Hastie and Tibshirani (1990, p. 282) described a study to determine risk factors for kyphosis, which is severe forward flexion of the spine following corrective spinal surgery. The Kyphosis data file at the text website shows the 40 observations on y = whether kyphosis is present (1 = yes), with x = age as the explanatory variable.

a. Fit a logistic regression model. Test the effect of age.

b. Plot the data. Note the difference in dispersion of age at the two levels of kyphosis.

c. Fit the model logit[P(Y = 1)] = α + β1x + β2x2. Test the significance of the squared age term, plot the fit, and interpret. (The final paragraph of Section 4.1.5 is relevant to these results.)