an introduction to generalized linear models - GDM@FUDAN ...

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The next three chapters provide the theoretical background. Chapter 3 is about the exponential family of distributions, which includes the Normal, Poisson and binomial distributions. It also covers generalized linear models (as defined by Nelder and Wedderburn, 1972). Linear regression and many other models are special cases ofgeneralized linear models. In Chapter 4 methods ofestimation and model fitting are described. Chapter 5 outlines methods of statistical inference for generalized linear models. Most ofthese are based on how well a model describes the set ofdata. For example, hypothesis testing is carried out by first specifying alternative models (one corresponding to the null hypothesis and the other to a more general hypothesis). Then test statistics are calculated which measure the ‘goodness offit’ ofeach model and these are compared. Typically the model corresponding to the null hypothesis is simpler, so ifit fits the data about as well as a more complex model it is usually preferred on the grounds of parsimony (i.e., we retain the null hypothesis). Chapter 6 is about multiple linear regression and analysis of variance (ANOVA). Regression is the standard method for relating a continuous response variable to several continuous explanatory (or predictor) variables. ANOVA is used for a continuous response variable and categorical or qualitative explanatory variables (factors). Analysis of covariance (ANCOVA) is used when at least one ofthe explanatory variables is continuous. Nowadays it is common to use the same computational tools for all such situations. The terms multiple regression or general linear model are used to cover the range ofmethods for analyzing one continuous response variable and multiple explanatory variables. Chapter 7 is about methods for analyzing binary response data. The most common one is logistic regression which is used to model relationships between the response variable and several explanatory variables which may be categorical or continuous. Methods for relating the response to a single continuous variable, the dose, are also considered; these include probit analysis which was originally developed for analyzing dose-response data from bioassays. Logistic regression has been generalized in recent years to include responses with more than two nominal categories (nominal, multinomial, polytomous or polychotomous logistic regression) or ordinal categories (ordinal logistic regression). These new methods are discussed in Chapter 8. Chapter 9 concerns count data. The counts may be frequencies displayed in a contingency table or numbers ofevents, such as traffic accidents, which need to be analyzed in relation to some ‘exposure’ variable such as the number ofmotor vehicles registered or the distances travelled by the drivers. Modelling methods are based on assuming that the distribution ofcounts can be described by the Poisson distribution, at least approximately. These methods include Poisson regression and log-linear models. Survival analysis is the usual term for methods of analyzing failure time data. The parametric methods described in Chapter 10 fit into the framework © 2002 by Chapman & Hall/CRC
ofgeneralized linear models although the probability distribution assumed for the failure times may not belong to the exponential family. Generalized linear models have been extended to situations where the responses are correlated rather than independent random variables. This may occur, for instance, if they are repeated measurements on the same subject or measurements on a group of related subjects obtained, for example, from clustered sampling. The method of generalized estimating equations (GEE’s) has been developed for analyzing such data using techniques analogous to those for generalized linear models. This method is outlined in Chapter 11 together with a different approach to correlated data, namely multilevel modelling. Further examples ofgeneralized linear models are discussed in the books by McCullagh and Nelder (1989), Aitkin et al. (1989) and Healy (1988). Also there are many books about specific generalized linear models such as Hosmer and Lemeshow (2000), Agresti (1990, 1996), Collett (1991, 1994), Diggle, Liang and Zeger (1994), and Goldstein (1995). 1.3 Notation Generally we follow the convention of denoting random variables by upper case italic letters and observed values by the corresponding lower case letters. For example, the observations y1,y2, ..., yn are regarded as realizations ofthe random variables Y1,Y2, ..., Yn. Greek letters are used to denote parameters and the corresponding lower case roman letters are used to denote estimators and estimates; occasionally the symbol � is used for estimators or estimates. For example, the parameter β is estimated by � β or b. Sometimes these conventions are not strictly adhered to, either to avoid excessive notation in cases where the meaning should be apparent from the context, or when there is a strong tradition ofalternative notation (e.g., e or ε for random error terms). Vectors and matrices, whether random or not, are denoted by bold face lower and upper case letters, respectively. Thus, y represents a vector ofobservations ⎡ ⎢ ⎣ or a vector ofrandom variables ⎡ ⎢ ⎣ y1 . yn Y1 ... Yn ⎤ ⎥ ⎦ ⎤ ⎥ ⎦ , β denotes a vector ofparameters and X is a matrix. The superscript T is used for a matrix transpose or when a column vector is written as a row, e.g., y =[Y1, ..., Yn] T . © 2002 by Chapman & Hall/CRC
Page 1 and 2: CHAPMAN & HALL/CRC Texts in Statist
Page 3 and 4: AN INTRODUCTION TO GENERALIZED LINE
Page 5 and 6: Preface Contents 1 Introduction 1.1
Page 7 and 8: 10 Survival Analysis 10.1 Introduct
Page 9 and 10: 1 Introduction 1.1 Background This
Page 11: Table 1.1 Major methods of statisti
Page 15 and 16: 3. Let Y1, ..., Yn denote Normally
Page 17 and 18: divided by its degrees offreedom, F
Page 19 and 20: l(θ; y) = log L(θ; y), since the
Page 21 and 22: (i.e., the matrix ofsecond derivati
Page 23 and 24: Table 1.3 Successive approximations
Page 25 and 26: 2 Model Fitting 2.1 Introduction Th
Page 27 and 28: IfH1is true, then the log-likelihoo
Page 29 and 30: estimated in order to calculate to
Page 31 and 32: where xjk is the gestational age of
Page 33 and 34: Table 2.4 Summary of data on birthw
Page 35 and 36: Residuals Residuals Percent 2 1 0 -
Page 37 and 38: sampling distributions ofthe corres
Page 39 and 40: is categorical how many categories
Page 41 and 42: Cox and Snell, 1968; Prigibon, 1981
Page 43 and 44: 2.4Notation and coding for explanat
Page 45 and 46: and the rows of X are as follows Gr
Page 47 and 48: (a) Conduct an exploratory analysis
Page 49 and 50: (d) List the assumptions made for (
Page 51 and 52: putation involving numerical optimi
Page 53 and 54: worthwhile trying to identify a tra
Page 55 and 56: We also need expressions for the ex
Page 57 and 58: 1. Response variables Y1,... ,YN wh
Page 59 and 60: Table 3.2 Numbers of deaths from co
Page 61 and 62: 3.4 Use results (3.9) and (3.12) to
