UNIVERSITY OF OTAGO EXAMINATIONS 2008

Transcription

1 UNIVERSITY OF OTAGO EXAMINATIONS 2008 STATISTICS Paper STAT 242/342 Multivariate Methods (TIME ALLOWED: THREE HOURS) This examination paper comprises 25 pages. Candidates should answer questions as follows: You should attempt all questions for full marks. All questions are of equal value. The following material is provided: 2 A table giving some percentage points of the distribution is on page 11. Appendices of computer printouts are in an attached booklet on pages 12 to 25. Candidates are permitted copies of: Nil. Other instructions: There is no restriction on the model of calculator that may be used, but no device with communication capability shall be accepted as a calculator. (subject to inspection by examiners)

2 2 STAT 242/ As part of an oil exploration programme, crude oil samples were collected from four regions of sandstone, (1) Wilhelm, (2) Sub-Mulinia, (3) Mulinia and (4) Upper Mulinia. Five variables, X1 to X5, were measured on 49 samples: fourteen samples from Wilhelm, eleven from Sub-Mulinia, ten from Mulinia and fourteen from Upper Mulinia. The variables X1, X2 and X3 were trace element measures while X4 and X5 were hydrocarbon measures, in appropriate units. The data are below. WILHELM (1) MULINIA (3) X 1 X 2 X 3 X 4 X 5 X 1 X 2 X 3 X 4 X UPPER MULINIA (4) X 1 X 2 X 3 X 4 X SUB-MULINIA (2) X 1 X 2 X 3 X 4 X Sections of an SPSS printout for a discriminant function analysis on these data collected from the four regions are in Appendix 1. (a) Write down the equations of the first two discriminant functions, Z 1 and Z 2, in terms of the five original variables. (Round to two decimal places.) (b) What percentage of the total discriminating power is located in each of Z 1 and Z 2? (c) What do you conclude from the first Wilk s Lambda value of 0.127? (d) Calculate the values of the test statistic 2 j m 1 1 n ( p m) log (1 ) i 1 i 2 e j with p + m 2j degrees of freedom for j = 1 and 2 directly from the Chi-square values in the printout. What do you conclude from these values? Continued

3 1. continued 3 STAT 242/342 (e) Write down the canonical correlation for Z 1. What percentage of the variability between the four sandstone groups is accounted for by the values of Z 1? What percentage is accounted for by the values of Z 2? (f) A further sample has X 1 = 6.3, X 2 = 26.0, X 3 = 0.50, X 4 = 4.25 and X 5 = 3.2. Calculate the values of Z 1 and Z 2 for this sample. Which of the four regions is this sample likely to belong to? (g) Overall, how successful has this analysis been in classifying the samples to the four regions? (h) One of the samples from Sub-Mulinia has X 1 = 4.2, X 2 = 35.0, X 3 = 0.50, X 4 = 5.69 and X 5 = (It is case number 20 in the data file.) By reference to the Casewise statistics table write down the probabilities that this sample is correctly allocated to Sub-Mulinia and is incorrectly allocated to Mulinia. (i) Explain what would be achieved by rotating the discriminant functions.

4 4 STAT 242/ The aim of a study of hospital-acquired infection was to determine whether infection surveillance in the form of increased cultures and X-rays have reduced the rate of hospital-acquired infection. The data consist of a random sample of 87 hospitals with the following variables measured on each. X1 : Average length of stay of patients in hospital (in days) X2 : Average number of cultures taken per patient to be tested for infection X3 : Average number of X-rays taken per patient to test for infection X4 : Number of hospital beds available X5 : Average number of patients per day in hospital X6 : Number of full time registered nurses X7 : Percentage of 35 specialist services provided by the hospital Risk : Average rate of hospital-acquired infection (l = low; h = high) X1 X2 X3 X4 X5 X6 X7 Risk h l l h h h h l An SPSS printout for a factor analysis using the principal component extraction method, two factors and a varimax rotation is in Appendix 2. (a) (b) (c) (d) (e) (f) (g) (h) Do the KMO and MSA values suggest the variables will be satisfactory for a factor analysis? What do the communalities measure? What percentage of the total variation is explained by the two factor solution? How does the variation for each of the factors change when rotated? How would you describe a hospital with (i) a high score on rotated factor 1 (ii) a high score on rotated factor 2? From the plot of the scores of factor 1 against factor 2 identify the features of the hospitals with a low (l) average rate of hospital-acquired infection. From the plot of the scores of factor 1 against factor 2 identify the features of the outliers. Using the result from (g) do you believe increased infection surveillance in the form of increased cultures and X-rays has reduced hospital-acquired infection? Explain.

5 5 STAT 242/ A study was set up to investigate the profiles of people in different types of employment. The outcomes were the work status classes (status) of 879 respondents where the values of status were 1 if unemployed, 2 if in part time work and 3 if in full time work. The sex (1 if male, 2 if female ) and age (in years) of each respondent were also recorded as well as the education level, edlevel, with value 1 for education to the national school leaving age, 2 for further secondary education completed to year 13 and 3 for tertiary education. The data had the following form. status sex age edlevel Use the SPSS Printout in Appendix 3 to answer the following: (a) (b) (c) (d) The model status = intercept + sex + edlevel was first fitted to the data. Write down the equations of the two logit models. State, with reasons, whether both sex and edlevel are needed to model the employment status of the respondents. Explain what the estimated parameters for the model in (a) tell us about the probabilities of people being in the different types of employment. Use the fitted equations in (a) to find the probabilities that (i) a man with education only to leaving age will be unemployed and (ii) a woman with education only to leaving age will be unemployed. The covariate age is added to the model in (a). The printout for this model is also in Appendix 3. Identify the changes to the two fitted equations in (a) as a result of adding age.

