comparing dependent correlations for ordinal...

Comparing Dependent Correlations forOrdinal Data

A ThesisSubmitted to the Faculty of Graduate Studies and Research

In Partial Fulfillment of the Requirementsfor the Degree ofMaster of Science

inStatistics

University of Regina

byYun Gao

Regina, SaskatchewanApril 2015

Copyright 2015: Yun Gao

UNIVERSITY OF REGINA

FACULTY OF GRADUATE STUDIES AND RESEARCH

SUPERVISORY AND EXAMINING COMMITTEE

Yun Gao, candidate for the degree of Master of Science in Statistics, has presented a thesis titled, Comparing Dependent Correlations for Ordinal Data, in an oral examination held on April 1, 2015. The following committee members have found the thesis acceptable in form and content, and that the candidate demonstrated satisfactory knowledge of the subject material. External Examiner: Dr. Jingtao Yao, Department of Computer Science

Supervisor: Dr. Dianliang Deng, Department of Mathematics & Statistics

Committee Member: Dr. Yang Y. Zhao, Department of Mathematics & Statistics

Committee Member: Dr. Andrei Volodin, Department of Mathematics & Statistics

Chair of Defense: Dr. Satish Sharma, Faculty of Engineering & Applied Science

ABSTRACT

The methods and application for analyzing categorical ordinal data have matured

in statistical inferences during recent decades of development. The methods include

logistic regression models, odds ratios, inferential methods by using chi-squared tests

of independence and conditional independence. On the basis, this thesis presents

an analysis of equality of dependent correlations with the longitudinal ordinal vari-

able. Eight test statistics, Dunn and Clark’s Z, Steriger’s Z, Meng’s Z, Hitter’s Z,

Hotelling’s t, William’s t and William’s modified t per Hendrickson, for comparing

dependent correlations are presented. The results via simulation studies indicate that

the choice as to which test statistics is relatively optimal, in terms of empirical level

and statistical power, depends not only on sample size but also on the magnitude of

the correlations and the effect size. On the other hand, this thesis suggests the meth-

ods of modification for some statistical tests when they performed unsatisfactory with

ordinal variables. The thesis also briefly discusses practicing the relatively efficient

test statistics for testing equality of the correlation coefficients in real medical data

and has achieved the good results.

ii

ACKNOWLEDGEMENTS

Grateful acknowledgment is made to my supervisor, Dr. Dianliang Deng, who

gave me considerable help by means of suggestion, comments and criticism. His

encouragement and unwavering support has sustained me through frustration and

depression. Without his pushing me ahead, the completion of this thesis would be

impossible. My heartfelt thanks also go to Professor Yang Zhao and Professor Andrei

Volodin and many other professors who have helped me, for their help in the making

of this thesis as well as their enlightening lectures from which I have benefited a great

deal.

This research is partially supported by Faculty of Graduate Studies and Research,

teaching assistantships and research assistantships provided by the Department of

Mathematics and Statistics .

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DEDICATION

This thesis is dedicated to my parents, for their loving considerations and great

confidence in me all through these years. It is also dedicated to my friend, Chenxu

Sun, and my fellow classmates who gave me their help and time in listening to me.

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Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

1 Introduction and Motivation 1

2 Literature Review 4

2.1 Ordinal Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 Measurement of Ordinal Data . . . . . . . . . . . . . . . . . . 4

2.1.2 The Classification of Ordinal Data . . . . . . . . . . . . . . . 5

2.1.3 The Difference Between Nominal and Ordinal Data . . . . . . 7

2.2 Models for Ordinal Responses . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Cumulative Logits Models . . . . . . . . . . . . . . . . . . . . 9

2.2.2 Adjacent - Categories Logits Models . . . . . . . . . . . . . . 11

2.2.3 Continuation Ratio Models . . . . . . . . . . . . . . . . . . . 12

2.2.4 Polytomous Logistic Models . . . . . . . . . . . . . . . . . . . 14

2.2.5 Stereotype Models . . . . . . . . . . . . . . . . . . . . . . . . 14

v

2.2.6 Other Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Statistics of Measuring dependent correlations in Contingency Table 17

3.1 Correlation with Ordinal Variables in Contingency Tables . . . . . . . 18

3.1.1 Ordinal Probabilities and Scores . . . . . . . . . . . . . . . . . 18

3.1.2 Contingency Tables . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1.3 Correlations with Ordinal Scores in Contingency Table . . . . 21

3.2 Test Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.1 Test Statistics with Student Distribution . . . . . . . . . . . . 23

3.2.2 Test Statistics with Standard Normal Distribution . . . . . . . 25

3.2.3 Exact Inference of Test Statistics . . . . . . . . . . . . . . . . 29

3.3 Modified Test Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4 Simulation Study 31

4.1 Data Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2 Hypotheses and the Criteria of Test Statistics . . . . . . . . . . . . . 34

4.2.1 Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2.2 The Criteria of Test Statistics . . . . . . . . . . . . . . . . . . 34

4.3 Results of Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3.1 Empirical Level . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.3.2 Statistical Power . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5 Applications 57

5.1 Data Sources and Features . . . . . . . . . . . . . . . . . . . . . . . . 57

vi

5.1.1 Data Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.1.2 Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.1.3 Modification of the real data in contingency tables . . . . . . . 61

5.2 Test Results and Conclusion of the Medical Data . . . . . . . . . . . 67

6 Conclusion and Future Work 76

6.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.1.1 Modification of Test Statistics by using Bootstrap Method . . 76

6.1.2 Testing the Equality of a Set of Correlated Correlations by using

Chi-square Statistics . . . . . . . . . . . . . . . . . . . . . . . 78

6.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

vii

List of Tables

3.1 An example of contingency table . . . . . . . . . . . . . . . . . . . . 20

3.2 Ordinal Variable Y through Times in Stratum k . . . . . . . . . . . . 22

4.1 Empirical levels of eight statistics for nominal level α = 0.01 with Σ1 37



4.4 Empirical levels of Zm and Zmmodifiedwith Σ1 . . . . . . . . . . . . . . 46


4.6 Empirical power of all eight statistics for nominal level α = 0.01 . . . 53

5.1 Deep Pain Sensation (X) at baseline time t0 and visit time t1 without

modification of data . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.2 Deep Pain Sensation (X) at baseline time t1 and visit time t2 without


5.3 Deep Pain Sensation (X) at visit time t2 and visit time t3 without


5.4 Deep Pain Sensation (X) at baseline time t0 and visit time t1 . . . . 62

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5.5 Deep Pain Sensation (X) at visit time t1 and visit time t2 . . . . . . 62

5.6 Deep Pain Sensation (X) at baseline time t2 and visit time t3 . . . . 62

5.7 Pain Intensity (Y ) at baseline time t0 and visit time t1 . . . . . . . . 63



5.10 Lack of Energy (A) at baseline time t0 and visit time t1 . . . . . . . . 64



5.13 Nausea (B) at baseline time t0 and visit time t1 . . . . . . . . . . . . 65



5.16 Joint Pain/Muscle Cramps (C) at baseline time t0 and visit time t1 . 66



5.19 The correlations coefficients for ordinal variables . . . . . . . . . . . . 68

5.20 Result of deep pain sensation . . . . . . . . . . . . . . . . . . . . . . 69

5.21 Result of pain intensity . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.22 Result of lack of energy . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.23 Result of nausea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.24 Result of joint pain/muscle cramps . . . . . . . . . . . . . . . . . . . 73

ix

List of Figures

4.1 Empirical levels at α = 0.01 . . . . . . . . . . . . . . . . . . . . . . . 40



4.4 Empirical levels of Meng’s Z at α = 0.01 . . . . . . . . . . . . . . . . 44

4.5 Empirical level of modified Meng’s Z at α = 0.01 . . . . . . . . . . . 47

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Chapter 1

Introduction and Motivation

A real data set which is used to illustrate the applicability and practicality of the

proposed approaches is from a project of cancer research center in the United States.

The data set involves the demographic information of the patients, such as age, gen-

der, race, history of several types of disease and medication records. It also contains

the recorded data from the cancer center, such as pain scale, changes in physiolo-

gy and psychology, chemotherapy, concomitant medication records and neuromeres

treatment. The data are classified as ordinal, nominal, string, interval, continuous

and ratio, and recorded on multiple times. We are interested in ordinal variables,

such as deep pain sensation, pain intensity and nausea, which have 11 categories at

most. The contingency tables are constructed by one of the ordinal variables between

adjacent visit times. The objective of this thesis is to test the homogeneity of a set of

correlations between the contingency tables with one ordinal variable. The research

in the medical sense is testing whether or not the consistent strength of treatment

for patients which embodied in specifying ordinal variables rather than judging the

1

patients who received an effective or ineffective treatment. The results as such should

have greater utility for applied researchers.

In order to compare the correlations of a common ordinal variable between ad-

jacent visit times, we arrange the data into contingency tables first. For instance,

in first contingency tables, the row data is an ordinal variable at visit time t1, and

column data is the same variable at visit time t2. The row data of the second con-

tingency table is the same variable at visit time t2, and column data is the variable

at visit time t3. By such process, we get several contingency tables and compare

the correlations from those tables. It is noted that the numbers of observations in a

sample should be specified, and we omit the observation who drop out of the project

over time. This thesis presents several test statistics for comparing the dependent

correlations with a sample from multivariate continuous normal distribution. They

are Olkin’s Z, Dunn and Clark’s Z, Steriger’s Z, Meng’s Z, Hitter’s Z, Hotelling’s t,

William’s t and William’s modified t per Hendrickson. In order to use them in ordinal

variables, the ordered scores could be assigned. Therefore, the statistical tests involve

the correlation coefficient where the categorical numbers are considered.

In this thesis, we ascertain the following proper test statistics for the ordinal

contingency tables through the simulation evaluation. One group including Dunn and

Clark’s Z, Hotelling’s t and William’s modified t per Hendrickson performs relatively

well in some cases in term of sample size and correlation coefficient. In other cases we

will discuss the test statistics in simulation study, another group including Steriger’s

Z, Meng’s Z, Hitter’s Z, William’s t is more appropriate. In addition, even thought

these test statistics work in many cases, they could be made a modification to perform

2

better in more situations.

The structure of the thesis is as follows. In Chapter 2, we present the measure-

ment and classification of ordinal data and compare the advantage of using ordinal

and nominal variables. The classical models for analysis of ordinal response, such as

logistic regression modeling and ordinal association structure modeling, are reviewed

in this chapter. In Chapter 3, we discuss several test statistics for comparing two or

more correlations using a sample from ordinal variables. Besides, a simple modifica-

tion of some test statistics is proposed. In Chapter 4, we generate a simulated sample

from a multivariate normal distribution and categorize the simulated variables into

ordinal data. Through the simulated results, we evaluate the test statistics and filter

the relatively appropriate several statistics used in ordinal variables. The application

to the medical real data is introduced in Chapter 5. Chapter 6 considers the possible

future work and conclusion.

3

Chapter 2

Literature Review

2.1 Ordinal Data

2.1.1 Measurement of Ordinal Data

The traditional theory of measurement is mainly popular in quantitative research, for

example, “length, capacity, weight, etc” are some variables which have units (such

as meter, liter, gram) of measurement. However, for the ordinal variable without the

standard unit, the measurement becomes invalid. Stevens (1951) redefined the theory

of measurement, so we can obtain the data for ordinal variable in a meaningful way.

According to the Dictionary of Statistics and Methodology (Vogt,1993), the or-

dinal scale of measurement is “... a scale of measurement that ranks subjects (puts

them in an order) on some variable, the difference between ranks does not need to be

equal as they are in an interval scale”. In general, the ordinal variable A is assumed

that (a) A ∈ a1, ..., aI , where ai ∈ R, i = (1, 2, ..., I) and I is the number of exclusive

4

and exhaustive categories. (b) The categories satisfy a1 < a2 < ... < aI . Obviously,

the measurement scale is usually used in social survey, market survey, financial sur-

vey and amount of social research activities, such as income range, educational level,

measuring attitude and other surveys.

According to the above-mentioned definition, we shall meet two questions. One is

to prove if sorting exists or not. Actually, it is hard to prove the existing of sorting

theoretically. However, we need not pay much attention to this point, because readers

already have a common understanding of its existing in practical life. What we need

to consider is the defined sorting standards on measuring ordered variable. We take

the survey of informants daily smoking as an example: 1 = seldom; 2 = normally; 3 =

usually, each of which means the amount range of daily smoking. If we do not provide

the sorting standards like the above-mentioned ones, each informant may have his or

her own standard about the amount of smoking, which causes that the investigative

result turns out to be ambiguous and its data is not comparable. Another question is

if classification has no objective standard as reference, we have to make measurement

by subjective judgment. Actually, the ordered measurement usually underlies some

subjective elements, which has immeasurable impact on the accuracy of the subjective

classification of variables. The above-mentioned fact, to some extent, is also the season

why the analysis of the ordered variable is less in statistical analysis.

2.1.2 The Classification of Ordinal Data

In fact, the ordered variable may match certain assumption. For example, if latent

variable exists and is classified by ordered variable, some parameterized methods shall

apply to this variable. On the basis of this thought, an ordered variable may come

5

from three aspects: measurable latent variables, unmeasurable latent variables and

nonexistent latent variables. If the ordered variable comes from nonexistent latent

variables, it may be divided into two categories, one has the objective standard with

classification, and another has no standard. In conclusion, ordered variable may be

divided into five types (J.Kampen and M.Swyngedouw, 2000):

1. The classification is on the measurable continual latent variable with known

threshold. For example, the classification of personal monthly income includes

the following categories: less than 1200, 1200 to 3000, 3000 to 6000, more than

6000.

2. The classification is on the measurable continual latent variable with unknown

threshold. For example, the classification of personal monthly income includes

the following categories: low, middle and high.

3. The classification is on the immeasurable continual latent variable with un-

known threshold. For example, the classification of personal intelligence in-

cludes the following categories: low, middle and high.

4. The latent variable does not exist and ordered variable is classified according to

a certain rule and with a standard as reference. For example, the classification

of the drug effect in clinical trials includes the following categories: invalid,

better, effective, significant.

5. The latent variable does not exist and ordered variable is classified according

to a certain rule and without any standard as reference. For example, the

6

classification of personal comments on certain expression includes the following

categories: disagree, neutral, agree.

In all five types, the classified standard on ordered data is an objective standard,

which is completely independent of informants and investigators under the situation

of the first classification (1). Whether the fourth classified standard is that objec-

tive depends on the correction on the study by investigators. The second and fifth

standards of classification are difficult. In practical situations, investigators can only

expect that informants match the classification standard of ordered variable. The

third classification has no appropriate standard and is totally determined by investi-

gators.

2.1.3 The Difference Between Nominal and Ordinal Data

Qualitative data has two kinds of measurement scale (Stevens, 1951). Qualitative

data will be called nominal data if there is no natural ordered relationship between

each data classification, such as sex (male, female), trip mode (train, car, plane, ship,

bicycle and walking), religious belief (Buddhism, Christianity, Islam and others).

For nominal data, the order of each classification is irrelevant and the sorting on

classification is not considered in statistical analysis.