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4 Estimation 4.1 Introduction This
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x (m - 1) x (m) Figure 4.3 Newton-R
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Table 4.2 Details of Newton-Raphson
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y differentiating (4.13) and substi
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Table 4.3 Data for Poisson regressi
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Table 4.4 Successive approximations
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5 Inference 5.1 Introduction The tw
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consistent with the general result
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approximated by its expected value
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Hence E � (b − β)(b − β) T
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For Yi’s with other distributions
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8, D has a chi-squared distribution
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Consider the null hypothesis ⎡
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(a) Find the Wald statistic (�π
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6.2.2 Least squares estimation IfE(
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Table 6.2 Multiple hypothesis tests
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the minimum value ofthe sum ofsquar
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and ⎡ X T ⎢ X = ⎢ ⎣ 20 923
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or ‘worst possible’ value of S.
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Table 6.6 Dried weights yi of plant
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The first row (or column) ofthe (J
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so For the plant weight data and Y
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4. The model formed by omitting eff
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Finally for the model with only a m
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6.5 Analysis of covariance Analysis
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For the reduced model (6.14) �
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6.7 Exercises 6.1 Table 6.15 shows
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Table 6.17 Cholesterol (CHOL), age
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6.8 Table 6.20 shows the data from
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Table 7.1 Frequencies for N binomia
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x x Figure 7.2 Normal distribution:
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and log(1 − πi) =− log [1 + ex
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Table 7.4 Comparison of observed nu
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Proportion germinated 0.7 0.6 0.5 4
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which is asymptotically equivalent
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From equation (7.5), �m residuals
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Proportion with symptoms of senilit
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Table 7.10 Hosmer-Lemeshow test for
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(a) Are the proportions of graduate
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category 2, and so on, then let ⎡
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(iii) Likelihood ratio chi-squared
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Women: preference for air condition
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Table 8.3 Results from fitting the
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π 1 π 2 π 3 π 4 C1 C2 C3 Figure
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The adjacent category logit model i
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Table 8.4 Results of proportional o
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(c) Use a Wald statistic to test th
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variables. The study design may mea
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series expansion given in Section 7
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for smokers and zero for non-smoker
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column totals. It appears that Hutc
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similar to the ulcer patients with
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� k θ.k =1. This hypothesis can
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9.6 Inference for log-linear models
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Table 9.10 Log-linear models for th
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9.9 Exercises 9.1 Let Y1, ..., YN b
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Table 9.14 Contingency table with 2
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1 2 D 3 A D 4 L 5 D D TO TL TC time
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ofthe distribution. The median surv
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10.2.3 Weibull distribution Another
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Table 10.1 Remission times of leuke
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log H(y) 1 0 -1 -2 0 1 2 3 log (y)
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As there are r uncensored observati
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small number ofcategorical explanat
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Cox Snell residuals 3 2 1 0 Devianc
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is sometimes used for modelling sur
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11 Clustered and Longitudinal Data
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data from the stroke example in Sec
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score 100 80 60 40 20 0 2 4 6 8 wee
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Table11.3 Results of naive analyses
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Table 11.6 Analysis of variance of
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1. All the off-diagonal elements ar
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These are also called the quasi-sco
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andom effect.This is an example ofa
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Table 11.7 Comparison of analyses o
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Table 11.8 Measurements of left ven
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Table 11.9 Numbers of ears clear of
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References Aitkin, M., Anderson, D.
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Diggle, P. J., Liang, K.-Y. and Zeg
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Roberts, G., Martyn, A. L., Dobson,
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