Hi Researchers, this appears to be a quite difficult question. As we know, if 20% of the smokers developed lung cancer and 10% of the non-smokers developed lung cancer, then we say that cigarette smoking is associated with a 2-time (20% divided by 10%) relative risk of developing lung cancer. But if 0% of the non-smokers developed lung cancer, what would the relative risk be? And how would you describe the risk of smoking in this situation for general readers? Thank you!
If the incidence in non smokers was actually zero (unlikely) the relative risk would be infinity.
In this case you should use risk difference. eg for general reader say the risk for smokers is 20% compared to zero, or a risk difference of 20%. If the risk in non smokers is 10%, and the risk in smokers is 20%, the risk difference is 20=10 = 10% (say the risk in smokers is 20% compared to 10% in non smokers and one can easily see a ratio effect as well).
NB these are not good example figures for smoking
Hi Ning, I like James' answer. I've seen various methods over the years to handle this situation; a commonly-taught (but imperfect) method is to add 1 to each cell in the 2X2 table; or to add 0.5 to the "zero" cell. Each of these can be criticized for various reasons.
You could conceivably compute a confidence interval for a binomial proportion, for example if you have zero out of 50 smokers getting lung cancer, then the 95% confidence interval would be 0.000 to 0.071; in other words your upper bound of the confidence interval around zero would be 7. You could then use that 7 for the "worst case" scenario in your relative risk calculation, i.e., the maximum number of non-smokers who would have lung cancer when you actually measured 0 out of 50. (you would use 7 out of 50 instead). The underlying idea is that there is always uncertainty around your sample and if you were to sample 50 people again from the same population, you might find more than zero people who get lung cancer.
Actually that's a 97.5% confidence interval, since the lower bound of the 95% confidence interval would be less than zero --- and you can't have fewer than zero people!
the way you asked the question it's actually very simple to answer and no calculation is needed at all... if the intervention group shows a non-zero incidence but the reference group has zero incidence then of course the relative risk is infinitely large.
but of course if you should not be interested in a mere point estimate, so the approach with confidence intervalls for risks makes a lot of sense. just don't forget to use confidence intervalls for the risks in both intervention and control groups, and combine them correctly.
The estimate itself would be undefined, since devision by zero is not defined. However in the real world, there would be a finite size of sample and sampling variation, and these would give us a handle on the situation. Suppose you had 20 out of 100 smokers who developed lung cancer, and 0 out of 100 in the non-smokers. You could then apply the "rule of 3" to say that, while the estimate of the risk in the smokers was 0, and the lower limit limit of the confidence interval remained zero, the upper limit would be 3/100. Thus the confidence interval for the relative risk would have a lower limit of 0.5 divided by this, i.e. 16.7, and so the confidence limit for the RR for smokers would be "16.7 and above" (or 16.7 to infinity, if you prefer).
But here's an open question: could Bayesians use epidemiological information for the incidence of lung cancer among non-smokers in similar populations to improve on the use of the "rule of 3", or adding 0.5 to zeros in a contigency table? If that were treated as a prior, I suppose the posterior for the chance of getting lung cancer for non-smokers from the current study would be some kind of beta distribution, of which the prior binomial would be the conjugate. I'll leave this to Bayesian experts.
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