However, lots of classified and ordered data are stipulated in advance, which is

called ordered data, such as quality of product (good, normal, bad), degree of educa-

tion (primary school and below, secondary school, high school or technical secondary

school, college or above) and physical condition (very good, good, normal, poor and

very poor). The classified order of ordered data is certain, but the distance between

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each classification is implicit. For example, a man or woman with good physical con-

dition has a better constitution than others with normal physical condition, but there

is no explicit and accurate data to describe how good it is. Therefore, the statistical

analysis of ordered data is needed to take advantage of the ordering relationship be-

tween each classification. The type of data depends on the measurement scale that

is called the classified method. For example, in the aspects of public education and

private education, the obtained data is nominal data according to the degree of educa-

tion. But in the aspect of the classification with primary school and below, secondary

school, high school or technical secondary school, college or above, the obtained data

is ordered data. However, the scale of measurement also determines its applicative

statistical method for data in general. Ordinal scale in the measurable levels is higher

than nominal scale. The applicative statistical method for high level shall not apply

to the method that is applicative to low level in general (Agresti, 2002). The statisti-

cal method which is applicative to the analysis of ordered data shall not apply to the

analysis of nominal data because the classification of nominal data is without order

and does not match the applicable condition of this method.

2.2 Models for Ordinal Responses

In practical work, we often encounter a kind of multivariate response data. Response

variable Y is divided according to level, the value of Y is 1, 2, ..., k, signifying k ordered

categories, and explanatory variable X, it can be discrete, but also can be continuous

or combined with both. “Order” that we talk about in this chapter is response variable

8

Y . And as to the analysis on this kind of materials, current analysis mainly adopts

linear model and multiplied discriminant model. These models have strict demands on

variables, and they always have assumption on normality and covariance matrix. In

medicine, many data are unable to satisfy the above-mentioned assumption, because

an explanatory variable generally composes of qualitative data and quantitative data

with unknown distribution, and its various kinds of variances may differ a lot. For

instance, when researching the relationship between the same “dangerous factor” of

a certain disease, the variance of the diseased group is distinctively bigger than the

normal group, so it is improper to handle this issue with traditional method. In

addition, using linear models to score Y has some arbitrariness, and it is hard to

obtain interpretation on multiplied discriminant model.

With regard to the above-mentioned issues, some models for analyzing ordered

category data are brought up in recent years. These methods without the limitation

of ordinary linear model can sufficiently “use” the information of order. Next we will

first of all give brief introduction on several models for ordinal response, and then

illustrate application upon each models and existing issues.

2.2.1 Cumulative Logits Models

We first present cumulative logits model with explanatory variables, which is one of

the most widely used models for ordinal response. For subject i, assume yi be the

outcome category for the response variable, and xi be a column vector of the values

of the explanatory variables. The cumulative logits model was originally proposed by

Walker and Duncan [29] and called the proportional odds model by McCullagh [16],

9

and the model has the form

logit [P (Yi ≤ j|x)] = αj + β′xi = αj + β1xi1 + β2xi2 + · · · (2.1)

where j= 1, 2, ..., c− 1, and β, which is a column vector of parameters, describes the

effects of the explanatory variables.

In model (2.1), the logit for cumulative probability j has its own intercept, αj.

We can easily find that P (Yi ≤ j | x) increases in j for each fixed value of x , so the

{αj} are increasing in j, and the logit is an increasing function of this probability.

The equivalent forms for model (2.1) can be re-expressed by

P (Y ≤ j|x) =exp(αj + β′x)

1 + exp(αj + β′x), j = 1, ...c− 1. (2.2)

or

P (Y ≤ j|x)

P (Y > j|x)=

P (Y ≤ j|x)

1− P (Y ≤ j|x)= exp(αj + β′xi) (2.3)

If we consider two populations which are characterized by explanatory variables

x1 and x2, the general model with multiple explanatory variables satisfies

logit [P (Y ≤ j | x1)]− logit [P (Y ≤ j | x2)] = logP (Y ≤ j | x1)/P (Y > j | x1)

P (Y ≤ j | x2)/P (Y > j | x2)

= β′(x1 − x2).

It is clear to see that the ratio is independent of category number j. That means

the ratio of the cumulative odds for two populations is the same for all of the cumu-

lative odds. It is only effected by the explanatory variables x1 and x2 and the vector

of parameters β.

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In 1980, McCullagh discussed this model as proportional odds model. He consid-

ered that how to categorize in a practical issue has a certain level of subjectivity and

hoped that the analysis result will not change because of particular category option.

Cumulative logits model can well solve this problem to some extent.

However, when applying this model, it shall firstly examine the assumption con-

ditions of proportional odds; if this condition is untenable, then we need other model

to analyze. Peterson and Harrell (1990) proposed unrestricted partial proportional

odds model to unleash the restriction on this condition. Regarding this situation,

Bender and Grouven (1998) also discussed the situation of using binary logits regres-

sion model to replace cumulative logits regression model to analyze. However, Torra

and Domingo Ferrer (2006) pointed out that using several binary logits regression

models to handle ordered variable will make the model lack efficiency because of not

making use of the important information of ordering between variables.

2.2.2 Adjacent - Categories Logits Models

In next two sections, we will introduce alternative logits models using the adjacent-

categories logits and the continuation-ratio logits. These two models have interpreta-

tions that can use individual categories rather than the cumulative probabilities.

The adjacent-categories logits with multinomial probabilities P (Y = j) = πj are

logit [P (Y = j | Y = j or Y = j + 1)] = logπjπj+1

, j = 1, ..., c− 1. (2.4)

thus, the general adjacent-categories logit model with a set of explanatory variable x

has the form

logπj(x)

πj+1(x)= log

P (Y = j|x)

P (Y = j + 1|x)= αj + β′jx, j = 1, ..., c− 1. (2.5)

11

From the models, we figure out that the parameter β1 corresponds to the regression

coefficients for the log odds of π1 relative to π2; the parameter β2 corresponds to the

regression coefficients for the log odds of π2 relative to π3, and so on. Therefore, the

effects of this model are presented with local odds ratios because the model uses pairs

of adjacent categories j and j+1, rather than the cumulative odds ratios. The effects

rely on the distance between categories, so the model recognizes the ordering of the

response scale.

2.2.3 Continuation Ratio Models

After presenting adjacent-categories logits models, we next introduce another logits

model which uses continuation-ratio logits. There are two types of logits. One set

forms the log odds for each category relative to the higher categories

logπj

πj+1 + · · ·+ πc, j = 1, ..., c− 1. (2.6)

which is useful when a sequential mechanism determines the response outcome, in

the sense that an observation must potentially occur in category j before it can occur

in a higher category (Tutz,1991).

An alternative set of continuation-ratio logits forms the log odds for each category

relative to the lower categories,

logπj+1

π1 + · · ·+ πj, j = 1, ..., c− 1. (2.7)

which is useful if the sequential mechanism works in the reverse direction.

The two forms of continuation-ratio logits are not equivalent. For example, there

12

are c = 4 categories, the first set of sequential continuation-ratio logits is

logπ1

π2 + π3 + π4. log

π2π3 + π4

. logπ3π4. (2.8)

however the second set is

logπ2π1. log

π3π1 + π2

. logπ4

π1 + π2 + π3. (2.9)

In this section, we only consider the first type of continuation-ratio logit. In this

case with duration and development scales, a subject passes through each category

in order before determining the response outcome. Continuation-ratio logit models

using sequential logits have the form

logit [ωj(x)] = logP (Y = j|x)

P (Y > j|x)= αj + β′jx, j = 1, ..., c− 1. (2.10)

This model was proposed by Cox on 1972 when he researched the survival life

table. In 1977, Thomposon discussed this model again. The left side of the model is

the log of the hazard function given by h(t) = f(t)/[1−F (t)] in survival time analysis,

where the f(t) = P (Y = j|x) is a probability density function and F (t) = P (Y ≤ j|x)

is a cumulative distribution function. Survival times are often measured with discrete

categories, with the response grouped into a set of categories. For example, how many

years that patients are survival after receiving a particular medical treatment ( < 1

year,1 to 5 years, 5 to 10 years, > 10 years ). Because survival time is strictly

ordered, sometimes continuation-ratio logits model is applied to grouped survival

data. Fienberg and McCullagh had introduced and used this model.

13

2.2.4 Polytomous Logistic Models

Anderson(1972) proposed the polytomous logistic model which is based on the logistic

models for binary response by Cox(1970). The model is then

logπj(x)

πj+1(x)= αj + β′jx, j = 1, ..., c− 1. (2.11)

In terms of the response probabilities, the polytomous logistic model has the following

representation:

πj(x) = P (Y = j|x) =exp(αj + β′jx)∑ck=1 exp(αk + β′kx)

, j = 1, 2, ..., c. (2.12)

where αc = 0 and βc = 0. The parameter vector β = (β1, β2, ..., βc−1)′ needs to be

estimated, and there are (c − 1) intercept parameters αj. The basic assumption of

this model is that the log odd ratio for arbitrary binary response is linear function of

explanatory variables x.

In fact, this model does not belong to the class of ordered response model, but it

still utilize with ordinal response and is the basic of stereotype models. We deal with

the ordinal data via analyzing the values βi1, βi2, ..., βi(k−1) of the ith variables, thereby

we can analyze the relationship between response Y and explanatorily variable xi.

2.2.5 Stereotype Models

Stereotype model is based on the aforementioned polytomous logistic model. When

Y is a categorized ordinal response, let the parameter vector β′j in polytomous logistic

model be φjβ′, then the model is presented as

πj(x) = P (Y = j|x) =exp(αj + φjβ

′x)∑ck=1 exp(αk + φkβ

′x), j = 1, 2, ..., c, (2.13)

14

Except for the parameters β = (β1, β2, ..., βc−1)′ and αj, (c−2) parameters φj need to

be estimated. That is, with constraint on φ1 = 1 and φc = 0, the other φ2, φ3, ..., φc−1

can be estimated. Anderson imposed an additional order constraint on the φj with

φ1 ≥ φ2 ≥ ... ≥ φc. However, we can ignore this constrain in practical applications

and observe the order via φj.

2.2.6 Other Models

Probit and log-log are other ordered regression models. Probit model is similar to

accumulative odds model. When Log-log model is asymmetric long-tailed distribution

in response variable Y, in fitting it may show some advantage, but these models are

less simple and easy to explain than the above-mentioned models, so the application

is relatively few.

With regard to the assumption of normal distribution and homogeneity of variance

going against traditional variance analysis, Piepho and Kalka (2003) proposed to

conduct analysis on ordered variable by using threshold model containing random

effect and fixed effect. At the same time, they pointed out that although threshold

model has strong flexibility, the model increases the number of parameters to be

estimated, increases the danger of excess fitting, and this model is unsuitable for

small sample analysis.

Regarding general linear model, Gautam and Kimeldorf (1996) proposed to estab-

lish score function of F testing statistics to ordered variable, the method of through

optimizing testing statistics to obtain ordered variable’s best assignment.

In addition, Kottas and Muller (2003), Agresti and Hitchcock (2005) research on

the application of parameteric and nonparametric Bayesian model in ordered variable

15

analysis; Jaime S. Cardoso (2006) concludes and proposes to apply neural network

methods to categorize and analyze on ordered materials.

16

Chapter 3

Statistics of Measuring

dependent correlations in

Contingency Table

Since one of the objective we are focusing on is testing the equality of correlations

from ordinal variables where the test statistics are based on two-way contingency

tables. It is noted that the ordinal variables from individuals which we are interested

are measured repeatedly through time. So we will test the equality of correlation

with one ordinal variables from adjacent times. In this chapter, we first arrange an

ordinal variable based on time into several contingency tables, and then introduce

several test statistics which are applied to test the homogeneity of correlations from

one ordinal variables between adjacent times. In the last section of this chapter, we

do some modification of the test statistics which we introduce in previous section.

17

3.1 Correlation with Ordinal Variables in Contin-

gency Tables

3.1.1 Ordinal Probabilities and Scores

1. Probabilities for ordered categories

We already use the basic ideal of probabilities for ordinal response Y in the ordinal

models in last chapter. Suppose Y that is an ordinal response, and let c denote the

number of categories for ordinal response Y , n denote sample size, n1, n2, ..., nc denote

the frequencies in the categories with n =∑

j nj, and {pj = nj/n} denote the sample

proportions. Let P (Y ≤ j) = πj denote the probability of response in category j,

then the cumulative probability for ordinal response Y is

Fj = P (Y ≤ j) = π1 + π2 + ...+ πj, j = 1, 2, ..., c.

and the cumulative probabilities should reflect the ordering of the categories, that is,

0 < F1 < F2 < ... < Fc = 1

2. Scores for an ordinal variable

In this part we will summary how to utilize the ordinal nature of the categorical scale.

One simple way is to use the cumulative probabilities to identity the median response,

that is, finding the smallest category j such that Fj ≥ 0.50. The median will move

from one category to the next one if the probability is changed a little bit by using

this method.

18

The second way to utilize the ordinal nature of the categorical scale is assigning

ordered scores. For an ordinal variable Y having c categories, we assign the scores

satisfying the order v1 ≤ v2 ≤ ... ≤ vc which is the same order as the categories. By

using the method, the scores can summarize the observations with ordinary measures

for quantitative data and treat the ordinal score as an interval scale. Selecting scores

is not an unique way. For example, when an ordinal variable Y has c = 5 categories,

if we compares the correlations for two groups using the scores (1,2,3,4,5) which could

yields the same conclusion as using the scores (2,4,6,8,10), (0,5,10,15,20) or any set

of linearly transformed scores, but we can get different conclusions from using the

scores such as (1,2,5,7,9) or (0,3,8,10,11). Therefore, the key point of assigning scores

is the choice for the relative distances between pairs of adjacent categories.

The last but not least, the method of selecting scores is using the data themselves

to determine the scores. Bross(1958) introduced the average cumulative proportion

scores and called the term ridits. In the term of sample proportions {pj}, the average

cumulative proportion in category j is

aj =

j−1∑k=1

pk +1

2pj, j = 1, 2, ..., c,

and in the term of the sample cumulative proportions Fj = p1 +p2 + ...+pj, the ridits

scores present as

aj =Fj−1 + Fj

2,

with F0 = 0. The ridits have the same ordering as the categories, that is, a1 ≤ a2 ≤

... ≤ ac.

19

Table 3.1: An example of contingency table

Gender

HandednessRight-handed Left-handed Total

Males n11 = 43 n12 = 9 n1+ = 52

Females n21 = 44 n22 = 4 n2+ = 48

Total n+1 = 87 n+2 = 13 n = 100

3.1.2 Contingency Tables

1. A 2× 2 contingency table

A two-way table which is also called a contingency table is a useful tool for examining

relationships between categorical variables. The entries in the cells of a two-way table

can be frequency counts or relative frequencies. Table 3.1 cross classified n = 100

respondents to a General Social Survey by their gender and by their handedness.

The table illustrated the cell count notation for these data. For example, n11 = 43,

and the related sample joint proportion is p11 = 43/100 = 0.43. It is noted that the

categories of the two factors in this contingency tables are not ordered. In addition,

the observations are recorded once at this study.

20

2. Sorting an ordinal variables based on time in several stratums of two-

way contingency tables

With regard to the nature of ordinal variables in the real medical data which we

use in this thesis, we consider one factor which is categorized by order and recorded

by several visit times. We examine the relationships between adjacent times for

the ordinal variable. The visit times are termed strata, and it is assumed that the

response are dependent from stratum to stratum. For an ordinal variable Y with

c categories, the variable Y is collected from a sample with n observations through

times t1, t2, ..., ts. Let Y (tk) denote the row data in stratum k, and Y (tk+1) denote

the column data in stratum k, k = 1, 2, ..., s − 1. Let nij(k) denote the relative

frequency that the number of observations is classified with i-th category in row and

j-th category in column at stratum k, i, j = 1, 2, ..., c, and k = 1, 2, ..., s − 1. Table

3.2 shows that ordinal variable Y through times in stratum k of two-way contingency

tables.

