Introduction to Categorical Data Analysis

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PROBLEMS 267 Table 8.10. Data from General Social Survey for Problem 8.2 Believe in Believe in Hell Heaven Yes No Yes 833 125 No 2 160 8.3 Refer to the previous exercise. Estimate and interpret the odds ratio for a logistic model for the probability of a “yes” response as a function of the item (heaven or hell), using (a) the marginal model (8.3) and (b) the conditional model (8.4). 8.4 Explain the following analogy: The McNemar test is to binary data as the paired difference t test is to normally distributed data. 8.5 Section 8.1.1 gave the large-sample z or chi-squared McNemar test for comparing dependent proportions. The exact P -value, needed for small samples, uses the binomial distribution. For Table 8.1, consider Ha: π1+ >π+1, or equivalently, Ha: π12 >π21. a. The exact P -value is the binomial probability of at least 132 successes out of 239 trials, when the parameter is 0.50. Explain why. (Software reports P -value = 0.060.) b. For these data, how is the mid P -value (Section 1.4.5) defined in terms of binomial probabilities? (This P -value = 0.053.) c. Explain why Ha: π1+ �= π+1 has ordinary P -value = 0.120 and mid P -value = 0.106. (The large-sample McNemar test has P -value that is an approximation for the binomial mid P -value. It is also 0.106 for these data.) 8.6 For Table 7.19 on opinions about measures to deal with AIDS, treat the data as matched pairs on opinion, stratified by gender. a. For females, test the equality of the true proportions supporting government action for the two items. b. Refer to (a). Construct a 90% confidence interval for the difference between the true proportions of support. Interpret. c. For females, estimate the odds ratio exp(β) for (i) marginal model (8.3), (ii) conditional model (8.4). Interpret. d. Explain how you could construct a 90% confidence interval for the difference between males and females in their differences of proportions of support for a particular item. (Hint: The gender samples are independent.)
268 MODELS FOR MATCHED PAIRS 8.7 Refer to Table 8.1 on ways to help the environment. Suppose sample proportions of approval of 0.314 and 0.292 were based on independent samples of size 1144 each. Construct a 95% confidence interval for the true difference of proportions. Compare with the result in Section 8.1.2, and comment on how the use of dependent samples can improve precision. 8.8 A crossover experiment with 100 subjects compares two treatments for migraine headaches. The response scale is success (+) or failure (−). Half the study subjects, randomly selected, used drug A the first time they got a migraine headache and drug B the next time. For them, six had responses (A+,B+), 25 had responses (A+,B−), 10 had responses (A−,B+), and nine had responses (A−,B−). The other 50 subjects took the drugs in the reverse order. For them, 10 were (A+,B+), 20 were (A+,B−), 12 were (A−,B+), and eight were (A−,B−). a. Ignoring treatment order, use the McNemar test to compare the success probabilities for the two treatments. Interpret. b. The McNemar test uses only the pairs of responses that differ. For this study, Table 8.11 shows such data from both treatment orders. Explain why a test of independence for this table tests the hypothesis that success rates are identical for the two treatments. Analyze these data, and interpret. Table 8.11. Data for Problem 8.8 Treatment Treatment That is Better Order First Second A then B 25 10 B then A 12 20 8.9 Estimate β in model (8.4) applied to Table 8.1 on helping the environment. Interpret. 8.10 A case–control study has eight pairs of subjects. The cases have colon cancer, and the controls are matched with the cases on gender and age. A possible explanatory variable is the extent of red meat in a subject’s diet, measured as “low” or “high.” For three pairs, both the case and the control were high; for one pair, both the case and the control were low; for three pairs, the case was high and the control was low; for one pair, the case was low and the control was high. a. Display the data in a 2 × 2 cross-classification of diet for the case against diet for the control. Display the 2 × 2 × 8 table with partial tables relating diet to response (case, control) for the matched pairs. Successive parts refer to these as Table A and Table B.
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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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318 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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