ABSTRACT
Multivariate parametric and nonparametric techniques are two broad statistical
methods for testing independence among continuous random variables. In this
thesis, multivariate parametric and nonparametric techniques were performed to
tests for independence using multivariate data collected from Human
Development Report 2005. Three parametric tests, Hotelling’s T2, Wilks Lambda
and Canonical correlation analysis were performed. Wilcoxon, Friedman,
Kendall’s and Cochran are the nonparametric analysis performed. The results
show that parametric and nonparametric tests of hypothesis yield the same results.
TABLE OF CONTENTS
Content Page No.
TITLE PAGE i
DECLARATION
iii
CERTIFICATION iv
DEDICATION v
ACKNOWLEDGEMENT
vi
ABSTRACT
viii
TABLE OF CONTENTS ix
LIST OF TABLES x
CHAPTER ONE INTRODUCTION
1.01 INTRODUCTION 1
1.02 Multivariate Parametric Test 2
1.03 Multivariate Nonparametric Tests 4
1.04 Advantages of Nonparametric Test. 6
1.05 Disadvantages of Nonparametric Tests. 6
1.06 Purpose of the Study. 7
1.07 Limitation of the Study 7
1.08 Aims and Objectives
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CHAPTER TWO LITERATURE REVIEW
2.01 INTRODUCTION 9
2.02 Multinormality Theory 9
2.03 Parametric versus non-parametric statistics in the analysis of randomized
trials with non-normally distributed data. 10
2.04 Background of the study
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2.05 A review of method of comparison using Mann-Whitney and ANCOVA 15
2.06 Their Results 19
2.07 Their-Observations 19
2.08 Testing for Independence in High Dimension 20
2.09 Testing Independence of Multivariate data using Boot strap 22
CHAPTER THREE MATERIALS USED FOR THE STUDY
3.01 Data Set 24
3.02 Software Used. 26
3.03 Methodology 27
x
3.04 Wilks’ lambda Test 27
3.05 Canonical Correlation Analysis 29
3.06 Wilcoxon signed-ranks test 29
3.07 Friedman two way analysis of variance test. 30
3.08 Kendall’s W Test. 32
3.09 Cochran Q Test 33
CHAPTER FOUR ANALYSIS
4.01 Hoteling’s-T2-Test 35
4.02 Wilks’ Lasmbda test 36
4.03 Canonical correlation analysis 37
4.04 The Wilcoxon signed-ranks sum test 39
4.05 The Friedman two way analysis of variance test 41
4.06 Kendall’s W test 42
4.07 Cochran Q test 43
CHAPTER FIVE CONCLUSION AND RECOMMENDATION
5.01 SUM MARY AND CONCLUSION 44
5.02 Summary 44
5.03 Conclusion 44
5.04 Recommendation 45
References 46
APPENDIX-A 49
Appendix-B 50
LIST OF TABLES
TABLE TITLE PAGE
1: Hotelling’s T2 test of independence 37
2: Wilks Lambda test of independence 38
3: Canonical correlation analysis of independence 39
4: Wilcoxon signed-rank test of independence 41
5: Friedman test of independence 43
6: Kendall test of independence 44
7: Cochran test of independence 45
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CHAPTER ONE
1.01 INTRODUCTION
Testing hypothesis is very important in any statistical analysis. However,
many considerations enter into the choice of statistical inference. Statistical
testing often refers to the concepts of parameter (mean, standard deviation,
correlation) and reference population (multinomial distribution). However,
the multinormal distribution and others play a key role in using parametric
tests. If the data are not normal, however, it does not make sense to
compute statistics such as the mean, or standard deviation. To make a good
choice we must consider the manner in which the sample of scores were
drown, the nature of the measurement or scaling which was employed in
the definitions of the variables involved (i.e. in the scores). All the above
assumptions are useful in determining which statistical test is most
appropriate for analyzing one set of data.
In practice, we often encounter more complex situations when measures of
associations for more variables are required. In that situations, the
multivariate test procedures are very useful for us. However, in some cases
measurements may represent attributes of psychological characteristics
while others represent attributes of physical characteristics. We then require
a test of independence between pairs of vectors, where the vectors have
different measurement scales and dimensions. For instance if xi
T = (xi xj) i =
2
1, 2, . . . . . . . . . n, j = 1, 2 . . . . . . . . . . n for n denotes a sample size of
vector pairs, where x1
2 and x1
2 were continuous vector of dimension p and q
respectively. The test is Hо xi
1 and xj
2 are independent against the
alternative. Hı xi
1 and xi
2 are not independent.
Testing for independence between continuous random variables is an old
approach but important and has been the subject of much interest. Many
thorough investigations on their efficiency have been given by many
authors. Gieser and Randles (1997) Takinen, kanlainen and Oja (2003a)
taskinen et al (2003b) Genest and Verret (2003) Chernoff Savage and
Hodges-Lehrnann (1956) Hallin and Puri (1958) Hallin and Paindanceine
(2002a, b and 2003) on multivariate inferences using both or either
parametric or nonparametric methods.
All these reviewed seems to accord much interest on an adequate
generalization of rank and signs. The reason for this interest is the
confusion people make since ranks and sign are independent and intimately
related and exchangeable. They (ranks and signs) were inherently confined
to the analysis of independent observation.
