Your solution to the coursework should be fairly concise (maximum of about 12 pages) and it should take, on average, about 20 hours to complete.
Please read all the instructions and advice given below carefully.
Your submitted solutions should all be in one document which must be prepared using LaTeX. For each part of the project you should provide explanations as to how youAdd 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.
Do not use screenshots of R code/output in your report. 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.
This coursework should hopefully help to reinforce some of the methodology you have been studying, 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 important in order to achieve a high mark for the coursework.
Add
The data for this work is related to auto insurance claims in Sweden over a particular period of time. The data was collected by the Swedish Committee on Analysis of Risk Premium in Motor Insurance. They are in the file sweden_ins_data.txt on Blackboard which contains two columns:
claims = number of claims – this will be the predictor / independent variable (or covariate)
in the regression model. We can denote it by x.
payment = total payment for all the claims in thousands of Swedish Kronor for geographical zones in Sweden – this will be the dependent / response variable in the regression model. We denote this variable by y.
There are n = 63 observations in the dataset. The simple linear regression model for the data is given by:
yi = α + βxi + ϵi i = 1,…,n
where α and β are parameters with unknown values, and the random errors ϵ1,…,ϵn are assumed to be independent N(0,σ2) random variables where the value of σ2 is also unknown.
1. Produce a scatterplot of the data and comment on any evident features, and also the apparent suitability (or not) of the above regression model for the data. [3]
2. Write your own function in R to fit, using least squares, a simple linear regression model to a set of data comprising a response variable, y and a single covariate, x. Your function should have only two arguments – a vector of data fory and a vector of data for x.
The output from your function should be in the form of a list object which comprises:
a vector containing the parameter estimates ˆAdd α and βˆ; a vector containing the fitted values; a vector of the estimated errors (or residuals); a scalar giving the value of the residual degrees of freedom.
The four items in the output should all be calculated manually (DIY) within the function itself using the input data.
To complete this part, run you function using the Swedish insurance data. [8]
3. Using the output generated in part 2:
(i) Report the estimated values of α and β that have been calculated. Superimpose the fitted regression line on to a scatterplot of the data and comment on the results. [3] (ii) Using an unbiased estimator, estimate the value of the error variance, σ2. [2] (iii) Use an F-test to test
H0 : E[Y |x] = α vs H1 : E[Y |x] = α + βx at a 5% significance
level. Report your conclusions. [2]
(iv) Calculate a 95% confidence interval for E[Y |x = 80]. [2]
In the next parts we will use the residuals to examine the plausibility of the assumptions made about our regression model. 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 so that they are all on the same scale. To obtain the standardised residuals from the fitted model object created in R we can use therstandard function which performs the standardisation in a special way that recognises the unequal variances.
4. Using diagnostic plots of the standardised residuals a) against the fitted values and bAdd ) against the predictor, make comments about the validity of the assumptions that were made regarding the model that has been fitted to the data. [4]
5. Manually construct (rather than using an existing R function for this purpose) a Normal quantile-quantile plot of the standardised residuals and superimpose a suitable reference 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. [3]
6. We now wish to carry out a Kolmogorov-Smirnov (KS) test to assess whether the distributionof the standardised residuals is N(0,1). Find the value of the KS test statistic and also that standardised residual value where the absolute difference between the empirical and N(0,1) cdfs is a maximum. [3]
7. Produce a plot containing the empirical cdf of the standardised residuals and the N(0,1) cdf and indicate on it the point at which the maximum difference between the curves occurs.
[3]
8. 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 using the same sample size as that of the
Swedish insurance data. [3] Run your function and use the results to plot a histogram of the estimated sampling distribution with a superimposed kernel density estimate of this 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]
Coursework 3 – Regression and Goodness of fit Add
