Introduction to Categorical Data Analysis

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CHAPTER 8: MODELS FOR MATCHED PAIRS 339 Table A.11. SAS Code for Testing Marginal Homogeneity with Coffee Data of Table 8.5 data migrate; input first $ second $ count m11 m12 m13 m14 m21 m22 m23 m24 m31 m32 m33 m34 m41 m42 m43 m44 m55 m1 m2 m3 m4; datalines; high high 93 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 high tast 17 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 high sank 44 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 high nesc 7 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 high brim 10 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 ... nesc nesc 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 nesc brim 2 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 1 brim high 10 -1 0 0 0 -1 0 0 0 -1 0 0 0 -1 0 0 0 0 1 0 0 0 brim tast 4 0 -1 0 0 0 -1 0 0 0 -1 0 0 0 -1 0 0 0 0 1 0 0 brim sank 12 0 0 -1 0 0 0 -1 0 0 0 -1 0 0 0 -1 0 0 0 0 1 0 brim nesc 2 0 0 0 -1 0 0 0 -1 0 0 0 -1 0 0 0 -1 0 0 0 0 1 brim brim 27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ; proc genmod; model count = m11 m12 m13 m14 m21 m22 m23 m24 m31 m32 m33 m34 m41 m42 m43 m44 m55 m1 m2 m3 m4 / dist=poi link=identity obstats residuals; and columns). Here, m11 denotes expected frequency μ11, m1 denotes μ1+ = μ+1, and so forth. This parameterization uses formulas such as μ15 = μ1+ − μ11 − μ12 − μ13 − μ14 for terms in the last column or last row. The likelihood-ratio test statistic for testing marginal homogeneity is the deviance statistic for this model. Table A.12 shows analyses of Table 8.6. First the data are entered as a 4 × 4 table, and the loglinear model fitted is quasi independence. The “qi” factor invokes Table A.12. SAS Code Showing Square-table Analyses of Tables 8.6 data square; input recycle drive qi count ©©; datalines; 1 1 1 12 1 2 5 43 1 3 5 163 1 4 5 233 2 1 5 4 2 2 2 21 2 3 5 99 2 4 5 185 3 1 5 4 3 2 5 8 3 3 3 77 3 4 5 230 4 1 5 0 4 2 5 1 4 3 5 18 4 4 4 132 ; proc genmod; class drive recycle qi; model count = drive recycle qi / dist=poi link=log; ∗ quasi indep; data square2; input score below above ©©; trials = below + above; datalines; 1 4 43 1 8 99 1 18 230 2 4 163 2 1 185 3 0 233 ; proc genmod data=square2; model above/trials = / dist=bin link=logit noint; proc genmod data=square2; model above/trials = score / dist=bin link=logit noint;
340 APPENDIX A: SOFTWARE FOR CATEGORICAL DATA ANALYSIS the δi parameters in equation (8.12). It takes a separate level for each cell on the main diagonal, and a common value for all other cells. The bottom of Table A.12 fits logit models for the data entered in the form of pairs of cell counts (nij ,nji). These six sets of binomial count are labeled as “above” and “below” with reference to the main diagonal. The variable defined as “score” is the distance (uj − ui) = j − i. The first model is symmetry and the second is ordinal quasi symmetry. Neither model contains an intercept (NOINT). The quasi-symmetry model can be fitted using the approach shown next for the equivalent Bradley–Terry model. TableA.13 uses GENMOD for logit fitting of the Bradley–Terry model to Table 8.9 by forming an artificial explanatory variable for each player. For a given observation, the variable for player i is 1 if he wins, −1 if he loses, and 0 if he is not one of the players for that match. Each observation lists the number of wins (“wins”) for the player with variate-level equal to 1 out of the number of matches (“n”) against the player with variate-level equal to −1. The model has these artificial variates, one of which is redundant, as explanatory variables with no intercept term. The COVB option provides the estimated covariance matrix of parameter estimators. Table A.13. SAS Code for Fitting Bradley–Terry Model to Tennis Data in Table 8.9 data tennnis; input win n agassi federer henman hewitt roddick ; datalines; 0 6 1 -1 0 0 0 0 0 1 0 -1 0 0 ... 3 5 0 0 0 1 -1 ; proc genmod; model win/n = agassi federer henman hewitt roddick / dist=bin link=logit noint covb; CHAPTER 9: MODELING CORRELATED, CLUSTERED RESPONSES TableA.14 uses GENMOD to analyze the depression data in Table 9.1 using GEE. The REPEATED statement specifies the variable name (here, “case”) that identifies the subjects for each cluster. Possible working correlation structures are TYPE=EXCH for exchangeable, TYPE=AR for autoregressive, TYPE=INDEP for independence, and TYPE=UNSTR for unstructured. Output shows estimates and standard errors under the naive working correlation and incorporating the empirical dependence. Alternatively, the working association structure in the binary case can use the log odds ratio (e.g., using LOGOR=EXCH for exchangeability). The type3 option with the GEE approach provides score-type tests about effects. See Stokes et al. (2000, Section 15.11) for the use of GEE with missing data. See Table A.22 in Agresti (2002) for using GENMOD to implement GEE for a cumulative logit model for the insomnia data in Table 9.6.
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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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190 MULTICATEGORY LOGIT MODELS 6.3.
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194 MULTICATEGORY LOGIT MODELS With
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196 MULTICATEGORY LOGIT MODELS high
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CHAPTER 7 Loglinear Models for Cont
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206 LOGLINEAR MODELS FOR CONTINGENC
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216 Table 7.9. Injury (I) by Gender
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CHAPTER 8 Models for Matched Pairs
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246 MODELS FOR MATCHED PAIRS are nu
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248 MODELS FOR MATCHED PAIRS An alt
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250 MODELS FOR MATCHED PAIRS This c
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252 MODELS FOR MATCHED PAIRS only a
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254 MODELS FOR MATCHED PAIRS Table
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258 MODELS FOR MATCHED PAIRS from t
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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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CHAPTER 9 Modeling Correlated, Clus
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278 MODELING CORRELATED, CLUSTERED
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Page 367 and 368: Index of Examples abortion, opinion
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Page 373 and 374: 352 SUBJECT INDEX Exact inference (
Page 375 and 376: 354 SUBJECT INDEX Loglinear models
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Introduction to Categorical Data Analysis

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