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CONFOUNDING BETWEEN DISEASE RISK FACTORS 401The weights d i are proportional to the precision of the logarithm of the odds ratiounder the null association hypothesis. For the data in Table 10.1 the estimator (10.1)for the overall odds ratio is 1.57, as compared to 0.95 and 1.87 in the two agegroups.A stratified analysis may become impractical if the data are dispersed over manysubcategories or multiple confounders, or if the impact of the risk factor is distorted bycategorisation, and multivariate methods such as logistic regression are the only practicalapproach. For example, logit regression methods are applicable to pooling oddsratios over a series of 2 2 tables from a case-control study, even if case/control statusdoes not result from random sampling of a defined population (Selvin, 1998, Chapter 4).In the case of <strong>Bayesian</strong> estimation, credible intervals on the resulting effect estimateswill be obtained without requiring Normality assumptions. This is especially importantfor small cell counts {a, b, c, d}, including the case when a cell count is zero such thatthe estimate ad/(bc) is undefined.Multivariate methods may be applied when there is intentional matching of a casewith one or more controls on the confounders. The precision of a risk estimate (e.g.odds ratio or relative risk) will be increased if the control to case ratio M exceeds 1. Aswell as providing control for confounding per se, matching on risk factors with anestablished effect (e.g. age and cardiovascular outcomes, or smoking and lung cancer)may enhance the power of observational studies to detect impacts of risk factors of asyet uncertain effect (Sturmer and Brenner, 2000). Matched studies with a dichotomousoutcome may be handled by conditional logistic regression, conditioning on the observedcovariates in each matched set (each matched set of case and controls becomes astratum with its own intercept). For 1:1 matching, this reduces to the standard logisticregression.Example 10.1 Alcohol consumption and oesophageal cancer with age stratification Onepossible analysis of studies such as that in Table 10.1 is provided by a logit or log-linearmodel for the predictor combinations produced by the confounder and the exposure. Forinstance, in Table 10.1 the assumption of a common odds ratio over the two age groupscorresponds to a model with main effects in age and blood pressure only, but without aninteraction between them.This example follows Breslow and Day (1980) and Zelterman (1999) in considering acase control study of oesophageal cancer (Y) in relation to alcohol consumption (X ),where age (Z) is a confounding factor. Table 10.2 shows the age banded and all agesstudy data.The unstandardised estimate of the overall odds ratio (from the all ages sub-table)96 666=(109 104) ˆ 5:64However, the association appears to vary by age (the corresponding estimate for the65±74 group being only 2.59). One might apply the Mantel±Haenszel procedure toobtain an aggregate effect pooling over the age specific odds ratios, though this iscomplicated by undefined odds ratios (from classical procedures) in the lowest andhighest age groups.An alternative is a log-linear model based on Poisson sampling for the frequenciesf YXZ , with case-control status being denoted Y (ˆ1 for control and 2 for case). A loglinear model (Model A) corresponding to a common odds ratio over the six age groupsis then specified as
402 MODELLING AND ESTABLISHING CAUSAL RELATIONSTable 10.2Case control data on oesophageal cancerAnnual alcohol consumptionAge group Over 80 g Under 80 g25±34 Case 1 0Control 9 10635±44 Case 4 5Control 26 16445±54 Case 25 21Control 29 13855±64 Case 42 34Control 27 13965±74 Case 19 36Control 18 8875 Case 5 8Control 0 31All ages Case 96 104Control 109 666f YXZ Poi(m YXZ )log (m YXZ ) ˆ a b Y g X d Z e YX k YZ Z XZand the common odds ratio across sub-tables is estimated as f ˆ exp (e YX ). Fairlydiffuse N(0, 1000) priors are adopted for all the effects in this model. A two chain runwith null starting values in one chain, and values based on a trial run in the other, showsconvergence from 5000 iterations: the scale reduction factors for b 2 only settle down towithin [0.9, 1.1] after then.The <strong>Bayesian</strong> estimation has the benefit of providing a full distributional profile forf; see Figure 10.1 with the positive skew in f apparent. Tests on the coefficient (e.g. the0.40.350.30.250.20.150.10.0504.5 4.7 4.9 5.1 5.3 5.5OR5.7 5.9 6.1 6.3 6.5Figure 10.1Posterior density of common odds ratio
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Applied Bayesian ModellingApplied B
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Applied Bayesian ModellingPETER CON