3.1.3 Correlations with Ordinal Scores in Contingency Table

After sorting ordinal variables in contingency table, we define the correlation coeffi-

cient for ordinal variables. With regard to Table 3.1, let u1 ≤ u2 ≤ ... ≤ ui ≤ ... ≤ uc,

i = 1, 2, ..., c denote the scores for the rows, let v1 ≤ v2 ≤ ... ≤ vj ≤ ... ≤ vc ,

j = 1, 2, ..., c denote the scores for the columns. Since we are interested in one ordinal

variables through times, the scores for rows equal the scores for columns, that is,

ui = vj when i = j. Let {pi.} = ni./n be the probability in row i, {p.j} = n.j/n

be the probability in column j, and {pij} = nij/n be the probability in the cell ij.

21

Table 3.2: Ordinal Variable Y through Times in Stratum k

Y (tk)

Y (tk+1)1 2 · · · j · · · c Row Totals

1 n11(k) n12(k) · · · n1j(k) · · · n1c(k) n1.(k)

2 n21(k) n22(k) · · · n2j(k) · · · n2c(k) n2.(k)

......

......

......

......

i ni1(k) ni2(k) · · · nij(k) · · · nic(k) ni.(k)

......

......

......

......

c nc1(k) nc2(k) · · · ncj(k) · · · ncc(k) nc.(k)

Column Total n.1(k) n.2(k) · · · n.j(k) · · · n.c(k) n

For simplicity of notation, we ignore the particular values of the stratum k in the

notations above, such as {pi.(k)} = {pi.}. Therefore the sample correlation is given

by

r =

∑ci,j=1(ui − u)(vj − v)pij√

[∑c

i=1(ui − u)2pi.][∑c

j=1(ui − v)2p.j]

where u =∑

i uipi. is the sample mean of the row scores, v =∑

j vjp.j is the sample

mean of the column scores, and∑

i,j(ui − u)(vj − v)pij weights cross-products of

deviation scores by their relative frequency.

The correlation falls between −1 and +1. The larger the correlation r for stratum

k is in absolute value, the stronger strength of relationship about the ordinal variable

Y (tk+1) relaying on the variable Y (tk−1), k = 1, 2, ..., s − 1. In order to test the

equality of correlations between adjacent times, we totally have (s− 1) strata. That

22

is, let rk,k+1 denote the correlations between ordinal variables Y (tk) and Y (tk+1),

k = 1, 2, ..., s− 1. And also there are total (s− 1) correlations to compare, and null

hypotheses could be r12 = r23 = ... = rs−1,s.

3.2 Test Statistics

In this section, we introduce eight statistics for testing the equality of two dependent

correlations in a common sample. Suppose X = (X1, X2, X3) has a trivariate distri-

bution with covariance symmetric matrix Σ, where σij = ρijσiσj be the ij-th element

of Σ, with ρii = 1(i = 1, 2, 3) and σi = σj = 1. Our hypothesis test is

H0 : ρ12 = ρ23;

Ha : ρ12 6= ρ23.

Let r12 be a number specifying the sample correlation between variable X1 and X2, r23

be a number specifying the correlation between variable X2 and X3, r13 be a number

specifying the correlation between variable X1 and X3, n be an integer defining the

size of the group in the following equation of test statistics.

3.2.1 Test Statistics with Student Distribution

We first present three test statistics which are compared with the 1 − α/2 point of

Student T distribution with n − 3 degrees of freedom. The three t test statistics

should be appropriate for small to moderate sample size.

23

1.Hotelling’s t Test Statistics

In 1940, Hotelling suggested it as a test statistic based on the difference r12 − r23

divided by some estimate of the asymptotic standard deviation of r12− r23. The test

statistic t is given by

t =(r12 − r23)

√(n− 3)(1 + r13)√2|R|

with df = n− 3

where R is the determinant of the sample correlation matrix

|R| = 1 + 2r12r23r13 − r212 − r223 − r213

Because of the nuisance parameter ρ13 in the expressions of the covariance matrix Σ,

Hotelling suggested the asymptotic standard deviation with considering the nuisance

parameter.

2.William’s t Test Statistics

Williams’ test statistic is proposed on 1959, which is a modification of Hotelling’s t

statistic. This statistic also depends on a standardized version of r12− r23 and differs

from Hotelling’ t only by the term in the denominator. The test statistics is given by

t = (r12 − r23)√

(n− 1)(1 + r13)

2(n−1n−3)|R|+ r2(1− r13)3

=(r12 − r23)

√(n− 3)(1 + r13)√

2|R|+ (r12−r23)2(1−r13)3(n−3)4(n−1)

with df = n− 3, where

r =r12 + r23

2

and

|R| = 1 + 2r12r23r13 − r212 − r223 − r213

24

From the second equation of William’s t test statistic, we can easily find that William’s

t only add (r12−r23)2(1−r13)3(n−3)4(n−1) in the denominator under square roots. That is, the

difference of r12− r23, the nuisance parameter r13, or the sample size n will affect the

conclusion by the test statistic.

3.Hendrickson’s t Test Statistic

William’s t per Hendrickson’s statistic is also a modification of Hotelling’s t by

William, and is written as

t =(r12 − r23)

√(n− 3)(1 + r13)√

2|R|+ (r12−r23)2(1−r13)34(n−1)

with df = n− 3

where

|R| = 1 + 2r12r23r13 − r212 − r223 − r213

This test statistic crossed out (n − 3) in the part of modification in Williams’ t test

statistic. So the second modification of Hotelling’s t is more accurate for effect of

sample size, the difference of r12− r23 and the predictor of nuisance parameter r13 on

statistics. From the equation, we can find the effect of this modification is small if

one or more the following three events occur: n is large; r12 − r23 is small, and r13 is

close to one.

3.2.2 Test Statistics with Standard Normal Distribution

The next five test statistics we are interested in are to be compared with the 1−α/2

point of the standard normal distribution.

25

1.Olkin’s Z Test Statistic

The first one statistic with standard normal distribution for testing equality of corre-

lations was proposed by Pearson and Filon(1898). Then Olkin(1967)transformed the

test statistics on 1967,

z =(r12 − r23)

√n√

(1− r212)2 + (1− r223)2 − 2r313 − (2r13 − r12r23)(1− r213 − r212 − r223)

=(r12 − r23)

√n√

(1− r212)2 + (1− r223)2 − 2k

where

k = r13(1− r212 − r223)−1

2(r12r23)(1− r212 − r223 − r213)

which has the asymptotic standard-normal distribution for large sample size. The

first equation presents Olkin’s Z test statistic, and the second is Pearson and Filon’s

Z. That is, Olkin’s Z = Pearson and Filon’s Z. It is noted that Pearson and Filon’s

Z could also test the difference between two correlations with no variables in common

within the same sample, that is H0 : ρ12 = ρ34.

Using Fisher’s r-to-Z transformation

The next following test statistics make use of Fisher’s r to Z transform. We use

the transform in the case if the sample is not large and population correlations have

extreme values which would increase for these statistics far away from the nominal

levels, then the Fisher r-to-Z transform

Z =1

2(ln(1 + r)− ln(1− r))

helps to eliminate this problem because it transforms a sample correlation to a variable

that is close to normal distribution, even with small to moderate sample size and

26

extreme sample correlations.

2.Dunn and Clark’s Z Test Statistic

The first test statistic using Fisher’s r − to − Z is Dunn and Clark’s Z, this test is

calculated as

z =(Z12 − Z23)

√n− 3√

2− 2c

where

c =r13(1− r212 − r223)− 1

2r12r23(1− r213 − r212 − r223)

(1− r212)(1− r223)

To obtain the asymptotic variance of Z12−Z23, Dunn and Clark used the expression

for the asymptotic correlations, therefore c denotes the asymptotic correlation of Z12

and Z23.

3.Steiger’s Z Test Statistic

This test was proposed by Steiger (1980) and is a modification of Dunn and Clark’s

Z. The test statistic Z is defined as

z =(Z12 − Z23)

√n− 3√

2− 2c

where

r =r12 + r23

2

and

c =r13(1− 2r2)− 1

2r2(1− 2r2 − r213)

(1− r2)2

In an effort to further improve the control of empirical level, Steiger arithmetically

averages the correlation r12 and r23, which would be instead of the individual corre-

lations r12 and r23 in the equation. However, Steiger’s method using the arithmetic

27

average the correlation r12 and r23 could be further modified by using the backtrans-

formed average Fisher’s Z, it is Hittner, May, and Silver’s Z Test Statistics.

4.Meng, Rosenthal, and Rubin’s Z Test Statistic

Meng’s Z is equivalent to Dunn and Clark’s test statistic asymptotically but is in a

rather simple and easy-to-use form. The following equation yields Meng’s Z,

z = (Z12 − Z23)

√(n− 3)

2(1− r13)h

where

h =1− f r2

1− r2

f =1− r13

2(1− r2),which must be ≤ 1,

r2 =r12

2 + r232

2

The bound on f is derived from constraints that the covariance matrix for correlation

coefficients must be nonnegative, so, f should be equal to 1 if (1−r13)/(2(1−r2)) > 1.

In addition, from the equation of Meng’s Z test statistic, we can obtain a (1 −

α/2)100% confidence interval of r12 and r23:

L,U = (Z12 − Z23)± zα2

√2(1− r13)hn− 3

5.Hittner, May, and Silver’s Z Test Statistic

Silver and Dunlap on 1987 first proposed the approach to backtransform averaged

Fisher’s r − to − Z, and the method was applied to the comparison of overlapping

correlations by Hittner et al on 2003. Hitter’s Z is based on Steiger’s Z, and is

28

calculated as,

z =(Z12 − Z23)

√n− 3√

2− 2c

where

c =r13(1− 2r2z)− 1

2r2z(1− 2r2z − r213)

(1− r2z)2,

rz =exp(2Z − 1)

exp(2Z + 1)

and

Z =Z12 + Z23

2

Silver and Dunlap shown that the backtransformed average Z is generally less biased

than using the average r in Steiger’s Z when sample size is very small, and the average

Z becomes increasingly less biased for large values of average r.[13]

3.2.3 Exact Inference of Test Statistics

In fact, the test statistics which we introduced above were all proposed in past studies.

However, they were used to compare the strength of association between a variable

X1, and each two potential predictor variables, X2 and X3, so the statistics were

used to test whether ρ12 = ρ13, and ρ23 is a nuisance parameter. But in this thesis,

through using scores in correlation coefficients and transforming the formula of test

statistics, the tests will be applied to comparing ρ12 = ρ23 for three dependent ordinal

variables. On the other hand, the test statistics in the past studies were used to

compare correlations between dependent continuous variables, but we try to measure

whether they can be used to compare correlations for dependent ordinal variables

and to choose more appropriate test statistics to apply. Based on the two points, we

29

simulated a sample to test whether the transformed test statistics we mentioned in

this chapter can be used in real medical data.

3.3 Modified Test Statistics

Suppose the estimated test statistics θ when they are used to test correlations with an

ordinal variables between adjacent times do not accurately follow standard normal

distribution. Therefore we need to modify the test statistics. It means the test

statistics need to be re-normalized as,

θmodified =θ − E(θ)√V ar(θ)

Based on the test statistics which we introduced in last section, E(θ) and V ar(θ)

should be the functions of sample size and the correlation coefficients, and denote

them as functions f1(n, r12, r23, r13) and f2(n, r12, r23, r13) respectively. For simplicity

of notation, we replace them to f1 and f2.

θmodified =θ − f1√f2

=1√f2θ − f1√

f2= θ + θf3 = θ +B

Therefore, our objective of modification is to find the function f3(n, r12, r23, r13) from

a theoretical and empirical points of view. Because of the complicated equations of

test statistics we list above which include sample size and three parameters, it is

difficult to find E(θ) and V ar(θ). Therefore, we try to modify the test statistics in

the empirical way, and it will show in simulation study on Chapter 4.

30

Chapter 4

Simulation Study

4.1 Data Generation

A limited study for the evaluation of eight test statistics for comparing dependent

correlations with ordinal longitudinal variables based on common sample is conducted.

For the purpose of simulation, we generate a sample of ordinal random variables that

meet the following conditions.

• Samples are generated with U = (U1, U2, U3) following the continuous multi-

variate normal distribution N(0,Σ);

Σ =

1 ρ12 ρ13

ρ12 1 ρ23

ρ13 ρ23 1

(4.1)

31

• The ordinal variables are categorized by U. Each variable consists of 5 cate-

gories;

• Sample size is 30, 60, 90, and 120 respectively;

• Replications are 10, 000 times.

• The appropriate two-tailed test are performed at nominal levels of 0.01, 0.05

and 0.10.

Here is the description of the conditions.

First, we generated a sample of U = (U1, U2, U3) having the continuous multi-

variate normal distribution N(0,Σ). Under the null hypotheses, we tried to use 29

parameter configurations, which adequately cover the possibilities for ρ12 = ρ23 that

the covariance matrix is a positive definite. We summarized the 29 parameter con-

figuration as these two patterns of Σ. That is,

Σ1 =

1 ρ ρ2

ρ 1 ρ

ρ2 ρ 1

Σ2 =

1 ρ1 ρ2

ρ1 1 ρ1

ρ2 ρ1 1

In Σ1, we choose the correlation coefficients ρ = 0, 0.1, 0.2, ..., 0.9. The propose

of setting Σ1 is to measure the effect of ρ12 and ρ23 in Σ on the result. Besides, in

order to measure the rule of ρ13 in Σ, we set the second pattern of Σ, i.e. Σ2, where

ρ1 = 0, 0.1, 0.3, 0.5, 0.7 and ρ2 = 0.1, 0.3, 0.5, 0.7. We note that the parameter ρ13 is

nuisance parameter which must be handled. We set ρ12 and ρ23 are all positive because

we tried some early runs including configurations where ρ12 or both parameters were

32

negative, but all the results were consistent with the case where all parameters are

positive. Therefore, those situations were not used in subsequent runs.

On the other hand, for the power analysis, we want to examine the extent to which

different degrees of discrepancy between ρ12 and ρ23 (hereinafter refereed to as effect

sizes) affect the results. So we try 16 parameter configurations under alternative

hypotheses. Both of the effect size and the nuisance parameter ρ13 in Σ that we

examined were 0.1, 0.15, 0.3, 0.5.

Secondly, we categorize the dependent variables Umn into 5 levels, where m =

1, 2, 3 and n = 1, 2, 3, ..., N , N is sample size. Then we could get the ordinal variable

Xmn. We assume that the probabilities of numbers falling in each ordinal category

are equal, so the range of each category equals to 1 over 5 base on standard normal

distribution. So if Umn falls in the interval −∞ to −0.84162, it is under category 1,

etc. Thus, we get

Xmn =

1, if Umn ∈ (−∞,−0.84162];

2, if Umn ∈ (−0.84162,−0.25335];

3, if Umn ∈ (−0.25335, 0.25335];

4, if Umn ∈ (0.25335, 0.84162];

5, if Umn ∈ (0.84162,+∞);

where m = 1, 2, 3, and n = 1, 2, 3, ..., N,N is the sample size.

In this way, we obtain a sample that contains N observations. Next, we split the

sample on a contingency table. For instance, if X1n = r, X2n = s, we count once for

the cell on row r and column s in contingency table with correlation coefficient r12.

33

We performe all of the simulation using R data step language.