6 6 STAT 242/ Cluster analysis is a multivariate technique for identifying unknown groups of cases. (a) (i) Illustrate the nearest neighbour (single linkage) and furthest neighbour (complete linkage) methods of clustering using the following matrix of distances between cases A, B, C and D. A B C D A 0 B 3 0 C D (ii) Sketch the dendrogram in both cases, indicating the levels at which clusters merge. Describe an advantage and a disadvantage of standardising the original data before establishing a matrix of distances. (b) The table below has the record times for seven athletic events for women in 32 countries. The events are the 100m, 200m, 400m, 800m, 1500m, 3000m runs and the marathon with times in seconds for the first three events and in minutes for the next four. An SPSS printout for two cluster analyses based on standardised data is in Appendix 4. The question of interest is whether the countries can be grouped into informative sport activity clusters on the basis of these athletic records. Country 100(s) 200(s) 400(s) 800(m) 1500(m) 3000(m) marathon(m) Argentina Australia Belgium Brazil Burma Canada Chile China Colombia Czechoslovakia Denmark Finland France Germany Great Britain Hungary India Indonesia Italy Japan Kenya Malaysia Mexico New Zealand Poland Romania Russia Singapore Spain Sweden Thailand USA Continued

7 4. (b) continued 7 STAT 242/342 (i) Explain why it is necessary to standardise these seven variables prior to the cluster analysis. (ii) List the countries in the four clusters suggested by the furthest neighbour (complete linkage) dendrogram. (iii) What cluster feature is illustrated with the nearest neighbour dendrogram? Why is this feature a problem? (iv) Which country is closest to the USA (using the furthest neighbour (v) dendrogram)? What are the features of the cluster which includes New Zealand (using the furthest neighbour dendrogram)? (vi) Which cluster of countries has the fastest record times (using the furthest neighbour dendrogram)? (c) With large data sets it can be difficult identifying the features of each cluster in a cluster analysis. Describe briefly how you might use discriminant function analysis to identify the features of clusters in terms of the original variables.

8 8 STAT 242/ (a) Multidimensional scaling is an ordination or scaling procedure for investigating the relationships between a set of objects. (i) Explain the iteration procedure for developing an optimal relationship between the objects. You should discuss the initial distances ( ij ), the configuration distances (d ij ), the disparities ( d ˆij ) and the stress measure (without quoting the formula). (ii) Explain the difference between the direct and indirect methods of obtaining the initial distances ij. (iii) Distinguish between metric and non metric scaling. (b) A market research firm conducted a study concerning the perceived similarity for different types of food. The average values (over 50 respondents) for each of the foods on the following three bipolar adjectives: Bland (1) Spicy (7), Light (1) Heavy (7) and Low calories (1) High calories (7) are in the table below: Food Bland/Spicy Light/Heavy Low/High Calories Japanese Cantonese Szechuan French Mexican Mandarin American Spanish Italian Greek Thai Indian An SPSS printout for a metric multidimensional scaling in two dimensions is in Appendix 5(b). (i) How good is the constructed map of foods? (ii) Interpret the two dimensions (possibly rotating axes). How would you characterise a food with a high score on your dimension 1? (iii) Identify groupings of common types of food. (c) A botanist is comparing vegetation at 10 sites. The data take the following form where 1 indicates presence and 0 indicates absence of a plant species at a site. Continued Species Site Site Site Site Site Site Site Site Site Site The data for any two sites can be summarised in a table of counts (shown on the next page) of the number of times two species are present (a), only one species is present (b or c) or both species are absent (d).

9 9 STAT 242/ (c) continued Species i Species j Present Absent Total Present a b a + b Absent c d c + d Total a + c b + d p Two measures of distance between any two sites based on the plant species are: Simple Match measure: (a + d)/p Lance-Williams measure: (b + c)/(2a + b + c) (i) Calculate the values of these measures for sites 2 and 3. (ii) Explain why one of these measures is a dissimilarity measure and the other is a similarity measure. What are the ranges of the possible values for these measures? (iii) An SPSS printout for a two-dimensional metric multidimensional scaling on the sites using the Lance-Williams measure is in Appendix 5(c). Identify groups of sites with similar vegetation.

10 10 STAT 242/ (a) Describe the objectives of a correspondence analysis, including a brief description of the type of data analysed by this procedure. (b) The data in the table below are from a lifestyle and leisure survey in Norway. The two classifications reported are occupation of respondent with six categories and activities engaged in during the previous year with ten categories. Occupation Activity Manual White collar employees Professional Farmer Student Retired Total Sport event Cinema Dance/disco Café/restaurant Theatre Classical concert Pop concert Art exhibition Library Church service Total The SPSS printout for a correspondence analysis in Appendix 6 should be used to answer the following questions. (i) (ii) List conclusions from the summary table. Your report should discuss the implication of the chi-square value, the inertia values, and the number of dimensions which should be investigated in the analysis. Interpret dimension 1 for both the activities and the occupations. (iii) Interpret dimension 2 for both the activities and the occupations. (iv) Summarise the main features of the association between the activity classes and the occupation classes.

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12 12 STAT 242/342 APPENDIX: SPSS Printouts for Questions 1 to 6

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