ST221 Assessed coursework 2
Deadline: 02 May 2023, 1 pm
Please read these instructions carefully!
This assignment counts for 20% of your final module mark. The maximum score for this
coursework is 50 marks.
Your solutions must be produced using a word processor, R Markdown, LaTeX or similar
and must be converted to a pdf file before submission. Please do not add your name on your
submission to allow for anonymous marking. You may cut-and-paste R-output. Handwritten
answers will not be awarded marks. Use a font size of 11pt or larger. Question sub-sections
must be clearly labelled for ease of marking.
If you do not submit your solutions in a typed format, then this will not be
accepted as a submission. You should convert your solutions into one PDF file to be
submitted on the ST221 moodle page.
Please read Chapter 5 in the course guide which gives details around the procedures regarding
coursework including applying for extensions and lateness penalties. Please ensure that you
submit in good time before the deadline. Penalties will apply if work is submitted more
than 1 minute after the deadline unless an extension or waiver is granted. Coursework is
not eligible for mitigating circumstances due to the loss of work in progress. The penalty
for late submission is 5% per 24 hour period encompassing a working day. However no
submission will be accepted more than 5 working days after the original deadline unless there
is a pre-approved extension extending past the cut off period.
If you have any queries about the coursework, please post them on the ST221 forum, but
do not post any part of your solutions. You can also submit questions to the anonymous
question form on moodle.
Please be aware that your work will be submitted to TurnItIn, a piece of
plagiarism-detection software. Cases of suspected collusion or plagiarism will be fol-
lowed up as outlined in Section 5.3 of the course guide. Note that detailed discussions of
the assignment or comparisons of numerical/graphical results or computer code are not
permitted.
Make sure to read questions carefully. If asked to produce a plot, then please include the
plot in your report. Make sure it is of appropriate scale and the axes are clearly labelled.
Include R code only if requested to do so.
Good luck with the assignment!
1
Download the file houses.csv from the ST221 moodle page and load it into R.1 The original
dataset consists of information about the value and various characteristics of a sample of
houses in Saratoga County, USA, from 2006.
The variables (adapted from the original dataset) are:
• price: the value of the property (in 1000 US dollars);
• livingArea: the living area of the property (in square feet);
• bathrooms: the number of bathrooms;
(a bathroom that has no shower or bathtub is counted as a half bathroom);
• bedrooms: the number of bedrooms;
• fuel: the type of fuel used for heating (gas, electric or oil).
We aim to develop a normal linear model that predicts the value of the property from its
characteristics.
Question 1 - Pairwise relationships
[Total 20 marks]
(a) [2 marks] Create a new variable log2livingArea by transforming livingArea using a
logarithm of base 2. Fit a simple linear regression of price on log2livingArea. Report the
R model summary.
(b) [2 marks] Give a quantitative interpretation of the estimated coefficient for
log2livingArea.
(c) [3 marks] Briefly explain why it is not sensible to give an interpretation of the estimated
intercept for the model in (a).
(d) [3 marks] Implement fuel as a factor variable. Produce a graphical illustration of the
relationship between price and fuel and give a brief description of the salient features in
the plot.
(e) [1 mark] In the linear model defined by the R model formula price ~ 0 + factor(fuel),
what do the estimated coefficients correspond to?
(f) [3 marks] The variable bathrooms is discrete taking only a small number of values. Fit
the two alternative linear models specified below:
M0 : price ∼ bathrooms and M1 : price ∼ 0+ factor(bathrooms)
Produce a plot of the house prices (in US Dollars) predicted by model M0 against bathrooms.
Using a different colour and plotting symbol add to the plot the predicted house prices (in
US Dollars) produced by model M1. Make sure to include a legend.
1The data is adapted from the Saratoga Houses which is, for example, available from DASL, the Data and
Story library.
2
(g) [6 marks] Perform a test for non-linearity for the simple linear regression of price on
bathrooms using a significance level of 5%. Include the relevant R output and clearly state
• the model under the null hypothesis and the model under the alternative hypothesis,
(you may use an R formula notation to specify the models);
• the test statistic and its distribution under the null hypothesis;
• the observed value of the test statistic;
• the p-value of the test;
• the outcome of the test and
• the conclusion that can be drawn from the outcome of the test.
Question 2 - Multiple regression
[Total 20 marks]
(a) [1 mark] Consider the R summary output below for a linear model with response variable
price and explanatory variables: fuel, bathrooms and log2livingArea. Which category
of fuel has been used as reference category?
