Evaluation of Secular Equilibrium using Equivalence Testing. Paul Black, Mark Fitzgerald, David Gratson

1. Equivalence Testing for Secular Equilibrium Paul Black, Ph.D. Mark Fitzgerald, Ph.D. Dave Gratson, M.S. Neptune & Company, Inc.

2. Background Examining a Site with potential radionuclide contamination in near-Surface Soils Radionuclides of interest are naturally-occurring Uranium (U), thorium (Th), radium (Ra) Problems seen with analysis of radionuclide data Background not consistent for radionuclides in the same chain (analytical problem) Lack of (approximate) secular equilibrium in background (statistical problem)

3. What is Secular Equilibrium? A definition:The half-life of the precursor (parent) radioisotope is so much longer than that of the product (daughter) that the radioactivity of the daughter becomes equal to that of the parent with sufficient time Implication:Concentration activities are the same for some radionuclides in the same chain

4. In reality…..Approximate Secular Equilibrium The open nature of the system The relative geochemical mobility of each radioisotope The system environment (e.g., soil, rock, water) The passage of sufficient time for the buildup of daughters (ingrowth) post-contamination The effects of laboratory radiochemical analysis.

5. Evaluation Issues - Analytical Statistically compare Site and Background data If the Site is not contaminated then statistical background comparisons should “pass” Site data showed unexpected differences from Background data because of analytical issues Preparation methods involved HF for Background, but not for all the Site data (some HNO3) Led to consideration of secular equilibrium (SE) for evaluation of the radionuclide data

6. Evaluation Issues - Statistical Statistically test radionuclide data for SE….. Background should exhibit approximate SE Lack of Site contamination implies approximate SE However, standard statistical methods showed a lack of SE in these cases Statistical differences driven mostly by Analytical method differences

7. A Statistical Hypothesis Under the assumption of SE, activity concentrations should be the same for some radionuclides in a decay chain Since the analysis endpoint of human-health or ecological risk is typically based on the mean concentration….. SE holds only if: μ1 = μ2 = μ3 = … = μk for radionuclides 1 through k

8. “Standard” Statistical Test Test null hypothesis “H” vs. alternative “not H” For SE testing, “H” assumes: All the radionuclide means are exactly equal Analysis of Variance (ANOVA) method applies This is a “point” null hypothesis Burden of proof is on demonstrating SE does not hold With sufficient data lack of SE will always be shown Statistically This is a problem when Site = Background

9. Practical Considerations Under SE, the mean radioactivity for the radionuclides will be identical But, will the measured radioactivity be identical? Different radionuclides might be measured by different methods E.g. alpha spectroscopy vs. beta emissions Natural effects (geochemical, etc.) Small differences can be expected, even in Background, but ANOVA cannot accommodate them (“point” null problem)

10. Disincentive to Data Collection Also - under this standard classical ANOVA set-up … Since SE is the null hypothesis, the less data collected, the better the chance that SE will be “accepted” If there is no contamination, but there are slight differences in the measured means, then the more data collected, the more likely SE will be “rejected” Responsible party has no incentive to collect more data – not a desirable situation

11. Equivalence Testing Changes hypothesis of interest to: μ1 ≈ μ2 ≈ μ3 ≈ … ≈ μk Doesn’t expect the means to be exactly equal but rather “practically the same” – or “practically equivalent” Switches null and alternative hypotheses Assume that the means are not “practically equivalent” (null hypothesis) Burden of proof is now on the data to demonstrate approximate SE (alternative hypothesis)

12. Multivariate Analysis Under SE, correlation between the radionuclide measurements can be expected (not good for ANOVA) If a radionuclide is naturally at greater concentration for a given sample, then SE says that all radionuclides will have higher radioactivity for that sample In practice, can often see outliers in activity concentrations, but they appear simultaneously for all radionuclides (in the same chain) How to address strong correlation? Data transformation

13. Proportional Activity Analysis aided by conversion to relative proportional activity Tends to remove outliers Correlation between radionuclides normalized (now small and negative) Universal scale, making equivalence easier to define (in the null hypothesis)

14. Equivalence Test Set-Up Testing: versus The value Δ determines the level of deviation from equal proportions of radioactivity that will be considered equivalent Defines a spherical region in k-dimensions

15. Example for 3 dimensions (radionuclides) Equivalence region is the colored circle Δ is the radius of the circle

16. The Test Statistic The equivalence test is based on the F statistic: Calculation of p-value is somewhat complex

17. The Test Statistic The intuitive notion is the following: Construct a confidence region for the mean proportions (an ellipse in k-dimensions) If the entire confidence region lies within the equivalence region, then reject the null hypothesis and declare the mean proportions equivalent – in secular equilibrium That is, are the means close enough! Otherwise, approximate SE is not proven

18. Example confidence region that does not pass equivalence test (3-D)

19. Example confidence region that does pass equivalence test (3-D)

20. Example Data Site with potential uranium chain contamination Data collected on 4 radioisotopes in the decay chain U-238 (half-life of 4.5 billion years, α decay) U-234 (half-life of 245,500 years, α decay) Th-230 (half-life of 75,380 years, α decay) Ra-226 (half-life of 1,602 years, α decay)

21. Boxplots

22. Scatterplots

23. Boxplots of Proportions

24. Scatterplot of Proportions

25. Results of ANOVA Tests

26. Results of Equivalence Test

27. Site Application Evaluation of radionuclide analytical data Comparison of Site and Background data Coupled with statistical Background comparison tests Lack of SE implies contamination Or something else is wrong (analytical, background) However, SE does not necessarily imply Background

28. Closing Remarks Equivalence Testing is a statistical method that accommodates small differences that are expected but are practically unimportant Small differences due to natural or analytical effects This method could also be applied to Background comparisons in general (for metals, etc.)

29. Resources Guided Interactive Statistical and Decision ToolsGiSdT – Open Source, based on R (www.r-project.org) Web-based (free access) EnviroGiSdT Pc-based (free download) E-mail for more information:pblack@neptuneinc.org