STAT823- Homework 11: Logistic Regression Solved

Directions
Using RMarkdown in RStudio, complete the following questions. Launch RStudio and open a new RMarkdown file or use the class RMarkdown template provided and save it on your working directory as a .Rmd file. At the end of the activity, save your pdf generated from RMarkdown+Knitr and submit your homework on the Blackboard.

Only question 1 is required. Question 2 is optional.

If you have questions, please post them on the lesson discussion board.

Some R-codes and output from the code have been provided for you. R codes and output must be clearly shown.

Homework submitted after the due date will attract a penalty of 10 points per day after the due date.

 

1           Logistic Regression Analyses
1. Use R to complete the following:

(a) Plot the mean response function of a logistic regression model,

exp(β0 + β1Xi)

Pr(Yi = 1) =  

1 + exp(β0 + β1Xi)

when β0 = −25 and β1 = 0.2. Hint: generate values of X over the range ∼ 90 ≤ X ≤∼ 160, then plug X into the mean response function and use the plot(x,y,type = "l") command.

x <- seq(90, 160, 1) b0 <- -25 b1 <- 0.2 y <- exp(b0 + b1 * x)/(1 + exp(b0 + b1 * x))

# ilogit is a function that does: exp(x)/(1+exp(x))
(b)     For what value of X is Pr(Y ) = 0.5?

(c)      Find the odds:

Pr(Y = 1)

 

1 − Pr(Y = 1)

 

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when X = 150 and when X = 151, and the ratio of the odds when X = 151 (numerator) to the odds when X = 150 (denominator). Is this odds ratio equal to exp(β1) as it should be?

Optional 2. A marketing research firm was engaged by an automobile manufacturer to conduct a pilot study to examine the feasibility of using logistic regression for predicting whether a family will purchase a new car during the next year. A random sample of 33 suburban families was selected. Data on annual family income (X1, in thousand dollars) and the current age of the oldest family automobile (X2, in years) were obtained. A follow-up interview conducted 12 months later was used to determine whether the family actually purchased a new car (Y = 1 or did not purchase a new car (Y = 0) during the year. Use the attached dataset Q2 to answer the following. Assume that a multiple logistic regression model with two predictor variables is appropriate:

(a)      Fit the model and find the estimates of β0, β1, and β2. Plot the estimated function over the data.

rm(list = ls(all = TRUE)[!grepl("global.var.A", ls(all = TRUE))]) q2 <- read.table("data/Q2.txt", quote = "\"", comment.char = "") str(q2) colnames(q2) <- c("y", "income", "age") require(epiDisplay) summ(q2) mod2 <- glm(y ~ income + age, family = binomial(link = "logit"), data = q2)

summary(mod2) library(epiDisplay)
(b)     Find and interpret estimates of exp(β1) and exp(β2).

(c)      What is the estimated probability that a family with annual income of 50 thousand and an oldest car of 3 years will purchase a new car next year? Compute a 95% interval estimate for this probability.

# predicted probability for income = 50, age = 3 library(faraway)

ilogit(b0 + b1 * 50 + b2 * 3) # Approach 1 # A 60.9% predicted chance of buying a car x0 <-

# c(1, 50, 3) eta0 <- sum(x0*coef(mod2))

# ilogit(eta0)

predict(mod2, newdata = data.frame(income = 50, age = 3), type = "response")

(pr1 <- predict(mod2, newdata = data.frame(income = 50, age = 3), se = T))

# predicted 95% interval for this probability ilogit(c(pr1$fit - 1.96 * pr1$se.fit, pr1$fit + 1.96 * pr1$se.fit))
(d)     Calculate the confidence intervals for exp(β1) and exp(β2) and interpret.

(e)      Assess model goodness of fit. Explain your results.