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STA 138 Winter 2019

Homework 4 - Due Friday, Feb 15th

Book Portion (does not require R)

Note: This may be hand written or typed. Answers

should be clearly marked. Please put your name in

the upper right corner.

1. A logistic regression model was fit, where Y = 1 indicates

they defaulted on a loan, and Y = 0 indicates they did

not. The explanatory variable was X = loan balance (in

dollars). The estimated regression model is:

ln(π) = 10.4522 + 0.005368X

Further, SE(β1) = 0.000306, and you may assume the

minimum value for the loan was 0, and the maximum was

3000.

(a) Interpret exp(?10.4522) in terms of the problem.

(b) Interpret exp(0.005368) in terms of the problem.

when their balance is 2000.

(d) Does the probability that a subject defaults on a loan

increase or decrease with loan balance? Explain your

answer.

2. Continue with problem 1.

(a) Find the 95% Wald confidence interval for the value of

exp(β1).

(b) Interpret your interval in terms of the problem.

would expect when the loan amount increases by 500?

(d) Would a confidence interval for exp(β0) be useful in

this case? Explain your answer.

3. A logistic regression model was fit, where Y = 0 indicates

the subject does not smoke, and Y = 1 indicates they

subject did smoke. The explanatory variable was X =

smoking status of parents (Yes if at least one parent

smoked, No if no parents smoked). The estimated regression

model is:

ln(π) = 1.8266 + 0.4592XY es

Further, SE(1) = 0.08782.

(a) Interpret exp(?1.8266) in terms of the problem.

(b) Interpret exp(0.4592) in terms of the problem.

categorical variable.

(d) Predict the probability of the subject smoking if their

parent did not smoke.

4. Continue with problem 3.

(a) Test if H0 : β1 = 0, and list the test-statistic, p-value,

and conclusion

(b) Interpret the p-value in terms of the problem.

(d) Interpret the interval in terms of the problem.

5. Answer the following with TRUE or FALSE. It is good

practice to explain your answer.

(a) If the confidence interval for exp(β1) contains 1, it

suggests no influence of X1 on the odds of the trait

(Y = 1).

(b) exp(β1) gives the estimate for the odds of the trait

(Y = 1).

0 always has a practical interpretation.

(d) If we fail to reject H0 : β1 = 0, we conclude that X1

does not effect the probability of the trait.

R Portion (requires some use of R)

Note: You do not have to use R Markdown to turn

in the homework, but the homework must be turned

in in a reasonable format. The answers to the questions

should be in the body of the homework, and the

code used to obtain those answers should be in an appendix.

There should be no code in the body of the

homework. You can accomplish this in R, Word, LaTex,

Google Docs, etc. This portion should be printed

out and turned in with the hand-written portion.

I. In the file college.csv on Canvas, you will find data

on education aspirations of high school students based

on family income. The columns are values of X (family

income), values of Y (aspirations), and scores for X, Y

labeled ui and vj respectively.

(a) Using R, find the p-value for Pearsons test statistic

for testing independence.

(b) Using R, find the p-value of the MantelYates/Mantel-Haenzel

test statistic for testing

independence.

appropriate? Explain.

(d) What would your conclusion be for the test-statistic

you deemed most appropriate?

II. Online you will find a dataset flu.csv. The columns

we are interested in are shot (1 indicates flu shot, 0

indicates no flu shot), and age (the age of the subject).

(a) Fit the logistic regression model and write down the

estimated logistic-regression function.

(b) Using (a), estimate the probability that a 55 year

old will get the flu shot.

the probability of a flu shot go up or down?

(d) Interpret the value of exp(β?

1).

(e) Interpret the value of exp(β?

0), if appropriate.

III. Continue with question II.

(a) Find the Wald and LR test-statistic for testing if β1

= 0.

(b) Find the p-values for the test-statistics in (a).

problem.

(d) State the conclusion of the hypothesis test in terms

of the problem.

(e) Find the 99% LR confidence interval for β1, and

interpret it in terms of the problem.

IV. Online you will find a dataset flu2.csv. The columns

we are interested in are shot (1 indicates flu shot, 0

indicates no flu shot), and gender (M or F).

(a) Find the estimated logistic regression function.

(b) Using (a), write down the two separate models suggested

by the categorical variable.

(d) Interpret the value of exp(β0).

(e) Test to see if Gender can be dropped from the model.

State the test-statistic, p-value, and conclusion.

V. Online you will find a dataset heart.csv. The columns

we are interested in are CHD (1 indicates coronary heart

disease (CHD), 0 indicates no CHD), and age (the age

of the subject). These data come from Hosmer, D.W.,

Lemeshow, S. and Sturdivant, R.X. (2013) Applied Logistic

Regression: Third Edition.

(a) Fit the logistic regression model and write down the

estimated logistic-regression function.

(b) Using (a), estimate the probability that a 69 year

old will have CHD.

the probability of CHD go up or down?