Applied Bayesian Modelling - Free

400 MODELLING AND ESTABLISHING CAUSAL RELATIONS10.2.1 Stratification vs. multivariate methodsOne method to reduce the effect of a confounder Z is to stratify according to its levels(Z 1 , : : Z m }, and then combine effect measures such as odds ratios over strata, accordingto their precisions. Data from an Israeli cross-sectional prevalence study reported byKahn and Sempos (1989) illustrate the basic questions (Table 10.1). Cases and noncases(in terms of previous myocardial infarction) are classified by age (Z) and systolicblood pressure (X ).Age is related to the outcome because the odds ratio for MI among persons over 60 asagainst younger subjects is clearly above 1. The empirical estimate is15 1767=(188 41) ˆ 3:44with log (OR) ˆ 1:24 having a standard deviation 2 of (1=15 1=1767 1=1881=41) 0:5 ˆ 0:31. Moreover, age is related to SBP since with age over 60 as the`outcome' and SBP over 140 as the `exposure', the empirical odds ratio is124 1192=(616 79) ˆ 3:04Providing there is no pronounced effect modification, it is legitimate to seek anoverall odds ratio association controlling for the confounding effect of age. Supposethe cells in each age group sub-table are denoted {a, b, c, d} and the total ast ˆ a b c d.To combine odds ratios OR i (or possibly log OR i ) over tables, the Mantel±Haenszel(MH) estimator sums n i ˆ a i d i =t i and d i ˆ b i c i =t i to give an overall odds ratioXn i = X d i (10:1)i iThis is a weighted average of the stratum (i.e. age band) specific odds ratios, with weightfor each stratum equal to d i ˆ b i c i =t i , since{a i d i =(b i c i )}d i ˆ a i d i =t i ˆ n iTable 10.1Myocardial infarction by age and SBPAge Over 60 MI Cases No MI All in SBP groupSBP > ˆ 140 9 115 124SBP < 140 6 73 79All in Age Band 15 188 203Age Under 60SBP > ˆ 140 20 596 616SBP < 140 21 1171 1192All in Age Band 41 1767 18082 The standard error estimate is provide by the Woolf method which relies on the Normality of log(OR).