Introduction to Categorical Data Analysis

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PROBLEMS 165 c. The concordance index c equals 0.658 for the model with the four main effects and the six interaction terms, 0.640 for the model with only the four main effect terms, and 0.568 for the model with only T/F as a predictor. According to this criterion, which model would you choose (i) if you want to maximize sample predictive power (ii) if you think model parsimony is important? 5.7 From the same survey referred to in Problem 4.16, Table 5.11 cross-classifies whether a person smokes frequently with the four scales of the MBTI personality test. SAS reports model −2 log likelihood values of 1130.23 with only an intercept term, 1124.86 with also the main effect predictors, 1119.87 with also all the two-factor interactions, and 1116.47 with also all the three-factor interactions. a. Write the model for each case, and show that the numbers of parameters are 1, 5, 11, and 15. b. According to AIC, which of these four models is preferable? c. When a classification table for the model containing the four main effect terms uses the sample proportion of frequent smokers of 0.23 as the cutoff, sensitivity = 0.48 and specificity = 0.55. The area under the ROC curve is c = 0.55. Does knowledge of personality type help you predict well whether someone is a frequent smoker? Explain. Table 5.11. Data on Smoking Frequently and Four Scales of Myers–Briggs Personality Test Extroversion/Introversion E I Sensing/iNtuitive S N S N Smoking Frequently Thinking/ Judging/ Feeling Perceiving Yes No Yes No Yes No Yes No T J 13 64 6 17 32 108 4 9 P 11 31 4 14 9 43 9 26 F J 16 89 6 25 34 104 4 27 P 19 60 23 57 29 76 22 57 Source: Reproduced with special permission of CPP Inc., Mountain View, CA 94043. Copyright 1996 by CPP Inc. All rights reserved. Further reproduction is prohibited without the Publisher’s written consent. 5.8 Refer to the classification table in Table 5.3 with π0 = 0.50. a. Explain how this table was constructed. b. Estimate the sensitivity and specificity, and interpret. 5.9 Problem 4.1 with Table 4.8 used a labeling index (LI) to predict π = the probability of remission in cancer patients.
166 BUILDING AND APPLYING LOGISTIC REGRESSION MODELS a. When the data for the 27 subjects are 14 binomial observations (for the 14 distinct levels of LI), the deviance for this model is 15.7 with df = 12. Is it appropriate to use this to check the fit of the model? Why or why not? b. The model that also has a quadratic term for LI has deviance = 11.8. Conduct a test comparing the two models. c. The model in (b) has fit, logit( ˆπ) =−13.096 + 0.9625(LI ) − 0.0160(LI) 2 , with SE = 0.0095 for ˆβ2 =−0.0160. If you know basic calculus, explain why ˆπ is increasing for LI between 0 and 30. Since LI varies between 8 and 38 in this sample, the estimated effect of LI is positive over most of its observed values. d. For the model with only the linear term, the Hosmer–Lemeshow test statistic = 6.6 with df = 6. Interpret. 5.10 For the horseshoe crab data, fit the logistic regression model with x = weight as the sole predictor of the presence of satellites. a. For a classification table using the sample proportion of 0.642 as the cutoff, report the sensitivity and specificity. Interpret. b. Form a ROC curve, and report and interpret the area under it. c. Investigate the model goodness-of-fit using the Hosmer–Lemeshow statistic or some other model-checking approach. Interpret. d. Inferentially compare the model to the model with x and x 2 as predictors. Interpret. e. Compare the models in (d) using the AIC. Interpret. 5.11 Here is an alternative to the Hosmer–Lemeshow goodness-of-fit test when at least one predictor is continuous: Partition values for the explanatory variables into a set of regions. Add these regions as a predictor in the model by setting up dummy variables for the regions. The test statistic compares the fit of this model to the simpler one, testing that the extra parameters are not needed. Doing this for model (4.11) by partitioning according to the eight width regions in Table 4.11, the likelihood-ratio statistic for testing that the extra parameters are unneeded equals 7.5, based on df = 7. Interpret. 5.12 Refer to Table 7.27 in Chapter 7 with opinion about premarital sex as the response variable. Use a process (such as backward elimination) or criterion (such as AIC) to select a model. Interpret the parameter estimates for that model. 5.13 Logistic regression is often applied to large financial databases. For example, credit scoring is a method of modeling the influence of predictors on the probability that a consumer is credit worthy. The data archive found under the index at www.stat.uni-muenchen.de for a textbook by L. Fahrmeir and G. Tutz (Multivariate Statistical Modelling Based on Generalized Linear Models,
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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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216 Table 7.9. Injury (I) by Gender
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218 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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Introduction to Categorical Data Analysis

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