	29
group size	50	49	50	50	50	49

Consider the logistic regression model

logit[P(death at concentration C)] = β0 + β1 X C

1. Under the assumption β1 = 0, write down the log-likelihood function (of _0 alone).
2. Write a SAS program to draw the above log-likelihood function.
3. Use SAS to find the ML estimate β0 under the assumption β1 = 0.
4. Use your own SAS program to find the maximized log-likelihood function value at the above ML estimate of _0. Compare your results with what SAS reports.
5. Using results from Homework 2, calculate the likelihood ratio test statistics for testing H0 : β1 = 0 vs. HA : β1 ≠0.
6. What is the p-value of the above?
7. Calculate the OR comparing two concentrations, one of which is 0:4 units higher than the other.
8. Calculate the estimated probability of beetle death at concentration C = 12:4.
9. Use SAS to draw a graph relating probability of beetle death to concentration.

Answer 1

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Page 2

Note: Only Question 1 is answered above until part four of the question as mentioned in the guidelines.