Introduction to Categorical Data Analysis

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7.4 INDEPENDENCE GRAPHS AND COLLAPSIBILITY 223 Such models focus on the effects of G and L on S and on I as well as the association between S and I. 7.4 INDEPENDENCE GRAPHS AND COLLAPSIBILITY We next present a graphical representation for conditional independences in loglinear models. The graph indicates which pairs of variables are conditionally independent, given the others. This representation is helpful for revealing implications of models, such as determining when marginal and conditional odds ratios are identical. 7.4.1 Independence Graphs An independence graph for a loglinear model has a set of vertices, each vertex representing a variable. There are as many vertices as dimensions of the contingency table. Any two vertices either are or are not connected by an edge. A missing edge between two vertices represents a conditional independence between the corresponding two variables. For example, for a four-way table, the loglinear model (WX, WY, WZ, YZ) lacks XY and XZ association terms. It assumes that X and Y are independent and that X and Z are independent, conditional on the other two variables. The independence graph portrays this model. Edges connect W with X, W with Y , W with Z, and Y with Z. These represent pairwise conditional associations. Edges do not connect X with Y or X with Z, because those pairs are conditionally independent. Two loglinear models that have the same conditional independences have the same independence graph. For instance, the independence graph just portrayed for model (WX, WY, WZ, YZ) is also the one for model (WX, WYZ) that also contains a threefactor WYZ term. A path in an independence graph is a sequence of edges leading from one variable to another. Two variables X and Y are said to be separated by a subset of variables if all paths connecting X and Y intersect that subset. In the above graph, W separates X and Y , since any path connecting X with Y goes through W . The subset {W,Z} also separates X and Y . A fundamental result states that two variables are conditionally independent given any subset of variables that separates them. Thus, not only are X and Y conditionally independent given W and Z, but also given W alone. Similarly, X and Z are conditionally independent given W alone.
224 LOGLINEAR MODELS FOR CONTINGENCY TABLES The loglinear model (WX, XY, YZ) has independence graph W ——– X ——– Y ——– Z Here, W and Z are separated by X, byY , and by X and Y . So, W and Z are independent given X alone or given Y alone or given both X and Y . Also, W and Y are independent, given X alone or given X and Z, and X and Z are independent, given Y alone or given Y and W. 7.4.2 Collapsibility Conditions for Three-Way Tables Sometimes researchers collapse multiway contingency tables to make them simpler to describe and analyze. However, Section 2.7.5 showed that marginal associations may differ from conditional associations. For example, if X and Y are conditionally independent, given Z, they are not necessarily marginally independent. Under the following collapsibility conditions, a model’s odds ratios are identical in partial tables as in the marginal table: For three-way tables, XY marginal and conditional odds ratios are identical if either Z and X are conditionally independent or if Z and Y are conditionally independent. The conditions state that the variable treated as the control (Z) is conditionally independent of X or Y , or both. These conditions correspond to loglinear models (XY, YZ) and (XY, XZ). That is, the XY association is identical in the partial tables and the marginal table for models with independence graphs X ——– Y ——– Z and Y ——– X ——– Z or even simpler models. For Table 7.3 from Section 7.1.6 with A = alcohol use, C = cigarette use, and M = marijuana use, the model (AM, CM) of AC conditional independence has independence graph A ——– M ——– C Consider the AM association, identifying C with Z in the collapsibility conditions. In this model, since C is conditionally independent of A, the AM conditional odds ratios are the same as the AM marginal odds ratio collapsed over C. In fact, from Table 7.5, both the fitted marginal and conditional AM odds ratios equal 61.9. Similarly, the CM association is collapsible. The AC association is not, however. The collapsibility conditions are not satisfied, because M is conditionally dependent with both A and C in model (AM, CM). Thus, A and C may be marginally dependent, even though they are conditionally independent in this model. In fact, from Table 7.5, the model’s fitted AC marginal odds ratio equals 2.7, not 1.0.
