A Single Interval Isn’t Enough

Interval estimates such as confidence consonance intervals are now widely reported in many journals alongside the exact P-value of a statistical test and point estimate. While this is a large improvement over what constituted statistical reporting in the past two decades, it is still largely inadequate.

Take for example, the 95% consonance interval. As many have stated before, there is nothing special about 95%, yet we rarely see intervals of any other level. Choosing to compute a 95% interval is as mindless as choosing a 5% alpha level for hypothesis testing.

A single consonance interval is only a slice of a wide range of consonance intervals at different levels. Mindlessly reporting 95% intervals only promotes cargo-cult statistics since there is not much thought behind the choice. (1)

Cargo-cult statisticsthe ritualistic miming of statistics rather than conscientious practice.” - Stark & Saltelli, 2018

Thus, we propose that instead of only calculating one interval estimate, every interval associated with a consonance level be calculated, along with its corresponding P-value and S-value, and plotted to form a function. (2-8)

This can be accomplished using the concurve package in R and on Stata.

For R:

Install the Package From CRAN


Install the Developer Version


Check out the Examples.

For Stata:

Check out the Article on Using Stata for concurve.

"Statistical software enables and promotes cargo-cult statistics. Marketing and adoption of statistical software are driven by ease of use and the range of statistical routines the software implements. Offering complex and “modern” methods provides a competitive advantage. And some disciplines have in effect standardised on particular statistical software, often proprietary software.

Statistical software does not help you know what to compute, nor how to interpret the result. It does not offer to explain the assumptions behind methods, nor does it flag delicate or dubious assumptions. It does not warn you about multiplicity or p-hacking. It does not check whether you picked the hypothesis or analysis after looking at the data, nor track the number of analyses you tried before arriving at the one you sought to publish – another form of multiplicity. The more “powerful” and “user-friendly” the software is, the more it invites cargo-cult statistics." - Stark & Saltelli, 2018


  1. Stark PB, Saltelli A. Cargo-cult statistics and scientific crisis. Significance. 2018;15(4):40-43.
  2. Poole C. Beyond the confidence interval. Am J Public Health. 1987;77(2):195-199.
  3. Sullivan KM, Foster DA. Use of the confidence interval function. Epidemiology. 1990;1(1):39-42.
  4. Rothman KJ, Greenland S, Lash TL. Modern epidemiology. 2012.
  5. Singh K, Xie M, Strawderman WE. Confidence distribution (CD) – distribution estimator of a parameter. arXiv [mathST]. 2007.
  6. Schweder T, Hjort NL. Confidence and Likelihood*. Scand J Stat. 2002;29(2):309-332.
  7. Amrhein V, Trafimow D, Greenland S. Inferential Statistics as Descriptive Statistics: There is No Replication Crisis if We Don’t Expect Replication. Am Stat. 2019
  8. Greenland S. Valid P-values Behave Exactly As They Should. Some misleading criticisms of P-values and their resolution with S-values. Am Stat. 2019;18(136).