Doubly Classified Model with R

270 Pages·2018·5.319 MB·english

by  Teck Kiang Tan

Teck Kiang Tan
fi
Doubly Classi ed Model
with R
123
Teck KiangTan
Centrefor Skills, Performance
andProductivity
Institute for AdultLearning
Singapore
Singapore
ISBN978-981-10-6994-9 ISBN978-981-10-6995-6 (eBook)
https://doi.org/10.1007/978-981-10-6995-6
LibraryofCongressControlNumber:2017955661
©SpringerNatureSingaporePteLtd.2017
ThisSpringerimprintispublishedbySpringerNature
TheregisteredcompanyisSpringerNatureSingaporePteLtd.
Theregisteredcompanyaddressis:
152BeachRoad,#21-01/04GatewayEast,Singapore189721,Singapore
Foreword
RichardBransononcesaid,“Thereisnogreaterthingyoucandowithyourlifeand
work than to follow your passions in a way that serves the world and you.” As I
supervised Teck as a graduate student and later observed his contributions to the
world of research, I can truly see how this book is his heartfelt effort to help
aspiring statisticians leverage on the affordances of statistics.
He has done this in two profound ways. First, he noticed that despite the use-
fulness of doubly classified models for categorical data analysis, it was
under-discussedintextbooks.Thisledhimtoelaborateonawiderrangeofrelevant
models;goingbeyondthecommonlydiscussedmodelslikeindependenceandquasi
symmetry models. Second, as an active researcher, he realized that beyond under-
standing the range of models available, the next step for any interested statistician
would be to generate models for practical applications. Unfortunately, this would
involvetheuseofstatistical software like SASorSPSS that isboth expensiveand
not easily accessible. Cognizant that not everyone has access to research funds or
software, Teck has taken pains to develop new syntax for generating model on
Package R, a free statistical software that isgainingwidespread popularity.
IamgladthatTeckhaswrittenthisbookbecauseitisnotjustaboutexplaining
theintricaciesofonedoubleclassifiedmodeloveranotherbutitistohelpinterested
students get an entry pass into this world and its possibilities. For example, Teck
understands that not everyone is comfortable with mathematical symbols and has
developed a “symbolic table” to help represent notations that are more complex.
This symbolic table is a graphical tool to help readers understand the various
models discussed.
Although this book gives many tips and guidelines on how to develop double
classified models, I can see how this book may also be useful for readers who are
interested in learning about Package R and use the guided steps as a lesson of R’s
functionalities. Whatever your purpose, I am sure that you would appreciate the
examples included to illustrate concepts and its applications.
Trivina Kang
Office of Teacher Education (OTE),
Policy & Leadership Studies (PLS),
National Institute of Education
Nanyang Technological University
Singapore
Preface
ThemotivationinwritingthisbookcomesafterthepresentationofaposterattheR
conference 2015 held at Denmark, Alborg, titled “Extending the Quasi-Symmetry
Model: Quasi-Symmetry with n Degree.” I found that modeling doubly classified
table is quite unknown to most conference attendees. Although there has been a
dramatic growth in the development and application of doubly classified models,
there is not a single book that is written in R on this subject. The applications on
doubly classified modeling are also restricted to illustrate it using software like
SPSSandSAS,appearedmostofteninjournalarticles(e.g.,Lawalandcolleagues).
Doubly classified models are not commonly appeared in statistic textbooks. Even
therearebookswrittenonthissubject,itisoftenrestrictedtoasectionoratmosta
chapter.Thisisprobablytherighttimetoputthesemodelsinanorganizedwayinto
a book for sharing.
This book focuses on doubly classified models. The main aim of the book is to
describedoublyclassifiedmodelsinawayreaderscaneasilyunderstand.Although
mathematicalrepresentationsareunavoidableforpurposeofclarity,theyarealways
accompanied with explanation in plain language. As doubly classified is probably
not a familiar topic to a lot of people, including statisticians and data analysts, for
those who have not heard of it, this book serves you well. You will find it a good
startingpointtohaveabasicunderstandingaboutdoublyclassifiedmodels.Alotof
examples accompanied the explanation are given in the text to illustrate the char-
acteristics of the various doubly classified models. It is applied oriented however
not losing its function as a basic textbook and reference. The formulas for the
various doubly classified models are given, together with examples illustrating the
concepts and usages. For those who are not familiar with doubly classified mod-
elingwill findthisbookeasiertounderstand.Anewpresentation form,referredto
assymbolic table, isused throughoutthetext asasummarized tabletoexplainthe
main characteristics of doubly classified model. Researchers, data analysts, and
undergraduateandpostgraduatestudentsaresuitedaudiencesandmostsuitablefor
applied statisticians and researchers aim to use doubly classified model as an
analytical tool for their studies.
