Inference by eye: reading the overlap of independent confidence intervals

When 95 per cent confidence intervals (CIs) on independent means do not overlap, the two-tailed p-value is less than 0.05 and there is a statistically significant difference between the means. However, p for non-overlapping 95 per cent CIs is actually considerably smaller than 0.05: If the two CIs just touch, p is about 0.01, and the intervals can overlap by as much as about half the length of one CI arm before p becomes as large as 0.05. Keeping in mind this rule-that overlap of half the length of one arm corresponds approximately to statistical significance at p = 0.05-can be helpful for a quick appreciation of figures that display CIs, especially if precise p-values are not reported. The author investigated the robustness of this and similar rules, and found them sufficiently accurate when sample sizes are at least 10, and the two intervals do not differ in width by more than a factor of 2. The author reviewed previous discussions of CI overlap and extended the investigation to p-values other than 0.05 and 0.01. He also studied 95 per cent CIs on two proportions, and on two Pearson correlations, and found similar rules apply to overlap of these asymmetric CIs, for a very broad range of cases. Wider use of figures with 95 per cent CIs is desirable, and these rules may assist easy and appropriate understanding of such figures.