Coursework 3 – Regression and Goodness of fit
MATH20811 Practical Statistics : Coursework 3
The marks awarded for this coursework constitute 40% of the total assessment for the module.
Your solution to the coursework should be a consice report (max 10 pages) and it should take, on average, about 20 hours to complete.
The submission deadline is 10am on Monday 23 December 2019.
Late Submission of Work: Any student’s work that is submitted after the given deadline will be classed as late, unless an extension has already been agreed via mitigating circumstances or a DASS extension. The following rules for the application of penalties for late submission are quoted from the University guidance on late submission document, version 1.3 (dated July 2019):
“Any work submitted at any time within the first 24 hours following the published submission deadline will receive a penalty of 10% of the maximum amount of marks available. Any work submitted at any time between 24 hours and up to 48 hours late will receive a deduction of 20% of the marks available, and so on, at the rate of an additional 10% of available marks deducted per 24 hours, until the assignment is submitted or no marks remain.”
Your submitted solutions should all be in one document. This must be prepared using LaTeX. For each part of the question you should provide explanations as to how you completed what is required, show your workings and also comment on computational results, where applicable.
When you include a plot, be sure to give it a title and label the axes correctly.
When you have written or used R code to answer any of the parts, then you should list this R code after the particular written answer to which it applies. This may be the R code for a function you have written and/or code you have used to produce numerical results, plots and tables. R code should also be clearly annotated.
Avoid using screenshots of R code/output. Instead, to include R code use the verbatim environment and summarise R output in tables using the table environment, as demonstrated in the solution of Example Sheet 2.
Your file should be submitted through the module site on Blackboard to the Turnitin assessment in the Coursework folder entitled “MATH20811 CW3” by the above time and date. The work will be marked anonymously on Blackboard so please ensure that your filename is clear but that it does not contain your name and student id number. Similarly, do not include your name and id number in the document itself.
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This coursework should hopefully help to reinforce some of the methodology you have been study- ing, as well as the skills in R you have been developing in the module. Correct interpretation and meaningful discussion of the results (i.e. attempt to put the results into context) are as important as correct calculation of the results, in order to achieve a high mark for the coursework.
Coursework 3 – Regression and Goodness of fit
1. The data for this question were taken from the public records. They comprise the prices of a random sample of 1728 homes in Saratoga County, New York, USA in 2006, together with the values of 15 other variables which may be influential in explaining the price of a house in this area.
To obtain the data, firstly install the R package mosaicData using the install command in the Packages menu in R. Then run the following R code to load the data into your R workspace:
library(mosaicData)
data(SaratogaHouses)
names(SaratogaHouses)
Read the data into R using the above commands.
The first part of the coursework asks you to fit multiple regression models to the data. The house prices in the SaratogaHouses data object are in the variable price, while the 15 covariates are:
lotSize, age, landValue, livingArea, pctCollege, bedrooms, fireplaces, bathrooms, rooms, heating, fuel, sewer, waterfront, newConstruction, centralAir.
However, we propose to use the following variables instead of the original variables price, lotSize, age and landValue in regression models:
tprice=price/1000
tlotSize=sqrt(lotSize)
tage=sqrt(age)
tlandValue=landValue/1000
Please note that no other transformations to any of the variables should be used.
(a) Fitaregressionmodel(callthislm1)topredictthetransformedhousepricesin(tprice) using the 12 original predictors and the 3 transformed predictors which are defined above.
Using the relevant results from the summary of the model, informally comment (with justifications) on which variables appear to be significant in predicting the response (you may assume a 5% significance level).
Note: For categorical variables, if at least one of the categories is significantly different to the reference category, then the variable should be retained in the model. [3]
Coursework 3 – Regression and Goodness of fit
(b) Remove the variables which you identified as non-significant in (a) and refit the model (call this lm2).
Using dummy variables to represent the categorical variables, write down the model in algebraic form, explaining your notation. [4]
(c) Define the hypotheses for an appropriate test to compare models lm1 and lm2, in terms of βd, where βd is a vector containing the coefficients of the variables not included in lm2 (βd needs to be defined explicitly). Carry out the test (you may use an R built-in function) and interpret the results. [4]
(d) Based on summary(lm2), explain how each variable affects the response and discuss any other findings about the model. [5]
(e) Using diagnostic plots of the residuals against the fitted values and the various predic- tors, assess whether any of the model assumptions appear to be violated. [3]
In the next parts we will examine the assumption that the errors in our model lm2 are Normally distributed. It can be shown theoretically that the estimated errors from a fitted model (the residuals) have differing variances which depend on the values of the covariates. Consequently, we will work with the standardised residuals and check if their underlying distribution is consistent with a N(0,1) probability model. To obtain the standardised residuals from model lm2 we can use the following code which performs the standardisation in a special way that recognises the unequal variances.
std.res=rstandard(lm2)
(f) Manually construct (rather than using an existing R function for this purpose) a quantile-quantile plot of the standardised residuals and superimpose a suitable refer- ence line to help gauge Normality. Comment on the form of your plot and say whether you think that Normality was a tenable assumption or not. [4]
(g) We now wish to carry out a Kolmogorov-Smirnov test to assess whether the distribution of the standardised residuals is N(0,1). Write code in R to calculate (manually) the value of the KS test statistic and find the std residual value where the difference between the empirical and N(0,1) cdf’s is a maximum. [4]
(h) ProduceaplotcontainingtheempiricalcdfofthestandardisedresidualsandtheN(0,1) cdf and indicate the point at which the maximum difference between the curves occurs. [4]
(i) Write a function in R to simulate the sampling distribution of the Kolmogorov-Smirnov test statistic when the N(0,1) null distribution is true. [4]
Run your function and use the results to plot a histogram of the estimated sampling distribution. [2]
Use your simulated test statistic values to obtain an estimated 5% critical value for your test. Compare your observed value with this and report your conclusions. [2]
(j) If you found significant evidence against the assumption of Normally distributed errors in (i), how does this impact your conclusions about the significance of the predictors in (a)? [1]
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