4.2 Hypotheses and the Criteria of Test Statistics

4.2.1 Hypotheses

The hypotheses for simulation tests are

H0 : ρ12 = ρ23;

Ha : ρ12 6= ρ23.

4.2.2 The Criteria of Test Statistics

Given the 10,000 iterations, the empirical levels of test statistics should be near the

nominal levels. It means that the 0.01, 0.05, 0.10 levels used here would imply that

the particular test being considered ought to reject H0 approximately 100, 500, 1000

times respectively, at any of these null parameter values. The empirical level is more

close to nominal level, the test statistic is relatively more appropriate. However, the

choice of the test statistics depends not only on sample size but also the magnitude

of the correlations. Besides, under the alternative hypotheses, we choose the optimal

test statistic which should yield the highest empirical power.

4.3 Results of Simulation

In compliance with the above notations and rules of generating the simulated data,

the simulation results are given below.

34

4.3.1 Empirical Level

1. Results with Σ1

We first evaluate the result of simulation based on the generated data with Σ1. Tables

4.1, 4.2 and 4.3 show the empirical levels of all eight statistics for ρ = 0, 0.1, ..., 0.9

and n = 30, 60, 90, 120 based on 10,000 replication at nominal level α = 0.01, 0.05 and

0.10 respectively. We denote that Zo = Olkin’s Z, Zd = Dunn and Clark’s Z, Zs = S-

teriger’s Z, Zm = Meng’s Z, Zh = Hitter’s Z, Th = Hotelling’s t, Tw = William’s t, Tm

= William’s modified t per Hendrickson in the tables and figures below. Observations

from the tables are that, there are considerable variability in empirical levels across

the eight statistics for the various sample sizes and the magnitude of correlation coef-

ficients ρ. Nevertheless, there were some trends that emerged despite the difference.

First, across all sample sizes and nominal levels, Olkin’s Z has substantially greater

empirical level than do the other seven test statistics when the magnitude of ρ are

small to moderate (i.e, 0 to 0.4). Whereas for the medium and large correlation ρ

(i.e, 0.5 to 0.9), the discrepancy between Olkin’s Z and the other seven test statistics

becomes less pronounced, especially for the large sample sizes (i.e, n = 90 and 120).

In addition, across all the sample sizes, for small and moderate correlation ρ (i.e,

0 to 0.4) the remaining seven ones of the eight statistical tests (here are Dunn and

Clark’s Z, Steriger’s Z, Meng’s Z, Hitter’s Z, Hotelling’s t, William’s t, William’s

modified t per Hendrickson) yield the empirical levels that hovered around the nominal

levels. However, there were some instances in which the empirical level was somewhat

liberal. For example, we now focus on the result in Table 4.2. For the small correlation

35

ρ = 0.3 and 0.4, the empirical levels for Dunn and Clsrk’s Z, Hotelling’s t and

William’s modified t per Hendrickson for all the sample sizes exceeded 0.05 and range

from 0.054 to 0.067. By comparing the analysis, it is interesting to note that the

remaining four of seven statistics are conservative with respect to control of empirical

level for this situation. It is showed that Dunn and Clark’s Z, Hotelling’s t and

William’s modified t per Hendrickson perform more appropriately empirical levels

when correlation values are range from 0 to 0.2, however Steriger’s Z, Meng’s Z,

Hitter’s Z and William’s t perform well at ρ = 0.3 and 0.4. On the other hand, with

regard to the remaining seven test statistics, we find that they control their empirical

levels fairly effectively for medium and large correlation ρ (i.e. 0.5 to 0.9), and

empirical levels for these seven test statistics trend up as the correlation ρ increased

from 0 to 0.9 cross all the sample size.

We plotted the empirical levels to intuitively distinguish and compare the eight

test statistics. The figures 4.1, 4.2, 4.3 show the empirical levels of all eight test

statistics at the nominal levels α = 0.01, 0.05 and 0.10 respectively for n = 30, 60, 90,

and 120. The figures 4.1, 4.2, 4.3 show the empirical levels of statistics at the three

nominal levels respectively.

From the plots we can easily find some trends that we discussed above. For

instance, the empirical level of Olkin’s Z trend to decrease or reasonable stabilization

in a range given the small and medium sample sizes (i.s,N=30 and 60) from ρ = 0 to

0.9. However, the plots are increased for the large sample sizes (i.e, N=90 and 120).

In addition, the empirical levels for other seven test statistics are overall trend up as

correlation ρ increases. Moreover, the empirical levels for the seven test statistics are

36

Table 4.1: Empirical levels of eight statistics for nominal level α = 0.01 with Σ1

N ρ Zo Zd Zs Zm Zh Th Tw Tm

30

0 0.0218 0.0110 0.0089 0.0073 0.0073 0.0096 0.0092 0.00940.1 0.0197 0.0098 0.0075 0.0059 0.0059 0.0092 0.0086 0.00910.2 0.0228 0.0119 0.0100 0.0082 0.0082 0.0119 0.0106 0.01190.3 0.0226 0.0131 0.0101 0.0090 0.0090 0.0141 0.0113 0.01400.4 0.0202 0.0143 0.0125 0.0109 0.0109 0.0158 0.0127 0.01570.5 0.0162 0.0152 0.0141 0.0125 0.0125 0.0165 0.0134 0.01650.6 0.0136 0.0165 0.0155 0.0142 0.0142 0.0198 0.0144 0.01980.7 0.0100 0.0193 0.0176 0.0166 0.0166 0.0245 0.0159 0.02450.8 0.0054 0.0235 0.0225 0.0218 0.0219 0.0274 0.0205 0.02740.9 0.0016 0.0288 0.0273 0.0264 0.0264 0.0306 0.0229 0.0306

60

0 0.0158 0.0108 0.0098 0.0091 0.0091 0.0103 0.0101 0.01030.1 0.0153 0.0111 0.0101 0.0088 0.0088 0.0108 0.0105 0.01080.2 0.0148 0.0105 0.0096 0.0088 0.0088 0.0106 0.0102 0.01050.3 0.0167 0.0128 0.0122 0.0113 0.0113 0.0145 0.0126 0.01430.4 0.0171 0.0141 0.0133 0.0125 0.0125 0.0160 0.0134 0.01600.5 0.0150 0.0144 0.0133 0.0128 0.0128 0.0180 0.0130 0.01800.6 0.0145 0.0161 0.0152 0.0143 0.0143 0.0204 0.0145 0.02040.7 0.0140 0.0179 0.0173 0.0169 0.0169 0.0238 0.0166 0.02380.8 0.0118 0.0214 0.0210 0.0207 0.0207 0.0270 0.0197 0.02700.9 0.0112 0.0268 0.0260 0.0253 0.0253 0.0302 0.0238 0.0302

90

0 0.0161 0.0106 0.0093 0.0089 0.0089 0.0101 0.0100 0.01010.1 0.0125 0.0101 0.0094 0.0088 0.0088 0.0098 0.0096 0.00980.2 0.0147 0.0119 0.0113 0.0105 0.0105 0.0120 0.0116 0.01200.3 0.0156 0.0126 0.0118 0.0116 0.0116 0.0139 0.0120 0.01390.4 0.0149 0.0134 0.0125 0.0118 0.0118 0.0158 0.0125 0.01580.5 0.0160 0.0156 0.0151 0.0143 0.0143 0.0198 0.0150 0.01980.6 0.0162 0.0170 0.0164 0.0159 0.0159 0.0215 0.0162 0.02150.7 0.0181 0.0217 0.0211 0.0205 0.0205 0.0286 0.0202 0.02860.8 0.0151 0.0230 0.0226 0.0223 0.0223 0.0280 0.0220 0.02800.9 0.0136 0.0228 0.0225 0.0222 0.0223 0.0256 0.0215 0.0256

120

0 0.0122 0.0100 0.0096 0.0091 0.0091 0.0099 0.0098 0.00990.1 0.0124 0.0098 0.0095 0.0092 0.0092 0.0098 0.0097 0.00980.2 0.0131 0.0108 0.0104 0.0101 0.0101 0.0110 0.0107 0.01100.3 0.0145 0.0120 0.0119 0.0115 0.0115 0.0136 0.0119 0.01360.4 0.0136 0.0120 0.0119 0.0114 0.0114 0.0145 0.0119 0.01450.5 0.0144 0.0136 0.0131 0.0125 0.0125 0.0170 0.0129 0.01700.6 0.0132 0.0142 0.0137 0.0134 0.0134 0.0182 0.0134 0.01820.7 0.0188 0.0218 0.0215 0.0209 0.0209 0.0286 0.0209 0.02860.8 0.0177 0.0226 0.0226 0.0223 0.0223 0.0281 0.0220 0.02810.9 0.0158 0.0234 0.0231 0.0231 0.0232 0.0277 0.0225 0.0277

37



30

0 0.0751 0.0527 0.0498 0.0461 0.0461 0.0506 0.0500 0.05040.1 0.0755 0.0520 0.0485 0.0443 0.0443 0.0509 0.0490 0.05090.2 0.0796 0.0561 0.0519 0.0470 0.0470 0.0559 0.0522 0.05580.3 0.0713 0.0535 0.0510 0.0476 0.0476 0.0562 0.0505 0.05600.4 0.0740 0.0595 0.0569 0.0535 0.0535 0.0640 0.0560 0.06400.5 0.0766 0.0677 0.0641 0.0616 0.0616 0.0747 0.0629 0.07460.6 0.0701 0.0680 0.0657 0.0637 0.0637 0.0784 0.0641 0.07840.7 0.0632 0.0704 0.0683 0.0673 0.0673 0.0828 0.0658 0.08280.8 0.0615 0.0811 0.0795 0.0788 0.0789 0.0933 0.0755 0.09330.9 0.0509 0.0913 0.0902 0.0898 0.0901 0.0981 0.0853 0.0981

60

0 0.0599 0.0484 0.0458 0.0437 0.0437 0.0471 0.0459 0.04700.1 0.0619 0.0502 0.0481 0.0462 0.0462 0.0493 0.0484 0.04930.2 0.0682 0.0582 0.0562 0.0539 0.0539 0.0589 0.0565 0.05880.3 0.0632 0.0546 0.0532 0.0503 0.0503 0.0579 0.0529 0.05790.4 0.0646 0.0568 0.0553 0.0538 0.0538 0.0631 0.0548 0.06310.5 0.0684 0.0642 0.0625 0.0613 0.0613 0.0713 0.0621 0.07130.6 0.0658 0.0657 0.0643 0.0634 0.0634 0.0772 0.0637 0.07720.7 0.0655 0.0702 0.0692 0.0684 0.0684 0.0848 0.0681 0.08480.8 0.0660 0.0741 0.0734 0.0731 0.0731 0.0873 0.0716 0.08720.9 0.0695 0.0884 0.0879 0.0874 0.0874 0.0970 0.0844 0.0970

90

0 0.0591 0.0514 0.0502 0.0482 0.0482 0.0507 0.0503 0.05070.1 0.0551 0.0491 0.0483 0.0469 0.0469 0.0490 0.0484 0.04890.2 0.0555 0.0508 0.0498 0.0490 0.0490 0.0520 0.0500 0.05200.3 0.0627 0.0567 0.0555 0.0537 0.0537 0.0605 0.0556 0.06050.4 0.0649 0.0591 0.0577 0.0568 0.0568 0.0667 0.0575 0.06670.5 0.0670 0.0637 0.0629 0.0622 0.0622 0.0723 0.0626 0.07230.6 0.0706 0.0699 0.0694 0.0687 0.0687 0.0820 0.0686 0.08200.7 0.0741 0.0775 0.0768 0.0757 0.0757 0.0913 0.0754 0.09130.8 0.0723 0.0808 0.0801 0.0801 0.0802 0.0943 0.0785 0.09430.9 0.0701 0.0829 0.0826 0.0825 0.0825 0.0942 0.0810 0.0942

120

0 0.0551 0.0495 0.0485 0.0478 0.0478 0.0489 0.0487 0.04890.1 0.0548 0.0499 0.0489 0.0481 0.0481 0.0495 0.0490 0.04950.2 0.0549 0.0498 0.0491 0.0481 0.0481 0.0508 0.0491 0.05080.3 0.0599 0.0549 0.0545 0.0534 0.0534 0.0590 0.0544 0.05900.4 0.0603 0.0558 0.0553 0.0544 0.0545 0.0632 0.0550 0.06320.5 0.0589 0.0565 0.0560 0.0554 0.0555 0.0681 0.0557 0.06810.6 0.0631 0.0628 0.0622 0.0619 0.0619 0.0746 0.0619 0.07460.7 0.0745 0.0764 0.0759 0.0755 0.0755 0.0907 0.0754 0.09070.8 0.0765 0.0811 0.0808 0.0807 0.0807 0.0941 0.0797 0.09410.9 0.0753 0.0818 0.0815 0.0815 0.0818 0.0922 0.0802 0.0922

38



30

0 0.1277 0.1020 0.0985 0.0940 0.0940 0.0998 0.0980 0.09940.1 0.1308 0.1047 0.1008 0.0966 0.0967 0.1031 0.1003 0.10280.2 0.1370 0.1096 0.1053 0.1023 0.1023 0.1098 0.1052 0.10960.3 0.1274 0.1032 0.0996 0.0954 0.0954 0.1063 0.0983 0.10620.4 0.1322 0.1118 0.1087 0.1055 0.1055 0.1178 0.1072 0.11770.5 0.1369 0.1220 0.1192 0.1164 0.1164 0.1334 0.1170 0.13340.6 0.1338 0.1267 0.1246 0.1230 0.1230 0.1403 0.1226 0.14010.7 0.1289 0.1300 0.1280 0.1269 0.1269 0.1466 0.1248 0.14660.8 0.1284 0.1423 0.1410 0.1399 0.1399 0.1591 0.1357 0.15910.9 0.1263 0.1561 0.1555 0.1552 0.1555 0.1649 0.1503 0.1649

60

0 0.1152 0.1007 0.0987 0.0972 0.0972 0.0995 0.0986 0.09950.1 0.1177 0.1025 0.1010 0.0988 0.0988 0.1022 0.1001 0.10210.2 0.1277 0.1126 0.1109 0.1092 0.1092 0.1137 0.1106 0.11370.3 0.1182 0.1072 0.1056 0.1043 0.1043 0.1102 0.1054 0.11020.4 0.1220 0.1122 0.1106 0.1084 0.1084 0.1216 0.1096 0.12160.5 0.1218 0.1145 0.1134 0.1128 0.1128 0.1265 0.1127 0.12650.6 0.1253 0.1212 0.1204 0.1189 0.1189 0.1374 0.1194 0.13740.7 0.1263 0.1269 0.1258 0.1254 0.1254 0.1441 0.1242 0.14410.8 0.1271 0.1329 0.1318 0.1312 0.1312 0.1511 0.1300 0.15110.9 0.1349 0.1487 0.1483 0.1480 0.1480 0.1616 0.1445 0.1616

90

0 0.1095 0.1004 0.0990 0.0979 0.0979 0.0992 0.0987 0.09910.1 0.1070 0.0989 0.0978 0.0964 0.0964 0.0984 0.0977 0.09840.2 0.1062 0.0991 0.0985 0.0973 0.0973 0.1002 0.0983 0.10020.3 0.1149 0.1073 0.1067 0.1058 0.1058 0.1121 0.1064 0.11210.4 0.1213 0.1141 0.1133 0.1126 0.1126 0.1239 0.1131 0.12390.5 0.1237 0.1176 0.1168 0.1157 0.1157 0.1311 0.1161 0.13110.6 0.1287 0.1256 0.1246 0.1239 0.1239 0.1432 0.1240 0.14320.7 0.1353 0.1362 0.1355 0.1351 0.1352 0.1519 0.1345 0.15190.8 0.1369 0.1412 0.1406 0.1404 0.1405 0.1580 0.1395 0.15800.9 0.1358 0.1437 0.1434 0.1432 0.1432 0.1574 0.1424 0.1574

120

0 0.1087 0.1007 0.0997 0.0991 0.0991 0.0998 0.0997 0.09980.1 0.1073 0.1012 0.1006 0.0994 0.0994 0.1016 0.1005 0.10160.2 0.1086 0.1018 0.1008 0.0992 0.0992 0.1046 0.1007 0.10460.3 0.1121 0.1061 0.1045 0.1038 0.1038 0.1113 0.1043 0.11130.4 0.1165 0.1119 0.1112 0.1106 0.1106 0.1206 0.1110 0.12060.5 0.1189 0.1163 0.1158 0.1153 0.1153 0.1286 0.1155 0.12860.6 0.1206 0.1177 0.1169 0.1161 0.1161 0.1353 0.1161 0.13530.7 0.1383 0.1389 0.1383 0.1382 0.1382 0.1554 0.1377 0.15540.8 0.1430 0.1468 0.1465 0.1462 0.1462 0.1663 0.1457 0.16630.9 0.1397 0.1471 0.1470 0.1469 0.1471 0.1603 0.1452 0.1603

39

far away from the nominal levels when the correlation ρ is greater than 0.5. Besides,

Hotelling’s t and William’s modified t per Hendrickson are relatively poor to control

the empirical levels at that situation. Therefore, we need to do some modification for

test statistics as the correlation ρ increases.

