1. Robert W. L. Thomas, Ph.D.; AECOM; Bowie, Maryland, USA
NAVSEA WARFARE CENTERS
Risk Reduction Distributions Relating to the Benefits of Safety Efforts
NAVAL SURFACE WARFARE CENTER DAHLGREN DIVISION
DAM NECK ACTIVITY
August 2018
Robert W. L. Thomas, Ph.D.; AECOM; Bowie, Maryland, USA
Marilyn J. Eichelberger, BA; Department of the Navy, Naval Surface Warfare Center Dahlgren Division Dam Neck Activity, Dahlgren, Virginia, USA
Missey Lee, MS; Department of the Navy, Naval Surface Warfare Center Dahlgren Division Dam Neck Activity, Dahlgren, Virginia, USA
Distribution Statement A: Approved for Public Release
NAVSEA WARFARE CENTERS

2. Topics
Objective
Incremental Path to Improve Our Model
2018 Risk Reduction Model
Mathematical Equations
Results
Conclusion
References

3. Objective
Today’s objective is to compute the mathematically modeled reduction in risk, i.e., severity multiplied by probability, for general mishaps before and after system safety efforts
For today’s brief, mishap probability and consequence data before and after the safety efforts are used to:
Compute the initial risk distribution
Estimate the risk reduction distribution that might be generated by safety efforts

4. Incremental Path to Improve Our Model
In our 2017 paper, risk became a quantitative item with a known probability distribution, providing a new metric for safety program effectiveness
Our 2017 objective was to examine the character of safety programs to not only reduce risk, but also to reduce the relative risk uncertainty
Approximate the distributions of both the probability and severity of a mishap as lognormal
Define the risk plane as the graph of consequence or severity as a function of the probability
Examine the likely behavior of the co-distribution as the risk reduction process is executed
Show how differential forces across the risk plane reduce both the risk itself and the relative uncertainty in the risk at the same time

5. How Confidence Regions Move and Distort

6. Conceptual Illustration of the Movement and Distortion of Confidence Regions in the Risk
Stretching The Confidence Region So That Negative Correlation
Is Introduced Results In Reduced Relative Error In Risk

7. How Negative Correlation Reduces Risk Uncertainty
Distribution of Risk in Log-Log Plot
Risk = Probability * Severity, so
Log(Risk) = Log(Probability) + Log(Severity) or LR = LP + LS
Variance, V(LR) = V(LP) + V(LS) + 2*Corr(LP,LS)*√(V(LP)V(LS))
Assumption: If LP and LS are Normally distributed then so is LR,
i.e. Risk distribution is lognormal
Negative Correlation Between Probability
And Severity Reduces Uncertainty In Risk

8. Origin of Our Data
For our data sets, two years ago using Monte Carlo Method, we simulated random accident claims and distance driven since last accident value pairs to illustrate the problem
Last year we used the same data sets to examine risk distributions
We noted that both “Before” and “After” the Safety Effort, the distribution could reasonably be approximated by Lognormal form
We showed that the risk distribution “After” had a lower median and a smaller shape factor

9. Comparison of Claims Data Before and After Safety Improvement Program

10. Risk Distributions “Before” and “After”
Point where the benefit of the safety program presents itself through the reduced probability of high risk
After The Safety Program, Risk Is Concentrated At Lower Values

11. 2018 Risk Reduction Model
This year we used the distributions computed from the same data sets
We calculated the distribution of the risk reductions using the 10,201 pairs (.5%, 1-99%, and 99.5%) for both the cases “Before” and “After”
We discovered that the risk reduction distribution can be approximated by Lognormal distribution with a negative offset parameter
This produces a better fit to the difference distribution

12. Lognormal Probability Density Function (PDF) with Offset
For location, z, median parameter, µ, and shape parameter, σ, Lognormal PDF = [zσ√(2π)]-1.exp([ln(z) - µ]/[2σ2]) For an offset, δ, we set z = x – δ, so that Lognormal PDF = [(x – δ)σ√(2π)]-1.exp([ln(x - δ) - µ]/[2σ2]) for x > δ = 0 for x <= δ where µ = median parameter (median of x is δ + exp(µ)) σ = shape parameter (standard deviation of ln(x – δ)) Note: δ is usually, but not always selected to be smaller than the minimum value of x
Previously δ Was Zero
For Describing Risk Reduction, A Small Negative Value For This Parameter Often Required
Lognormal Definition Has Now Been Expanded To Include A Non-Zero Offset

13. Using a Lognormal Approximation to Any Distribution
We define a lognormal distributed parameter, X: X= δ + exp(µ + σ Z) (1) Where Z = standard Normally distributed parameter δ = offset parameter µ = median parameter σ = shape parameter The mean and variance of X in terms of µ, σ and δ are m = δ + exp(µ + σ2/2) (2), V = [exp(σ2) – 1].exp(2µ + σ2) (3), Given general values of m, V and δ, we can solve for µ and σ2 as follows: µ = ln[(m-δ)/√{1+V/(m-δ)²}] (4), and σ2 = ln(1+V/(m-δ)²) (5)
For Any General Distribution With A Minimum Abscissa of δ, Mean Value, m,
and Variance, V, A corresponding Lognormal Distribution Approximation
Can Obtained By Solving Equations (4) and (5) For µ And σ2

14. Estimated Risk Reduction Distribution Using Lognormal Approximation

15. Conclusion
Lognormal assumption with offset for risk reduction is an approximation whose validity is reasonable for real-world situations
Use of our Model allows management to estimate the percentile probability of returns on investment for planned safety or performance improvement efforts

16. References
Kady, R.A., Ranasinghe, A., Zemore, M.G. and Eller, R.A., “The Challenges of a Quantitative Approach to Risk Assessment”, 32nd ISSC, St Louis MO, August 4-8, 2014.
Military Standard 882E, "System Safety." May 11, 2012, , retrieved March 24, 2016.
Thomas, R.W.L., Eichelberger, M., Lee, M. and Hahn, J, “An Innovative Approach to Hypothesis Testing for System Safety Assessment”, 33rd ISSC, San Diego, August 21-24, 2015.
Thomas, R.W.L., Eichelberger, M., Lee, M. and Hahn, J, “An Innovative Approach to Hypothesis Testing for System Safety Assessment”, International Systems Safety Journal, Fall 2015, Pages
Thomas, R.W.L., Eichelberger, M., Lee, M., “Setting Testing Requirements for Simulation Based System Safety Assessments”, 34th ISSC, Orlando, August 8-12, 2016.
Thomas, R.W.L., Eichelberger, M., Lee, M., “The Theory of Risk Uncertainty Reduction”, 35th ISSC, Albuquerque, August 21-25, 2017.