Introduction to Categorical Data Analysis

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PROBLEMS 241 7.17 For a multiway contingency table, when is a logistic model more appropriate than a loglinear model, and when is a loglinear model more appropriate? 7.18 For a three-way table, consider the independence graph, X ——– Z Y a. Write the corresponding loglinear model. b. Which, if any, pairs of variables are conditionally independent? c. If Y is a binary response, what is the corresponding logistic model? d. Which pairs of variables have the same marginal association as their conditional association? 7.19 Consider loglinear model (WXZ,WYZ). a. Draw its independence graph, and identify variables that are conditionally independent. b. Explain why this is the most general loglinear model for a four-way table for which X and Y are conditionally independent. 7.20 For a four-way table, are X and Y independent, given Z alone, for model (a) (WX, XZ, YZ, WZ), (b) (WX, XZ, YZ, WY)? 7.21 Refer to Problem 7.13 with Table 7.25. a. Show that model (CE,CH,CL,EH,EL,HL) fits well. Show that model (CEH,CEL,CHL,EHL) also fits well but does not provide a significant improvement. Beginning with (CE,CH,CL,EH,EL,HL), show that backward elimination yields (CE,CL,EH,HL). Interpret its fit. b. Based on the independence graph for (CE,CL,EH,HL), show that: (i) every path between C and H involves a variable in {E,L}; (ii) collapsing over H , one obtains the same associations between C and E and between C and L, and collapsing over C, one obtains the same associations between H and E and between H and L; (iii) the conditional independence patterns between C and H and between E and L are not collapsible. 7.22 Consider model (AC,AM,CM,AG,AR,GM,GR) for the drug use data in Section 7.4.5. a. Explain why the AM conditional odds ratio is unchanged by collapsing over race, but it is not unchanged by collapsing over gender. b. Suppose we remove the GM term. Construct the independence graph, and show that {G, R} are separated from {C, M} by A. c. For the model in (b), explain why all conditional associations among A, C, and M are identical to those in model (AC, AM, CM), collapsing over G and R.
242 LOGLINEAR MODELS FOR CONTINGENCY TABLES 7.23 Consider logit models for a four-way table in which X1,X2, and X3 are predictors of Y . When the table is collapsed over X3, indicate whether the association between X1 and Y remains unchanged, for the model (a) that has main effects of all predictors, (b) that has main effects of X1 and X2 but assumes no effect for X3. 7.24 Table 7.28 is from a General Social Survey. Subjects were asked whether methods of birth control should be available to teenagers between the ages of 14 and 16, and how often they attend religious services. a. Fit the independence model, and use residuals to describe lack of fit. b. Using equally spaced scores, fit the linear-by-linear association model. Describe the association. c. Test goodness of fit of the model in (b); test independence in a way that uses the ordinality, and interpret. d. Fit the L × L model using column scores {1, 2, 4, 5}. Repeat (b), and indicate whether results are substantively different with these scores. e. Using formula (7.12) with the model in (d), explain why a fitted local log odds ratio using columns 2 and 3 is double a fitted local log odds ratio using columns 1 and 2 or columns 3 and 4. What is the relationship between the odds ratios? Table 7.28. Data for Problem 7.24 Teenage Birth Control Religious Strongly Strongly Attendance Agree Agree Disagree Disagree Never 49 49 19 9 Less than once a year 31 27 11 11 Once or twice a year 46 55 25 8 Several times a year 34 37 19 7 About once a month 21 22 14 16 2–3 times a month 26 36 16 16 Nearly every week 8 16 15 11 Every week 32 65 57 61 Several times a week 4 17 16 20 Source: General Social Survey. 7.25 Generalizations of the linear-by-linear model (7.11) analyze association between ordinal variables X and Y while controlling for a categorical variable that may be nominal or ordinal. The model log μij k = λ + λ X i + λY j + λZ k + βuivj + λ XZ ik + λYZ jk
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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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174 MULTICATEGORY LOGIT MODELS are
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176 MULTICATEGORY LOGIT MODELS By e
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178 MULTICATEGORY LOGIT MODELS 6.1.
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180 MULTICATEGORY LOGIT MODELS 6.2
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182 MULTICATEGORY LOGIT MODELS Then
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184 MULTICATEGORY LOGIT MODELS Like
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186 MULTICATEGORY LOGIT MODELS Tabl
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188 MULTICATEGORY LOGIT MODELS only
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292 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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