Use

>install.packages("boot")

>library(boot)

>data(downs.bc)

in r to access the data.

1. The data downs.bc is found in the boot package. It describes the proportion of births with Down's syndrome by the ages of the mothers and was collected in British Columbia. a. Plot the proportion of Down's births against age and comment on the relationship. b. Fit a logistic regression model to predict the probability of Down's syndrome with age as the linear predictor. Plot the predicted probability of Down's syndrome as a function of age along with the observed probabilities in the same plot. Comment on the observed quality of fit as well as the fit based on the residual deviance.

2. Now add a quadratic term to age in the linear predictor. a. Plot the predicted probability of Down's syndrome as a function of age along with the observed probabilities in the same plot. Comment on the change in the quality of fit based both on the plot and on the residual deviance after adding the quadratic term. b. Find the p-value for the likelihood ratio test with the linear predictor being a linear function of age as the null hypothesis and the quadratic model as the alternative hypothesis.

3. Using the quadratic model predict the probability of a Down's birth at age 30 and at age 40. What is the odds ratio taking the odds at age 40 in the numerator? What is the relative risk?