Call:
lm(formula = price ~ fuel + bathrooms + log2livingArea, data = houses)
Residuals:
Min 1Q Median 3Q Max
-53.432 -11.862 0.149 12.104 58.212
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -307.701 20.688 -14.873 < 2e-16 ***
fuelelectric -5.861 1.703 -3.442 0.000608 ***
fueloil -6.269 1.952 -3.212 0.001375 **
bathrooms 14.512 1.497 9.695 < 2e-16 ***
log2livingArea 44.932 2.058 21.833 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 17.34 on 763 degrees of freedom
Multiple R-squared: 0.6421, Adjusted R-squared: 0.6403
F-statistic: 342.3 on 4 and 763 DF, p-value: < 2.2e-16
(b) [2 marks] Give a quantitative interpretation of the coefficient reported for fueloil.
(c) [1 mark] Suppose a house has a living area of 1600 square feet, 2 bathrooms and gas
fueled heating. Using the model in Question 2 (a) predict the price of the house in US Dollars.
3
(d) [3 marks] Below is the R model summary of a linear model fitted to the same data and
using the same explanatory variables as the model in Question 2 (a). Explain why it is not
a contradiction that the coefficient for electric fuel is significant in Question 2 (a) [at a 5%
significance level] but not significant in the R output below.
Call:
lm(formula = price ~ fuel + bathrooms + log2livingArea, data = houses)
Residuals:
Min 1Q Median 3Q Max
-53.432 -11.862 0.149 12.104 58.212
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -313.9700 20.6811 -15.181 < 2e-16 ***
fuelgas 6.2695 1.9521 3.212 0.00137 **
fuelelectric 0.4089 2.2500 0.182 0.85586
bathrooms 14.5120 1.4969 9.695 < 2e-16 ***
log2livingArea 44.9317 2.0580 21.833 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 17.34 on 763 degrees of freedom
Multiple R-squared: 0.6421, Adjusted R-squared: 0.6403
F-statistic: 342.3 on 4 and 763 DF, p-value: < 2.2e-16
(e) [4 marks] For the model in Question 2 (a), identify the datapoint with the highest
influence and report its Cook’s distance. Is this point influential due to being a regression
outlier, due to having a high leverage or due to both, having a high leverage and being a
regression outlier? Support your answer with appropriate evidence.
(f) [6 marks] Below are the results of a sequential ANOVA for the model in Question 2 (a),
that is for the model
LM2 : price ∼ log2livingArea+ bathrooms+ fuel
Analysis of Variance Table
Response: price
Df Sum Sq Mean Sq F value Pr(>F)
log2livingArea 1 369280 369280 1228.7391 < 2.2e-16 ***
bathrooms 1 36829 36829 122.5461 < 2.2e-16 ***
fuel 2 5343 2671 8.8885 0.0001528 ***
Residuals 763 229309 301
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
4
Consider a test for the existence of regression for the model
LM1 : price ∼ log2livingArea+ bathrooms
(Note the difference between model LM1 and model LM2.) Explain how to use the information
provided in the sequential ANOVA above to produce an ANOVA table for the test of the
existence of regression for the model LM1. Write out this ANOVA table.
(g) [3 marks] An estate agent suggests that, all else being equal, the increase in the average
value of a house when doubling the living area is dependent on the type of fuel used by its
heating system. Perform an appropriate hypothesis test at a 5% significance level to examine
whether there is evidence for this in the data set given here. Present the R output of the test
and state the conclusion that can be drawn from the outcome of the test.
Question 3 - Adding bedrooms as a predictor variable
[Total 10 marks]
(a) [3 marks] Another suggestion by the estate agent is to add bedrooms as an additional
predictor variable to the model. Produce boxplots of price grouped by bedrooms. Briefly
comment on the plot.
(b) [3 marks] Fit a linear model with response variable price and explanatory variables:
fuel, bathrooms, log2livingArea and bedrooms, where bedrooms is implemented as a
quantitative predictor, not a factor. Report the R model summary output. Assuming a 5%
significance value, what can you conclude about the explanatory variable bedrooms from the
model summary?
(c) [4 marks] Consider the model in Question 3 (b). The variance inflation factor of the jth
explanatory variable Xj is defined as V IFj = 11−R2j where R
2
j is the coefficient of determination
from a regression of Xj on the other predictors. By fitting the appropriate linear models
compute the VIF for bedrooms and for log2livingArea. Include your R code in your answer
and comment on the results. Hint: the coefficient of determination can be extracted from a
fitted model using the command summary(model)$r.squared, where model is the R object
containing the fitted model.