0.0 0.5 1.0 1.5

0.00

0.01

0.02

0.03

0.04

α=0.01

ZoZdZsZmZhthtwtm

(a)Correlation Coefficient

Em

pirc

al S

izes

Rat

es N

=30

0.0 0.5 1.0 1.5

0.00

0.01

0.02

0.03

0.04

α=0.01

ZoZdZsZmZhthtwtm

(b)Correlation Coefficient

Em

pirc

al S

izes

Rat

es a

t N=

60

0.0 0.5 1.0 1.5

0.00

0.01

0.02

0.03

0.04

α=0.01

ZoZdZsZmZhthtwtm

(c)Correlation Coefficient

Em

pirc

al S

izes

Rat

es a

t N=

90

0.0 0.5 1.0 1.5

0.00

0.01

0.02

0.03

0.04

α=0.01

ZoZdZsZmZhthtwtm

(d)Correlation Coefficient

Em

pirc

al S

izes

Rat

es a

t N=

120

Figure 4.1: Empirical levels at α = 0.01

40

0.0 0.5 1.0 1.5

0.05

0.06

0.07

0.08

0.09

0.10

α=0.05

ZoZdZsZmZhthtwtm


Em

pirc

al S

izes

Rat

es a

t N=

30

0.0 0.5 1.0 1.5

0.05

0.06

0.07

0.08

0.09

0.10

α=0.05

ZoZdZsZmZhthtwtm


Em

pirc

al S

izes

Rat

es a

t N=

60

0.0 0.5 1.0 1.5

0.05

0.06

0.07

0.08

0.09

0.10

α=0.05

ZoZdZsZmZhthtwtm


Em

pirc

al S

izes

Rat

es a

t N=

90

0.0 0.5 1.0 1.5

0.05

0.06

0.07

0.08

0.09

0.10

α=0.05

ZoZdZsZmZhthtwtm


Em

pirc

al S

izes

Rat

es a

t N=

120


41

0.0 0.5 1.0 1.5

0.10

0.12

0.14

0.16

α=0.10

ZoZdZsZmZhthtwtm


Em

pirc

al S

izes

Rat

es a

t N=

30

0.0 0.5 1.0 1.5

0.10

0.12

0.14

0.16

α=0.10

ZoZdZsZmZhthtwtm


Em

pirc

al S

izes

Rat

es a

t N=

60

0.0 0.5 1.0 1.5

0.10

0.12

0.14

0.16

α=0.10

ZoZdZsZmZhthtwtm


Em

pirc

al S

izes

Rat

es a

t N=

90

0.0 0.5 1.0 1.5

0.10

0.12

0.14

0.16

α=0.10

ZoZdZsZmZhthtwtm


Em

pirc

al S

izes

Rat

es a

t N=

120


42

With regard to the result of simulation in the first pattern, Σ1, and based on the

idea of modification we mentioned in Section 3.3, we try to modify Meng’s Z test

statistics. From the Figures 4.1 to 4.3, the variational trend of empirical size for

Meng’s Z keeps consistent at difference sample sizes. For instance, at nominal level

α = 0.01, Figure.4.4 shows the empirical levels of Zm and regression lines at sample

size N = 30, 60, 90, 120 respectively. We can see the slope of the regression lines for

each empirical level are broadly similar at different sample sizes. From this point of

view, we modify Zm such that decreases as the correlation ρ increases, and the most

ideal effect of the modification is that the regression line for Zmmodifiedparallel and

close to the nominal level line.

It means that we can modify Meng’s Z by using the idea in Section 3.3 where the

function f3 is a function of ρ in pattern Σ1, and it does not depend on sample size.

Zmmodified= Zm + Zmf3

= Zm + Zma(ρ− ρ0)

where ρ0 in this function is the average values of point of intersection for regression

line and the line of the nominal level, and a is the empirical value and satisfies a < 1

so that the regression line for Zmmodifiedis closer to the nominal level line. According

to arithmetic and several simulation test, we modify the Meng’s Z to a relatively

ideal test statistics with a = 0.1625 and ρ0 = 0.2. Table 4.4 compares the empirical

levels of Zm and Zmmodifiedbased on generated data with pattern Σ1. Through the

adjustment, we find that the empirical levels of the modified Zm obviously yielded

values hovering around the nominal levels that did better than the empirical levels

of Zm, especially when correlation coefficient ρ is moderate to large (i.e, 0.5 to 0.9).

43

●

●

● ●

●

●

●

●

●

●

0.0 0.2 0.4 0.6 0.8

0.01

00.

015

0.02

00.

025

Correlation Coefficient

Em

pirc

al S

izes

of Z

m a

t N=

30

(a)

●

●

●

●

●

●

●

●

● ●

0.0 0.2 0.4 0.6 0.8

0.01

00.

015

0.02

00.

025


Em

pirc

al S

izes

of Z

m a

t N=

60

(b)

●

●

●

●●

●

●

●

●

●

0.0 0.2 0.4 0.6 0.8

0.01

00.

015

0.02

00.

025


Em

pirc

al S

izes

of Z

m a

t N=

90

(c)

●

●●

●

●

●

●

●

●

●

0.0 0.2 0.4 0.6 0.8

0.01

00.

015

0.02

00.

025


Em

pirc

al S

izes

of Z

m a

t N=

120

(d)

Figure 4.4: Empirical levels of Meng’s Z at α = 0.01

44

Figure 4.5 shows the empirical size of modified Zm and regression line. We can easily

compare the regression lines of empirical levels for modified Zm (see Figure 4.5) and

original Zm (see Figure 4.4), and the regression lines of empirical levels for modified

Zm is more paralleled and closer to the nominal level lines at four settled sample sizes.

It is shown that the method of modification we mentioned in Section 3.3 effectively

re-normalize the test statistics when we generated data with the pattern Σ1. However,

there still exist some problems by using the method of modification because it cannot

apply to general cases. The test statistics should be modified by estimating their

expected values and variances. We suggest that researchers who work with the same

issue might consider this direction to study.

45

Table 4.4: Empirical levels of Zm and Zmmodifiedwith Σ1

α = 0.01 α = 0.05 α = 0.10N ρ Zm Zmmodified

Zm ZmmodifiedZm Zmmodified

30

0 0.0073 0.0094 0.0461 0.0533 0.0940 0.10670.1 0.0059 0.0091 0.0443 0.0498 0.0966 0.10340.2 0.0082 0.0090 0.0470 0.0471 0.1023 0.09660.3 0.0090 0.0089 0.0476 0.0494 0.0954 0.09810.4 0.0109 0.0089 0.0535 0.0494 0.1055 0.09690.5 0.0125 0.0100 0.0616 0.0468 0.1164 0.09890.6 0.0142 0.0095 0.0637 0.0454 0.1230 0.10130.7 0.0166 0.0095 0.0673 0.0487 0.1269 0.09980.8 0.0218 0.0102 0.0788 0.0517 0.1399 0.10280.9 0.0264 0.0103 0.0898 0.0504 0.1552 0.0998

60

0 0.0091 0.0102 0.0437 0.0552 0.0972 0.10970.1 0.0088 0.0096 0.0462 0.0480 0.0988 0.09960.2 0.0088 0.0100 0.0539 0.0491 0.1092 0.10050.3 0.0113 0.0101 0.0503 0.0480 0.1043 0.09590.4 0.0125 0.0095 0.0538 0.0494 0.1084 0.09710.5 0.0128 0.0097 0.0613 0.0467 0.1128 0.09610.6 0.0143 0.0109 0.0634 0.0470 0.1189 0.09670.7 0.0169 0.0097 0.0684 0.0506 0.1254 0.09940.8 0.0207 0.0107 0.0731 0.0502 0.1312 0.10510.9 0.0253 0.0103 0.0874 0.0499 0.1480 0.1021

90

0 0.0089 0.0105 0.0482 0.0546 0.0979 0.10470.1 0.0088 0.0102 0.0469 0.0511 0.0964 0.10210.2 0.0105 0.0094 0.0490 0.0512 0.0973 0.10230.3 0.0116 0.0092 0.0537 0.0509 0.1058 0.10090.4 0.0118 0.0091 0.0568 0.0493 0.1126 0.10110.5 0.0143 0.0080 0.0622 0.0501 0.1157 0.10020.6 0.0159 0.0097 0.0687 0.0499 0.1239 0.09860.7 0.0205 0.0101 0.0757 0.0489 0.1351 0.10110.8 0.0223 0.0096 0.0801 0.0501 0.1404 0.09910.9 0.0222 0.0089 0.0825 0.0491 0.1432 0.0990

120

0 0.0091 0.0095 0.0478 0.0558 0.0991 0.11110.1 0.0092 0.0102 0.0481 0.0507 0.0994 0.10040.2 0.0101 0.0089 0.0481 0.0474 0.0992 0.09920.3 0.0115 0.0092 0.0534 0.0501 0.1038 0.10120.4 0.0114 0.0090 0.0544 0.0474 0.1106 0.09890.5 0.0125 0.0081 0.0554 0.0492 0.1153 0.09910.6 0.0134 0.0096 0.0619 0.0482 0.1161 0.09930.7 0.0209 0.0107 0.0755 0.0494 0.1382 0.10180.8 0.0223 0.0102 0.0807 0.0501 0.1462 0.10090.9 0.0231 0.0091 0.0815 0.0474 0.1469 0.1008

46

●● ● ● ●

●

● ●

● ●

0.0 0.2 0.4 0.6 0.8

0.00

80.

012

0.01

60.

020


Em

pirc

al S

izes

of m

odifi

ed Z

m a

t N=

30

(a)

●

●● ●

●●

●

●

●●

0.0 0.2 0.4 0.6 0.8

0.00

80.

012

0.01

60.

020


Em

pirc

al S

izes

of m

odifi

ed Z

m a

t N=

60

(b)

●●

●● ●

●

●●

●

●

0.0 0.2 0.4 0.6 0.8

0.00

80.

012

0.01

60.

020


Em

pirc

al S

izes

of m

odifi

ed Z

m a

t N=

90

(c)

●

●

●●

●

●

●

●

●

●

0.0 0.2 0.4 0.6 0.8

0.00

80.

012

0.01

60.

020


Em

pirc

al S

izes

of m

odifi

ed Z

m a

t N=

120

(d)

Figure 4.5: Empirical level of modified Meng’s Z at α = 0.01

47

2. Result with Σ2

Table 4.5 presents the result of the simulation study based on the generated data

with Σ2 consistent with H0 : ρ12 = ρ23 for sample size 30, 60, 90 and 120 respectively

at nominal level α = 0.01. We generated a sample with Σ2 to compare the eight test

statistics according to changes of correlation coefficient ρ13 in Σ.

With regard to the empirical levels finding in Table 4.5, we discover that the

empirical level for each test statistic is changed by the magnitude of ρ13. As we

mentioned in the result with Σ1, the seven of the eight test statistics, Dunn and

Clark’s Z , Steriger’s Z, Meng’s Z , Hitter’s Z, Hotelling’s t, William’s t, William’s

modified t per Hendrickson, are conservative with respect to the value of empirical

levels when correlation coefficients ρ1 are small to moderate. From Table 4.5, the

magnitude of ρ2 have had little effect on the empirical levels for test statistics when

ρ1 are small (i.e, 0 to 0.3). When ρ2 increases, the empirical levels are far away from

nominal level and are smaller than it. The effect of change for ρ2 is not in a good

way. It means when researchers need to choose one optimal test statistic to apply to

the real data, they do not pay much attention to deal with the correlation ρ2 when

ρ1 are small.

However, when ρ1 are medium to large (i.e, 0.5 and 0.7), the magnitude of ρ2

played a role in the control of the empirical levels. The effect of ρ2 is more pronounced

when ρ1 equals 0.5. For example, consider the small sample size in which ρ = 0.5

based on the generated data with Σ1 (i.e, n = 30, ρ12 = ρ23 = 0.5, and ρ13 = 0.52

= 0.25), the empirical levels for Steriger’s Z, Meng’s Z, Hitter’s Z and William’s

t shown in Table 4.1 are 0.0141, 0.0125, 0.0125 and 0.0134 respectively which all

48

rates are over the nominal level α = 0.01. Nevertheless, in the same situation but

the sample were generated with Σ2, the same four test statistics demonstrate the

relatively optimal empirical levels ranging from 0.0099 to 0.0111, except ρ2 = 0.5. On

the other hand, although the effect of ρ2 is not obvious when ρ1 are large (i.e ρ = 0.7),

the test statistics performed better due to the change of ρ2. Comparing the result in

Table 4.1 and Table 4.5, we also pay attention to Steriger’s Z, Meng’s Z, Hitter’s Z

and William’s t, the empirical levels for them are more closed to the nominal level as

ρ13 in Σ increases.

Even thought the force of ρ13 in Σ is not consistent for all the situations, we can

do the modification to test statistics follow the method above. In order to deal with

the inflated empirical levels as ρ12 = ρ23 increases, we can modify the test statistics

by multiplying an empirical function of correlation coefficients which is similar to last

part we discussed.

