Introduction to Categorical Data Analysis

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7.3 THE LOGLINEAR–LOGISTIC CONNECTION 219 of G, L, and S is similar at both levels of the third variable. The significantly better fit of model (GLS, GI, IL, IS) mainly reflects the enormous sample size. Although the three-factor interaction is weak, it is significant because the large sample provides small standard errors. A comparison of fitted odds ratios for the two models suggests that the simpler model (GI, GL, GS, IL, IS, LS) is adequate for practical purposes. Simpler models are easier to summarize. A goodness-of-fit test should not be the sole criterion for selecting a model. For large samples, it is helpful to summarize the closeness of a model fit to the sample data in a way that, unlike a test statistic, is not affected by the sample size. For a table of arbitrary dimension with cell counts {ni = npi} and fitted values {ˆμi = n ˆπi}, one such measure is the dissimilarity index, D = � |ni −ˆμi|/2n = � |pi −ˆπi|/2 This index takes values between 0 and 1. Smaller values represent a better fit. It represents the proportion of sample cases that must move to different cells for the model to achieve a perfect fit. The dissimilarity index helps indicate whether the lack of fit is important in a practical sense. A very small D value suggests that the sample data follow the model pattern closely, even though the model is not perfect. For Table 7.9, model (GI, GL, GS, IL, IS, LS) has D = 0.008, and model (GLS, GI, IL, IS) has D = 0.003. These values are very small. For either model, moving less than 1% of the data yields a perfect fit. The relatively large value of G 2 for model (GI, GL, GS, IL, IS, LS) indicated that the model does not truly hold. Nevertheless, the small value for D suggests that, in practical terms, the model provides a decent fit. 7.3 THE LOGLINEAR–LOGISTIC CONNECTION Loglinear models for contingency tables focus on associations between categorical response variables. Logistic regression models, on the other hand, describe how a categorical response depends on a set of explanatory variables. Though the model types seem distinct, connections exist between them. For a loglinear model, one can construct logits for one response to help interpret the model. Moreover, logistic models with categorical explanatory variables have equivalent loglinear models. 7.3.1 Using Logistic Models to Interpret Loglinear Models To understand implications of a loglinear model formula, it can help to form a logit for one of the variables. We illustrate with the homogeneous association model for three-way tables, log μij k = λ + λ X i + λY j + λZ k + λXY ij + λXZ ik + λYZ jk
220 LOGLINEAR MODELS FOR CONTINGENCY TABLES Suppose Y is binary. We treat it as a response and X and Z as explanatory. When X is at level i and Z is at level k, � � � � P(Y = 1) P(Y = 1 | X = i, Z = k) logit[P(Y = 1)] =log = log 1 − P(Y = 1) P(Y = 2 | X = i, Z = k) � � μi1k = log = log(μi1k) − log(μi2k) μi2k = (λ + λ X i + λY 1 + λZ k − (λ + λ X i + λY 2 + λZ k = (λ Y 1 − λY2 ) + (λXY i1 + λXY i1 + λXY i2 + λXZ ik + λXZ ik + λYZ 1k ) + λYZ 2k ) − λXY i2 ) + (λYZ 1k − λYZ 2k ) The first parenthetical term does not depend on i or k. The second parenthetical term depends on the level i of X. The third parenthetical term depends on the level k of Z. The logit has the additive form logit[P(Y = 1)] =α + β X i + βZ k (7.7) Section 4.3.3 discussed this model, in which the logit depends on X and Z in an additive manner. Additivity on the logit scale is the standard definition of “no interaction” for categorical variables. When Y is binary, the loglinear model of homogeneous association is equivalent to this logistic regression model. When X is also binary, model (7.7) and loglinear model (XY, XZ, YZ) are characterized by equal odds ratios between X and Y at each of the K levels of Z. 7.3.2 Example: Auto Accident Data Revisited For the data on Maine auto accidents (Table 7.9), Section 7.2.6 showed that loglinear model (GLS, GI, LI, IS) fits well. That model is log μgiℓs = λ + λ G g + λI i + λL ℓ + λS s + λ LS ℓs + λGLS gℓs + λGI gi + λGL gℓ + λGS gs + λIL iℓ + λIS is (7.8) We could treat injury (I) as a response variable, and gender (G), location (L), and seat-belt use (S) as explanatory variables. You can check that this loglinear model implies a logistic model of the form logit[P(I = 1)] =α + β G g + βL ℓ + βS s (7.9) Here, G, L, and S all affect I, but without interacting. The parameters in the two models are related by β G g = λGI g1 − λGI g2 ,βL ℓ = λIL 1ℓ − λIL 2ℓ ,βS s = λIS 1s − λIS 2s
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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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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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