Quantile–Quantile Toolbox
Douglas M. Hawkins · The Journal of Applied Laboratory Medicine · 2024
We thank the editors for the positive comments (1) on our paper (2) on a QQ (quantile-quantile) toolbox for reference intervals. We agree that it would have been helpful to discuss Hoffmann’s work (3), but do not see our toolbox as a refinement of Hoffmann’s approach, despite the use of linear fits to portions of QQ plots in both. Rather, we see the two as addressing different problems. While he mentioned a direct study briefly, Hoffmann, and subsequent users of the methodology, have been primarily interested in the indirect data study—measurements for a mixed bag of subjects in which a majority are eligible members of the reference population, along with a not-necessarily-small minority who are not eligible. This means that the data follow a mixture distribution with 2 or more components, one representing the population of eligible subjects, and one or more describing the ineligibles, who may be ineligible for more than one distinct reason. The components of the mixture might be normal, but for the typical right-skew analyte the lognormal and gamma distributions may be more attractive. Hoffmann’s algorithm relies on the idea that under suitable circumstances, a normal QQ plot of the data—before or after transformation—may be approximately linear in a range corresponding to the eligible majority, breaking away at the end(s) where eligibles are scarce and ineligibles take over. We aim at a direct data set, in which every subject is believed to be eligible, and see our work as enriching, streamlining, and rendering more rigorous the IFCC framework, a detailed outline of which is in (4); see also (5). Direct data sets are generally harder to get and more expensive than indirect since they require that each and every subject’s eligibility be verified, and therefore tend to be smaller. Their virtue is that the reference ranges they give are guaranteed to be appropriate for that reference population. A single common statistical distribution is appropriate for a direct data set. Our choice of this distribution is normal, or normal-in-the-middle. The latter, a generalization of the distribution underpinning the winsorized mean (6), holds that the central 95% of the distribution is normal, but that the 2.5% tails follow an arbitrary distribution. This normal (or normal-in-the-middle) distribution could apply to the original scale of the data, or following a Box–Cox transformation, creating the 4 compartments in which we try to accommodate the data. The 2.5% unspecified tail(s) could be a reflection of a true departure from gaussian at the edge, or the result of a few ineligible subjects sneaking into the data set. This single fixed distribution is conceptually different from the mixture assumption underlying Hoffmann’s work, and it is not obvious to us that our methods are suitable for indirect studies. The AST (aspartate transferase) data set in (1), if treated as a direct study, would give a nonparametric reference limit of 49 90% confidence interval (46,52) IU/L, a little higher than that reported for the winsorized QQ correlation coefficient, as the latter’s provision for 2.5% ignorable readings would lead one to expect. Analyzing it as an indirect study with potentially many more than 2.5% ineligibles gives the lower estimates the editorial reports. We did all development in R but think it important that the methods are accessible to laboratorians who do not use R, and we wrote code in Excel to satisfy ourselves that any likely reader should be able to implement them. With scripting languages such as R, SAS IML, or JMP JSL, coding is even less of an obstacle. Finally, the editorial says that parametric methods need larger samples than nonparametric. We think the general statistical consensus is opposite to this. In particular, the relevant CLSI guidline EP28-A3c notes it as an advantage that, whereas their nonparametric approach requires a minimum of 120 samples, parametric analyses can be done on much smaller samples. This is in no way an endorsement of skimping on sample size but a recognition that it is not always possible to get a direct data set with triple-digit sample size. We agree that some form of QQ methodology should be a valuable tool for indirect, as well as direct studies, and look forward to further developments. Author Contributions:The corresponding author takes full responsibility that all authors on this publication have met the following required criteria of eligibility for authorship: (a) significant contributions to the conception and design, acquisition of data, or analysis and interpretation of data; (b) drafting or revising the article for intellectual content; (c) final approval of the published article; and (d) agreement to be accountable for all aspects of the article thus ensuring that questions related to the accuracy or integrity of any part of the article are appropriately investigated and resolved. Nobody who qualifies for authorship has been omitted from the list. Authors’ Disclosures or Potential Conflicts of Interest:Upon manuscript submission, all authors completed the author disclosure form. Research Funding: None declared. Disclosures: D.M. Hawkins has received consulting fees from Scottsdale Scientific LLC.