Question

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Consider the following data from a study examining the relationship between smoking behaviour and 20-year survival status of 1314 women (Appleton et al. 1996). Status after 20 years Dead Alive Smoker 139 443 Non-smoker 230 502 Total 369 945 Total 582 732 1314 (a) Determine each of the following: (i) The percentage of the smoker group who died in 20 years. (ii) The proportion of the non-smoker group who died in 20 years. (iii) The risk of death for the smoker group. (iv) The odds of death to alive in the non-smoker group. (b) What is the relative risk of death for smokers compared to non-smokers? Interpret your answer in the form of a sentence that could be understood by someone with no training in statistics. (c) Repeat part (b) using increased risk instead of relative risk. (d) Determine the odds ratio of death for smokers compared to non-smokers. (e) Name one possible confounding variable in this analysis. Explain briefly how this con- founding variable would lead to the surprising results found in part (b) and (c).

Answer 1

a.i The percentage of the smoker group who died in years is

$\frac{139}{582} \cdot 100 \approx 23.9\%$

a.ii The proportion of the non-smoker group who died in years is

$\frac{230}{732} \approx 0.314$

a.iii The risk of death for the smoker group is over the years.

a.iv The odds of death to alive in the non-smoker group are

$\frac{230}{502} \approx 0.46$

b. The relative risk of death for smokers can be calculated as

$RR = \frac{139/582}{230/732} \approx 0.76$

Layman interpretation: Women who smoke are less likely to die over the next years, when compared to women who don't smoke.

c. Increased risk of death for smokers is

$RR - 1 \approx -0.24$

The negative sign indicates that the risk is lower for smokers.

d. The odds ratio of death for smokers compared to non-smokers is

$\frac{139/443}{230/502} \approx 0.69$

e. Smokers having a lower risk than non-smokers very obviously hints at a spurious analysis. One possible confounding variable in this analysis could be the age of participants. If the median age of the two groups is significantly different, then you'd expect them to have varying remaining life expectancy. In this study, if the non-smokers sampled are already older, then they are more likely to die of other causes in the next years. This would then decrease the calculated risk for the smokers, as compared to the actual risk.