Problem 2 A study was conducted on children who had corrective spinal surgery. We are interested in factors that might result in kyphosis (a kind of deformation) after surgery. The data can be loaded by data (kyphosis, package = "rpart") Consult the help page on the data for further details. (a) Make plots of the response as it relates to each of the three predictors. You may find a jittered scatterplot more effective than the interleaved histogram for a dataset of this size. Comment on how the predictors appear to be related to the response. (6 pts) (b) Fit a GLM with the kyphosis indicator as the response and the other three variables as predictors. Investigate if you can find any predictor that can be dropped from the current model. If you can find one, fit a reduced model. Repeat this process until you find a model with all the predictors being significant. (5 pts) Answer the questions (c)-(f) using the final model you obtained in (b). (c) Interpret the effects of the predictor(s) in the model to the response. (5 pts) d) Plot the relationship between the probability of y = 1 and the predicted linear predictor using the inverse logit function. Validate your interpretation in (c) with the plot. (5 pts) (e) Produce the 95% confidence interval based on the profile likelihood of the coefficient estimate of the predictor(s). Using the lower and upper limits, compute the corresponding confidence intervals (CIs) for the probability of y = 1. Add those CIs to the previous plot. (8 pts) (f) Plot the deviance residuals against the Start predictor, using binning by values of Start. Comment on the plot. (5 pts) (g) Starting from the model with all three predictor variables, determine the best subset of variables using AIC as the criterion (Use the step function). Compare the selected final model with one obtained in (b). What do you observe? (5 pts) Answer the questions (h)-(j) using the model you obtained in (g). (h) Make a plot of the leverages. Interpret the plot. (5 pts) (i) Let's check the goodness of fit for this model. a. Using binning by values of Start, create a plot comparing the predicted probability and observed probability with 95% confidence intervals. Comment on the plot. (7 pts) b. Compute the Hosmer-Lemeshow statistic and associated p-value. What do you conclude? (5 pts) (j) Use the model to classify the subjects into predicted outcomes using a 0.5 cutoff. Produce cross- tabulation of these predicted outcomes with the actual outcomes. Find the sensitivity and specificity. (6 pts)\f