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Table 2.3 Birthweight and gestational age for boys and girls. Boys Girls Age Birthweight Age Birthweight 40 2968 40 3317 38 2795 36 2729 40 3163 40 2935 35 2925 38 2754 36 2625 42 3210 37 2847 39 2817 41 3292 40 3126 40 3473 37 2539 37 2628 36 2412 38 3176 38 2991 40 3421 39 2875 38 2975 40 3231 Means 38.33 3024.00 38.75 2911.33 Birth weight 3500 3000 2500 34 36 38 40 Gestational age Figure 2.2 Birthweight plotted against gestational age for boys (open circles) and girls (solid circles); data in Table 2.3. © 2002 by Chapman & Hall/CRC 42
where xjk is the gestational age ofthe kth baby in group j. The intercept parameters α1 and α2 are likely to differ because, on average, the boys were heavier than the girls. The slope parameters β1 and β2 represent the average increases in birthweight for each additional week of gestational age. The question ofinterest can be formulated in terms oftesting the null hypothesis H0 : β1 = β2 = β (that is, the growth rates are equal and so the lines are parallel), against the alternative hypothesis H1 : β1 �= β2. We can test H0 against H1 by fitting two models E(Yjk) =µjk = αj + βxjk; Yjk ∼ N(µjk,σ 2 ), (2.6) E(Yjk) =µjk = αj + βjxjk; Yjk ∼ N(µjk,σ 2 ). (2.7) The probability density function for Yjk is f(yjk; µjk) = 1 1 √ exp[− 2πσ2 2σ2 (yjk − µjk) 2 ]. We begin by fitting the more general model (2.7). The log-likelihood function is l1(α1,α2,β1,β2; y) = J� j=1 k=1 K� [− 1 2 log(2πσ2 ) − 1 = − 1 2 JK log(2πσ2 ) − 1 2σ 2 2σ 2 (yjk − µjk) 2 ] J� j=1 k=1 K� (yjk − αj − βjxjk) 2 where J = 2 and K = 12 in this case. When obtaining maximum likelihood estimates of α1,α2,β1 and β2 we treat the parameter σ 2 as a known constant, or nuisance parameter, and we do not estimate it. The maximum likelihood estimates are the solutions ofthe simultaneous equations ∂l1 ∂αj ∂l1 ∂βj = 1 σ 2 = 1 σ 2 � (yjk − αj − βjxjk) =0, k � xjk(yjk − αj − βjxjk) =0, (2.8) k where j = 1 or 2. An alternative to maximum likelihood estimation is least squares estimation. For model (2.7), this involves minimizing the expression S1 = J� K� j=1 k=1 © 2002 by Chapman & Hall/CRC (yjk − µjk) 2 = J� j=1 k=1 K� (yjk − αj − βjxjk) 2 . (2.9)
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: estimated in order to calculate to
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
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
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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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