The Empirical Bayes Method for Before and After Analysis

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. 126-131. National Academies Press, Washington, D.C.. 2002 http://www.engr.uky.edu/~rsouley/CE%20635/docs/Bayes_tutor_hauer.pdf Open This Document and read through as you go along on PPT

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

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

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”

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)

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

Example 1: road segment, 1 yr. of data

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

Example 1 (cont): = 4.71 ± 1.19 accidents/km/year Note that the variance of the EB estimate, 2.142 = 4.58, is less than the variance of the accident counts on the segment, 9.44 = 4.71 ± 1.19 accidents/km/year

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 = 7.97 ± 1.44 accidents/year for the section (compare to previous estimate and reliability) From ex 1) 8.48 ± 2.14

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

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

Large for fatals (helps you not to “chase” them Example 5: Severity 2.41 x 0.019 = 0.046 … 1.8 x 3 x 0.046 = 0.247 Note: φ stays same (mult dist. by constant); Large for fatals (helps you not to “chase” them Note: 20.357 ≠ 23.9 (from prob 2) … why? What is the suggested an ad hoc solution?

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?

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

Example 7: (cont)

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?

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

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

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?

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

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

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). http://www.r-project.org/ (info on “R”) Download R 2.15.0 for Windows (47 megabytes, 32/64 bit) Installation and other instructions New features in this version: Windows specific, all platforms.

EB estimate

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

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