# Question about combined relative risk interpretation

I am looking at a review of the effects of cigar smoking, and I am having trouble interpreting some of the results. Study linked here: Study

Specifically I have questions about table 3, below:

In the Primay Cigar line, the RR is 1.1 for people age 50-64 who use 1-2 daily. Yet, the Combined is 1.02 (0.97-1.07) (95% CI).

I know that looking at the combined CI for people who use 1-2 daily can be interpreted as an 2% increase in all cause mortality relative to never smokers, but it is not statistically significant.

Can I extrapolate the combined CI to the 50-64 group alone? In other words, can I say "people 50-64 years old who consume 1-2 cigars daily have a 10% increase in all cause mortality relative to never smokers, but the risk is not statistically significant."?

No, you can't extrapolate the confidence interval like that -- it's possible (though not that likely in this example) that a confidence interval calculated for the 50-64 group would exclude the null.

As an example of where that sort of interpretation would more obviously be wrong, consider the 3-4 cigars 80+ estimate of 0.95. You can't use the combined CI 1.02-1.15 there because it doesn't even contain the estimate!

Unless there's a very good reason to expect an association only in 50-64 year olds, I would say something like "The best single estimate of the risk is that it's 10% higher in people 55-64 who consume 1-2 cigars, but (taking data for all ages into account) this estimate is smaller than the uncertainty and there's no convincing evidence of an increase in this group". Or something. The 1-2 cigars line is extremely unconvincing.

• What makes the 1-2 line unconvincing? My gut feeling agrees with you, but I am curious as to more solid reasoning. Commented May 22 at 0:44
• The variation with age group and the fact that combining over ages gives basically no association. Commented May 22 at 2:14
• But if you combine all the ages, and follow the cohorts for long enough, ultimately, the OR for all cause mortality should be 1, no? Or am I brainfarting? Commented May 22 at 18:10
• If you average the OR over all ages you get an average age-specific OR, and that can quite easily be greater than 1. A value >1 means that at every specific age your chance of dying (if you get to that age) is higher. Commented May 23 at 0:49

In this case you might be able to get some crude estimates of other confidence intervals, but not by "extrapolat[ing] the combined CI" in the way you seem to suggest. You certainly shouldn't use those estimates for inference on the published study, but they might be helpful if you wanted to design a similar study.

You will note that the CI for all "combined" values are essentially symmetric about the point estimates. That won't always be the case. For example, reported CI for hazard ratios in survival analysis are not symmetric about the point estimates; symmetry is typically in the log-hazard scale. In this table, however, it might be reasonable to assume that there would be symmetry of CI about the other point estimates.

The question then is how wide would the CI be about a different point estimate.

One reason why you can't just "extrapolate" the combined CI value (even if you re-center it around a different point estimate) is that the width of a CI depends on the number of observations. It tends to decrease with the square root of the number of observations. For example, if the 50-64 age group represented 1/4 of the combined observations, you would expect the CI for that age group to be about twice as wide as that for the "combined" CI. Note that the relevant "number of observations" also depends on the nature of the outcome variable. In this type of study it's the number of deaths, not the total number of individuals, that probably matters.

This study seems just to be a report of deaths as a function of age range and smoking history, restricted to white males. In a more complicated study that used regression methods to adjust for other variables like race and sex, that simple approach would be suspect; the CI might also depend on the distributions of the other variables.

There's a problem with applying even that simplified approach to this type of table: the multiple comparisons problem. The more comparisons that you make, the more likely that one will appear "statistically significant" must by chance. If you choose to evaluate a particular point estimate in part because it seems "big" you run even a larger chance of misleading yourself.

This type of approach can be useful if you are trying to design a new study based on information in the literature. In designing a new study you need to know the size of the net effect you want to detect and the variability of that effect. If you don't have any more direct information based on your own work, others' reports of similar investigations might be the best you can do.