Table 4.5: Empirical levels of eight statistics for nominal level α =

0.01 with Σ2

ρ1 ρ2 Zo Zd Zs Zm Zh Th Tw Tm

N = 30

0

0.1 0.0183 0.0093 0.0069 0.0057 0.0057 0.0089 0.0085 0.00890.3 0.0156 0.0079 0.0064 0.0053 0.0053 0.0080 0.0078 0.00800.5 0.0138 0.0065 0.0054 0.0040 0.0040 0.0070 0.0068 0.00700.7 0.0114 0.0062 0.0053 0.0043 0.0043 0.0086 0.0086 0.0086

0.1

0.1 0.0237 0.0115 0.0091 0.0079 0.0079 0.0114 0.0106 0.01130.3 0.0188 0.0096 0.0081 0.0066 0.0066 0.0098 0.0094 0.00970.5 0.0164 0.0091 0.0074 0.0061 0.0061 0.0096 0.0094 0.00960.7 0.0131 0.0072 0.0064 0.0051 0.0051 0.0090 0.0088 0.0089

0.3

0.1 0.0204 0.0128 0.0103 0.0086 0.0086 0.0131 0.0109 0.01280.3 0.0194 0.0130 0.0112 0.0098 0.0098 0.0134 0.0124 0.01340.5 0.0154 0.0098 0.0084 0.0070 0.0070 0.0109 0.0102 0.01080.7 0.0121 0.0090 0.0074 0.0064 0.0064 0.0104 0.0101 0.0104

0.5

0.1 0.0158 0.0121 0.0110 0.0104 0.0104 0.0161 0.0103 0.0161

(see next page)

49

Table Continued(see last page)


0.3 0.0138 0.0122 0.0104 0.0099 0.0099 0.0139 0.0105 0.01390.5 0.0154 0.0098 0.0084 0.0070 0.0070 0.0109 0.0102 0.01080.7 0.0112 0.0134 0.0114 0.0100 0.0100 0.0139 0.0103 0.0139

0.7

0.1 0.0113 0.0187 0.0178 0.0187 0.0187 0.0617 0.0149 0.06160.3 0.0108 0.0195 0.0183 0.0180 0.0180 0.0335 0.0162 0.03330.5 0.0092 0.0193 0.0175 0.0162 0.0162 0.0249 0.0155 0.02480.7 0.0078 0.0205 0.0180 0.0165 0.0165 0.0211 0.0186 0.0211

N = 60

0

0.1 0.0136 0.0093 0.0084 0.0069 0.0069 0.0090 0.0089 0.00900.3 0.0136 0.0089 0.0079 0.0069 0.0069 0.0089 0.0088 0.00890.5 0.0131 0.0092 0.0077 0.0074 0.0074 0.0097 0.0096 0.00970.7 0.0107 0.0076 0.0072 0.0069 0.0069 0.0082 0.0082 0.0082

0.1

0.1 0.0161 0.0110 0.0102 0.0090 0.0090 0.0109 0.0107 0.01090.3 0.0144 0.0093 0.0081 0.0072 0.0072 0.0092 0.0091 0.00920.5 0.0128 0.0094 0.0086 0.0077 0.0077 0.0094 0.0094 0.00940.7 0.0115 0.0088 0.0080 0.0076 0.0076 0.0100 0.0099 0.0100

0.3

0.1 0.0174 0.0131 0.0122 0.0114 0.0114 0.0141 0.0126 0.01410.3 0.0155 0.0123 0.0112 0.0105 0.0105 0.0129 0.0120 0.01290.5 0.0137 0.0107 0.0103 0.0097 0.0097 0.0113 0.0107 0.01130.7 0.012 0.0104 0.0093 0.0088 0.0088 0.0113 0.0110 0.0113

0.5

0.1 0.0138 0.0124 0.0119 0.0114 0.0114 0.0179 0.0117 0.01790.3 0.0126 0.0119 0.0110 0.0102 0.0102 0.0150 0.0109 0.01500.5 0.0149 0.0146 0.0139 0.0131 0.0131 0.0158 0.0141 0.01580.7 0.0136 0.0142 0.0134 0.0126 0.0126 0.0148 0.0142 0.0148

0.7

0.1 0.0132 0.0169 0.0166 0.0168 0.0168 0.0637 0.0145 0.06370.3 0.0134 0.0175 0.0167 0.0165 0.0165 0.0356 0.0156 0.03560.5 0.0124 0.0163 0.0159 0.0156 0.0156 0.0224 0.0154 0.02240.7 0.0130 0.0190 0.0177 0.0168 0.0168 0.0203 0.0177 0.0203

N = 90

0

0.1 0.0124 0.0092 0.0088 0.0085 0.0085 0.0091 0.0090 0.00910.3 0.0117 0.0090 0.0084 0.0075 0.0075 0.0089 0.0089 0.00890.5 0.0109 0.0088 0.0085 0.0081 0.0081 0.0089 0.0089 0.00890.7 0.0112 0.0100 0.0099 0.0093 0.0093 0.0104 0.0104 0.0104

0.1

0.1 0.0140 0.0102 0.0094 0.0090 0.0090 0.0099 0.0097 0.00990.3 0.0143 0.0117 0.0107 0.0093 0.0093 0.0116 0.0116 0.01160.5 0.0130 0.0102 0.0097 0.0093 0.0093 0.0107 0.0104 0.01070.7 0.0110 0.0095 0.0090 0.0085 0.0085 0.0101 0.0101 0.0101

0.3

0.1 0.0172 0.0141 0.0135 0.0127 0.0127 0.0157 0.0135 0.01570.3 0.0135 0.0116 0.0112 0.0110 0.0110 0.0122 0.0114 0.01220.5 0.0123 0.0109 0.0105 0.0103 0.0103 0.0114 0.0110 0.01140.7 0.0141 0.0128 0.0119 0.0115 0.0115 0.0135 0.0133 0.0135

0.5

0.1 0.0155 0.0147 0.0140 0.0136 0.0136 0.0197 0.0138 0.01970.3 0.0134 0.0133 0.0131 0.0127 0.0127 0.0159 0.0130 0.01590.5 0.0148 0.0148 0.0142 0.0136 0.0136 0.0163 0.0143 0.01630.7 0.0124 0.0127 0.0119 0.0116 0.0116 0.0134 0.0127 0.0134

0.7

0.1 0.0153 0.0180 0.0174 0.0177 0.0177 0.0657 0.0164 0.06570.3 0.0159 0.0180 0.0179 0.0178 0.0178 0.0344 0.0172 0.03430.5 0.0157 0.0179 0.0174 0.0172 0.0172 0.0233 0.0171 0.0233

(see next page)

50



0.7 0.0111 0.0155 0.0150 0.0142 0.0142 0.0164 0.0153 0.0164N = 120

0

0.1 0.0120 0.0099 0.0092 0.0088 0.0088 0.0095 0.0095 0.00950.3 0.0107 0.0096 0.0090 0.0085 0.0085 0.0096 0.0095 0.00960.5 0.0125 0.0102 0.0099 0.0098 0.0098 0.0106 0.0105 0.01060.7 0.0123 0.0113 0.0110 0.0109 0.0109 0.0114 0.0114 0.0114

0.1

0.1 0.0140 0.0119 0.0112 0.0103 0.0103 0.0119 0.0117 0.01190.3 0.0129 0.0114 0.0109 0.0098 0.0098 0.0114 0.0114 0.01140.5 0.0118 0.0095 0.0090 0.0088 0.0088 0.0098 0.0098 0.00980.7 0.0113 0.0101 0.0100 0.0097 0.0097 0.0105 0.0105 0.0105

0.3

0.1 0.0153 0.0132 0.0126 0.0122 0.0122 0.0142 0.0128 0.01420.3 0.0137 0.0125 0.0122 0.0114 0.0114 0.0131 0.0125 0.01310.5 0.0134 0.0119 0.0113 0.0111 0.0111 0.0123 0.0118 0.01230.7 0.0135 0.0127 0.0124 0.0121 0.0121 0.0131 0.0130 0.0131

0.5

0.1 0.0149 0.0142 0.0141 0.0139 0.0139 0.0201 0.0140 0.02010.3 0.0140 0.0135 0.0132 0.0129 0.0129 0.0165 0.0132 0.01650.5 0.0120 0.0120 0.0115 0.0112 0.0112 0.0128 0.0119 0.01280.7 0.0127 0.0130 0.0122 0.0118 0.0118 0.0133 0.0129 0.0133

0.7

0.1 0.0145 0.0162 0.0159 0.0160 0.0160 0.0615 0.0156 0.06150.3 0.0156 0.0181 0.0179 0.0178 0.0178 0.0315 0.0173 0.03150.5 0.0148 0.0172 0.0166 0.0165 0.0165 0.0226 0.0165 0.02260.7 0.0142 0.0170 0.0162 0.0160 0.0160 0.0186 0.0162 0.0186

51

4.3.2 Statistical Power

The simulated power estimates at nominal level α = 0.05 for all eight test statistics are

presented in Table 4.6. We denote that ES = effect size which refers to the magnitude

of difference between two correlations ρ12 and ρ23. The specified correlations used to

generate each effect size are as follows: for ES = 0.1, ρ12 = 0.5, ρ23 = 0.4; for ES =

0.15, ρ12 = 0.55, ρ23 = 0.4; for ES = 0.3, ρ12 = 0.5, ρ23 = 0.2; for ES = 0.5, ρ12 = 0.7,

ρ23 = 0.2. As the data indicates, we obtain acceptable levels of power (approximately

0.8 and higher) for all the moderate and large sample sizes (i.e, n = 60, 90, 120) with

an effect size of 0.5 regardless of the value of ρ13. For the smallest sample size, the

acceptable levels of power occur only at effect size of 0.5 and ρ13 = 0.5.

Besides, there are some trends in the estimated power across the statistical tests.

First, there is a general tendency across all situations for Olkin’s Z. Olkin’s Z yields

the highest empirical power for all the cases and is also accompanied by aforemen-

tioned inflated empirical level. As we know the view is that a optimal test statistic

should yield empirical level that are closer to the nominal level and higher power

estimates. We cannot recommend that applied researchers use Olkin’s Z because the

test statistic are contradictory on this point with regard to empirical level and power.

In general, the other seven test statistics being considered, Dunn and Clark’s

Z, Hotelling’s t and William’s modified t per Hendrickson are essentially similar in

aspect of empirical power. On the other hand, Steriger’s Z, Meng’s Z , Hitter’s Z

and William’s t yield equivalently in aspect of empirical power. It is noted that the

empirical powers for the first three test statistics are comparatively appropriate than

other four in all cases. In addition, the power simulation reveals a similar pattern of

52

finding across the different sample size, that is, we can discover that the empirical

power for statistical tests increases as the sample size becomes large. We should

consider the rule of the sample size when we do modification of test statistics.

Table 4.6: Empirical power of all eight statistics for nominal level

α = 0.01

ES ρ13 Zo Zd Zs Zm Zh Th Tw Tm

N = 30

0.1

0.1 0.0949 0.0795 0.0760 0.0735 0.0735 0.0899 0.0743 0.08990.15 0.0915 0.0763 0.0727 0.0706 0.0706 0.0845 0.0714 0.08440.3 0.0985 0.0855 0.0817 0.0788 0.0788 0.0904 0.0816 0.09020.5 0.0981 0.0886 0.0845 0.0809 0.0809 0.0908 0.0865 0.0907

0.15

0.1 0.1263 0.1116 0.1080 0.1054 0.1054 0.1263 0.1049 0.12620.15 0.1268 0.1102 0.1053 0.1020 0.1020 0.1232 0.1029 0.12310.3 0.1325 0.1199 0.1155 0.1115 0.1115 0.1257 0.1148 0.12560.5 0.1488 0.1354 0.1293 0.1242 0.1242 0.1392 0.1313 0.1391

0.3

0.1 0.2568 0.2211 0.2152 0.2088 0.2088 0.2288 0.2141 0.22870.15 0.2675 0.2323 0.2242 0.2164 0.2164 0.2391 0.2240 0.23900.3 0.3024 0.2675 0.2573 0.2492 0.2492 0.2711 0.2607 0.27100.5 0.3630 0.3263 0.3174 0.3091 0.3091 0.3282 0.3224 0.3282

0.5

0.1 0.6600 0.6259 0.6190 0.6125 0.6125 0.6509 0.6149 0.65030.15 0.6581 0.6290 0.6213 0.6135 0.6136 0.6489 0.6180 0.64840.3 0.7396 0.7169 0.7082 0.6997 0.6997 0.7265 0.7086 0.72650.5 0.8469 0.8374 0.8320 0.8268 0.8268 0.8397 0.8335 0.8397

N = 60

0.1

0.1 0.1156 0.1049 0.1033 0.1007 0.1007 0.1199 0.1016 0.11990.15 0.1131 0.1041 0.1023 0.1003 0.1003 0.1160 0.1018 0.11590.3 0.1215 0.1134 0.1102 0.1076 0.1076 0.1207 0.1099 0.12070.5 0.1352 0.1294 0.1256 0.1237 0.1237 0.1327 0.1268 0.1327

0.15

0.1 0.1817 0.1703 0.1683 0.1660 0.1660 0.1926 0.1670 0.19260.15 0.1844 0.1747 0.1719 0.1695 0.1695 0.1928 0.1701 0.19280.3 0.2051 0.1950 0.1910 0.1878 0.1878 0.2084 0.1900 0.20840.5 0.2387 0.2284 0.2249 0.2209 0.2209 0.2342 0.2262 0.2342

0.3

0.1 0.4287 0.4062 0.4008 0.3965 0.3965 0.4183 0.4001 0.41820.15 0.4441 0.4215 0.4169 0.4111 0.4111 0.4321 0.4169 0.43210.3 0.5025 0.4811 0.4765 0.4708 0.4708 0.4872 0.4776 0.48720.5 0.6212 0.6009 0.5955 0.5912 0.5912 0.6043 0.5985 0.6043

0.5

0.1 0.9001 0.8927 0.8908 0.8892 0.8892 0.9050 0.8902 0.90500.15 0.9144 0.9085 0.9069 0.9054 0.9054 0.9183 0.9060 0.91830.3 0.9562 0.9537 0.9529 0.9520 0.9520 0.9570 0.9527 0.95700.5 0.9891 0.9889 0.9888 0.9887 0.9887 0.9891 0.9888 0.9891

(see next page)

53


ES ρ13 Zo Zd Zs Zm Zh Th Tw Tm

N = 90

0.1

0.1 0.1268 0.1193 0.1179 0.1164 0.1164 0.1379 0.1173 0.13790.15 0.1367 0.1302 0.1287 0.1270 0.1270 0.1457 0.1279 0.14570.3 0.1460 0.1404 0.1390 0.1374 0.1374 0.1496 0.1390 0.14960.5 0.1671 0.1620 0.1598 0.1581 0.1581 0.1670 0.1605 0.1670

0.15

0.1 0.2311 0.2222 0.2205 0.2183 0.2183 0.2498 0.2189 0.24970.15 0.2351 0.2269 0.2252 0.2239 0.2239 0.2503 0.2245 0.25030.3 0.2540 0.2468 0.2442 0.2421 0.2421 0.2606 0.2440 0.26060.5 0.3351 0.3269 0.3243 0.3219 0.3219 0.3337 0.3248 0.3337

0.3

0.1 0.5763 0.5594 0.5567 0.5530 0.5530 0.5756 0.5562 0.57560.15 0.5932 0.5771 0.5736 0.5704 0.5704 0.5900 0.5734 0.59000.3 0.6555 0.6416 0.6390 0.6361 0.6361 0.6487 0.6395 0.64870.5 0.7815 0.7716 0.7676 0.7648 0.7648 0.7742 0.7699 0.7742

0.5

0.1 0.9802 0.9792 0.9789 0.9786 0.9786 0.9823 0.9786 0.98230.15 0.9831 0.9819 0.9814 0.9808 0.9809 0.9848 0.9813 0.98480.3 0.9934 0.9932 0.9931 0.9931 0.9932 0.9936 0.9931 0.99360.5 0.9991 0.9991 0.9991 0.9991 0.9992 0.9991 0.9991 0.9991

N = 120

0.1

0.1 0.1502 0.1455 0.1445 0.1429 0.1429 0.1626 0.1436 0.16250.15 0.1593 0.1530 0.1520 0.1512 0.1512 0.1711 0.1515 0.17110.3 0.1722 0.1664 0.1654 0.1647 0.1647 0.1764 0.1653 0.17640.5 0.2030 0.1978 0.1960 0.1948 0.1948 0.2027 0.1962 0.2027

0.15

0.1 0.2813 0.2745 0.2728 0.2713 0.2713 0.3041 0.2719 0.30410.15 0.2874 0.2796 0.2778 0.2762 0.2762 0.3061 0.2767 0.30610.3 0.3292 0.3231 0.3212 0.3191 0.3191 0.3396 0.3207 0.33960.5 0.3973 0.3910 0.3888 0.3873 0.3873 0.3995 0.3894 0.3995

0.3

0.1 0.6898 0.6793 0.6773 0.6745 0.6745 0.6925 0.6771 0.69250.15 0.7134 0.7037 0.7017 0.6998 0.6998 0.7148 0.7017 0.71480.3 0.7790 0.7705 0.7688 0.7671 0.7671 0.7767 0.7693 0.77670.5 0.8829 0.8769 0.8755 0.8743 0.8743 0.8796 0.8764 0.8796

0.5

0.1 0.9951 0.9948 0.9947 0.9947 0.9947 0.9958 0.9947 0.99580.15 0.9963 0.9960 0.9958 0.9958 0.9959 0.9970 0.9958 0.99700.3 0.9991 0.9990 0.9990 0.9990 0.9990 0.9991 0.9990 0.99910.5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

54

4.4 Discussion

To sum up this chapter, according to the simulation results, our data support as-

sertions that Olkin’s Z demonstrated relatively the higher empirical level and the

highest empirical power than did other test statistics, especially when ρ12 and ρ23 are

small to moderate, so we cannot recommend Olkin’s Z to apply in medical data.

