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Table 2.2 Observed values and standardized residuals for the data on chronic medical conditions (Table 2.1), with estimates obtained from models (2.1) and (2.2). Value Frequency Standardized residuals Standardized residuals of Y from (2.1); from (2.2); �θ =1.184 �θ1 =1.423 and �θ2 =0.913 Town 0 6 -1.088 -1.193 1 10 -0.169 -0.355 2 4 0.750 0.484 3 5 1.669 1.322 4 1 2.589 2.160 Country 0 9 -1.088 -0.956 1 8 -0.169 0.091 2 5 0.750 1.138 3 1 1.669 2.184 Residuals for model (2.1) Residuals for model (2.2) town country town country -1 Figure 2.1 Plots of residuals for models (2.1) and (2.2) for the data in Table 2.2 on chronic medical conditions. © 2002 by Chapman & Hall/CRC 0 1 2
estimated in order to calculate to fitted values � θi (for example, see Agresti, 1990, page 479). Expression (2.5) is, in fact, the usual chi-squared goodness of fit statistic for count data which is often written as X 2 = � (oi − ei) 2 ∼ χ 2 (m) ei where oi denotes the observed frequency and ei denotes the corresponding expected frequency. In this case oi = Yi , ei = � θi and � r 2 i = X2 . For the data on chronic medical conditions, for model (2.1) � r 2 i =6× (−1.088) 2 +10× (−0.169) 2 + ...+1× 1.669 2 =46.759. This value is consistent with � r2 i being an observation from the central chisquared distribution with m =23+26−1 = 48 degrees offreedom. (Recall from Section 1.4.2, that if X2 ∼ χ2 (m) then E(X2 )=m and notice that the calculated value X2 = � r2 i =46.759 is near the expected value of48.) Similarly, for model (2.2) � 2 ri =6× (−1.193) 2 + ...+1× 2.184 2 =43.659 which is consistent with the central chi-squared distribution with m =49−2 = 47 degrees offreedom. The difference between the values of � r2 i from models (2.1) and (2.2) is small: 46.759−43.659 = 3.10. This suggests that model (2.2) with two parameters, may not describe the data much better than the simpler model (2.1). Ifthis is so, then the data provide evidence supporting the null hypothesis H0 : θ1 = θ2. More formal testing of the hypothesis is discussed in Chapter 4. The next example illustrates steps ofthe model fitting process with continuous data. 2.2.2 Birthweight and gestational age The data in Table 2.3 are the birthweights (in grams) and estimated gestational ages (in weeks) of12 male and female babies born in a certain hospital. The mean ages are almost the same for both sexes but the mean birthweight for boys is higher than the mean birthweight for girls. The data are shown in the scatter plot in Figure 2.2. There is a linear trend ofbirthweight increasing with gestational age and the girls tend to weigh less than the boys ofthe same gestational age. The question ofinterest is whether the rate ofincrease ofbirthweight with gestational age is the same for boys and girls. Let Yjk be a random variable representing the birthweight ofthe kth baby in group j where j = 1 for boys and j = 2 for girls and k =1,... ,12. Suppose that the Yjk’s are all independent and are Normally distributed with means µjk =E(Yjk), which may differ among babies, and variance σ 2 which is the same for all of them. A fairly general model relating birthweight to gestational age is © 2002 by Chapman & Hall/CRC E(Yjk) =µjk = αj + βjxjk
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: IfH1is true, then the log-likelihoo
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
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
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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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