MATH3772/5772
Multivariate Analysis Practical
A data file athlrecs .txt containing country record times for men’s track events for 55 coun- tries immediately prior to the 1984 Olympic Games can be found in Minerva. It contains the following variables:
Country: Name of country
m100: Record time for 100m race in seconds
m200: Record time for 200m race in seconds
m400: Record time for 400m race in seconds
m800: Record time for 800m race in minutes
m1500: Record time for 1500m race in minutes
km5: Record time for 5000m race in minutes
km10: Record time for 10000m race in minutes
mara: Record time for the Marathon (approx. 26 miles) in minutes
status: 1 for developed countries; 3 for third world countries
For the purposes of this practical, just concentrate on the 4 races m100, m200, m400, m800. Simultaneous confidence intervals may be helpful for parts 2 and 3.
1. Examine whether it is reasonable to assume that the data can be described as multivariate normal.
2. For the whole set of 55 countries, investigate the hypothesis μ800 = 2μ400 = 4μ200 = 8μ100 . This hypothesis says that the speed of the record runs over that range of distances is constant (after first ensuring the units of time are the same for all races). To carry out this test you may find it convenient to make a linear transformation of the data. Let X denote a 55 × 4 data matrix for the races of interest. Find a matrix A(3 × 4) such that if the above hypothesis holds, then the mean of the data matrix Y = XAT is 0.
3. The countries have been split (somewhat arbitrarily) into developed countries (status = 1) and third world countries (status = 3). Next investigate the hypothesis that the 4-dimensional mean vector for race times is the same for the two groups of countries.
4. [Level 5 only.] Carry out a kmeans clustering of the data into k = 2 clusters. Compare the resulting clusters to the partitioning of the data by the status variable.
Some useful commands in R
ath=read.table("athlrecs.txt",header=T)
attach(ath)
x=cbind(m100, m200, m400, m800) # create a data matrix for the 4 races
# using all 55 countries
x1=x[status==1,] # define a 23 x 4 submatrix of developed countries
x2=x[status==3,] # define a 32 x 4 submatrix of third world countries
General assessment information
This practical is included in the assessment for MATH3772/5772. It comprises 20% of your overall mark for the module.
Your report should be submitted by 5pm on Monday 2 December 2024 in Gradescope (in Min- erva go to Assessment and Feedback - > Submit My Work - > Gradescope, and look for “Practical”).
You are encouraged to collaborate with other students, but the work that you submit must be done independently. Furthermore, the use of artificial intelligence (e.g. ChatGPT) is prohibited. Serious consequences will result if copying or use of AI is detected.
There will be two available practical sessions to give opportunity for you to ask me questions. This will be on Thursday 21 November 2024 at 9-11am in the Psychology Computer Cluster 1.43, and on the same day at 12noon-2pm in the EC Stoner Computer Cluster 6.61. I anticipate that students will bring questions about both statistical questions (what they are supposed to be doing) and computing problems (how to get an R program to work).
The analyses in the practical should be performed using the statistics program R.
Writing up
Write a short report (in Word or Latex, must be typed) outlining the analyses you have per- formed, including discussions of how appropriate the techniques were and explaining the results.
The report should be aimed at someone who has a basic knowledge of statistics and hypothesis tests. The report should contain relevant plots (which should be explained in the report) that can be copied from R. The report should not contain any R commands or output directly copied and pasted from the R console. The aim of this practical is to explain the analyses that you have performed and giving R commands does not do this. The R commands and relevant outputs should be put in the appendix (below). The report should be word processed and should not exceed 6 pages of A4 in total, plus two extra pages for Level 5, including any plots that you wish to show. The texts in the report should be single spaced with at least 11 pt font size.
In addition to the report, attach an appendix that includes all of the R commands that you have used and the associated output. There is no need to reproduce any plots in the appendix. You also should include an Academic Integrity Form (available in Minerva) before submitting your report. The appendix and the academic integrity form do not count towards the number of pages in your report.