Confidence intervals for dependent data: Equating non-overlap with statistical significance

We revisit the problem of determining confidence interval widths for the comparison of means. For the independent two-sample (two-sided) case, Goldstein and Healy (1995) draw attention to the fact that comparisons based on 95% error bars are not very effective in assessing the statistical significance of the difference in means and derive the correct confidence interval for such a comparison. We provide an extension to Goldstein and Healy (1995) to account for the correlation structure and unequal variances. We use the results to develop rules of thumb for evaluating differences, in an exploratory manner, like Moses (1987) and Cumming (2009), from the independent case. We illustrate the method for the simple comparison of two means in a real data set, provide R code that may be easily implemented in practice, and discuss the extension of the method to other applied problems.