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ContentsPrefacexiChapter 1The Basis
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CONTENTSvii5.3.2 INAR models for co
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CONTENTSix9.4.2 Gamma process prior
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xiiPREFACEoptions (one of the benef
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2 BAYESIAN MODEL ESTIMATION VIA REP
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10 BAYESIAN MODEL ESTIMATION VIA RE
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32 HIERARCHICAL MIXTURE MODELSprinc
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34 HIERARCHICAL MIXTURE MODELS…P(
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36 HIERARCHICAL MIXTURE MODELS^m( y
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38 HIERARCHICAL MIXTURE MODELSent e
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40 HIERARCHICAL MIXTURE MODELSratio
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42 HIERARCHICAL MIXTURE MODELSence
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44 HIERARCHICAL MIXTURE MODELS^t 2
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46 HIERARCHICAL MIXTURE MODELSTable
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50 HIERARCHICAL MIXTURE MODELSwith
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52 HIERARCHICAL MIXTURE MODELSWhile
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56 HIERARCHICAL MIXTURE MODELSY ij
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58 HIERARCHICAL MIXTURE MODELSA thr
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60 HIERARCHICAL MIXTURE MODELSremai
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62 HIERARCHICAL MIXTURE MODELSin te
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64 HIERARCHICAL MIXTURE MODELSwheth
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66 HIERARCHICAL MIXTURE MODELSUsing
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68 HIERARCHICAL MIXTURE MODELS2.5.1
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70 HIERARCHICAL MIXTURE MODELSThe o
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72 HIERARCHICAL MIXTURE MODELSalone
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76 HIERARCHICAL MIXTURE MODELSDe Fi
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78 HIERARCHICAL MIXTURE MODELSEXERC
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80 REGRESSION MODELSmethodology for
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82 REGRESSION MODELS3.1.3 Regressio
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84 REGRESSION MODELSscaling factor
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86 REGRESSION MODELSThen in a predi
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88 REGRESSION MODELSgroup out at a
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90 REGRESSION MODELSTable 3.1 Probi
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92 REGRESSION MODELSBetter models w
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94 REGRESSION MODELSmodel enhances
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96 REGRESSION MODELSThe last 15 000
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98 REGRESSION MODELSOmitting consta
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100 REGRESSION MODELS(3.13) would i
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102 REGRESSION MODELSorF 1 (g ij )
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104 REGRESSION MODELSa birth increa
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106 REGRESSION MODELSThe coefficien
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108 REGRESSION MODELSbut the analys
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110 REGRESSION MODELSThe logit link
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112 REGRESSION MODELSseparation, an
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114 REGRESSION MODELS4.543.53Precis
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116 REGRESSION MODELSand a second o
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118 REGRESSION MODELSTable 3.14Kyph
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120 REGRESSION MODELSstandard devia
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122 REGRESSION MODELSTable 3.15Data
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124 REGRESSION MODELSTable 3.20 Leu
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126 REGRESSION MODELSNote that eith
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128 REGRESSION MODELSTable 2 (conti
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130 REGRESSION MODELSDellaportas, P
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132 REGRESSION MODELSWinkelmann, R.
