Introduction to Categorical Data Analysis

More documents

Recommendations

Info

8.2 LOGISTIC REGRESSION FOR MATCHED PAIRS 247 For Table 8.1, the difference of sample proportions equals 0.314 − 0.292 = 0.022. For n = 1144 observations with n12 = 132 and n21 = 107, SE = � (132 + 107) − (132 − 107) 2 /1144/1144 = 0.0135 A 95% confidence interval equals 0.022 ± 1.96(0.0135), or(−0.004, 0.048). We infer that the probability of a “yes” response was between 0.004 less and 0.048 higher for paying higher taxes than for accepting a cut in living standards. If the probabilities differ, the difference is apparently small. This is a Wald confidence interval. For small samples the actual coverage probability is closer to the nominal confidence level if you add 0.50 to every cell before finding the standard error. The parenthetical part of the last term in the estimated variance (8.2) represents the effect of the dependence of the marginal proportions through their covariance. Matched-pairs data usually exhibit a strong positive association. Responses for most subjects are the same for the column and the row classification. A sample odds ratio exceeding 1.0 corresponds to p11p22 >p12p21 and hence a negative contribution from the third term in this variance expression. Thus, an advantage of using dependent samples, rather than independent samples, is a smaller variance for the estimated difference in proportions. 8.2 LOGISTIC REGRESSION FOR MATCHED PAIRS Logistic regression extends to handle matched-pairs responses. This section presents such models for binary data. 8.2.1 Marginal Models for Marginal Proportions Let us first see how the analyses of the previous section relate to models. Let (Y1,Y2) denote the two responses, where a “1” denotes category 1 (success) and “0” denotes category 2. In Table 8.1, suppose Y1 is opinion about raising taxes and Y2 is opinion about accepting cuts in living standards. Then, P(Y1 = 1) = π1+ and P(Y2 = 1) = π+1 are marginal probabilities estimated by 359/1144 = 0.31 and 334/1144 = 0.29. The difference between the marginal probabilities occurs as a parameter in a model using identity link function. For the model P(Y1 = 1) = α + δ, P(Y2 = 1) = α δ = P(Y1 = 1) − P(Y2 = 1). The ML estimate of δ is the difference between the sample marginal proportions. For Table 8.1, ˆδ = 0.31 − 0.29 = 0.02. The hypothesis of equal marginal probabilities for the McNemar test is, in terms of this model, H0: δ = 0.
248 MODELS FOR MATCHED PAIRS An alternative model applies the logit link, This is equivalent to logit[P(Y1 = 1)] =α + β, logit[P(Y2 = 1)] =α (8.3) logit[P(Yt = 1)] =α + βxt where xt is an indicator variable that equals 1 when t = 1 and 0 when t = 2. The parameter e β is the odds ratio comparing the marginal distributions. Its ML estimate is the odds ratio for the sample marginal distributions. For Table 8.1, exp( ˆβ) = [(359/785)/(334/810)] =1.11. The population odds of willingness to pay higher taxes are estimated to be 11% higher than the population odds of willingness to accept cuts in living standards. These models are called marginal models. They focus on the marginal distributions of responses for the two observations. 8.2.2 Subject-Specific and Population-Averaged Tables Next, we show a three-way representation of binary matched-pairs data that motivates a different type of model. This display presents the data as n separate 2 × 2 partial tables, one for each matched pair. The kth partial table shows the responses (Y1,Y2) for the kth matched pair. It has columns that are the two possible outcomes (e.g., “yes” and “no”) for each observation. It shows the outcome of Y1 (e.g., response to question 1) in row 1 and the outcome of Y2 (e.g., response to question 2) in row 2. Table 8.1 cross-classified results on two environmental questions for 1144 subjects. Table 8.2 shows a partial table for a subject who answers “yes” on both