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Percent Residuals Residuals 99 90 50 10 1 2 1 0 -1 2 1 0 -1 Model (2.7) Model (2.7) 34 36 38 40 42 Model (2.7) -2 2600 2800 3000 3200 3400 Fittted values -1 Gestational age 0 Residuals Figure 2.4 Plots of standardized residuals for Model (2.7) for the data on birthweight and gestational age (Table 2.3); for the top and middle plots, open circles correspond to data from boys and solid circles correspond to data from girls. © 2002 by Chapman & Hall/CRC 1 2
sampling distributions ofthe corresponding random variables and �S1 = �S0 = J� j=1 k=1 J� j=1 k=1 It can be shown (see Exercise 2.3) that �S1 = J� j=1 k=1 − K� (Yjk − aj − bjxjk) 2 K� (Yjk − aj − bxjk) 2 . K� [Yjk − (αj + βjxjk)] 2 − K J� (bj − βj) 2 K� ( x 2 jk − Kx 2 j) j=1 k=1 J� (Y j − αj − βjxj) 2 and that the random variables Yjk, Y j and bj are all independent and have the following distributions: Yjk ∼ N(αj + βjxjk,σ 2 ), j=1 Y j ∼ N(αj + βjxj,σ 2 /K), bj ∼ N(βj,σ 2 K� /( x 2 jk − Kx 2 j)). Therefore � S1/σ2 is a linear combination ofsums ofsquares ofrandom variables with Normal distributions. In general, there are JK random variables (Yjk − αj − βjxjk) 2 /σ2 , J random variables (Y j − αj − βjxj) 2K/σ2 and J random variables (bj − βj) 2 ( � k x2 jk − Kx2j)/σ2 . They are all independent and each has the χ2 (1) distribution. From the properties ofthe chi-squared distribution in Section 1.5, it follows that � S1/σ2 ∼ χ2 (JK − 2J). Similarly, ifH0 is correct then � S0/σ2 ∼ χ2 [JK − (J + 1)]. In this example J =2so �S1/σ2 ∼ χ2 (2K − 4) and � S0/σ2 ∼ χ2 (2K − 3). In each case the value for the degrees offreedom is the number ofobservations minus the number of parameters estimated. If β1 and β2 are not equal (corresponding to H1), then � S0/σ2 will have a non-central chi-squared distribution with JK − (J + 1) degrees offreedom. On the other hand, provided that model (2.7) describes the data well, � S1/σ2 will have a central chi-squared distribution with JK − 2J degrees offreedom. The statistic � S0 − � S1 represents the improvement in fit of(2.7) compared to (2.6). IfH0 is correct, then 1 σ2 ( � S0 − � S1) ∼ χ 2 (J − 1). IfH0 is not correct then ( � S0 − � S1)/σ 2 has a non-central chi-squared distribu- © 2002 by Chapman & Hall/CRC k=1
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: Residuals Residuals Percent 2 1 0 -
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
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
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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
Page 221:
Roberts, G., Martyn, A. L., Dobson,
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