In addition, based on all the result of simulation above, we advise that the re-

searchers who are interested in this topic when they meet different situations, they

should choose different relatively appropriate test statistics to use. When correla-

tions are small to moderate (i.e, 0 to 0.4), the other seven test statistics, Dunn and

Clark’s Z , Steriger’s Z, Meng’s Z , Hitter’s Z, Hotelling’s t, William’s t, William’s

modified t per Hendrickson, perform well and are appropriate for applying to real

medical data. Especially when sample size is small, Steriger’s Z, Meng’s Z , Hitter’s

Z and William’s t are relatively optimal to use. More specifically, when correlation is

small (i.e, 0 to 0.2), Dunn and Clark’s Z, Hotelling’s t and William’s modified t per

Hendrickson perform well; when correlation is moderate (i.e, 0.3 to 0.4), Steriger’s Z,

Meng’s Z , Hitter’s Z and William’s t were more appropriate.

Moreover, we do not recommend the other seven test statistics when correlation

coefficients are large (i.e, 0.5 to 0.9) before doing modification. The method of mod-

ification we mentioned in Chapter 3 is used in generated data with pattern Σ1 and

effectively re-normalized the test statistics even thought the methods cannot exten-

sively use in general case and is complicated to apply in real medical data. Especially,

through simulation, we found starting point of modification, that is, ρ13 played a role

55

in Σ. In that case, the change of ρ13 could improve the test such that the empirical

levels hover around the nominal levels. Researchers can extend the idea of modifica-

tion when they deal with the same issues with ordinal variables. Therefore, we apply

all the test statistics expect Olkin’s Z to the real medical data in next chapter, and

compare the results from them.

56

Chapter 5

Applications

5.1 Data Sources and Features

5.1.1 Data Sources

As it is mentioned in the introduction, the medical data is from a project of cancer

patient study of a medical institution in the United States. It involves the patients’

pain scale data, history of certain disease treatments, chemotherapy and medica-

tion treatments on multiple time points, concomitant medication record, neuromeres

treatment, and individual’s demographics, etc. The data information of patients were

recorded several times, and the first visit time is a baseline time without any medical

treatment in the medical institution. The other visit times are study visit times when

the patients have had chemotherapy or medical treatment. However, some patient

drop out or absent a few times due to death or moving. The types of variables in the

medical data are ordinal, nominal, string, interval, continuous and ratio. In order to

57

test the equality of two or more correlation with a single ordinal longitudinal variable

based on a common sample, the correlations between three pairs of adjacent visit

times based on the five ordinal variables are what we concern about. The five ordinal

variables in a common sample are deep pain sensation (X), pain intensity (Y ), lack

of energy (A), nausea (B), and joint pain/muscle cramps (C). The first two ordinal

variables are neuropathic pain data, and they are ordinal variables ranging from 0

to 10. The remaining three of the five show patient’s performance status during the

medical treatment. They are nominal variables, but we treat them as ordinal ranging

from 0 to 4, where 0 = not at all, 1 = a little bit; 2 = somewhat; 3 = quite a bit; 4 =

very much.

5.1.2 Features

We first consider deep pain sensation at baseline time t0, visit time t1, visit time t2,

and visit time t3 to explain the features of the ordinal data set. After sorting out the

data set to three contingency tables, that is, Table 5.1: X(t0) vs X(t1), Table 5.2:

X(t1) vs X(t2) and Table 5.3: X(t2) vs X(t3), several features of the data appear.

58

Table 5.1: Deep Pain Sensation (X) at baseline time t0 and visit time t1 withoutmodification of data

X(t0)X(t1) 0 1 2 3 4 5 6 7 8 9 10 Total

0 14 4 1 2 211 1 1 1 32 1 1 23 04 1 15 1 16 1 17 1 18 09 0

10 0

Total 16 5 2 0 0 3 2 1 1 0 0 30

Note: The empty grids are filled with “0”.

Table 5.2: Deep Pain Sensation (X) at baseline time t1 and visit time t2 withoutmodification of data

X(t1)X(t2) 0 1 2 3 4 5 6 7 8 9 10 Total

0 10 2 1 1 1 151 3 1 42 1 1 23 04 05 1 2 36 1 17 1 18 1 19 0

10 0

Total 14 2 0 1 4 1 1 2 1 1 0 27


59

Table 5.3: Deep Pain Sensation (X) at visit time t2 and visit time t3 without modi-fication of data

X(t2)X(t3) 0 1 2 3 4 5 6 7 8 9 10 Total

0 8 1 1 101 1 1 22 03 1 14 1 1 1 1 45 1 16 1 1 27 1 18 09 1 1

10 0

Total 11 0 1 2 1 1 3 2 1 0 0 22


First of all, the deep pain sensation from the sample in the medical data is an

ordinal variable. The scales of the variable are divided into 11 levels from 0 to 10

which are order of categories. Even the deep pain scale cannot describe the difference

between categories in numerical way, the difference does exist. Secondly, it is an

extremely sparse data. Since the variable is set into 11 ordinal categories, so there

are total 121 grids. But only 30 patients are studied in this project. So the most

of the entries on the table has no data falling into them so that it is an extremely

sparse ordinal table. Last but not least, the total number of observations on each

contingency table is not the same because the patients drop out from the project over

times. The first table (Table 5.1) shows X at the baseline time t0 and visit time t1

with 30 patients. Table 5.2 displays X at the baseline time t1 and visit time t2 with

27 patients. In the third contingency table (Table 5.3), there are only 22 patients

left.

60

For now, based on the features of data, we cannot analyze the correlations based

on the ordinal variable from a common sample. Besides, it will effect the results with

extremely sparse contingency table. That is, we need to do some modification of the

data.

5.1.3 Modification of the real data in contingency tables

The data set in contingency tables are modified in the following steps. We still take

the ordinal variable, deep pain sensation, as an example to explain. To begin with, in

order to solve the problem of the extremely sparse ordinal contingency table, we re-set

the ordinal categories of the variable into 5 levels. We combine pain scales 0 and 1 as

0, 2 and 3 as 1, 4 and 5 as 2, 6 and 7 as 3, and 8, 9 and 10 as 5. The three categories

8, 9 and 10 are combined as one categories because only few observations belong to

the range. In addition, the objective in the chapter is to compare the correlations

based on one ordinal variable between adjacent visit times (i.e, t0, t1, t2 and t3). It

means that we need to guarantee the specified sample size N of the groups that the

correlations are based on. By recollecting the data, we found 22 valid observations

which involved in the project over four visit times. After doing the modification of

the data, we obtain the following three new contingency tables (Table 5.4, 5.5, 5.6)

with deep pain sensation.

With regard to the idea of the modification for the data set, the contingency tables

(Table 5.7 to Table 5.18) for the remaining four ordinal variables, pain intensity (Y ),

lack of energy (A), nausea (B), and joint pain/muscle cramps (C), are shown below.

Since the three ordinal variables, lack of energy, nausea, and joint pain/muscle cramps,

are categorized as 5 scales, we do not modify the categories of three variables.

61

Table 5.4: Deep Pain Sensation (X) at baseline time t0 and visit time t1

X(t0)X(t1) 0 1 2 3 4 Total

0 14 2 2 181 1 12 1 1 23 1 14 0

Total 15 2 3 1 1 22

Table 5.5: Deep Pain Sensation (X) at visit time t1 and visit time t2

X(t1)X(t2) 0 1 2 3 4 Total

0 11 1 2 1 151 1 1 22 1 2 33 1 14 1 1

Total 12 1 5 3 1 22

Table 5.6: Deep Pain Sensation (X) at baseline time t2 and visit time t3

X(t2)X(t3) 0 1 2 3 4 Total

0 9 1 2 121 1 12 1 2 1 1 53 1 1 1 34 1 1

Total 11 3 2 5 1 22


62

Table 5.7: Pain Intensity (Y ) at baseline time t0 and visit time t1

Y (t0)Y (t1) 0 1 2 3 4 Total

0 14 2 1 1 1 191 1 1 22 03 1 14 0

Total 15 2 1 3 1 22



Y (t1)Y (t2) 0 1 2 3 4 Total

0 10 1 2 1 1 151 1 1 22 1 13 1 1 1 34 1 1

Total 12 3 4 2 1 22


Y (t2)Y (t3) 0 1 2 3 4 Total

0 10 2 121 1 1 1 32 1 2 1 43 1 1 24 1 1

Total 12 5 2 2 1 22

63

Table 5.10: Lack of Energy (A) at baseline time t0 and visit time t1

A(t0)A(t1) 0 1 2 3 4 Total

0 2 3 3 1 91 3 5 82 1 1 1 33 2 24 0

Total 3 6 9 4 0 22


A(t1)A(t2) 0 1 2 3 4 Total

0 1 1 1 31 4 1 1 62 3 2 2 2 93 1 2 1 44 0

Total 2 8 6 4 2 22


A(t2)A(t3) 0 1 2 3 4 Total

0 1 1 21 1 3 3 1 82 1 3 1 1 63 1 2 1 44 1 1 2

Total 4 3 10 3 2 22


64

Table 5.13: Nausea (B) at baseline time t0 and visit time t1

B(t0)B(t1) 0 1 2 3 4 Total

0 10 3 2 1 161 3 2 52 1 13 04 0

Total 10 6 5 0 1 22


B(t1)B(t2) 0 1 2 3 4 Total

0 9 1 101 2 2 2 62 1 2 1 1 53 04 1 1

Total 12 5 1 3 1 22


B(t2)B(t3) 0 1 2 3 4 Total

0 8 1 2 1 121 2 3 52 1 13 2 1 34 1 1

Total 10 7 3 1 1 22


65

Table 5.16: Joint Pain/Muscle Cramps (C) at baseline time t0 and visit time t1

C(t0)C(t1) 0 1 2 3 4 Total

0 11 1 1 1 141 2 1 2 1 62 1 1 23 04 0

Total 14 2 3 3 0 22


C(t1)C(t2) 0 1 2 3 4 Total

0 11 1 2 141 2 22 1 1 1 33 1 2 34 0

Total 12 2 6 2 0 22


C(t2)C(t3) 0 1 2 3 4 Total

0 11 1 121 1 1 22 3 2 1 63 1 1 24 0

Total 15 3 2 2 0 22


66

Therefore, as the tables show above, the scales of all five ordinal variables are

re-divided into 5 levels from 0 to 4. There are total 25 grids, and 22 observations

are studied in this project. That is, we improve the contingency tables and solve the

potential problem which might arises from extremely sparse table.

5.2 Test Results and Conclusion of the Medical

Data

According to the simulation study, we compared the eight test statistics which can be

used to test equality of correlation based on common ordinal longitudinal variable.

As it is discussed in the previous chapter, the remaining seven of the eight statistical

tests are efficient for sample sizes are small or large.

The hypotheses for the medical data are

H0: r12 = r23 = r34;

Ha: At least one pair of rij are different, where i, j = 1, 2, 3, 4 and i 6= j.

we cannot test the above hypotheses by the test statistics, so we do the hypothesis

test separately, that is,

H0 : r12 = r23 vs Ha: r12 6= r23

H0 : r23 = r34 vs Ha: r23 6= r34

where,

r12 : the correlation coefficient between an ordinal variable at baseline time t0 and

the ordinal variable at visit time t1.

r23 : the correlation coefficient between the ordinal variable at baseline time t1 and

67


r34: the correlation coefficient between the ordinal variable at baseline time t2 and


After processing the data sets and testing by the appropriate statistics, we obtain

the correlations coefficients for the five ordinal variables in Table 5.19, and the test

results in Table 5.20 to Table 5.24.

Table 5.19: The correlations coefficients for ordinal variables

Ordinal variables r12 r13 r23 r24 r34

Deep pain sensation 0.7460 0.3044 0.4437 0.1530 0.4970

Pain intensity 0.4326 0.2647 0.1679 0.5207 0.6886

Lack of energy 0.4253 -0.059 0.2868 0.5730 0.3636

Nausea 0.3650 -0.1562 0.6983 0.5572 0.5015

Pain joint/muscle cramps 0.3865 0.3142 0.6914 0.6956 0.6045

68

Table 5.20: Result of deep pain sensation

Hypothesis

Tests

Test

Statisticsα = 0.01 α = 0.05 α = 0.10

H0:r12 = r23

Ha:r12 6= r23

Zd = 1.6183 NR NR NR

Zs = 1.6083 NR NR NR

Zm = 1.5992 NR NR NR

Zh = 1.5976 NR NR NR

Th = 1.7851 NR NR R

Tw = 1.6635 NR NR NR

Tm = 1.7847 NR NR R

H0:r23 = r34

Ha:r23 6= r34

Zd = −0.2159 NR NR NR

Zs = −0.2159 NR NR NR

Zm = −0.2159 NR NR NR

Zh = −0.2159 NR NR NR

Th = −0.2278 NR NR NR

Tw = −0.2171 NR NR NR

Tm = −0.2278 NR NR NR

Note in Table 5.20 to Table 5.24: 1. “R” in brackets denotes re-

jecting null hypothesis; “NR” in brackets denotes doing not re-

ject null hypothesis. 2. The critical values are Z0.005 = ±2.575,

Z0.025 = ±1.96, Z0.05 = ±1.645, t(19),0.01 = ±2.861, t(19),0.05 =

±2.093 and t(19),0.10 = ±1.729.