Singapore, Singapore Teck Kiang Tan
September 2017
Contents
1 Introduction to R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 What Is R?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Libraries in R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Installing R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Running R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5 Installing Add-on Packages and Libraries. . . . . . . . . . . . . . . . . 4
1.6 Basic Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.6.1 Numeric. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6.2 Integer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6.3 Logical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.6.4 Character. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.7 Data Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.7.1 Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.7.2 Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.7.3 Data Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.7.4 List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.7.5 Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.8 Read File into R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1 Contingency, Square, and Doubly Classified Table. . . . . . . . . . 21
2.2 Generating Frequency Tables. . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.1 Tables from Vector . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.2 Tables from Data Frame. . . . . . . . . . . . . . . . . . . . . . . 23
2.2.3 Margin Total and Proportion. . . . . . . . . . . . . . . . . . . . 24
2.2.4 xtab Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.5 Package gmodels, CrossTable Function . . . . . . . . . . . . 26
2.2.6 Package descr, CrossTable Function . . . . . . . . . . . . . . 26
2.3 Graphics for Tabulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.1 Bar Plot—Base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.2 Package ggplot2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.3 Package lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4 Odds, Odds Ratios, Local Odds Ratios, and Margin . . . . . . . . . 30
2.4.1 Odds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4.2 Properties and Interpretation of Odds. . . . . . . . . . . . . . 31
2.4.3 Odds Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4.4 Log Odds Ratio and Confidence Interval . . . . . . . . . . . 35
2.4.5 Margin Total . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5 Applications of Doubly Classified Data . . . . . . . . . . . . . . . . . . 39
2.5.1 Studies on Pairs of Matched Individuals . . . . . . . . . . . 39
2.5.2 Association Between Two Essentially Similar
Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5.3 Two Point Longitudinal Study for a Common
Variable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.5.4 Inter-rater Agreement . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.5.5 Two Indicators from a Scale. . . . . . . . . . . . . . . . . . . . 44
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3 Examining Symmetry of Doubly Classified Table . . . . . . . . . . . . . . 49
3.1 McNemar’s Test of Symmetry. . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2 Variations of McNemar’s Test. . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2.1 Exact Binomial Test. . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2.2 Mid-P McNemar’s Test . . . . . . . . . . . . . . . . . . . . . . . 51
3.2.3 Which Test to Use? . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3 Bower’s Test of Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4 Marginal Symmetry/Homogeneity Model . . . . . . . . . . . . . . . . . 56
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4 Symmetry and Asymmetry Models. . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1 Complete Symmetry Model. . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1.1 Symbolic Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.1.2 Nonstandard Log-Linear Approach in Generating
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.1.3 Complete Symmetry Model—Nonstandard Log-
Linear Specification . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.1.4 Properties of Complete Symmetry Model. . . . . . . . . . . 67
4.1.5 Function Model.Summary. . . . . . . . . . . . . . . . . . . . . . 68
4.1.6 A Detailed Illustration—The Dummy Specification . . . 73
4.2 Conditional Symmetry Model . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.3 Odds Symmetry Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3.1 Relationships of Complete Symmetry, Conditional
Symmetry, Odds Symmetry I and II . . . . . . . . . . . . . . 86
4.4 Diagonal Parameters Symmetry Model. . . . . . . . . . . . . . . . . . . 87
4.4.1 Relationships of Complete Symmetry, Conditional
Symmetry, and Diagonal Parameters Symmetry
Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.5 Linear Diagonal Parameters Symmetry Model . . . . . . . . . . . . . 91
4.6 Quasi Symmetry Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.6.1 Relationship of Quasi Symmetry, Conditional
Symmetry, and Complete Symmetry Models . . . . . . . . 97
4.7 Quasi Diagonal Parameters Symmetry Model. . . . . . . . . . . . . . 97
4.7.1 Relationships of Complete Symmetry, Conditional
Symmetry, Quasi Symmetry, Diagonal Parameters
Symmetry,andQuasiDiagonalParametersSymmetry
Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.8 The 2-Ratios Parameters Symmetry Model. . . . . . . . . . . . . . . . 103
4.8.1 Relationships of Conditional Symmetry, Linear
Diagonal Parameters Symmetry, and 2-Ratios
Parameters Symmetry Models. . . . . . . . . . . . . . . . . . . 107
