Introduction to Categorical Data Analysis

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7.1 LOGLINEAR MODELS FOR TWO-WAY AND THREE-WAY TABLES 209 For it, the next subsection shows that conditional odds ratios between any two variables are the same at each level of the third variable. This is the property of homogeneous association (Section 2.7.6). This loglinear model is called the homogeneous association model and symbolized by (XY, XZ, YZ). The most general loglinear model for three-way tables is log μij k = λ + λ X i + λY j + λZ k + λXY ij + λXZ ik + λYZ jk Denoted by (XYZ), it is the saturated model. It provides a perfect fit. + λXYZ ij k 7.1.5 Two-Factor Parameters Describe Conditional Associations Model interpretations refer to the highest-order parameters. For instance, consider the homogeneous association model (7.5). Its parameters relate directly to conditional odds ratios. We illustrate this for 2 × 2 × K tables. The XY conditional odds ratio θXY (k) describes association between X and Y in partial table k (recall Section 2.7.4). From an argument similar to that in Section 7.1.3, � � μ11kμ22k log θXY (k) = log = λ XY 11 + λXY 22 − λXY 12 − λXY 21 (7.6) μ12kμ21k The right-hand side of this expression does not depend on k, so the odds ratio is the same at every level of Z. Similarly, model (XY, XZ, YZ) also has equal XZ odds ratios at different levels of Y , and it has equal YZ odds ratios at different levels of X. Any model not having the three-factor term λ XYZ ij k satisfies homogeneous association. 7.1.6 Example: Alcohol, Cigarette, and Marijuana Use Table 7.3 is from a survey conducted by the Wright State University School of Medicine and the United Health Services in Dayton, Ohio. The survey asked students in their final year of a high school near Dayton, Ohio whether they had ever Table 7.3. Alcohol (A), Cigarette (C), and Marijuana (M) Use for High School Seniors Alcohol Cigarette Marijuana Use Use Use Yes No Yes Yes 911 538 No 44 456 No Yes 3 43 No 2 279 Source: I am grateful to Professor Harry Khamis, Wright State University, for supplying these data.
210 LOGLINEAR MODELS FOR CONTINGENCY TABLES used alcohol, cigarettes, or marijuana. Denote the variables in this 2 × 2 × 2 table by A for alcohol use, C for cigarette use, and M for marijuana use. Loglinear models are simple to fit with software. Table 7.4 shows fitted values for several models. The fit for model (AC, AM, CM) is close to the observed data, which are the fitted values for the saturated model (ACM). The other models fit poorly. Table 7.4. Fitted Values for Loglinear Models Applied to Table 7.3 Alcohol Cigarette Marijuana Loglinear Model Use Use Use (A, C, M) (AC,M) (AM, CM) (AC, AM, CM) (ACM) Yes Yes Yes 540.0 611.2 909.24 910.4 911 No 740.2 837.8 438.84 538.6 538 No Yes 282.1 210.9 45.76 44.6 44 No 386.7 289.1 555.16 455.4 456 No Yes Yes 90.6 19.4 4.76 3.6 3 No 124.2 26.6 142.16 42.4 43 No Yes 47.3 118.5 0.24 1.4 2 No 64.9 162.5 179.84 279.6 279 Table 7.5 illustrates association patterns for these models by presenting estimated marginal and conditional odds ratios. For example, the entry 1.0 for the AC conditional odds ratio for model (AM, CM) is the common value of the AC fitted odds ratios at the two levels of M, 1.0 = 909.24 × 0.24 45.76 × 4.76 = 438.84 × 179.84 555.16 × 142.16 This model implies conditional independence between alcohol use and cigarette use, controlling for marijuana use. The entry 2.7 for the AC marginal association for this model is the odds ratio for the marginal AC fitted table, 2.7 = (909.24 + 438.84)(0.24 + 179.84) (45.76 + 555.16)(4.76 + 142.16) Table 7.5. Estimated Odds Ratios for Loglinear Models in Table 7.4 Conditional Association Marginal Association Model AC AM CM AC AM CM (A, C, M) 1.0 1.0 1.0 1.0 1.0 1.0 (AC,M) 17.7 1.0 1.0 17.7 1.0 1.0 (AM, CM) 1.0 61.9 25.1 2.7 61.9 25.1 (AC, AM, CM) 7.8 19.8 17.3 17.7 61.9 25.1 (ACM) level 1 13.8 24.3 17.5 17.7 61.9 25.1 (ACM) level 2 7.7 13.5 9.7
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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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260 MODELS FOR MATCHED PAIRS likeli
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262 MODELS FOR MATCHED PAIRS the ad
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264 MODELS FOR MATCHED PAIRS 8.5.5
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266 MODELS FOR MATCHED PAIRS ˆβ4
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268 MODELS FOR MATCHED PAIRS 8.7 Re
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270 MODELS FOR MATCHED PAIRS b. The
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272 MODELS FOR MATCHED PAIRS Table
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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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