Winter, 2014 Wednesday, Feb. 26Stat 418 – Day 27Poisson Regression (3.3)Recap: Probit Regression and Gombit regression are examples of using cumulative distribution functions to provide alternative link functions for binomial probabilities in a generalized linear model.- Probit and logistic regression generally provide very similar fits- Logistic regression is far more popularNext we want to consider other underlying probability distributions for the random component Y.Example 1: Suppose you want to predict the number of parties attended in a month by college students based on their major, gpa, and family income. Or you want to predict the number of defectives in a silicon wafer based on characteristics of the manufacturing process such as protocol or employee shift. (a) What is the main difference in these research questions and what we have examined before?(b) How would you characterize the response variable in each case?Example 2: Joyce Pool studied a group of 41 African elephants in Amboseli National Park, Kenya, for 8 years. One research question was whether there is a relationship between mating success and age and whether males have diminished success after reaching some optimal age. The data in elelphants.jmp include the age of the elephant at the start of the study and the number of successful matings during the eight year period.(a) Produce a graph of the number of successful matings vs. the ages of the elephants. How would you describe the relationship? (Add a smoother.)(b) So E(Y) has what form? How do we find a linear relationship? What is the link function?Note: When you want to take logs, if you have 0s you can add .5 to the values first.(c) What do we know about the probability distribution for Y?(d) Using this general form, how will you interpret the β1 coefficient?Winter, 2014 Wednesday, Feb. 26(e) What is the form of the log likelihood?(f) What is the form of the deviance, deviance residual?(g) What is the form of the Pearson residual?Deviance residuals are somewhat more reliable for detecting outliers, Pearson residuals are easier to interpret. If Poisson means are above 5, these are approximately normal. (h) In JMP, select Analyze > Fit Model. Specify matings in the Y box and age in the Construct Model Effects box. Change the personality to Generalized Linear Model, a Poisson distribution, and a Log link.Check the Keep dialog open box and press Run.Is this model appropriate? (Goodness of Fit? Residual Plot?) The Deviance Goodness of Fit test can be unreliable in detecting model inadequacies, also look at plots and tests of model terms. A small p-value indicates either missing EVs, Poisson isn’t right, or severe outliers.Is this model useful?Interpret the effect of age.Predict the number of matings for a 40 year old elephant.(i) How do we consider whether they have “diminished success” after they reach a certain