STAT0030 Assessment 2 — Instructions 1. Answer both questions. 2. For Question 1 you should submit online – on the course Moodle page – an electronic file containing your report for part (d). You are NOT required to submit your R script for this question. 3. For Question 2 you should submit online – on the course Moodle page: • An electronic copy of your StudentNumber.r file, containing your R script (see below), • an electronic file containing the graph in part (b) that is produced by your script, and • an electronic copy of your StudentNumber_out.txt file (see below) containing output to parts (d), (e) and (f) produced by your R script. Your R script for Question 2 should be saved in a file named StudentNumber.r, where StudentNumber is your student number. For example, if your student number is 18239004, your R script should be saved in the file 18239004.r. Your script should create an output file called StudentNumber_out.txt, where again StudentNumber is your student number. This file should contain the output from parts (d), (e) and (f) and it should include text and comments indicating what the results are (produced via appropriate use of the cat() function in your code). Any output should correspond exactly to what appears on the screen when sourcing your script file. Your program should be well commented. It should consist of a header section summarising the logical structure, followed by the main body of the script. The main body should itself contain comments. 4. All required results for Question 1, and your script for Question 2, should be submit- ted via the Moodle page for the course — use the link “ICA2: Click here to submit your assignment”. You can submit all results requested above on the Moodle via four separate electronic files: one file for Question 1, and three files for Question 2. Make sure none of the files contains your surname, as the marking must be anonymous. STAT0030 Assessment 2 — Hints 1. In general, there is not a single ‘right’ answer to each question. To obtain a good mark you should approach the questions sensibly and justify what you’re doing. Credit will be given for code that is clear and readable, while code that is inadequately commented will be penalised. You might like to use scripts cosapprox.r (Lab 1) and tablet.r (Lab 3) as models. 2. Question 1 is designed to test your ability to use the computer to learn about a real data set. This will be assessed not only on your computing skills, but also on your ability to carry out a sensible statistical analysis: material from your other courses (in particular STAT0028 or STAT0032) will be relevant here. To earn high marks for this question, you need to take a structured and critical approach to the analysis and to demonstrate appropriate judgement in your choice of material to present. 3. In Question 2, make sure that the output for parts (d)–(f) is labelled appropriately so that the individual analyses can be identified. See question 2 at the end of Lab 3 for an example of how to do this. 4. Do not edit your StudentNumber_out.txt file in any way before submitting it on Moodle. Marks will be deducted if this file does not correspond exactly to the results we obtain when we run the electronic files containing the R scripts. 5. More credit will usually be given for code that is more generally applicable, rather than tailored to a particular situation or set of data. For example, if you were asked to print out the mean age of a group of people, you could do either of the following: • Calculate the mean before you write your final script, and then insert a line cat(“Mean age is 25.3\n”) (or whatever the mean happens to be) into your script. • In your script, create an object (say xbar) that holds the mean age, and then insert the line cat(paste(“Mean age is”,xbar,”\n”)) into your script. The second approach is clearly more general and will earn more credit, since it will work for other similar data also. 6. All graphs should be clearly and appropriately labelled (giving units of quantitative variables), titled and formatted. By ‘appropriately formatted’ we mean, for example, that axis scales should be well chosen. 7. Both questions carry equal marks. 8. Refer to the feedback you received on in-course assessment 1. 2 STAT0030 Assessment 2 – Marking guidelines Questions 1 and 2 are each marked out of 30. The marks for Question 1 are roughly subdivided into the following components. 1. Exploratory analysis (10 marks): investigation and commentary of initial statistical properties, relationships, and anything of note which helps justify your choice of graphs and modelling strategy. 2. Graphical presentation (5 marks): appropriate choice of graphs and formatting. 3. Modelling strategy (10 marks): marks here will be based on a structured, justified, well-principled approach with clear and concise discussion. 4. Interpretation of final model (5 marks): commentary on how good the model is and what it means in reality in the context of the third part of Question 1d. The marks for Question 2 are roughly allocated as follows: 1. File handling and plotting, Parts (a) and (b) (5 marks): read in file; calculate and print quantities; be able to produce and format graph according to instructions. 2. Negative log-likelihood function, Parts (c) and (d) (10 marks): write negll() func- tion that works and follows good programming practice (is usable, extensible, etc); has appropriate inputs and outputs; carry out appropriate testing. 3. Optimisation, Parts (e) and (f) (10 marks): considerate and correct use of nlm() function, correctly compute standard errors. 