questions. The full three-way table corresponding to Table 8.1 has 1144 partial tables. Of them, 227 look like Table 8.2, 132 have first row (1, 0) and second row (0, 1), representing “yes” on question 1 and “no” on question 2, 107 have first row (0, 1) and second row (1, 0), and 678 have (0, 1) in each row. Each subject has a partial table, displaying the two matched observations. The 1144 subjects provide 2288 observations in a 2 × 2 × 1144 contingency table. Collapsing this table over the 1144 partial tables yields a 2 × 2 table with first row equal to Table 8.2. Representation of Matched Pair Contributing to Count n11 in Table 8.1 Response Issue Yes No Pay higher taxes 1 0 Cut living standards 1 0
Page 1 and 2:
www.dbeBooks.com - An Ebook Library
Page 4:
An Introduction to Categorical Data
Page 7 and 8:
Copyright © 2007 by John Wiley & S
Page 9 and 10:
vi CONTENTS 2.1.3 Sensitivity and S
Page 11 and 12:
viii CONTENTS 4.1.5 Logistic Regres
Page 13 and 14:
x CONTENTS 6.3.1 Adjacent-Categorie
Page 15 and 16:
xii CONTENTS 9. Modeling Correlated
Page 18 and 19:
Preface to the Second Edition In re
Page 20:
PREFACE TO THE SECOND EDITION xvii
Page 23 and 24:
2 INTRODUCTION sciences (e.g., cate
Page 25 and 26:
4 INTRODUCTION models for continuou
Page 27 and 28:
6 INTRODUCTION The multinomial is a
Page 29 and 30:
8 INTRODUCTION precise, in terms of
Page 31 and 32:
10 INTRODUCTION that are judged pla
Page 33 and 34:
12 INTRODUCTION In this text, we us
Page 35 and 36:
14 INTRODUCTION 1.4.4 Small-Sample
Page 37 and 38:
16 INTRODUCTION For small samples,
Page 39 and 40:
18 INTRODUCTION the outcome y equal
Page 41 and 42:
20 INTRODUCTION a. Calculate the lo
Page 43 and 44:
22 CONTINGENCY TABLES Suppose there
Page 45 and 46:
24 CONTINGENCY TABLES Figure 2.1. T
Page 47 and 48:
26 CONTINGENCY TABLES compare two g
Page 49 and 50:
28 CONTINGENCY TABLES It can be any
Page 51 and 52:
30 CONTINGENCY TABLES The sample od
Page 53 and 54:
32 CONTINGENCY TABLES For Table 2.3
Page 55 and 56:
34 CONTINGENCY TABLES the binomial
Page 57 and 58:
36 CONTINGENCY TABLES The df value
Page 59 and 60:
38 CONTINGENCY TABLES Table 2.5. Cr
Page 61 and 62:
40 CONTINGENCY TABLES that combines
Page 63 and 64:
42 CONTINGENCY TABLES Large values
Page 65 and 66:
44 CONTINGENCY TABLES when the data
Page 67 and 68:
46 CONTINGENCY TABLES n11 equals P(
Page 69 and 70:
48 CONTINGENCY TABLES the observed
Page 71 and 72:
50 CONTINGENCY TABLES Table 2.10. D
Page 73 and 74:
52 CONTINGENCY TABLES Figure 2.4. P
Page 75 and 76:
54 CONTINGENCY TABLES marginal tabl
Page 77 and 78:
56 CONTINGENCY TABLES 2.4 A newspap
Page 79 and 80:
58 CONTINGENCY TABLES a. Construct
Page 81 and 82:
60 CONTINGENCY TABLES Table 2.14. D
Page 83 and 84:
62 CONTINGENCY TABLES a. Show that
Page 85 and 86:
64 CONTINGENCY TABLES 2.36 Give a
Page 87 and 88:
66 GENERALIZED LINEAR MODELS 3.1 CO
Page 89 and 90:
68 GENERALIZED LINEAR MODELS The ne
Page 91 and 92:
70 GENERALIZED LINEAR MODELS Figure
Page 93 and 94:
72 GENERALIZED LINEAR MODELS fitted
Page 95 and 96:
74 GENERALIZED LINEAR MODELS 3.3 GE
Page 97 and 98:
76 Table 3.2. Number of Crab Satell
Page 99 and 100:
Page 101 and 102:
Page 103 and 104:
82 GENERALIZED LINEAR MODELS with S
Page 105 and 106:
84 GENERALIZED LINEAR MODELS Simila
Page 107 and 108:
86 GENERALIZED LINEAR MODELS provid
Page 109 and 110:
88 GENERALIZED LINEAR MODELS 3.5 FI
Page 111 and 112:
90 GENERALIZED LINEAR MODELS Table
Page 113 and 114:
92 GENERALIZED LINEAR MODELS 37 obs
Page 115 and 116:
94 GENERALIZED LINEAR MODELS Table
Page 117 and 118:
96 GENERALIZED LINEAR MODELS 3.18 T
Page 119 and 120:
98 GENERALIZED LINEAR MODELS 3.22 T
Page 121 and 122:
100 LOGISTIC REGRESSION The logisti
Page 123 and 124:
102 LOGISTIC REGRESSION of observat
Page 125 and 126:
104 LOGISTIC REGRESSION for crabs i
Page 127 and 128:
106 LOGISTIC REGRESSION that P(Y =
Page 129 and 130:
108 LOGISTIC REGRESSION −2(L0 −
