Introduction to Categorical Data Analysis

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10.5 MODEL FITTING AND INFERENCE FOR GLMMS 317 a particular number of bars. The approximation depends on the choice of the number of terms, which is the number q of quadrature points at which the function to be integrated is evaluated. As q increases, the approximation improves: The estimates and standard errors get closer to the actual ML estimates and their standard errors. Be careful not to use too few quadrature points. Most software will pick a default value for q, often small such as q = 5. As a check you can then increase q above that value to make sure the estimates and standard errors have stabilized to the desired degree of precision (e.g., that they do not change in the first few significant digits). Gauss–Hermite quadrature is feasible when there is only a random intercept in the model or only a random intercept and a random slope. With complex random effect structure, other approaches are needed. Monte Carlo methods simulate in order to approximate the relevant integral. The Gauss–Hermite and Monte Carlo methods approximate the ML parameter estimates but converge to the ML estimates as they are applied more finely – for example, as the number of quadrature points increases for numerical integration. This is preferable to other approximate methods that are simpler but need not yield estimates near the ML estimates. Two such methods are penalized quasi-likelihood (PQL) and Laplace approximation. When true variance components are large, PQL can produce variance component estimates with substantial negative bias. Bias also occurs when the response distribution is far from normal (e.g., binary). The Laplace approximation replaces the function to be integrated by an approximation for which the integral has closed form. Another approach to fitting of GLMMs is Bayesian. With it, the distinction between fixed and random effects no longer occurs. A probability distribution (the prior distribution) is assumed for each effect of either type. Prior distributions are usually chosen to be very spread out. Then, the data have the primary influence in determining the estimates. 10.5.2 Inference for Model Parameters For GLMMs, inference about fixed effects proceeds in the usual way. For instance, likelihood-ratio tests can compare models when one model is the special case of the other in which one or more fixed effects equal 0. Inference about random effects, such as whether a variance component equals zero, is more complex. For example, the basketball free-throw shooting example in Section 10.2.2 used a model logit(πi) = ui + α that let the probability of success vary among players. The model logit(πi) = α in which the probability of success is the same for each player is the special case in which σ = 0 for the N(0,σ) distribution of {ui}. Since σ cannot be negative, the
318 RANDOM EFFECTS: GENERALIZED LINEAR MIXED MODELS simpler model falls on the boundary of the parameter space for the more complex model. When this happens, the usual likelihood-ratio chi-squared test for comparing models is not valid. Likewise, a Wald statistic such as ˆσ/SE does not have an approximate standard normal null distribution. (When σ = 0, because ˆσ 0 for a model containing a single random effect term. The null distribution has probability 1/2at0and1/2 following the shape of a chi-squared distribution with df = 1. The test statistic value of 0 occurs when ˆσ = 0, in which case the maximum of the likelihood function is identical under H0 and Ha. When ˆσ >0 and the observed test statistic equals t, the P -value for this test is half the right-tail probability above t for a chi-squared distribution with df = 1. For the basketball free-throw shooting example, the random effect has ˆσ = 0.42, with SE = 0.39. The likelihood-ratio test statistic comparing this model to the simpler model that assumes the same probability of success for each player equals 0.42. As usual, this equals double the difference between the maximized log likelihood values. The P -value is half the right-tail probability above 0.42 for a chi-squared distribution with df = 1, which is 0.26. It is plausible that all players have the same chance of success. However, the sample size was small, which is why an implausibly simplistic model seems adequate. PROBLEMS 10.1 Refer back to Table 8.10 from a recent General Social Survey that asked subjects whether they believe in heaven and whether they believe in hell. a. Fit model (10.3). If your software uses numerical integration, report ˆβ, ˆσ , and their standard errors for 2, 10, 100, 400, and 1000 quadrature points. Comment on convergence. b. Interpret ˆβ. c. Compare ˆβ and its SE for this approach to their values for the conditional ML approach of Section 8.2.3. 10.2 You plan to apply the matched-pairs model (10.3) to a data set for which yi1 is whether the subject agrees that abortion should be legal if the woman cannot afford the child (1 = yes, 0 = no), and yi2 is whether the subject opposes abortion if a woman wants it because she is unmarried (1 = yes, 0 = no). a. Indicate a way in which this model would probably be inappropriate. (Hint: Do you think these variables would have a positive, or negative, log odds ratio?) b. How could you reword the second question so the model would be more appropriate? 10.3 A dataset on pregnancy rates among girls in 13 north central Florida counties has information on the total in 2005 for each county i on Ti = number of births
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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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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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256 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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Page 367 and 368: Index of Examples abortion, opinion
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Page 373 and 374: 352 SUBJECT INDEX Exact inference (
Page 375 and 376: 354 SUBJECT INDEX Loglinear models
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368 BRIEF SOLUTIONS TO SOME ODD-NUM
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