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# Secular Equilibrium Presentation For Aehs

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

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

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• Some contamination potential – background comparison problems are analytical problem related – lack of secular equilibrium is statistically related
• P_i here represents the proportion of radioactivity due to radioisotope i for a single sample.Negative correlation since proportions – roughly -1/k.
• Mu_{p_i} is the mean proportion of activity for radioisotope i.
• Constrained to 3-D simplex
• S_p is the sample covariance matrix
• S_p is the sample covariance matrix
• Note: 2-D example is actually just a t-test, but for illustrative purposes, showing a confidence region in 2-D
• Note outliers – and effect on something like ANOVA (will overestimate sigma, lead to easy acceptance)
• Note high correlations – correspondence of outliers
• Note: after adjusting for correlation … not quite as clear if they’re the same, though th-232 is not highest.
• Correlation low and slightly negative – Outliers mostly gone
• Note:univariate confidence bounds, but real confidence interval is elliptical in 4-space
• Note:univariate confidence bounds, but real confidence interval is elliptical in 4-space
• ### Transcript

• 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
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
• 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
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)
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
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