69

Table 5.21: Result of pain intensity

Hypothesis

Tests

Test

Statisticsα = 0.01 α = 0.05 α = 0.10

H0:r12 = r23

Ha:r12 6= r23





Th = 1.1690 NR NR NR


Tm = 1.1689 NR NR NR

H0:r23 = r34

Ha:r23 6= r34

Zd = −2.5906 R R R

Zs = −2.5202 NR R R

Zm = −2.4574 NR R R

Zh = −2.4871 NR R R

Th = −2.9015 R R R

Tw = −2.8433 NR R R

Tm = −2.9004 R R R

70

Table 5.22: Result of lack of energy

Hypothesis

Tests

Test

Statisticsα = 0.01 α = 0.05 α = 0.10

H0:r12 = r23

Ha:r12 6= r23








H0:r23 = r34

Ha:r23 6= r34

Zd = −0.3882 NR NR NR

Zs = −0.3879 NR NR NR

Zm = −0.3877 NR NR NR

Zh = −0.3878 NR NR NR

Th = −0.3905 NR NR NR

Tw = −0.3893 NR NR NR

Tm = −0.3905 NR NR NR

71

Table 5.23: Result of nausea

Hypothesis

Tests

Test

Statisticsα = 0.01 α = 0.05 α = 0.10

H0:r12 = r23

Ha:r12 6= r23

Zd = −1.3347 NR NR NR

Zs = −1.3296 NR NR NR

Zm = −1.3319 NR NR NR

Zh = −1.3314 NR NR NR

Th = −1.7988 NR NR R

Tw = −1.3723 NR NR NR

Tm = −1.7955 NR NR R

H0:r23 = r34

Ha:r23 6= r34








72

Table 5.24: Result of joint pain/muscle cramps

Hypothesis

Tests

Test

Statisticsα = 0.01 α = 0.05 α = 0.10

H0:r12 = r23

Ha:r12 6= r23

Zd = −1.5092 NR NR NR

Zs = −1.4982 NR NR NR

Zm = −1.4877 NR NR NR

Zh = −1.4890 NR NR NR

Th = −1.6213 NR NR NR

Tw = −1.5487 NR NR NR

Tm = −1.6210 NR NR NR

H0:r23 = r34

Ha:r23 6= r34








73

From Table 5.20 and Table 5.23, we can conclude that the set of correlations

for the two ordinal variables (i.e, deep pain sensation and nausea) are consistent.

The values of the seven statistical tests are located in the confidence interval either

−Zcv < Ztest < Zcv or −t(19),cv < Ttest < t(19),cv at significant level α = 0.01, 0.05 and

0.10, except that at the significant level α = 0.10, we reject H0 : r12 = r23 by using

Hotelling’s t and William’s modified t per Hendrickson. It indicates that the relation

between two adjacent visit times for both of deep pain sensation and nausea maintain

a consistency with chemotherapy and medical treatment in most of the cases. It also

supports that scale of the two ordinal variables at one visit time is dependent on

the scale at previous visit time, and the dependent intensity will not change with

time of therapy. It does not mean that the patients received an ineffective treatment,

the strength of the treatment for patients which embodied in the scale of deep pain

sensation and degree of nausea keep consistent with time.

The test result for other two ordinal variables, lack of energy and joint pain/muscle

cramps, are displayed in Table 5.22 and Table 5.24. It is stronger evidence that shows

the homogeneity of the two sets of correlated correlations because all statistical tests

for the corresponding hypothesis test do not fall in the reject region. Especially in

the Table 5.22, the values of the test are far away from the critical values. The results

for these two ordinal variables show the important clinical and medical significance

that we figured out in last part.

However, with regard to test the equality of correlations, the result for pain in-

tensity Table 5.21 comes to a different conclusion. We accept the null hypotheses

r12 = r23 by using the seven test statistics under all the three nominal level, while we

74

reject another null hypotheses r23 = r34 by using the same statistical tests at nominal

level α = 0.05 and α = 0.10. Besides, at the nominal level α = 0.01, H0 : r23 = r34

is rejected by using Dunn and Clark’s Z test, otherwise accepting H0. We can find

that the relationship between pain intensity at visit time t3 and pain intensity at visit

time t4 is stronger than the relationship between t2 and t3, or that between t1 and t2

because all the test statistics for testing whether r23 = r34 are negative. It indicates

that pain intensity at visit time t4 is much more dependent on pain intensity at visit

time t3 than that other adjacent times. The difference might be caused by the medical

treatment or even the patients’ psychological impact from the procedure, time and

the effect of the treatment.

To sum up, the correlations with deep pain sensation (X), lack of energy (A), nau-

sea (B) and pain joint/muscle cramps (C) between adjacent visit times are consistent,

that is, r12 = r23 = r34. However, there is the difference between the correlations r23

and r34 for pain intensity at the nominal levels α = 0.05 and 0.10. It means we

reject the null hypotheses and r23 = r23 6= r34. It is interesting to note that there

appears the different conclusion for the two kind of pain index, deep pain sensation

and pain intensity. We do not have to doubt the conclusion by our the method. This

inconsistency and variation may exist in different pain observations. In addition, it

might be caused by data collection and sorting. Researchers may be interested in this

difference and do further investigations.

75

Chapter 6

Conclusion and Future Work

6.1 Future Work

The possible future work involves modifying test statistics by using bootstrap method,

not only in simulation study but also in real data, testing the equality of a set of

correlated correlations simultaneously by using chi-square statistics.

6.1.1 Modification of Test Statistics by using Bootstrap Method

1.Simulation and resampling

According the simulation study, we found some test statistics are relatively appropri-

ate but there are not optimal, so we need to modify the test statistics. One idea of

the modification is using bootstrap method. The bootstrap procedure is based on the

idea of resampling the data. The data which resampling by original data may be used

as substitute for the population when the population distribution is unknown. We

76

use the bootstrap method in simulation study to modify the test statistics which we

mentioned in Chapter 3, thereby the empirical levels are closer to the nominal levels.

The steps for obtaining a bootstrap estimate modification of test statistics θmodified

are as follows:

1. Follow the simulation method in Chapter 4 to generating a sample U = (U1, U2, U3)

and categorizing it as 5 levels Xmn, where m = 1, 2, 3, and n is sample size which

need to be set. The generated sample is original sample, and compute θoriginal

from the original data.

2. Draw a resample or bootstrap sample of size n with replacement from the o-

riginal sample. We denote k be the number of such bootstrap samples, usually

k ≥ 1000. Compute θb,i, the estimated of θ obtained from the ith bootstrap

sample. Then we have the mean and variance of θb,i by the k bootstrap samples

as,

E(θb,i) =1

k

k∑i=1

θb,i

and

V ar(θb,i) =1

k

k∑i=1

(θb,i − E(θb,i))2

3. Obtain the bootstrap modification of θmodified as

θmodified =θoriginal − E(θb,i)√

V ar(θb,i)

4. Repeat step 1, 2 and 3 on 10,000 times over to obtain the empirical bootstrap

distribution of θmodified, then compare the bootstrap distribution with normal

distribution.

77

2.Bootstrap in real data

Modifying the test statistics by using Bootstrap methods in real data is similar with

that in simulation study part. Now we the original sample is known. So the steps for

obtaining a bootstrap modification of test statistics θmodified is simple and as follows:

1. Compute θoriginal from the original data

2. Draw a resample or bootstrap sample of size n with replacement from the orig-

inal sample, and compute θb by the bootstrap sample.

3. Repeat step 2 on k times over, k ≥ 1000, to obtain the boostrap esitimate of

mean and variance as,

E =1

k

k∑i=1

θb,i

V ar =1

K

k∑i=1

(θb,i − E)2

4. Obtain the bootstrap modification of test statistics as,

θmodified =θorginal − E√

V ar

6.1.2 Testing the Equality of a Set of Correlated Correlations

by using Chi-square Statistics

The objective of this thesis is to test the heterogeneity of correlations with an ordinal

variable between adjacent visit times t1, t2, ..., ts. If visit times are more than 3 times,

so that there are more than two correlations between adjacent visit times. The null

78

hypotheses test should be r1 = r2 = ... = rk, where k = 1, 2, ..., s− 1. We divide the

null hypotheses test into several parts, such as H0(1) : r1 = r2; H0(2) : r2 = r3; ...;

H0(k− 1) : rk−1 = rk, because the test statistics which were introduced in Chapter 3

can only use to test the equality of two correlations. However, Meng, Rosenthal and

Runbin proposed a χ2 test to readily test the significance of heterogeneity of a set

of correlations by means of a simple extension of their Z test statistics which named

Meng’s Z in Chapter 3. The hypothesis test for now is,

H0 : r1 = r2 = ... = ri = ... = rk, i ∈ k, k = 1, 2, ..., s− 1

Ha: At least one pair of ri 6= ri+1.

The χ2 test performed as,

χ2 =(N − 3)

∑ki (Zri − Zr)

2

(1− rx)h,with df = k − 1

where,

h =1− f r2

1− r2= 1 +

r2

1− r2(1− f)

f =1− rx

2(1− r2),which must be ≤ 1

r2 =1

k

k∑i

r2i

In this χ2 test equation, Zri = 12

ln 1+ri1−ri is Fisher r − to − Z transform, and Zr is

the mean of the Zri . rx is the median intercorrelation being tested for heterogeneity.

χ2 test statistic follows chi-square distribution on k − 1degree of freedom, where k is

number of correlations that need to compare.

79

6.2 Conclusion

In the present thesis, we first introduced ordinal data including measuring methods,

classification, difference and advantage using ordinal data. Then we reviewed several

models for ordinal responses, how to apply them and the restriction of the models in

different situations. In Chapter 3, we focused on test statistics of measuring dependent

correlations in contingency tables. Before that, we needed to sort an ordinal variables

based on time in contingency tables and compute the correlation coefficients with

scores. In some case, we also need to reorganize contingency tables of ordinal data

due to sparse tables and observations’ dropping off over time. Next we presented

eight test statistics for testing the equality of two or more dependent correlations in

a common sample. In addition, we did a modification of test statistics in last part

but there still are some issues to apply it in real data.

In Chapter 4, we evaluated the eight test statistics by simulation. In terms of

empirical level and empirical power, the results of simulation indicated that the choice

as to which test statistics is optimal, which depends not only on sample size but also

on the magnitude of the correlations. Through summarizing the results of simulation

and considering with the condition of real medical data, we chose the seven test

statistics, Dunn and Clark’s Z , Steriger’s Z, Meng’s Z , Hitter’s Z, Hotelling’s t,

William’s t, William’s modified t per Hendrickson to apply to the real medical data.

Especially, we paid attention to Steriger’s Z, Meng’s Z , Hitter’s Z and William’s

t. The four test are relatively optimal to use because of the small sample in the

medical data. In last part of Chapter 5, we summarized the result of application

80

in real data and analyzed important clinical significance of the studies in the thesis.

Unfortunately, the modification of test statistics we presented in Chapter 3 were not

ideally used in real data. So we came up with an idea of modification by using

Bootstrap method not only in simulation evaluation but also in real data in Chapter

6. In addition, testing the equality of a set of correlated correlations by using Chi-

square statistics were considered to suggest that researchers who work with the same

issue might consider this direction to study.

81

Bibliography

[1] Agresti, A. (2007). An Introduction to Categorical Data Analysis. New York :

John Wiley and Sons.

[2] Agresti, A. (2010). Analysis of Ordinal Categorical Data (2nd ed). New York :

John Wiley and Sons.

[3] Ananth, C. V.,and Kleinbaum, D. G. (1997). Regression models for ordinal re-

sponses: a review of methods and applications. Internat. J. Epidemiol, 26, 1323-

1333.

[4] Bender, B. and Grouven, U. (1998). Using binary logistic regression models for

ordinal data with non-proportional odds Journal of Clinical Epidemiology, 51,

809-816.

[5] Dunn, O. J. and Clark, V. A. (1969). Correlation coefficients measured on the

same individuals. Journal of the American Statistical Association, 64, 366-377.

[6] Dunn, O. J. and Clark, V. A. (1971). Comparison of tests of the equality of de-

pendent correlation coefficients. Journal of the American Statistical Association,

66, 904-908.

82

[7] Efron, B. and Tibshirani, R. J. (1994). An Introduction to the Boorstrap. Chap-

man and Hall/CRC.

[8] Fahrmeir, L. and Tutz, G. (2001). Multivariate Statistical Modelling Based on

Generalized Linear Models(2nd ed). New York: Springer.

[9] Fisher, R. A. (1921). On the probable error of a coefficient of correlation deduced

from a small sample. Metron, 1, 1-32.

[10] Gautam, S. and Kimeldorf, G. (1999). Some results on the maximal correlation

in 2× k contingency tables. The American Statistician, 53, 336-341.

[11] Hendrickson, G. F., Stanley J. C., and Hills, J. R. (1970). Olkins new formu-

la for significance of r13 vs. r23 compared with Hotellings method. American

Educational Research Journal, 7, 189-195.

[12] Higgins, J. J. (2004). Introduction to Modern Nonparametric Statistics. Califor-

nia: Thomson Learning.

[13] Hittner, J. B., May, K., and Silver, N. C. (2003). A Monte Carlo evaluation of

tests for comparing dependent correlations. The Journal of General Psychology,

130, 149-168.

[14] Hotelling, H. (1940). The selection of variates for use in prediction, with some

comments on the general problem of nuisance parameters. Annals of Mathemat-

ical Statistics, 11, 271-283.

[15] Kampen, J. and Swyngedouw, M. (2000). The Ordinal Controversy Revisted

Quality and Quantity, 34, 87-102.

83

[16] McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models (2nd ed).

London: Chapman and Hall.

[17] Meng, X. L., Rosenthal, R., and Rubin, D. B. (1992). Comparing correlated

correlation coefficients. Psychological Bulletin, 111, 172-175.

[18] Neill, J. J., and Dunn, O. J. (1975). Equality of dependent correlation coefficients.

Biometrics, 31, 531-543.

[19] Olkin, I. (1967). Correlations revisited. In J. C. Stanley (Ed.), Improving exper-

imental design and statistical analysis 102-128. Chicago, IL: Rand McNally.

[20] Pearson, K., and Filon, L. N. G. (1898). Mathematical contributions to theory

of evolution: IV. On the probable errors of frequency constants and on the

influence of random selection and correlation. Philosophical Transactions of the

Royal Society of London, Series A, 191, 229-311.

[21] Peterson, B. and Harrell, F. E. (1990). Partial proportional odds models for

ordinal response variables. Journal of the Royal Statistical Society. Series C, 39,

205-217.

[22] Piepho, H. and Kalka, E. (2003). Threshold models with fixed and random effects

for ordered categorical data. Food Quality and Preference, 14, 343-357.

[23] Silver, N. C. and Dunlap, W. P. (1987). Averaging correlation coefficients: Should

Fishers Z transformation be used? Journal of Applied Psychology, 72, 146-148.

[24] Steiger, J. H. (1980). Tests for comparing elements of a correlation matrix. Psy-

chological Bulletin, 87, 245-251.

84

[25] Stevens, S. S. (1951). Mathematics, measurement, and psychophysics. New York:

Wiley.

[26] Torra, V., Domingo-Ferrer, J., Mateo-Sanz, J. M. and Ng, M. (2006). Regres-

sion for ordinal variables without underlying continuous variables. Information

Sciences, 176, 465-474.

[27] Tutz, G. (1991). Sequential models in categorical regression. Comput.Statist.Data

Anal, 11, 275-295.

[28] Vogt, W. P. (1993). Dictionary of Statistics and Mathodology. London: Sage.

[29] Walker, S. H. and Duncan, D. B. (1967). Estimation of the probability of an

even as a function of several independent varianles. Biometrika, 54, 167-179.

[30] Williams, E. J. (1959). The comparison of regression variables. Journal of the

Royal Statistical Society, Series B, 21, 396-399.

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