Page 131 and 132:
110 LOGISTIC REGRESSION The estimat
Page 133 and 134:
112 LOGISTIC REGRESSION Table 4.4.
Page 135 and 136:
114 LOGISTIC REGRESSION 4.3.4 The C
Page 137 and 138:
116 LOGISTIC REGRESSION 4.4.1 Examp
Page 139 and 140:
118 LOGISTIC REGRESSION 4.4.2 Model
Page 141 and 142:
120 LOGISTIC REGRESSION 1.2, based
Page 143 and 144:
122 LOGISTIC REGRESSION activity of
Page 145 and 146:
Page 147 and 148:
Page 149 and 150:
128 LOGISTIC REGRESSION 0 = no) and
Page 151 and 152:
Page 153 and 154:
132 LOGISTIC REGRESSION b. In Table
Page 155 and 156:
Page 157 and 158:
136 LOGISTIC REGRESSION c. The lack
Page 159 and 160:
138 BUILDING AND APPLYING LOGISTIC
Page 161 and 162:
Page 163 and 164:
Page 165 and 166:
Page 167 and 168:
Page 169 and 170:
Page 171 and 172:
Page 173 and 174:
Page 175 and 176:
Page 177 and 178:
Page 179 and 180:
Page 181 and 182:
Page 183 and 184:
Page 185 and 186:
Page 187 and 188:
Page 189 and 190:
Page 191 and 192:
Page 193 and 194:
Page 195 and 196:
174 MULTICATEGORY LOGIT MODELS are
Page 197 and 198:
176 MULTICATEGORY LOGIT MODELS By e
Page 199 and 200:
178 MULTICATEGORY LOGIT MODELS 6.1.
Page 201 and 202:
180 MULTICATEGORY LOGIT MODELS 6.2
Page 203 and 204:
182 MULTICATEGORY LOGIT MODELS Then
Page 205 and 206:
184 MULTICATEGORY LOGIT MODELS Like
Page 207 and 208:
186 MULTICATEGORY LOGIT MODELS Tabl
Page 209 and 210:
188 MULTICATEGORY LOGIT MODELS only
Page 211 and 212:
190 MULTICATEGORY LOGIT MODELS 6.3.
Page 213 and 214:
192 MULTICATEGORY LOGIT MODELS Tabl
Page 215 and 216:
194 MULTICATEGORY LOGIT MODELS With
Page 217 and 218: 196 MULTICATEGORY LOGIT MODELS high
Page 219 and 220: 198 MULTICATEGORY LOGIT MODELS 6.6
Page 221 and 222: 200 MULTICATEGORY LOGIT MODELS 6.9
Page 223 and 224: 202 MULTICATEGORY LOGIT MODELS Tabl
Page 225 and 226: CHAPTER 7 Loglinear Models for Cont
Page 227 and 228: 206 LOGLINEAR MODELS FOR CONTINGENC
Page 237 and 238: 216 Table 7.9. Injury (I) by Gender
Page 265 and 266: CHAPTER 8 Models for Matched Pairs
Page 267: 246 MODELS FOR MATCHED PAIRS are nu
Page 271 and 272: 250 MODELS FOR MATCHED PAIRS This c
Page 273 and 274: 252 MODELS FOR MATCHED PAIRS only a
Page 275 and 276: 254 MODELS FOR MATCHED PAIRS Table
Page 279 and 280: 258 MODELS FOR MATCHED PAIRS from t
Page 281 and 282: 260 MODELS FOR MATCHED PAIRS likeli
Page 283 and 284: 262 MODELS FOR MATCHED PAIRS the ad
Page 285 and 286: 264 MODELS FOR MATCHED PAIRS 8.5.5
Page 287 and 288: 266 MODELS FOR MATCHED PAIRS ˆβ4
Page 289 and 290: 268 MODELS FOR MATCHED PAIRS 8.7 Re
Page 291 and 292: 270 MODELS FOR MATCHED PAIRS b. The
Page 297 and 298: CHAPTER 9 Modeling Correlated, Clus
Page 299 and 300: 278 MODELING CORRELATED, CLUSTERED
Page 319 and 320:
298 RANDOM EFFECTS: GENERALIZED LIN
Page 321 and 322:
Page 323 and 324:
Page 325 and 326:
Page 327 and 328:
Page 329 and 330:
Page 331 and 332:
Page 333 and 334:
Page 335 and 336:
Page 337 and 338:
Page 339 and 340:
Page 341 and 342:
Page 343 and 344:
Page 345 and 346:
Page 347 and 348:
326 A HISTORICAL TOUR OF CATEGORICA
Page 349 and 350:
Page 351 and 352:
Page 353 and 354:
Appendix A: Software for Categorica
Page 355 and 356:
334 APPENDIX A: SOFTWARE FOR CATEGO
Page 357 and 358:
Page 359 and 360:
Page 361 and 362:
Page 363 and 364:
Page 365 and 366:
Bibliography Agresti, A. (2002). Ca
Page 367 and 368:
Index of Examples abortion, opinion
Page 369 and 370:
348 INDEX OF EXAMPLES lung cancer a
Page 371 and 372:
Subject Index Adjacent categories l
Page 373 and 374:
352 SUBJECT INDEX Exact inference (
Page 375 and 376:
354 SUBJECT INDEX Loglinear models
Page 377 and 378:
356 SUBJECT INDEX Residuals binomia
Page 379 and 380:
358 BRIEF SOLUTIONS TO SOME ODD-NUM
Page 381 and 382:
Page 383 and 384:
Page 385 and 386:
Page 387 and 388:
Page 389 and 390:
Page 391 and 392:
Page 393:
show all

Introduction to Categorical Data Analysis

Create successful ePaper yourself

Delete template?

Save as template?