Homework # 1, Stat 526, Fall 2006 1. The swiss data can be found in the faraway package. You can load the data and check out a description of the variables by: > library(faraway) > data(swiss) > help(swiss) Consider a linear model to identify predictors to Fertility. Attach your R codes but not your R outputs. Your data analysis should be justified and consist of • An initial data analysis that explores the numerical and graphical characteristics of the data; • Variable selection to choose the best model; • An exploration of transformations to improve the fit of the model; • Diagnostics to check the assumtions of your model. 2. Suppose a logit and probit model for Binary data with one predictor are fitted. Show that if the predictor variable is the negative value of the intercept over the slope then the predicted proportion is 1/2 (median). This value of the predictor is denoted by LD50. Find LD50 for complementary log-log link. 3. Show that the logistic regression function π(x) = eα+βx 1 + eα+βx and the probit regression function π(x) = Φ(α + βx) have the steepest slopes at π(x) = 0.5. The complementary regression function π(x) = 1 − e−e α+βx has the steep slopes at π(x) = 1 − e−1 . Note: The steepest slopes tell us where the model is the most accurate for the fit. 1