This problem explores an alternative model for the beetle survival data from problem (7.1) in which the log-odds for survival are related to the natural logarithm of the carbon disulfide concentration to which the beetles are exposed. The log concentration appears in the second column of the data table.

a. Using the log concentration of carbon disulfide as the explanatory variable, fit a logistic regression model to the beetle survival data.

Construct a table with one column for the parameter estimates, one column for their standard errors, and two more columns for the lower and upper limits of 95% confidence intervals. Interpret the confidence intervals.

b. Plot the estimated logistic regression curve and use that curve to determine the concentration corresponding to a survival probability of 0.5.

c. Use a data step to add 141 lines with new concentration values to the data file. Start at 49 and go up to 77 by increments of 0.2. Then compute the natural logarithm of each of those concentrations. Designate the counts for the number of survivors and the number exposed as missing values for the new data lines. Use proc logistic to fit the logistic regression model to the combined data and with the log concentration as the explanatory variable. Output estimates of the survival probabilities at each concentration. Use those results to plot the estimated survival probability curve against the concentration of carbon disulfide. Include the observed survival probabilities on the same plot. Does it appear that this logistic regression model provides a good description of the decreasing trend in the survival probabilities as the concentration of carbon disulfide increases?

d. Plot the estimates of the survival probability curves from problems 1 and 2 on the same plot. How do these curves differ?

e. Compare the AIC values for those two models. Comment on the results.