# The (mis)use of overlap of confidence intervals to assess effect modification

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DOI: 10.1007/s10654-011-9563-8

- Cite this article as:
- Knol, M.J., Pestman, W.R. & Grobbee, D.E. Eur J Epidemiol (2011) 26: 253. doi:10.1007/s10654-011-9563-8

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In randomized controlled trials as well as in observational studies, researchers are often interested in effects of treatment or exposure in different subgroups, i.e. effect modification [1, 2]. There are several methods to assess effect modification and the debate on which method is best is still ongoing [2, 3, 4, 5]. In this article we focus on an invalid method to assess effect modification, which is often used in articles in health sciences journals [6], namely concluding that there is no effect modification if the confidence intervals of the subgroups are overlapping [7, 8, 9].

_{2}/σ

_{1}(Supplementary material, formula (3)). Figure. 1 shows the relation between ρ and the type 1 error probability if the effect estimates are independent. If the effect estimates are not independent, the correlation coefficient between the effect estimates can also be accounted for (Supplementary material, formula (3)).

The assumption used in the formulas presented in the appendices is that the effect estimators in the subgroups are normally distributed. Assuming that epidemiologic effect measures, such as the odds ratio, risk ratio, hazard ratio and risk difference, follow a normal distribution, the methods presented can also be used for these epidemiologic measures. Note that the assumption for normality is generally unreasonable in small samples, but a satisfactory approximation in large samples.

## Example

As an example, imagine a large randomized controlled trial that investigates the effect of some intervention on mortality and that includes 10,000 men and 5,000 women. Besides the main effect of treatment, the researchers are interested in assessing whether the treatment effect is different for men and women. Suppose that the risk ratio in men is 0.67 (95% CI: 0.59-0.75) and in women is 0.83 (95% CI: 0.71-0.98). The confidence intervals are partly overlapping, which the researchers may wrongly interpret as no effect modification by sex. Filling in formula (3) (Supplementary material) results in a probability of non-overlapping 95% confidence intervals under the null hypothesis of 0.006. A confidence level of 83.8% could have been calculated to arrive at a type 1 error probability of 0.05, resulting in a confidence interval of 0.61–0.73 for men and 0.74–0.93 for women. Now, the confidence intervals do not overlap, so the p-value is at least smaller than 0.05, indicating statistically significant effect modification. Calculating the difference in risk ratios with a 95% confidence interval results in a ratio of risk ratios of 0.80 with a 95% confidence interval of 0.66-0.98, corresponding to a p-value of 0.028. This confirms our earlier observation of statistically significant effect modification.

## Acknowledgement

This study was performed in the context of the Escher project (T6-202), a project of the Dutch Top Institute Pharma.

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