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Analysis of Multi-level DataApplied
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MULTI-LEVEL MODELS: UNIVARIATE CONT
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MODELLING HETEROSCEDASTICITY 145pos
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where W ij ˆ X ij b is the total l
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MODELLING HETEROSCEDASTICITY 149var
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ROBUSTNESS IN MULTI-LEVEL MODELLING
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MULTI-LEVEL DATA ON MULTIVARIATE IN
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SMALL DOMAIN ESTIMATION 163knowledg
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X X X n jky ijk = X XjeJ 0 keK 0 i
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REVIEW 167Table 4.10(continued)Okla
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EXERCISES 169Hulting, F. and Harvil
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Models for Time SeriesApplied Bayes
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AUTOREGRESSIVE AND MOVING AVERAGE M
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DISCRETE OUTCOMES 1911.2Unemploymen
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DISCRETE OUTCOMES 193series to y t
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DISCRETE OUTCOMES 195One might also
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DISCRETE OUTCOMES 197each model, th
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DISCRETE OUTCOMES 199We consider a
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ERROR CORRECTION MODELS 201disequil
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DYNAMIC LINEAR MODELS AND TIME VARY
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where e t ˆ y t bx t . If y t has
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STOCHASTIC VARIANCES AND STOCHASTIC
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MODELLING STRUCTURAL SHIFTS 2153.53
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MODELLING STRUCTURAL SHIFTS 217Lubr
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MODELLING STRUCTURAL SHIFTS 219and
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REVIEW 221with the mean of t 1 esti
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REFERENCES 223Christensen, R. (1989
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EXERCISES 225Tanizaki, H. and Maria
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228 ANALYSIS OF PANEL DATAwhere b i
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230 ANALYSIS OF PANEL DATAZ it ˆ b
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232 ANALYSIS OF PANEL DATAwhere e i
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234 ANALYSIS OF PANEL DATA6.2.2 The
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236 ANALYSIS OF PANEL DATAregressio
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238 ANALYSIS OF PANEL DATApost-trea
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240 ANALYSIS OF PANEL DATAwhere the
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242 ANALYSIS OF PANEL DATAclinics),
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244 ANALYSIS OF PANEL DATApermanent
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246 ANALYSIS OF PANEL DATAIf, howev
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248 ANALYSIS OF PANEL DATAwhere k t
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250 ANALYSIS OF PANEL DATATable 6.7
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252 ANALYSIS OF PANEL DATAWe obtain
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254 ANALYSIS OF PANEL DATAmodelling
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256 ANALYSIS OF PANEL DATATo furthe
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258 ANALYSIS OF PANEL DATAAs a part
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260 ANALYSIS OF PANEL DATATable 6.1
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262 ANALYSIS OF PANEL DATAcrude dea
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264 ANALYSIS OF PANEL DATAThese are
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266 ANALYSIS OF PANEL DATAstands at
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268 ANALYSIS OF PANEL DATAThe place
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270 ANALYSIS OF PANEL DATADavies, R
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272 ANALYSIS OF PANEL DATAthe lag i
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274 MODELS FOR SPATIAL OUTCOMESinve
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276 MODELS FOR SPATIAL OUTCOMESfreq
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278 MODELS FOR SPATIAL OUTCOMESspat
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280 MODELS FOR SPATIAL OUTCOMES" #p
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282 MODELS FOR SPATIAL OUTCOMES7.3.
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284 MODELS FOR SPATIAL OUTCOMESTabl
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286 MODELS FOR SPATIAL OUTCOMESof t
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304 MODELS FOR SPATIAL OUTCOMESTabl
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306 MODELS FOR SPATIAL OUTCOMES7.6.
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308 MODELS FOR SPATIAL OUTCOMESInfo
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320 MODELS FOR SPATIAL OUTCOMESOliv
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322 MODELS FOR SPATIAL OUTCOMES10.
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324 STRUCTURAL EQUATION AND LATENT
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Page 459 and 460: INDEX 451cumulative mortality in re
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INDEX 453latent variable effects, n
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INDEX 455of common odds ratio 402vs
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INDEX 457goals of 171time varying c
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