Application of Equal Local Levels to Improve Q-Q Plot Testing Bands with R Package qqconf

Quantile-Quantile plots (Q-Q plots) are often difficult to interpret because it is unclear how large the deviation from the theoretical distribution must be to indicate a lack of fit. Most Q-Q plots could benefit from the addition of meaningful global testing bands, but the use of such bands unfortunately remains rare because of the drawbacks of current packages and algorithms. These drawbacks include incorrect global Type I error rate, lack of power to deviations in the tails of the distribution, relatively slow computation for large data sets, and limited applicability. To solve these problems, we apply the equal local levels global testing method and develop faster recursive algorithms (supplemented by asymptotic approximations for sample sizes over 100K) to calculate the resulting testing bands. We present the R Package qqconf, a versatile tool to create Q-Q plots and P-P plots in a wide variety of settings, with simultaneous testing bands rapidly created using our methods. In addition to being quick to compute, these bands have a variety of desirable properties, including accurate global levels, equal sensitivity to deviations in all parts of the null distribution, including the tails, and applicability to a range of null distributions.