Instructions
RESEARCH SCHOOL OF FINANCE, ACTUARIAL STUDIES AND STATISTICS
College of Business & Economics, The Australian National University
REGRESSION MODELLING
(STAT2008/STAT4038/STAT6038)
Sample Assignment 1
• This is NOT your current assignment. It is a sample assignment using questions taken from assignments from earlier years. Assignment instructions tend to change slightly from year to year – read the full instructions on this year’s assignment sheet and follow those instructions carefully.
Data
The data to be used in this sample assignment come from the recommended text by Julian J. Faraway (Linear Models with R, 2nd Edn, Chapman & Hall/CRC, 2015) and are all stored in the Faraway library, which is available from CRAN (the Comprehensive R Archive Network, the original Australian mirror site for which is located here in Canberra at the CSIRO). You can access Faraway’s stored library of data and functions by starting R and typing the following commands:
install.packages(“faraway”) # installs the faraway package and dataset library.
# This can take some time, especially on an ANU InfoCommons computer, so show some patience.
library(faraway) # this attaches the faraway library to your search path search()
ls(pos=”package:faraway”) # lists the contents of the faraway package
help(prostate)
help(teengamb)
# Faraway has provided brief help files on all of the datasets, which # include a description of the variables and the original source
prostate
teengamb
# shows the contents of the data to be used in this assignment
attach(prostate)
attach(teengamb)
# attaches the data to your search path, so you can reference the variables
Further details (such as the other packages you will need to load if you wish to use all of the stored functions described in the Faraway text) are available in Appendix A on page 265 of the Faraway text.
Copies of the datasets (as .csv files) and help files are also available on Wattle, in case you have trouble loading the Faraway library.
SampleAssignment1.doc Page 1 of 3
Question 1 (20 marks)
The dataset prostate comes from a study on 97 men with prostate cancer who were due to receive a radical prostatectomy (a surgical procedure). Of the variables included in this dataset, lcavol (log of the cancer volume) is a measure of the size of the cancer tumour and lpsa (log of the prostate specific antigen measure) is the result of a diagnostic blood test for prostate cancer.
(a) Plot lpsa against lcavol. Is there a significant correlation between lpsa and lcavol? Use R to
conduct a suitable hypothesis test and present and interpret the results.
(4 marks)
(b) Fit a simple linear regression model with lcavol as the response variable and lpsa as the predictor. Construct a plot of the residuals against the fitted values, a Q-Q plot of the residuals and a bar plot of the leverages for each observation. Comment on the model assumptions and on any unusual data points. (4 marks)
(c) Produce the ANOVA (Analysis of Variance) table for the SLR model in part (b) and interpret the results of the F test. Are these results consistent with the hypothesis test you conducted in part (a)? (4 marks)
(d) What are the estimated coefficients of the SLR model in part (b) and the standard errors associated with these coefficients? Interpret the values of these estimated coefficients and perform t-tests to test whether or not these coefficients differ significantly from zero. What do you conclude as a result of these t-tests? (4 marks)
(e) Plot lcavol against lpsa. Include the fitted SLR model from part (b) as a line on the plot and also show 95% confidence intervals for the mean or expected value of lcavol (do NOT plot the 95% prediction intervals). Do the results of a PSA test appear to be a reliable predictor of the size of the prostate cancer tumour? (4 marks)
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Question 2 (20 marks)
The dataset teengamb concerns a study of teenage gambling in Britain. For this assignment, we are interested in whether a teenager’s income (measured in UK £ per week) can be used to predict the amount they will gamble (gambling expenditure measured in UK £ per year), at least for the teenagers who do regularly gamble.
(a) Plot gamble against income. Describe the correlation shown in the plot. Would you expect a simple linear regression model to be a reasonable model for the relationship shown in the plot? (4 marks)
(b) Fit a simple linear regression model with gamble as the response variable and income as the predictor. Construct a plot of the residuals against the fitted values, a Q-Q plot of the residuals and a bar plot of Cook’s Distances for each observation. Comment on the model assumptions and on any unusual data points. (4 marks)
(c) In question 1, a natural log (to the base e) transformation had already been applied to both the response and predictor variables and appeared to produce reasonable results.
In this example, there are a number of teenagers who are not regular gamblers (their annual expenditure on gambling is very small or even zero). What is the problem with applying a log transformation in this situation? Exclude any teenager who spends less than £1 per year on gambling and fit another simple linear regression model with log(gamble) as the response variable and log(income) as the predictor. Check the same plots you produced for the earlier model in part (b). Are the same problems still apparent? (4 marks)
(d) Produce the ANOVA table and the table of the estimated coefficients for the revised SLR model in part (c). Interpret the values of the estimated coefficients for this SLR model and the results of the overall F test and the t-tests on the estimated coefficients.
(4 marks)
(e) Use the revised SLR model from part (c) to predict the annual expenditure on gambling for three British teenagers, who were not included in the original study, but who have weekly incomes of £1, £5 and £20, respectively. Find 95% prediction intervals for these predictions. Do you think this revised SLR model is a good model for making all three
of these predictions?
(4 marks)
SampleAssignment1.doc
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