4. Style (5 marks): efficient, elegant, extensible, well-laid out, readable code. See also examples cosapprox.r (Lab 1) and tablet.r (Lab 3) for inspiration. Marks will be deducted for code that a user would find ‘difficult’ to use. The ‘user’ here can either be (i) someone who cannot code and only knows how to run an R script and expects something meaningful to be produced on their screen or written to file; (ii) a fellow developer who would like to not only run your code but also understand how it works with a view to maybe building some of their own code on top of it. Generally, both of these user types should find your code useful and easy to use in order for you to get good marks. 3 STAT0030 Assessment 2 — Questions 1. The file cars.dat contains data from 32 cars. There are three quantitative variables: the horsepower (denoted by hp), the weight of the car in pounds (denoted by wt) and the fuel efficiency in miles per gallon (denoted by mpg). In addition there is an indicator variable tr denoting the gear transmission type, where 0 denotes automatic and 1 denotes manual. Engineers are interested in how the fuel efficiency depends on the horsepower and weight of the car and on whether this also depends on the transmission type. (a) Download the file cars.dat from the STAT0030 Moodle page. Read the data into R using read.table with the argument header=TRUE. (b) Obtain summary statistics for each quantitative variable for each transmission type, and make useful plots of the data — i.e., that are relevant to the objectives of the study. Such plots might include, but are not necessarily restricted to, pairwise scatter plots with different plotting symbols for the two transmission types. Put plots together in a single figure where appropriate and consider the possibility of using log scales for the quantitative variables. (c) Find a linear model that enables mpg to be predicted from the other variables and that is not more complicated than necessary. You may wish to consider using log transformations of one or more of the explanatory variables or of the response variable. You should consider a wide enough range of models to make your choice of model convincing and use appropriate diagnostics to assess them. But ultimately you are required to recommend a single model that is suitable for use (by engineers, for example) and to justify your recommendation. (d) Write a brief report on your analysis in three sections: I Describe briefly what you
found in your exploratory analysis in part (b) II Describe briefly (without too many technical details) what models you con- sidered in part (c) and why you chose the model you did, and III State your final model clearly and describe it in words. Remember to in- clude an estimate of the error standard deviation and say what this means also. Use your model to describe how the fuel efficiency depends on the transmission type (taking any other relevant variables into account). Also give an estimate of what would be the effect on the average fuel efficiency of increasing a car’s horse power (e.g., by 10 units or by a factor of 1.10). Give an appropriate assessment of the uncertainty in your estimate. Your report should not include all of your R commands and output, but it should include some R commands and output (for example, relating to your final choice of model) and your most useful graphs. It should be limited to at most three pages of text (including any output) and two pages of graphs. Your report should be at a level that can be understood easily by somebody with an MSc in Statistics. 4 2. The file osl.dat contains estimates of “equivalent dose” (denoted by de) and their standard errors (denoted by se) for a number of individual grains of quartz. These were obtained by a technique called optically stimulated luminescence, which gives dose measurements in units called grays (Gy). An equivalent dose estimate yi for grain i is assumed to come from a Normal distribution with mean µ and variance σ2 + s2i , where µ and σ are unknown and si is the known standard error of the dose estimate yi. We wish to write a program to estimate µ and σ by maximum likelihood for data such as these. For data (yi, si), i = 1, 2, . . . , n, the log likelihood function is l(µ, σ) = −1 2 n∑ i=1 ( log (σ2 + s2i ) + (yi − µ)2 σ2 + s2i ) + constant , (1) which is defined for σ ≥ 0 and for any µ (though in practice µ will be positive or possibly zero). (a) Download the file osl.dat from the STAT0030 Moodle page. Read it into R using read.table with the argument header=TRUE. (b) Plot a figure with two panels, giving a scatter plot of se against de in the upper panel and a histogram of de in the lower panel. Use the same de scale in both panels. Calculate the number of grains n and the mean and standard deviation of the equivalent doses and print these as text on your histogram. Label your figure and axes informatively. (c) Write a function called negll that takes two arguments (i) params, a vector containing the values of the two parameters (µ, σ), and (ii) dat, a matrix of the data pairs, and returns the negative log-likelihood, −l(µ, σ), omitting the constant term from equation (1). (d) Use your function negll to evaluate and print out the negative log-likelihood for the data in osl.dat for a few sensible values of µ and σ. (e) Use the R function nlm to find and print out the maximum likelihood estimates of µ and σ for the data in osl.dat by minimising the negative log likelihood. (f) Obtain and print out approximate standard errors for these estimates. 5
