1. The Empirical Bayes Method for Before and After Analysis

2. Key Reference
Hauer, E., D.W. Harwood, F.M. Council, M.S. Griffith, “The Empirical Bayes method for estimating safety: A tutorial.” Transportation Research Record 1784, pp National Academies Press, Washington, D.C
Open This Document and read through as you go along on PPT

3. EB Procedures Abridged Full Last 2-3 years data Traffic volume
Can use more data
Includes other factors

4. Empirical Bayes
Weight should be based on sound logic and real data

5. The SPF – Safety Performance Function
So what is the expected number of crashes for facilities of this type? Develop a (negative binomial) regression model to fit all the data – must have data to do this.*
An example SPF:
μ=average crashes/km-yr (or /yr for intersections)
So, if ADT = 4000
Note: this SPF depends only on ADT … it needn’t
* Can also use equations from HSM, but need “phi”

6. The overdispersion parameter
The negative binomial is a generalized Poisson where the variance is larger than the mean (overdispersed)
The “standard deviation-type” parameter of the negative binomial is the overdispersion parameter φ
Variance of crash counts = η[1+η/(φL)]
Where …
μ=average crashes/km-yr (or /yr for intersections)
η=μYL (or μY for intersections) = number of crashes/time
φ=estimated by the regression (units must be complementary with L, for intersections, L is taken as one)

7. Example 1:
How many crashes should we expect next year???

8. Example 1: road segment, 1 yr. of data

9. Example 1: computing the weight
What happens when Y is large? When μ is small (compared to φ)?

10. Example 1 (cont): = 4.71 ± 1.19 accidents/km/year
Note that the variance of the EB estimate, = 4.58, is less than the variance of the accident counts on the segment, 9.44
= ± 1.19 accidents/km/year

11. Example 2: 3 years of data: 12, 7, 8 4000 vpd Step 1: Step 2: Step 3:
Note effect of more data… weight is smaller, placing less emphasis on the model estimate
As before
= ± 1.44 accidents/year for the section
(compare to previous estimate and reliability)
From ex 1) 8.48 ± 2.14

12. Example 3: AMFs 1.2 meter shoulders (instead of 1.5)
AMF (CMF) = 1.04 (4% increase in crashes)
Step 1:
Step 2:
Step 3:
Why is weight lower?
1) 4.71 ± 1.19
2) 4.43 ± 0.80
3) 4.47 ± 0.81

13. Example 4: subsections
Total length = 1.5km, 11 crashes in 2 years

14. Large for fatals (helps you not to “chase” them
Example 5: Severity
2.41 x = … 1.8 x 3 x = 0.247
Note: φ stays same (mult dist. by constant);
Large for fatals (helps you not to “chase” them
Note: ≠ 23.9 (from prob 2) … why? What is the suggested an ad hoc solution?

15. Example 6: intersection
ADT=4520
SPF = 6.54×10-5 ×ADTmainline0.82×ADTminor road0.51
ADT=230
AMF = 1.27
7 crashes in 3 years
Step 1:
Step 2:
Step 3:
So, what can you conclude about the site?

16. Example 7: group of intersections
11 crashes in 3 years
Applies if you don’t know what crashes happened at what intersection
Step 1:
Step 2: (simplistic)
However, not clear what to use

17. Example 7: (cont)

18. Example 7: (cont) Step 3: using w=0.088,
Why so much confidence in the actual number?
Is it because we have 3 yrs of data?
Is it because 11 is smaller than 20.7?
What would happen if 11 had been, say, 32?

19. Example 8 (The full procedure)
1.8 km, 9 yrs. Unchanged road
ADT varies, AMF = 0.95, 74 total crashes
μ =

20. Example 8, cont.
(If all μ are equal)
Why so small???

21. Example 9: Secular Trends
Yearly multipliers can be used like AMFs to account for weather, technology changes (must be able to get them)
Make much difference?

22. Example 10: Projections
Projections can be made by using a simple ratio of ADTs (raised to the appropriate power) multiplied by the corresponding ratio of AMFs or yearly multipliers

23. Some thought questions
Does EB eliminate RTM as stated?
What happened if the SPF is not appropriate for your site
What does appropriate mean?

24. Software for Homework
You will need some software to develop the NB regression model for your SPF – that is the “R project” program. Investigate that now (see HW).
(info on “R”)
Download R for Windows (47 megabytes, 32/64 bit)
Installation and other instructions
New features in this version: Windows specific, all platforms.

25. EB estimate

26. EB estimate
Shouldn't THIS be the true safety effect?

27. Professor, May I be excused? My brain is full.
Gary Larson, The Far Side, ©1986