Introduction to Categorical Data Analysis

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5.4 CONDITIONAL LOGISTIC REGRESSION AND EXACT INFERENCE 157 than true ones. When most fitted values are between about 0.50 and 5, G 2 tends to be too liberal; the reported P -value tends to be too small. For fixed values of n and the number of cells, the chi-squared approximation is better for tests with smaller values of df. For instance, it is better for a test comparing two models M0 and M1 when M1 has at most a few more parameters than M0 than it is for testing fit by comparing a model to the saturated model. In the latter case one is testing that every parameter equals 0 that could be in the model but is not, and df can be large when there are many cells. For models with only main effects, the adequacy of model-comparison tests depends more on the two-way marginal totals relating Y to each predictor than on cell counts. Cell counts can be small as long as most totals in these marginal tables exceed about 5. When cell counts are so small that chi-squared approximations may be inadequate, one could combine categories of variables to obtain larger counts. This is usually not advisable unless there is a natural way to combine them and little information loss in defining the variable more crudely. In any case, poor sparse-data performance of chi-squared tests is becoming less problematic because of the development of small-sample methods. The following section discusses this for parameters in logistic regression. 5.4 CONDITIONAL LOGISTIC REGRESSION AND EXACT INFERENCE For inference about logistic regression parameters, the ordinary sampling distributions are approximately normal or chi-squared. The approximation improves as the sample size increases. For small samples, it is better to use the exact sampling distributions. Methods that find and use the exact distribution are now feasible due to recent advances in computer power and software. For example, StatXact (Cytel Software) conducts many exact inferences for two-way and three-way contingency tables. LogXact (Cytel software) and PROC LOGISTIC (SAS) handle exact inference for logistic regression parameters. 5.4.1 Conditional Maximum Likelihood Inference The exact inference approach deals with the primary parameters of interest using a conditional likelihood function that eliminates the other parameters. The technique uses a conditional probability distribution defined for potential samples that provide the same information about the other parameters that occurs in the observed sample. The distribution and the related conditional likelihood function depend only on the parameters of interest. The conditional maximum likelihood estimate of a parameter is the value at which the conditional likelihood function achieves it maximum. Ordinary ML estimators of parameters work best when the sample size is large compared with the number of model parameters. When the sample size is small, or when there are many parameters
158 BUILDING AND APPLYING LOGISTIC REGRESSION MODELS relative to the sample size, conditional ML estimates of parameters work better than ordinary ML estimators. Exact inference for a parameter uses the conditional likelihood function that eliminates all the other parameters. Since that conditional likelihood does not involve unknown parameters, probabilities such as P -values use exact distributions rather than approximations. When the sample size is small, conditional likelihood-based exact inference in logistic regression is more reliable than the ordinary large-sample inferences. 5.4.2 Small-Sample Tests for Contingency Tables Consider first logistic regression with a single explanatory variable, logit[π(x)]=α + βx When x takes only two values, the model applies to 2 × 2 tables of counts {nij } for which the two columns are the levels of Y . The usual sampling model treats the responses on Y in the two rows as independent binomial variates. The row totals, which are the numbers of trials for those binomial variates, are naturally fixed. For this model, the hypothesis of independence is H0: β = 0. The unknown parameter α refers to the relative number of outcomes of y = 1 and y = 0, which are the column totals. Software eliminates α from the likelihood by conditioning also on the column totals, which are the information in the data about α. Fixing both sets of marginal totals yields a hypergeometric distribution for n11, for which the probabilities do not depend on unknown parameters. The resulting exact test of H0: β = 0 is the same as Fisher’s exact test (Section 2.6.1). Next, suppose the model also has a second explanatory factor, Z, with K levels. If Z is nominal-scale, a relevant model is logit(π) = α + βx + β Z k Section 4.3.4 presented this model for 2 × 2 × K contingency tables. The test of H0: β = 0 refers to the effect of X on Y , controlling for Z. The exact test eliminates the other parameters by conditioning on the marginal totals in each partial table. This gives an exact test of conditional independence between X and Y , controlling for Z. For 2 × 2 × K tables {nij k}, conditional on the marginal totals in each partial table, the Cochran–Mantel–Haenszel test of conditional independence (Section 4.3.4) is a large-sample approximate method that compares � k n11k to its null expected value. Exact tests use � k n11k in the way they use n11 in Fisher’s exact test for 2 × 2 tables. Hypergeometric distributions in each partial table determine the probability distribution of � k n11k. The P -value for Ha: β>0 equals the right-tail probability that � k n11k is at least as large as observed, for the fixed marginal totals. Two-sided alternatives can use a two-tail probability of those outcomes that are no more likely than the observed one.
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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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208 LOGLINEAR MODELS FOR CONTINGENC
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216 Table 7.9. Injury (I) by Gender
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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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