1.02 Multivariate Parametric Test
A parametric statistical test specifies conditions about the parameters of the
population from which the sample was drawn. This means that the
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parametric model considers that the population distribution should have a
particular form (e.g. a normal distribution) and also involves hypothesis
about population parameters. The meaningfulness of the results of a
parametric test depends on the validity of these assumptions.
However, the most powerful tests are those having most extensive
assumptions. The parametric tests, for example the T2 test, for it to be
applied, the underlying conditions should be satisfied.
The observations must be drawn from normally distributed
populations.
The observation must be independent
The mean effects must be additive
The populations must have the same variance-covariance matrices
and
The variables involved must have been measured in at least an
interval scale.
If these conditions are met in the data under analysis, one can choose T2 or
Wilks lambda test. In addition to the above conditions, data measured in
interval (data identify subjects by non-relative rank in a scale: that can be
represented along a number line, there is no Absolute – Zero and the
number are non-relative e.g. IQ achievement, Aptitude, etc), or ratio scales
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(data identified subjects by quantification, number that can represent equal
units, and the range from an absolute zero point e.g. time, weight, distance,
temperature) may be analyzed by parametric methods.
The meaningfulness of the results of a parametric test depends on the
validity of these assumptions. Therefore, when those assumptions are valid,
these tests (parametric) are most likely of all tests to reject H0 when it is
false. Parametric tests are also generally more flexible, capable for example
of dealing with a mixture of categorical and continuous variable, which is
not possible with nonparametric tests. Bralley JV (1968)
1.03 Multivariate Nonparametric Tests
Nonparametric tests are distribution free, that is to say they do not assume
that the samples were drawn from a population with a specific distribution
and because of this, nonparametric statistics are useful not only when
variables are categorical, but also when parametric data do not conform to
the multinomal distribution.
Nonparametric method is defined as a statistical method which can be used
on data that is:
a. ordinal scale of measurement
b. a nominal scale of measurement
c. an interval or ratio scale of measurement where the distribution
function of the random variable producing the data are unspecified
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(or specified except for an infinite number of unknown parametric).
Conover (1999).
Thus, a nonparametric statistical tests (otherwise called distribution-free
test) is a test whose model does not specify conditions about the parameters
of the population from which the sample was drawn, hence, it is not
meaningful to look for normalization or distribution on an interval scale.
Nevertheless, certain assumptions are associated with most nonparametric
statistical tests, that is, the observations are independent and the variables
under consideration have underlying continuity.
These assumptions are not as strong as those associated with parametric
tests. Moreover, nonparametric tests do not require strong measurement as
required for the parametric tests. Most nonparametric tests apply to data in
an ordinal scale (data identified subjects by position in a series range by
height, order finished in a race, academic standing in class, measurement of
average, mean) and some apply also in a nominal scale (data which identify
subjects by category or name, gender, ethnicity, class, hair colour).
Nonparametric data types are called nominal or ordinal. However, if one or
more of the assumptions (of parametric) have been violated, then it is
capital important to transform data into a format that is compatible with the
appropriate nonparametric test. This is because the parametric tests
generally provide a more powerful test of an alternative hypothesis than
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nonparametric counterparts. If one or more of the underlying parametric
test assumption is or (are) violated, the power advantage may be negated.
1.04 Advantages of Nonparametric Test.
The nonparametric tests have the following advantages among others.
Dallal GE (1988).
a. Nonparametric tests are stable for treating sample made up of
observations from several different populations.
b. Nonparametric tests are distribution free i.e. they do not require any
assumption to be made about population following normal or any
other distribution.
c. Generally they are simple to understand and easy to apply when the
sample sizes are small.
d. Many nonparametric methods make it possible to work with very
small samples. This is particularly helpful to the researcher
collecting pilot study data or to medical researcher working with a
rare disease.
1.05 Disadvantages of Nonparametric Tests.
Nonparametric tests have some disadvantages, and the disadvantages are as
follows:
i. The major disadvantages of nonparametric techniques is contained
in its name. Because the procedures are nonparametric, there are no
parameters to describe and it becomes more difficult to make
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quantitative statements about the actual difference between
populations.
ii. Nonparametric procedures throw away information. The sign test for
example, uses only the signs of the observations. Ranks preserve
information about the order of the data but discard, the actual values.
Because information is discarded, nonparametric procedures can
never be as powerful as their parametric counterparts when
parametric tests can be used.
1.06 Purpose of the Study.
i. To find out if there exist any relationship between indexes of human
development report using both parametric and nonparametric tests.
ii. To equally test for independence using both multivariate parametric
method and nonparametric method to see if it can produce the same result.
iii. To find out similarities/difference between the two statistical
methods based on the result of the analysis used in this thesis.
iv. To analyze statistically, multivariate data by using parametric and
nonparametric tests for independence
1.07 Limitation of the Study
The study contained in this thesis extends from these existing literatures
providing both parametric and nonparametric multivariate tests of
independence. Asymptotic tests based on person’s spearman’s Kendall’s
Coefficient, Wilks test, signed rank test, Wilcoxon rank sum test, kruskalwallis
test, serial rank tests, locally most powerful tests of independence for
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specific copula model, testing independence for multivariate data using
Bootstrap, testing for independence in high dimensions etc. Thus, while the
emphasis is on test of hypothesis, it is useful to consider their asymptotic
relative efficiency in order to demonstrate the excellent performance of
these methods.
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