Ignoring a Covariate: An Example of Simpson's Paradox
Appleton, David R., French, Joyce M., Vanderpump, Mark P. J., The American Statistician
All statisticians know of the dangers of ignoring a covariate that is correlated to an outcome variable and an explanatory one (Simpson 1951). Simple but convincing examples based on real data are, however, in rather short supply. We present one here in which a naive analysis suggests a beneficial effect of cigarette smoking.
In 1972-1974 a one-in-six survey of the electoral roll, largely concerned with thyroid disease and heart disease, was carried out in Whickham, a mixed urban and rural district near Newcastle upon Tyne, United Kingdom (Tunbridge et al. 1977). Twenty years later a follow-up study was conducted (Vanderpump et al. 1995). Some of the results make interesting teaching material, and we present here those dealing with smoking habits as reported at the original survey, and whether or not the individual survived until the second survey. For the sake of simplicity we have restricted ourselves to women, and within them to the 1,314 who were classified either as current smokers or as never having smoked; there were relatively few women at the first survey (162) who had smoked but stopped, and only 18 whose smoking habits were not recorded. The 20-year survival status was determined for all the women in the original survey.
Table 1. Relationship Between Smoking Habits and 20-Year Survival in 1314 Women Smoker Yes No Total Dead 139 230 369 Alive 443 502 945 582 732 1,314 [x.sup.2] = 9.12 on 1 df; P = .0025. Odds ratio = .68 (95% confidence limits .53-.88).
Results are shown in Table 1. They imply a significant protective effect of smoking because only 24% of smokers died compared to 31% of nonsmokers. Can this be the correct interpretation? No, it cannot, and Table 2 shows one variable that is strongly related to both smoking and survival - namely age. Few of the older women (over 65 at the original survey) were smokers, but many of them had died by the time of follow-up.
We do not wish to give too much consideration here to the proper analysis of these data. However, it is clear that any sensible weighted average of the odds ratios for the different age groups will have a value of greater than unity. For example, Woolf's test applied to the first six two-by-two tables gives an overall odds ratio of 1.53 with 95% confidence limits of 1.08 and 2.16. This should be enough to indicate that we do indeed have an example of Simpson's paradox. However we do not intend to suggest that such an analysis would be totally satisfactory: it does not take the ordering of the categories into account. …