Introduction to Categorical Data Analysis

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3 3 26.5 1.97 1 3 3 24.5 2.20 1 3 3 28.5 3.00 1 2 3 28.2 2.87 1 2 3 24.5 1.60 1 2 3 27.5 2.55 1 2 2 24.7 2.55 4 2 1 25.2 2.00 1 3 3 27.3 2.90 1 2 3 26.3 2.40 1 2 3 29.0 3.10 1 2 3 25.3 1.90 2 2 3 26.5 2.30 4 2 3 27.8 3.25 3 2 3 27.0 2.50 6 3 3 25.7 2.10 0 2 3 25.0 2.10 2 2 3 31.9 3.33 2 4 3 23.7 1.80 0 4 3 29.3 3.23 12 3 3 22.0 1.40 0 2 3 25.0 2.40 5 3 3 27.0 2.50 6 3 3 23.8 1.80 6 1 1 30.2 3.28 2 3 3 26.2 2.22 0 2 3 24.2 1.65 2 2 3 27.4 2.90 3 2 2 25.4 2.30 0 3 3 28.4 3.20 3 4 3 22.5 1.47 4 2 3 26.2 2.02 2 2 1 24.9 2.30 6 1 2 24.5 1.95 6 2 3 25.1 1.80 0 2 1 28.0 2.90 4 4 3 25.8 2.25 10 2 3 27.9 3.05 7 2 3 24.9 2.20 0 2 1 28.4 3.10 5 3 3 27.2 2.40 5 2 2 25.0 2.25 6 2 3 27.5 2.63 6 2 1 33.5 5.20 7 2 3 30.5 3.32 3 3 3 29.0 2.92 3 2 1 24.3 2.00 0 2 3 25.8 2.40 0 4 3 25.0 2.10 8 2 1 31.7 3.73 4 2 3 29.5 3.02 4 3 3 24.0 1.90 10 2 3 30.0 3.00 9 2 3 27.6 2.85 4 2 3 26.2 2.30 0 2 1 23.1 2.00 0 2 1 22.9 1.60 0 4 3 24.5 1.90 0 2 3 24.7 1.95 4 2 3 28.3 3.20 0 2 3 23.9 1.85 2 3 3 23.8 1.80 0 3 2 29.8 3.50 4 2 3 26.5 2.35 4 2 3 26.0 2.28 3 2 3 28.2 3.05 8 4 3 25.7 2.15 0 2 3 26.5 2.75 7 2 3 25.8 2.20 0 3 3 24.1 1.80 0 3 3 26.2 2.17 2 3 3 26.1 2.75 3 3 3 29.0 3.28 4 1 1 28.0 2.62 0 4 3 27.0 2.63 0 2 2 24.5 2.00 0 1 1 26.5 2.35 0 Note: C= Color (1 = light medium, 2 = medium, 3 = dark medium, 4 = dark), S = spine condition (1 = both good, 2 = one worn or broken, 3 = both worn or broken), W = carapace width (cm), Wt = weight (kg), Sa = number of satellites. Source: Data provided by Dr. Jane Brockmann, Zoology Department, University of Florida; study described in Ethology, 102: 1–21, 1996. 77
78 GENERALIZED LINEAR MODELS Figure 3.4. Number of satellites by width of female crab. Most software has more sophisticated ways of smoothing the data, revealing the trend without grouping the width values. Smoothing methods based on generalized additive models do this by providing even more general structural form than GLMs. They find possibly complex functions of the explanatory variables that serve as the best predictors of a certain type. Figure 3.5 also shows a curve based on smoothing the data using this method. The sample means and the smoothed curve both show a strong increasing trend. (The means tend to fall above the curve, since the response counts in a category tend to be skewed to the right. The smoothed curve is less susceptible to outlying observations.) The trend seems approximately linear, and we next discuss models for which the mean or the log of the mean is linear in width. Let μ denote the expected number of satellites for a female crab, and let x denote her width. From GLM software, the ML fit of the Poisson loglinear model (3.5) is log ˆμ =ˆα + ˆβx =−3.305 + 0.164x The effect ˆβ = 0.164 of width has SE = 0.020. Since ˆβ >0, width has a positive estimated effect on the number of satellites. The model fit yields an estimated mean number of satellites ˆμ, afitted value, at any width. For instance, from equation (3.6), the fitted value at the mean width of x = 26.3 is ˆμ = exp( ˆα + ˆβx) = exp[−3.305 + 0.164(26.3)] =e 1.01 = 2.7
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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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128 LOGISTIC REGRESSION 0 = no) and
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130 LOGISTIC REGRESSION Table 4.16.
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132 LOGISTIC REGRESSION b. In Table
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134 LOGISTIC REGRESSION Table 4.20.
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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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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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