4.9 Quasi Conditional Symmetry Model . . . . . . . . . . . . . . . . . . . . 108
4.10 Quasi Odds Symmetry Model . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.10.1 Relationships of Quasi Symmetry, Quasi Conditional
Symmetry, and Quasi Odds Symmetry Models . . . . . . 116
4.10.2 Relationships of Odds Symmetry I, Odds Symmetry
II, and Quasi Odds Symmetry Model . . . . . . . . . . . . . 116
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5 Point Symmetry Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.1 Complete Point Symmetry Model . . . . . . . . . . . . . . . . . . . . . . 121
5.2 Inclined Point Symmetry Model . . . . . . . . . . . . . . . . . . . . . . . 126
5.3 Quasi Point Symmetry Model . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.4 Quasi Inclined Point Symmetry Model. . . . . . . . . . . . . . . . . . . 131
5.5 Proportional Point Symmetry Model . . . . . . . . . . . . . . . . . . . . 135
5.5.1 Comparison of Proportional and Inclined Point
Symmetry Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.6 Local Point Symmetry Model . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.7 Reverse Local Point Symmetry Model . . . . . . . . . . . . . . . . . . . 140
5.8 Reverse Proportional Point Symmetry Model . . . . . . . . . . . . . . 142
5.9 Reverse Inclined Point Symmetry Model . . . . . . . . . . . . . . . . . 145
5.10 Quasi Reverse Inclined Point Symmetry Model . . . . . . . . . . . . 148
5.11 Reverse Conditional Symmetry . . . . . . . . . . . . . . . . . . . . . . . . 151
5.12 Quasi Reverse Conditional Symmetry Model . . . . . . . . . . . . . . 153
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6 Non-independence Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.1 Independence and Non-independence Model . . . . . . . . . . . . . . 157
6.2 Principal Diagonal Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6.2.1 Fixed Distance Model. . . . . . . . . . . . . . . . . . . . . . . . . 160
6.2.2 Variable Distance Model . . . . . . . . . . . . . . . . . . . . . . 162
6.3 Diagonal Band Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.3.1 Uniform Loyalty Model . . . . . . . . . . . . . . . . . . . . . . . 166
6.3.2 Quasi Independence Model. . . . . . . . . . . . . . . . . . . . . 169
6.3.3 Triangle Parameters Model . . . . . . . . . . . . . . . . . . . . . 172
6.4 Full Diagonal Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
6.4.1 Uniform Association Model . . . . . . . . . . . . . . . . . . . . 175
6.4.2 Diagonal D Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.4.3 Diagonal Absolute Model. . . . . . . . . . . . . . . . . . . . . . 180
6.4.4 Uniform Fixed Distance Association Model. . . . . . . . . 183
6.4.5 Uniform Variable Distance Association Model. . . . . . . 186
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
7 Asymmetry + Non-Independence Models . . . . . . . . . . . . . . . . . . . . 193
7.1 Non-Symmetry + Independence Model . . . . . . . . . . . . . . . . . . 193
7.2 Non-Symmetry + Independence Triangle Model. . . . . . . . . . . . 197
7.3 Non-Symmetry + Independence Diagonals Model. . . . . . . . . . . 199
7.4 Non-Symmetry + Independence Diagonals Absolute Model . . . 200
7.5 Non-Symmetry + Independence Diagonals Absolute
Triangle Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
7.6 Non-Symmetry + Independence Models—Without
Diagonal Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
8 Modeling Strategy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
8.1 Fit Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
8.2 Graphical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
8.2.1 Heat Map Square Table . . . . . . . . . . . . . . . . . . . . . . . 210
8.2.2 Three-Dimensional Heat Map Bar Plot . . . . . . . . . . . . 214
8.2.3 Local Odds Ratio Square Table Heat Map Plot . . . . . . 217
8.2.4 Local Odds Ratio Square Table Three-dimensional
Heat Map Bar Plot. . . . . . . . . . . . . . . . . . . . . . . . . . . 219
8.2.5 Package gplots Square Table Balloon Plot. . . . . . . . . . 220
8.2.6 Mosaic Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
9 Creating Doubly Classified Models . . . . . . . . . . . . . . . . . . . . . . . . . 225
9.1 Reverse Complete Symmetry Model . . . . . . . . . . . . . . . . . . . . 225
9.2 Parallel Diagonal Symmetry Model . . . . . . . . . . . . . . . . . . . . . 227
9.3 Quasi-Symmetry with N Degree Models . . . . . . . . . . . . . . . . . 229
Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
10.1 Symbolic Table Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
10.2 R Syntax Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
10.3 Hierarchical Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
10.3.1 Hierarchical Tree—Asymmetry Models . . . . . . . . . . . . 250
10.3.2 Hierarchical Tree—Point Symmetry Models. . . . . . . . . 256
10.4 Nested Models—Chi-Square Difference Test . . . . . . . . . . . . . . 256
10.4.1 Chi-Square Difference Test—Complete Symmetry
and Conditional Symmetry Model. . . . . . . . . . . . . . . . 256
10.4.2 Chi-Square Difference Test—Complete Point
Symmetry and Inclined Point Symmetry Model . . . . . . 258
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 263
Symbols
n Observed frequency for cell(i, j)
ij
m Expected frequency for cell(i, j)
ij
m^ Fitted frequency for cell(i, j)
ij (cid:1) (cid:3)
lij Log of estimated frequency, ln m^ij
p Probability for cell(i, j)
ij
D Expected group cells of upper diagonal
upper
D Expected group cells of lower diagonal
lower
X Odds 1(cid:1)pp(cid:4)
dk dk ¼pij pji where k ¼j(cid:1)i
s Log of odds, lnðdÞ
hij Odds ratio, ^hij ¼mm^^ii;jjm^þ(cid:1)i1þm^1i;jþ(cid:3)þ11;j
Uij Log of odds ratio ln hij
X H
ðij;stÞ ðij;stÞ
H
ðst;ijÞ
Hðij;stÞ ppisppjt
js it