## Simpson's Paradox: What Can Happen if You Ignore an Important Variable

by Dr. Mark Irwin, Department of Statistics, The Ohio State University

Let's consider data collected as part of a one-in-six survey of the electoral roll carried out in Whickham, United Kingdom, during 1972-74 and a follow-up study conducted 20 years later. While the original study was mainly concerned with thyroid disease and heart disease, we will look at something different. Instead, we will use the data from these two studies to examine the relationship between smoking and 20-year survival rates (Am. Stat., vol. 50, pp. 340-1). For simplicity, we will restrict ourselves to the 587 women aged 45 to 74 at the start of the study who were either current smokers or had never smoked. The 20-year survival information was determined for all of the women in the study.

The survival information broken down by smoking status is shown in Table 1. The data suggest that smoking might be beneficial as 43% of the nonsmokers died versus only 38% of the smokers. Could this surprising result be a correct interpretation of the data? No, it can't, as Table 2 illustrates. The analysis presented in Table 1 ignores the important confounding variable strongly related to smoking and survival, the womens' ages at the start of the study. When survival rates are determined for women in each of the 10 year age ranges (45-54, 55-64, 65-74), the nonsmoking group does better in each case. In this example, few of the older women were smokers but most of them had died at the time of follow-up 20 years later. When age is ignored, as in the first table, the death rates are more of a reflection that the smokers tended to be younger and the nonsmokers tended to be older, not the effects of smoking.

This data set illustrates what has come to be known as Simpson's paradox, a reversal of the direction of a comparison or an association when data from several groups are combined to form a single group. Another example where missing an important confounding factor leads to an incorrect conclusion involves early observational studies examining the use of ultrasound and the frequency of low birth weight babies. Babies examined in the womb by ultrasound tended to have lower birth weights on average than those that weren't examined. However in this case, the confounding factor of problem pregnancies was ignored. The babies that were more likely to be examined by ultrasound tended to have problems that would also lead to lower birth weights. Later, randomized controlled clinical trials showed that ultrasound didn't have an adverse effect on birth weight and that, if anything, it tended to have a positive effect.

Confounding factors and Simpson's paradox provide just one example of difficulties that need to be considered as part of analyzing data. If you would like assistance from the Biostatistics Program in the design of your study or the analysis of your data, please feel to contact us.

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