Introduction to Categorical Data Analysis

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7.2 INFERENCE FOR LOGLINEAR MODELS 217 Table 7.10. Goodness-of-fit Tests for Loglinear Models Relating Injury (I), Gender (G), Location (L), and Seat-Belt Use (S) Model G 2 df P -value (G, I, L, S) 2792.8 11
218 LOGLINEAR MODELS FOR CONTINGENCY TABLES which translates to (0.42, 0.47) for the odds ratio. The odds of injury for passengers wearing seat belts were less than half the odds for passengers not wearing them, for each gender–location combination. The fitted odds ratios in Table 7.11 also suggest that, other factors being fixed, injury was more likely in rural than urban accidents and more likely for females than males. Also, the estimated odds that males used seat belts are only 0.63 times the estimated odds for females. 7.2.7 Three-Factor Interaction Interpretations are more complicated when a model contains three-factor terms. Such terms refer to interactions, the association between two variables varying across levels of the third variable. Table 7.10 shows results of adding a single three-factor term to model (GI, GL, GS, IL, IS, LS). Of the four possible models, (GLS, GI, IL, IS) fits best. Table 7.9 also displays its fit. For model (GLS, GI, IL, IS), each pair of variables is conditionally dependent, and at each level of I the association between G and L or between G and S or between L and S varies across the levels of the remaining variable. For this model, it is inappropriate to interpret the GL, GS, and LS two-factor terms on their own. For example, the presence of the GLS term implies that the GS odds ratio varies across the levels of L. Because I does not occur in a three-factor term, the conditional odds ratio between I and each variable is the same at each combination of levels of the other two variables. The first three lines of Table 7.11 report the fitted odds ratios for the GI, IL, and IS associations. When a model has a three-factor term, to study the interaction, calculate fitted odds ratios between two variables at each level of the third. Do this at any levels of remaining variables not involved in the interaction. The bottom six lines of Table 7.11 illustrates this for model (GLS, GI, IL, IS). For example, the fitted GS odds ratio of 0.66 for (L = urban) refers to four fitted values for urban accidents, both the four with (injury = no) and the four with (injury = yes); that is, 0.66 = (7273.2)(10, 959.2)/(11, 632.6)(10, 358.9) = (1009.8)(389.8)/(713.4)(834.1) 7.2.8 Large Samples and Statistical Versus Practical Significance The sample size can strongly influence results of any inferential procedure. We are more likely to detect an effect as the sample size increases. This suggests a cautionary remark. For small sample sizes, reality may be more complex than indicated by the simplest model that passes a goodness-of-fit test. By contrast, for large sample sizes, statistically significant effects can be weak and unimportant. We saw above that model (GLS, GI, IL, IS) seems to fit much better than (GI, GL, GS, IL, IS, LS): The difference in G 2 values is 23.4 − 7.5 = 15.9, based on df = 5 − 4 = 1(P = 0.0001). The fitted odds ratios in Table 7.11, however, show that the three-factor interaction is weak. The fitted odds ratio between any two
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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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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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Introduction to Categorical Data Analysis

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