Introduction to Categorical Data Analysis

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4.1 INTERPRETING THE LOGISTIC REGRESSION MODEL 105 width; that is, there is a 64% increase. To illustrate, the mean width value of x = 26.3 has ˆπ(x) = 0.674, and odds = 0.674/0.326 = 2.07. At x = 27.3 = 26.3 + 1.0, you can check that ˆπ(x) = 0.773 and odds = 0.773/0.227 = 3.40. However, this is a 64% increase; that is, 3.40 = 2.07(1.64). 4.1.5 Logistic Regression with Retrospective Studies Another property of logistic regression relates to situations in which the explanatory variable X rather than the response variable Y is random. This occurs with retrospective sampling designs. Sometimes such designs are used because one of the response categories occurs rarely, and a prospective study might have too few cases to enable one to estimate effects of predictors well. For a given sample size, effect estimates have smaller SEs when the number of outcomes of the two types are similar than when they are very different. Most commonly, retrospective designs are used with biomedical case-control studies (Section 2.3.5). For samples of subjects having Y = 1 (cases) and having Y = 0 (controls), the value of X is observed. Evidence exists of an association between X and Y if the distribution of X values differs between cases and controls. For case– control studies, it was noted in Section 2.3.5 that it is possible to estimate odds ratios but not other summary measures. Logistic regression parameters refer to odds and odds ratios. One can fit logistic regression models with data from case–control studies and estimate effects of explanatory variables. The intercept term α in the model is not meaningful, because it relates to the relative numbers of outcomes of y = 1 and y = 0. We do not estimate this, because the sample frequencies for y = 1 and y = 0 are fixed by the nature of the case–control study. With case–control studies, it is not possible to estimate effects in binary models with link functions other than the logit. Unlike the odds ratio, the effect for the conditional distribution of X given Y does not then equal that for Y given X. This provides an important advantage of the logit link over links such as the probit. It is a major reason why logistic regression surpasses other models in popularity for biomedical studies. Many case–control studies employ matching. Each case is matched with one or more control subjects. The controls are like the case on key characteristics such as age. The model and subsequent analysis should take the matching into account. Section 8.2.4 discusses logistic regression for matched case–control studies. 4.1.6 Normally Distributed X Implies Logistic Regression for Y Regardless of the sampling mechanism, the logistic regression model may or may not describe a relationship well. In one special case, it does necessarily hold. Suppose the distribution of X for subjects having Y = 1 is normal N(μ1,σ), and suppose the distribution of X for subjects having Y = 0 is normal N(μ0,σ); that is, with different mean but the same standard deviation. Then, a Bayes theorem calculation converting from the distribution of X given Y = y to the distribution of Y given X = x shows
106 LOGISTIC REGRESSION that P(Y = 1|x) satisfies the logistic regression curve. For that curve, the effect of x is β = (μ1 − μ0)/σ 2 . In particular, β has the same sign as μ1 − μ0. For example, if those with y = 1 tend to have higher values of x, then β>0. For example, consider Y = heart disease (1 = yes, 0 = no) and X = cholesterol level. Suppose cholesterol levels have approximately a N(μ0 = 160,σ = 50) distribution for those without heart disease and a N(μ1 = 260,σ = 50) distribution for those with heart disease. Then, the probability of having heart disease satisfies the logistic regression function (4.2) with predictor x and β = (260 − 160)/50 2 = 0.04. If the distributions of X are bell-shaped but with highly different spreads, then a logistic model containing also a quadratic term (i.e., both x and x 2 ) often fits well. In that case, the relationship is not monotone. Instead, P(Y = 1) increases and then decreases, or the reverse (see Exercise 4.7). 4.2 INFERENCE FOR LOGISTIC REGRESSION We have studied how logistic regression helps describe the effects of a predictor on a binary response variable. We next present statistical inference for the model parameters, to help judge the significance and size of the effects. 4.2.1 Binary Data can be Grouped or Ungrouped Widely available software reports the ML estimates of parameters and their standard errors. Sometimes sets of observations have the same values of predictor variables, such as when explanatory variables are discrete. Then, ML model fitting can treat the observations as the binomial counts of successes out of certain sample sizes, at the various combinations of values of the predictors. We will refer to this case as grouped binary data and the case in which each observation is a single binary outcome as ungrouped binary data. In Table 3.1 on snoring and heart disease in the previous chapter, 254 subjects reported snoring every night, of whom 30 had heart disease. If the data file has grouped binary data, a line in the data file reports these data as 30 cases of heart disease out of a sample size of 254. If the data file has ungrouped binary data, each line in the data file refers to a separate subject, so 30 lines contain a 1 for heart disease and 224 lines contain a 0 for heart disease. The ML estimates and SE values are the same for either type of data file. When at least one explanatory variable is continuous, binary data are naturally ungrouped. An example is the data that Table 3.2 reports for the horseshoe crabs. 4.2.2 Confidence Intervals for Effects A large-sample Wald confidence interval for the parameter β in the logistic regression model, logit[π(x)]=α + βx,is ˆβ ± zα/2(SE)
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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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Page 121 and 122: 100 LOGISTIC REGRESSION The logisti
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156 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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