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2.4.1 Example: Means for two groups For the data on chronic medical conditions the equation in the model E(Yjk) =θj; Yjk ∼ P oisson(θj),j =1, 2 can be written in the form of (2.13) with g as the identity function, (i.e., g(θj) =θj), ⎡ ⎢ y = ⎢ ⎣ Y1,1 Y1,2 . Y1,26 Y2,1 . Y2,23 ⎤ ⎥ , β = ⎥ ⎦ � θ1 θ2 � ⎡ 1 0 ⎤ and ⎢ 1 ⎢ . ⎢ . X = ⎢ 1 ⎢ 0 ⎢ ⎣ . 0 . . 0 1 . . ⎥ ⎦ 0 1 The top part of X picks out the terms θ1 corresponding to E(Y1k) and the bottom part picks out θ2 for E(Y2k). With this model the group means θ1 and θ2 can be estimated and compared. 2.4.2 Example: Simple linear regression for two groups The more general model for the data on birthweight and gestational age is E(Yjk) =µjk = αj + βjxjk; Yjk ∼ N(µjk,σ 2 ). This can be written in the form of (2.13) if g is the identity function, ⎡ ⎢ y = ⎢ ⎣ Y11 Y12 . Y1K Y21 . Y2K ⎤ ⎥ ⎡ ⎥ ⎢ ⎥ , β = ⎢ ⎣ ⎥ ⎦ α1 α2 β1 β2 ⎤ ⎥ ⎦ and ⎡ 1 ⎢ 1 ⎢ . ⎢ . X = ⎢ 1 ⎢ 0 ⎢ ⎣ . 0 0 . 0 1 . x11 x12 . x1K 0 . 0 0 . 0 x21 . ⎤ ⎥ ⎦ 0 1 0 x2K 2.4.3 Example: Alternative formulations for comparing the means of two groups There are several alternative ways of formulating the linear components for comparing means oftwo groups: Y11, ..., Y1K1and Y21, ..., Y2K2. (a) E(Y1k) =β1, and E(Y2k) =β2. � � β1 This is the version used in Example 2.4.1 above. In this case β = © 2002 by Chapman & Hall/CRC β2
and the rows of X are as follows Group 1 : Group 2 : � 1 0 � � 0 1 � . (b) E(Y1k) =µ + α1, and E(Y2k) =µ + α2. In this version µ represents the overall ⎡ mean and α1 and α2 are the group differences from µ. In this case β = ⎣ µ ⎤ α1 ⎦ and the rows of X are Group 1 : Group 2 : α2 � � 1 1 0 � � 1 0 1 . This formulation, however, has too many parameters as only two parameters can be estimated from the two sets of observations. modification or constraint is needed. (c) E(Y1k) =µ and E(Y2k) =µ + α. Therefore some Here Group 1 is treated as the reference group� and�α represents the ad- µ ditional effect ofGroup 2. For this version β = and the rows of X α are Group 1 Group 2 : : � 1 � 1 � 0 � 1 . This is an example of corner point parameterization in which group effects are defined as differences from a reference category called the ‘corner point’. (d) E(Y1k) =µ + α, and E(Y2k) =µ − α. This version treats the two groups symmetrically; µ is the overall average effect and α represents the group differences. This is an example ofa sumto-zero constraint because [E(Y1k) − µ]+[E(Y2k) − µ] =α +(−α) =0. � µ In this case β = α � and the rows of X are Group 1 Group 2 : : � 1 � 1 � 1 � −1 . 2.4.4 Example: Ordinal explanatory variables Let Yjk denote a continuous measurement ofquality oflife. Data are collected for three groups of patients with mild, moderate or severe disease. The groups can be described by levels ofan ordinal variable. This can be specified by © 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 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: 2.4Notation and coding for explanat
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
Page 63 and 64: 4 Estimation 4.1 Introduction This
Page 65 and 66: x (m - 1) x (m) Figure 4.3 Newton-R
Page 67 and 68: Table 4.2 Details of Newton-Raphson
Page 69 and 70: y differentiating (4.13) and substi
Page 71 and 72: Table 4.3 Data for Poisson regressi
Page 73 and 74: Table 4.4 Successive approximations
Page 75 and 76: 5 Inference 5.1 Introduction The tw
Page 77 and 78: consistent with the general result
Page 79 and 80: approximated by its expected value
Page 81 and 82: Hence E � (b − β)(b − β) T
Page 83 and 84: For Yi’s with other distributions
Page 85 and 86: 8, D has a chi-squared distribution
Page 87 and 88: Consider the null hypothesis ⎡
Page 89 and 90: (a) Find the Wald statistic (�π
Page 91 and 92: 6.2.2 Least squares estimation IfE(
Page 93 and 94: 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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