Linköpings Universitet
IDA/Statistik
BW
732A37 Multivariate Statistical Methods, 6 hp
Exam in Multivariate Statistical Methods, 2013-12-18
Time allowed:
Allowed aids:
Examinator:
Grades:
8-12
Calculator, The book: Johnson, Wichern: Applied Multivariate
Statistical Analysis. Notes in the book and Copy of the book are
allowed.
Bertil Wegmann
A=22-24 points, B=19-21p, C=16-18p, D=13-15p, E=10-12p
Provide a detailed report that shows motivation of the results.
GOOD LUCK!
_________________________________________________________________________________________
1
Let the random vector X  ( X1, X 2 )' have sample covariance matrix
 2  2
S  
.
 2 8 
Derive the sample correlation matrix, R, from S. Then, by using the sample
correlation matrix R, determine the principal components and find the
proportion of the total sample variance of X explained by the first
component.
3p
2
Blood pressures were collected for racketlon athletes before and after a
certain fitness programme. Each blood pressure was measured by both the
systolic and diastolic blood pressures. Let X jk  ( X Sjk , X Djk ) be the systolic
and diastolic blood pressures for athlete j before ( k =1) and after ( k =2) the
fitness programme. The observation vectors x S , x D  are:
Athlete number
Before programme( k =1):
After programme( k =2):
1
2
3
4
5
(120, 70), (110, 65), (130, 80), (125, 70), (130, 80)
(115, 65), (110, 60), (120, 75), (120, 60), (125, 70)
Let D j  X j1  X j 2 be the difference between before and after the programme
for athlete j and assume that the differences for all athletes represent
independent observations from an N  ,  d2  distribution.
a) Evaluate T 2 , for testing, H 0 :  ’ = [0 0].
3p
b) Specify the distribution of T 2 for the situation in a) and test H 0 at
significance level   0.05. State your conclusions clearly.
2p
3
Observations on two responses (X1 = height of a seed, X2 = thickness of a
seed (in centimeters)), were collected from a farmer’s plant. The observation
vectors x1 , x2  are:
Treatment 1:
Treatment 2:
(7.2, 0.7), (6.8, 0.6), (7.4, 0.7), (7.0, 0.6)
(7.1, 0.6), (6.8, 0.5), (7.0, 0.5)
Assume for each treatment that the data on  X1, X 2  is a random sample from
a bivariate normal population.
a) Construct the one-way MANOVA table for comparing population mean
vectors between treatments.
4p
b) Evaluate Wilks’ lambda,  * , and use Table 6.3 to test for differences
in the population mean vectors between the treatments. Use
significance level   0.01 for the test.
2p
4
Let X be distributed as
where
(
and
)
a) Find the conditional distribution of  X 1 , X 2  , given that X 3  1 .
2p
b) Are  X 1 , X 2  and X 3 independent? Explain.
1p
5
Consider a bivariate normal distribution with mean vector
sample covariance matrix
and
 3
 3
.
S  


3
5


Determine the constant-density contour that contains 90 % of the
probability. Hence, specify the midpoint and the axes of the constant-density
ellipse, and sketch the resulting ellipse.
3p
6
In a study of liquor preference the preference rankings of p = 9 liquor types
were collected from n = 1442 individuals. A principal component factor
analysis of the 9*9 sample correlation matrix of rank orderings gave the
following estimated factor loadings:
Estimated factor loadings
Variable
Liquors
Kirsch
Mirabelle
Rum
Marc
Whiskey
Calvados
Cognac
Armagnac
F1
0.64
0.50
0.46
0.17
-0.29
-0.29
-0.49
-0.52
-0.60
F2
F3
0.02
-0.06
-0.24
0.74
0.66
-0.08
0.20
-0.03
-0.17
0.16
-0.10
-0.19
0.97
-0.39
0.09
-0.04
0.42
0.14
a) Motivate a reasonable interpretation of the first factor in terms of the
estimated factor loadings.
1p
b) Using the estimated factor loadings, determine the proportion of total
sample variance due to the first factor.
1.5p
c) Using the estimated factor loadings, determine the communalities for
the first four liquors and interpret your results.
1.5p