Introduction to Categorical Data Analysis

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6.2 CUMULATIVE LOGIT MODELS FOR ORDINAL RESPONSES 181 Figure 6.2. Depiction of cumulative probabilities in proportional odds model. others shifted to the right or shifted to the left. As in logistic regression, the size of |β| determines how quickly the curves climb or drop. At any fixed x value, the curves have the same ordering as the cumulative probabilities, the one for P(Y ≤ 1) being lowest. Figure 6.2 has β>0. Figure 6.3 shows corresponding curves for the category probabilities, P(Y = j) = P(Y ≤ j)− P(Y ≤ j − 1).Asx increases, the response on Y is more likely to fall at the low end of the ordinal scale. When β
182 MULTICATEGORY LOGIT MODELS Then, as x increases, Y is more likely to fall at the high end of the scale. 2 When the model holds with β = 0, the graph has a horizontal line for each cumulative probability. Then, X and Y are statistically independent. Model interpretations can use odds ratios for the cumulative probabilities and their complements. For two values x1 and x2 of x, an odds ratio comparing the cumulative probabilities is P(Y ≤ j | X = x2)/P (Y > j | X = x2) P(Y ≤ j | X = x1)/P (Y > j | X = x1) The log of this odds ratio is the difference between the cumulative logits at those two values of x. This equals β(x2 − x1), proportional to the distance between the x values. In particular, for x2 − x1 = 1, the odds of response below any given category multiply by e β for each unit increase in x. For this log odds ratio β(x2 − x1), the same proportionality constant (β) applies for each cumulative probability. This property is called the proportional odds assumption of model (6.4). Explanatory variables in cumulative logit models can be quantitative, categorical (with indicator variables), or of both types. The ML fitting process uses an iterative algorithm simultaneously for all j. When the categories are reversed in order, the same fit results but the sign of ˆβ reverses. 6.2.2 Example: Political Ideology and Party Affiliation Table 6.7, from a General Social Survey, relates political ideology to political party affiliation. Political ideology has a five-point ordinal scale, ranging from very liberal to very conservative. Let x be an indicator variable for political party, with x = 1 for Democrats and x = 0 for Republicans. Table 6.7. Political Ideology by Gender and Political Party Political Ideology Political Very Slightly Slightly Very Gender Party Liberal Liberal Moderate Conservative Conservative Female Democratic 44 47 118 23 32 Republican 18 28 86 39 48 Male Democratic 36 34 53 18 23 Republican 12 18 62 45 51 Source: General Social Survey. Table 6.8 shows output (from PROC LOGISTIC in SAS) for the ML fit of model (6.4). With J = 5 response categories, the model has four {αj } intercepts. Usually 2 The model is sometimes written instead as logit[P(Y ≤ j)]=αj − βx, so that β>0 corresponds to Y being more likely to fall at the high end of the scale as x increases.
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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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Page 159 and 160: 138 BUILDING AND APPLYING LOGISTIC
Page 195 and 196: 174 MULTICATEGORY LOGIT MODELS are
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232 LOGLINEAR MODELS FOR CONTINGENC
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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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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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