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www.dbeBooks.com - An Ebook Library
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An Introduction to Categorical Data
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Copyright © 2007 by John Wiley & S
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vi CONTENTS 2.1.3 Sensitivity and S
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viii CONTENTS 4.1.5 Logistic Regres
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x CONTENTS 6.3.1 Adjacent-Categorie
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xii CONTENTS 9. Modeling Correlated
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Preface to the Second Edition In re
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PREFACE TO THE SECOND EDITION xvii
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2 INTRODUCTION sciences (e.g., cate
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4 INTRODUCTION models for continuou
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6 INTRODUCTION The multinomial is a
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8 INTRODUCTION precise, in terms of
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10 INTRODUCTION that are judged pla
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12 INTRODUCTION In this text, we us
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14 INTRODUCTION 1.4.4 Small-Sample
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16 INTRODUCTION For small samples,
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18 INTRODUCTION the outcome y equal
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20 INTRODUCTION a. Calculate the lo
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22 CONTINGENCY TABLES Suppose there
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24 CONTINGENCY TABLES Figure 2.1. T
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26 CONTINGENCY TABLES compare two g
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28 CONTINGENCY TABLES It can be any
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30 CONTINGENCY TABLES The sample od
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32 CONTINGENCY TABLES For Table 2.3
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34 CONTINGENCY TABLES the binomial
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36 CONTINGENCY TABLES The df value
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38 CONTINGENCY TABLES Table 2.5. Cr
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40 CONTINGENCY TABLES that combines
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42 CONTINGENCY TABLES Large values
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44 CONTINGENCY TABLES when the data
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46 CONTINGENCY TABLES n11 equals P(
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48 CONTINGENCY TABLES the observed
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50 CONTINGENCY TABLES Table 2.10. D
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52 CONTINGENCY TABLES Figure 2.4. P
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54 CONTINGENCY TABLES marginal tabl
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56 CONTINGENCY TABLES 2.4 A newspap
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58 CONTINGENCY TABLES a. Construct
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60 CONTINGENCY TABLES Table 2.14. D
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62 CONTINGENCY TABLES a. Show that
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64 CONTINGENCY TABLES 2.36 Give a
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66 GENERALIZED LINEAR MODELS 3.1 CO
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68 GENERALIZED LINEAR MODELS The ne
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70 GENERALIZED LINEAR MODELS Figure
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72 GENERALIZED LINEAR MODELS fitted
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74 GENERALIZED LINEAR MODELS 3.3 GE
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76 Table 3.2. Number of Crab Satell
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82 GENERALIZED LINEAR MODELS with S
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84 GENERALIZED LINEAR MODELS Simila
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86 GENERALIZED LINEAR MODELS provid
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88 GENERALIZED LINEAR MODELS 3.5 FI
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90 GENERALIZED LINEAR MODELS Table
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92 GENERALIZED LINEAR MODELS 37 obs
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94 GENERALIZED LINEAR MODELS Table
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96 GENERALIZED LINEAR MODELS 3.18 T
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98 GENERALIZED LINEAR MODELS 3.22 T
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100 LOGISTIC REGRESSION The logisti
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102 LOGISTIC REGRESSION of observat
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104 LOGISTIC REGRESSION for crabs i
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106 LOGISTIC REGRESSION that P(Y =
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108 LOGISTIC REGRESSION −2(L0 −
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110 LOGISTIC REGRESSION The estimat
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112 LOGISTIC REGRESSION Table 4.4.
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114 LOGISTIC REGRESSION 4.3.4 The C
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116 LOGISTIC REGRESSION 4.4.1 Examp
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118 LOGISTIC REGRESSION 4.4.2 Model
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120 LOGISTIC REGRESSION 1.2, based
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122 LOGISTIC REGRESSION activity of
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128 LOGISTIC REGRESSION 0 = no) and
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132 LOGISTIC REGRESSION b. In Table
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136 LOGISTIC REGRESSION c. The lack
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138 BUILDING AND APPLYING LOGISTIC
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Page 193 and 194: 172 BUILDING AND APPLYING LOGISTIC
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Page 225 and 226: CHAPTER 7 Loglinear Models for Cont
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274 MODELS FOR MATCHED PAIRS Table
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CHAPTER 9 Modeling Correlated, Clus
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278 MODELING CORRELATED, CLUSTERED
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298 RANDOM EFFECTS: GENERALIZED LIN
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326 A HISTORICAL TOUR OF CATEGORICA
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Appendix A: Software for Categorica
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334 APPENDIX A: SOFTWARE FOR CATEGO
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Bibliography Agresti, A. (2002). Ca
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Index of Examples abortion, opinion
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348 INDEX OF EXAMPLES lung cancer a
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Subject Index Adjacent categories l
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352 SUBJECT INDEX Exact inference (
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354 SUBJECT INDEX Loglinear models
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356 SUBJECT INDEX Residuals binomia
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358 BRIEF SOLUTIONS TO SOME ODD-NUM
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