For the horseshoe crab data, fit the logistic regression model with x = weight as the sole predictor of the presence of satellites.
a. For a classification table using the sample proportion of 0.642 as the cutoff, report the sensitivity and specificity. Interpret.
b. Form a ROC curve, and report and interpret the area under it.
c. Investigate the model goodness-of-fit using the Hosmer-Lemeshow statistic or some other model-checking approach. Interpret.
d. Inferentially compare the model to the model with x and x2 as predictors.
e. Compare the models in (d) using the AIC. Interpret.
Problem 4.1 A study used logistic regression to determine characteristics associated with Y = whether a cancer patient achieved remission (1 = yes). The most important explanatory variable was a labeling index (LI) that measures proliferative activity of cells after a patient receives an injection of tritiated thymidine. It represents the percentage of cells that are "labeled." Table 4.8 shows the grouped data. Software reports Table 4.9 for a logistic regression model using LI to predict π = P(Y = 1).
a. Show how software obtained πˆ = 0.068 when LI = 8.
b. Show that = 0.50 when LI = 26.0.
c. Show that the rate of change in is 0.009 when LI = 8 and is 0.036 when LI = 26.
d. The lower quartile and upper quartile for LI are 14 and 28. Show that πˆ increases by 0.42, from 0.15 to 0.57, between those values. e. When LI increases by 1, show the estimated odds of remission multiply by 1.16.
