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M. PHIL. IN STATISTICAL SCIENCE
Tuesday 12 June 2007 9 to 12
APPLIED STATISTICS
Attempt THREE questions. There are FOUR questions in total. The questions carry equal weight.
This is an ‘Open Book’ examination, involving the use of the Statistical Laboratory’s network of workstations. Candidates will receive this paper at 9.00am on Tuesday 12 June, and must hand in their scripts to the Chairman of Examiners by 1.00pm on Friday 15 June.
The data sets will be emailed to candidates on Tuesday 12 June.
(The Statistical Laboratory Computer Officer and an Examiner will normally be available for consultation if required between 9.00am and 4.30pm on these four days.)
Each candidate should submit his/her script with a signed statement that the work has been carried out without any collaboration with others.
The scripts may be handwritten. Candidates are requested to submit at most 25 pages in total. They are advised that the total work set should take between 4 and 6 hours. STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS Cover sheet None Treasury Tag Script paper
You may not start to read the questions
printed on the subsequent pages until
instructed to do so by the Invigilator.
1 The Times, 5 June 2006, published their Good University Guide giving data on nine aspects of activity for 109 universities. The nine aspects are
Satis a score of student satisfaction with teaching taken from a national student survey (maximum possible score is 20) Res average quality of research per staff member (maximum possible score is 7) Entry average A level entry score Stud.staff student:staff ratio Lib library/computer spending in £ per student Facil facilities spending in £ per student GoodHons percentage of graduates with a First or Upper Second Class Honours degree Prosp percentage of graduates in “graduate” employment or further study Complet percentage of students who graduate
A few lines of the data are given below. The final column is the overall score which is calculated form the above nine aspects of activity
Rank Inst Satis Res Entry Stud:staff Lib Facil GoodHons Prosp Complet Total 1 Oxford NA 6.5 511.7 13 1656 364 88.4 74.8 97.7 1000 2 Cambridge NA 6.6 525.1 11.9 1129 425 84.6 86.9 98.9 973 3 ImperialCollege 14.4 6.4 468.2 9.4 1230 481 75 83.8 96 878 . . . 108 Luton 14.1 1.8 185.7 23.2 606 229 48.4 41.7 74.8 302 109 ThamesValley NA 0.5 202.5 23.7 398 57 48.8 60.1 68.6 281
(i) Summarise the data using appropriate plots and tables. (ii) Investigate how the proportion of graduates with a First or Upper Second Class Honours degree depends on the other eight aspects of activity, carefully explaining any model selection techniques that you use in your analysis and illustrating the use of boxcox( ). Summarise your conclusions.
(iii) Discuss briefly the choice of the nine variables included in the overall score for the above ranking of universities. What other variables might have been included?
Applied Statistics
Diastolic BP 81-85 177.3 187 Diastolic BP 86-90 286.0 319 Diastolic BP 91-112 401.6 549
BP=Blood Pressure
The aim is to investigate possible multiplicative effects of systolic and diastolic blood pressure on the mean number of deaths relative to E.
(i) Carry out exploratory plots and summaries of the data. (ii) Assume A is Poisson with mean μE. Treating systolic and diastolic bloody pressure as factors, fit a model for μ in terms of the levels of systolic and diastolic blood pressure. (iii) The mortality ratio is defined as A/E. Using your preferred model in (ii), give an estimate and standard error for the expected mortality ratio when systolic and diastolic blood pressures lie in the ranges 126-130 and 91-112 respectively, and comment briefly.
Applied Statistics
3 The tables below are taken (slightly edited) from the Healthcare Commission annual health check (October 2006). Table 1 shows the Quality of Service ratings (Excellent, Good, Fair, Weak) for Healthcare Trusts in four regions (North, Central, London and South East, South West)
TABLE 1
Excellent Good Fair Weak Total number of Trusts North 10 63 83 14 170 Central 3 74 86 9 172 London & South East 5 39 65 11 120 South West 5 30 54 16 105
(i) A Trust is satisfactory if it is classified as Excellent or Good, and is otherwise unsatisfactory. Investigate whether the proportion of satisfactory Trusts varies over the regions using (a) a χ^2 test and (b) binomial modelling. (ii) For the original four Quality of Service ratings in Table 1, use Poisson models to investigate whether there are differences in Quality of Service ratings between regions.
(iii) Suppose that further examination of the data revealed that Table 1 was obtained by adding together the results for four different types of Trust. For the North and Central regions these results are shown in the tables below.
Type of Trust Excellent Good Fair Weak
Acute & Specialist North 5 31 13 3 Central 1 19 26 1 Ambulance North 0 4 0 4 Central 0 5 1 3
Primary care North 3 24 59 6 Central 2 42 50 5
Mental Health North 2 4 11 1 Central 0 8 9 0
What do you conclude now about differences between the North and Central regions in the proportion of satisfactory Trusts?
Applied Statistics [TURN OVER
the data.
(a) Begin by exploring the data using summary statistics, cross-tabulations and plots. Note anything that is helpful in understanding the data. What are the average success rates overall, by branch, and by gender?
(b) By appropriately modelling the data and with the use of plots, investigate the functional relationships (possibly non-linear) of the variables on the probability of a successful transaction. Include generalised linear models and generalised additive models in your analysis.
(c) Determine whether the opinions of the panel are substantiated by the data collected. Interpret and make inferences on your findings. Note that you should compare and contrast the different models you have fitted and then decide on the “best” model to use to base your final inferences on. You need to justify statistically the choice of “best” model.
(d) Is the proportion of successful transactions made a good measure of trader’s performance? What other measures of performance could be used?
END OF PAPER
Applied Statistics
