You should prepare a report on the exercise of no more than FIVE A4 pages. It MUST be in the standard format; with NO extra pages added (i.e. any material after page 5 will NOT be marked). Statistical analyses should be carried out using R, and the report MUST be prepared as a LATEX document.
• The following points give guidance on presentation when preparing reports on statistical analyses and projects.
• A report should include the name of the course, a relevant report title, the date of submission and the name of the author.
• The presentation should be coherent and neat.
• Discussion with other students is encouraged, but the report should be your own work.
• Reports should be divided into clearly-titled sections, and include
(i) An introduction, describing the data and the purpose of the study for which they were collected;
(ii) An account, usually in more than one section, of the statistical methods you have applied, and any numerical or graphical results;
(iii) A statement of conclusions related to the purpose of the study and aspects of the data you have been asked to investigate.
• R and similar commands should not be included unless they are specifically requested: in that case, the sequence of commands might go in an appendix.
• Output from statistical packages should be reproduced very selectively and with appropriate explanation.
• Any notation required should be defined and be used consistently.
• Any statistical models you fit should be carefully defined.
• Graphs and tables should be used where appropriate, and be well labelled with titles, axes and units of measurement where relevant. It should be possible to understand them independently of the text. If more than one graph or table is included they should be numbered.
• In tables, horizontal lines are needed only at the top and bottom and below headings; vertical lines are seldom required, except perhaps at the left and right. Columns of text look best if aligned to the left (unless deliberately indented); columns of numbers should be aligned on their decimal points. The numbers of decimal places used should normally be the same for figures which are to be compared (such as sums of squares or estimates for levels of a factor). Avoid excessive numbers of significant figures: three (or at most four) is usually sufficient.
A- Use R or R Commander and WORD or LATEX to produce brief reports on the following sets of data.
1. A food company carried out a study into how well the vitamin niacin could be measured in bran products. Of 36 samples of its bran flakes, 12 were enriched with 8 mg of niacin, 12 were enriched with 4 mg of niacin, and 12 were left unenriched. Four laboratories took part in the study. The company was interested in differences in measurement between the laboratories, and whether they were consistent in estimating differences in the amounts of niacin. Three bran samples of each of the three types were sent to each laboratory, with no indication of how much each was enriched. Each laboratory measured the niacin in its nine samples in random order according to the company's instructions. The measurements are shown below, and are also in the file niacin.csv. The 'factors' laboratory numbers and amounts of enrichment will need to be created.
(a) Examine whether the laboratories are consistent in their measurements of the differ-ences in the amounts of niacin.
(b) Examine the validity of the assumptions made in your analysis.
(c) Tabulate the means for the laboratories and the amounts of niacin enrichment.
(d) Explain how to calculate a 95% confidence interval for the difference in effects for any two laboratories or any two amounts of niacin enrichment.
(e) Test the hypothesis that the method of measurement correctly estimates the difference between the amounts of niacin in samples with zero and 4 milligrams of added niacin.
|
Laboratory
|
0 mg
|
Niacin enrichment
4 mg
|
8 mg
|
|
1
|
8.03
|
7.35
|
7.66
|
11.50
|
10.10
|
11.70
|
15.10
|
14.80
|
15.70
|
|
2
|
8.50
|
8.46
|
8.53
|
11.75
|
12.88
|
12.64
|
16.20
|
16.16
|
16.48
|
|
3
|
7.31
|
7.85
|
7.92
|
11.11
|
11.00
|
11.67
|
15.00
|
17.00
|
15.50
|
|
4
|
8.82
|
8.76
|
8.52
|
12.90
|
12.00
|
13.50
|
17.30
|
17.60
|
18.40
|
2. The data below (and in the file crackgrowth.csv) come from a study to investigate the effects of cyclic loading and environmental conditions on the growth of fatigue cracks in a construction material at a constant stress: the response is the rate of crack growth.
(a) Investigate the effects of the loading frequencies and the environments on the growth of fatigue cracks.
(b) Examine the validity of the assumptions made in your analysis.
(c) Examine also whether using the logarithm of crack growth rate simplifies the analysis.
| Loading frequency |
Environment |
| Air |
Water |
Salt water |
| 10.0 |
2.29 |
2.47 |
2.48 |
2.12 |
2.06 |
2.05 |
2.23 |
2.03 |
1.90 |
1.93 |
1.75 |
2.06 |
| 1.0 |
2.65 |
2.68 |
2.06 |
2.38 |
3.20 |
3.18 |
3.96 |
3.64 |
3.10 |
3.24 |
3.98 |
3.24 |
| 0.1 |
2.24 |
2.71 |
2.81 |
2.08 |
11.00 |
11.00 |
9.06 |
11.3 |
9.96 |
10.01 |
9.36 |
10.4 |
B- Use R Commander (or R) and LATEX to produce a report on the following problem.
Problem concerned the relationship between the ages of female house mice and the total masses of their eye lenses. The file mouseeyes.csv contains the masses (in milligrams) of the pairs of lenses extracted from two female and two male house mice at each of 22 ages (in weeks). For the females, the relationship of total mass to In(age) appears to be approximately linear, but the variance of total mass increases with age. The fit of In(total mass) to In(In(age)) is also approximately linear, but with more constant variance.
(a) Do the transformations to 1n(total mass) and ln(ln(age)) appear to be suitable for the male mice?
(b) Assuming that these transformations are suitable for both sexes. fit regressions of trans-formed total eye-lens mass on transformed age with different intercepts but a common slope for males and females, and with different intercepts and slopes. (To allow different intercepts, include a dummy variable equal to 0 for females and 1 for males; to allow different intercepts and slopes, include also the product of this variable with ln(In(age)).]
(c) Examine whether the transformed data are consistent with (1) parallel linear regressions for males and females (ii) identical regressions for the two sexes.
(d) If the ages of the mice were expressed in days rather than weeks, would the fit of the models used in (b) be better, worse or essentially the same?
C- Use R Commander (or R) and LATEX to produce a report on the following problem.
Four methods for treating blood plasma were compared in a randomized block experiment by applying them to plasma samples taken from eight subjects and recording the clotting times (in minutes), with the results given below. The file bloodclot.csv contains the clotting times. Investigate whether there is evidence of differences between the effects of the treatments, and describe any differences you find.
| Subject |
Treatment |
|
1 |
2 |
3 |
4 |
| 1 |
8.4 |
9.4 |
9.8 |
12.2 |
| 2 |
12.8 |
15.2 |
12.9 |
14.4 |
| 3 |
9.6 |
9.1 |
11.2 |
9.8 |
| 4 |
9.8 |
8.8 |
9.9 |
12.0 |
| 5 |
8.4 |
8.2 |
8.5 |
8.5 |
| 6 |
8.6 |
9.9 |
9.8 |
10.9 |
| 7 |
8.9 |
9.0 |
9.2 |
10.4 |
| 8 |
7.9 |
8.1 |
8.2 |
10.0 |