Introduction to Categorical Data Analysis

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PROBLEMS 197 6.3 Table 6.14 displays primary food choice for a sample of alligators, classified by length (≤2.3 meters, >2.3 meters) and by the lake in Florida in which they were caught. a. Fit a model to describe effects of length and lake on primary food choice. Report the prediction equations. b. Using the fit of your model, estimate the probability that the primary food choice is “fish,” for each length in Lake Oklawaha. Interpret the effect of length. Table 6.14. Data on Alligators for Exercise 6.3 Primary Food Choice Lake Size Fish Invertebrate Reptile Bird Other Hancock ≤2.3 23 4 2 2 8 >2.3 7 0 1 3 5 Oklawaha ≤2.3 5 11 1 0 3 >2.3 13 8 6 1 0 Trafford ≤2.3 5 11 2 1 5 >2.3 8 7 6 3 5 George ≤2.3 16 19 1 2 3 >2.3 17 1 0 1 3 Source: Wildlife Research Laboratory, Florida Game and Fresh Water Fish Commission. 6.4 Refer to the belief in afterlife example in Section 6.1.4. a. Estimate the probability of response “yes” for black females. b. Describe the gender effect by reporting and interpreting the estimated conditional odds ratio for the (i) “undecided” and “no” pair of response categories, (ii) “yes” and “undecided” pair. 6.5 For a recent General Social Survey, a prediction equation relating Y = job satisfaction (four ordered categories; 1 = the least satisfied) to the subject’s report of x1 = earnings compared with others with similar positions (four ordered categories; 1 = much less, 4 = much more), x2 = freedom to make decisions about how to do job (four ordered categories; 1 = very true, 4 = not at all true), and x3 = work environment allows productivity (four ordered categories; 1 = strongly agree, 4 = strongly disagree), was logit[ P(Y ˆ ≤ j)]= ˆαj − 0.54x1 + 0.60x2 + 1.19x3. a. Summarize each partial effect by indicating whether subjects tend to be more satisfied, or less satisfied, as (i) x1, (ii) x2, (iii) x3, increases. b. Report the settings for x1,x2,x3 at which a subject is most likely to have highest job satisfaction.
198 MULTICATEGORY LOGIT MODELS 6.6 Does marital happiness depend on family income? For the 2002 General Social Survey, counts in the happiness categories (not, pretty, very) were (6, 43, 75) for below average income, (6, 113, 178) for average income, and (6, 57, 117) for above average income. Table 6.15 shows output for a baseline-category logit model with very happy as the baseline category and scores {1, 2, 3} for the income categories. a. Report the prediction equations from this table. b. Interpret the income effect in the first equation. c. Report the Wald test statistic and P -value for testing that marital happiness is independent of family income. Interpret. d. Does the model fit adequately? Justify your answer. e. Estimate the probability that a person with average family income reports a very happy marriage. Table 6.15. Output on Modeling Happiness for Problem 6.6 Deviance and Pearson Goodness-of-Fit Statistics Criterion Value DF Value/DF Pr > ChiSq Deviance 3.1909 2 1.5954 0.2028 Pearson 3.1510 2 1.5755 0.2069 Testing Global Null Hypothesis: BETA = 0 Test Chi-Square DF Pr > ChiSq Likelihood Ratio 0.9439 2 0.6238 Wald 0.9432 2 0.6240 Analysis of Maximum Likelihood Estimates Standard Wald Parameter happy DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 1 −2.5551 0.7256 12.4009 0.0004 Intercept 2 1 −0.3513 0.2684 1.7133 0.1906 income 1 1 −0.2275 0.3412 0.4446 0.5049 income 2 1 −0.0962 0.1220 0.6210 0.4307 6.7 Refer to the previous exercise. Table 6.16 shows output for a cumulative logit model with scores {1, 2, 3} for the income categories. a. Explain why the output reports two intercepts but one income effect. b. Interpret the income effect. c. Report a test statistic and P -value for testing that marital happiness is independent of family income. Interpret. d. Does the model fit adequately? Justify your answer. e. Estimate the probability that a person with average family income reports a very happy marriage.
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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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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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Introduction to Categorical Data Analysis

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