In the ICU data described in Section 1.6.1 the primary outcome variable is vital status at hospital discharge, STA. Clinicians associated with the study felt that a key determinant of survival was the patient’s age at admission, AGE.
(a) Write down the equation for the logistic regression model of STA on AGE. Write down the equation for the logit transformation of this logistic regression model. What characteristic of the outcome variable, STA, leads us to consider the logistic regression model as opposed to the usual linear regression model to describe the relationship between STA and AGE?
(b) Form a scatterplot of STA versus AGE.
(c) Using the intervals (15, 24), (25, 34), (35, 44), (45, 54), (55, 64), (65, 74), (75, 84), (85, 94) for age, compute the STA mean over subjects within each age interval. Plot these values of mean STA versus the midpoint of the age interval using the same set of axes as was used in 1(b). Note: this plot may done “by hand” on a printed copy of the plot from 1(b).
(d) Write down an expression for the likelihood and log-likelihood for the logistic regression model in Exercise 1(a) using the ungrouped, n = 200, data. Obtain expressions for the two likelihood equations.
(e) Using a logistic regression package of your choice obtain the maximum likelihood estimates of the parameters of the logistic regression model in Exercise 1(a). These estimates should be based on the ungrouped, n = 200, data. Using these estimates, write down the equation for the fitted values, that is, the estimated logistic probabilities. Plot the equation for the fitted values on the axes used in the scatterplots in 1(b) and 1(c).
(f) Using the results of the output from the logistic regression package used for 1(e), assess the significance of the slope coefficient for AGE using the likelihood ratio test, the Wald test, and if possible, the score test. What assumptions are needed for the p-values computed for each of these tests to be valid? Are the results of these tests consistent with one another? What is the value of the deviance for the fitted model?
(g) Using the results from 1(e) compute 95 percent confidence intervals for the slope coefficient for AGE. Write a sentence interpreting this confidence.
(h) Obtain from the package used to fit the model in 1(e) the estimated covariance matrix. Compute the logit and estimated logistic probability for a 60-year-old subject. Evaluate the endpoints of the 95 percent confidence intervals for the logit and estimated logistic probability. Write a sentence interpreting the estimated probability and its confidence interval.