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

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5.3 Taylor series approximations 5.4 Sampling distribution for maximum likelihood estimators 5.5 Log-likelihood ratio statistic 5.6 Sampling distribution for the deviance 5.7 Hypothesis testing 5.8 Exercises 6 Normal Linear Models 6.1 Introduction 6.2 Basic results 6.3 Multiple linear regression 6.4 Analysis ofvariance 6.5 Analysis ofcovariance 6.6 General linear models 6.7 Exercises 7 Binary Variables and Logistic Regression 7.1 Probability distributions 7.2 Generalized linear models 7.3 Dose response models 7.4 General logistic regression model 7.5 Goodness offit statistics 7.6 Residuals 7.7 Other diagnostics 7.8 Example: Senility and WAIS 7.9 Exercises 8 Nominal and Ordinal Logistic Regression 8.1 Introduction 8.2 Multinomial distribution 8.3 Nominal logistic regression 8.4 Ordinal logistic regression 8.5 General comments 8.6 Exercises 9 Count Data, Poisson Regression and Log-Linear Models 9.1 Introduction 9.2 Poisson regression 9.3 Examples ofcontingency tables 9.4 Probability models for contingency tables 9.5 Log-linear models 9.6 Inference for log-linear models 9.7 Numerical examples 9.8 Remarks 9.9 Exercises © 2002 by Chapman & Hall/CRC
10 Survival Analysis 10.1 Introduction 10.2 Survivor functions and hazard functions 10.3 Empirical survivor function 10.4 Estimation 10.5 Inference 10.6 Model checking 10.7 Example: remission times 10.8 Exercises 11 Clustered and Longitudinal Data 11.1 Introduction 11.2 Example: Recovery from stroke 11.3 Repeated measures models for Normal data 11.4 Repeated measures models for non-Normal data 11.5 Multilevel models 11.6 Stroke example continued 11.7 Comments 11.8 Exercises Software References © 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: Preface Contents 1 Introduction 1.1
Page 9 and 10: 1 Introduction 1.1 Background This
Page 11 and 12: Table 1.1 Major methods of statisti
Page 13 and 14: ofgeneralized linear models althoug
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
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1. Response variables Y1,... ,YN wh
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Table 3.2 Numbers of deaths from co
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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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