The Empirical Bayes Method for Safety Estimation Doug Harwood MRIGlobal Kansas City, MO

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 /docs/Bayes_tutor_hauer.pdf

The Problem  You are a safety engineer for a highway agency. The agency plans next year to implement a countermeasure that will reduce crashes by 35% over the next three years. To estimate the benefits of this countermeasure, what safety measure will you multiply by 0.35?

What Do We Need To Know?  You need to know – or, rather, estimate – what would be expected to happen in the future if no action is taken  Then, you can apply crash modification factors (CMFs) for the known effects of planned actions to estimate their effects quantitatively

Common Approach: Use Last 3 Years of Crash Data Observed Crashes

More Data Gives a Different Result Observed Crashes

RTM Example with Average Observed Crashes

“True Safety Impact of a Measure” Long-term average (m) 3-year average ‘before’ (Xa) True safety effect Observed safety effect 3-year average ‘after’

Regression to the mean problem …  High crash locations are chosen for one reason (high number of crashes!) – might be truly high or might be just random variation  Even with no treatment, we would expect, on average, for this high crash frequency to decrease  This needs to be accounted for, but is often not, e.g., reporting crash reductions after treatment by comparing before and after frequencies over short periods

The “imprecision” problem … Assume 100 crashes per year, and 3 years of data, we can reliably estimate the number of crashes per year with (Poisson) standard deviation of about… However, if there are relatively few crashes per time period (say, 1 crash per 10 years) the estimate varies greatly … or 5.7% of the mean or 180% of the mean!

Things change…  BEWARE about assuming that everything will remain the same ….  Future conditions will not be identical to past conditions  Most especially, traffic volumes will likely change  Past trends can help forecast future volume changes

Focus on Crash Frequency vs. AADT Relationships: Use of Crash Rates May Be Misleading

The Empirical Bayes Approach  Empirical Bayes: an approach to estimating what will crashes will occur in the future if no countermeasure is implemented (or what would have happened if no countermeasure had been implemented)  Simply assuming that what occurred in a recent short-term “before period” will happen again in the future is naïve and potentially very inaccurate  Yet, this assumption has been the norm for many years

The Empirical Bayes Approach  The observed crash history for the site being analyzed is one useful and important piece of information  What other information do we have available?

The Empirical Bayes Approach  We know the short-term crash history for the site  The long-term average crash history for that site would be even better, BUT…  Long-term crash records may not available  If the average crash frequency is low, even the long- term average crash frequency may be imprecise  Geometrics, traffic control, lane use, and other site conditions change over time  We can get the crash history for other similar sites, referred to as a REFERENCE GROUP

Empirical Bayes  Increases precision  Reduced RTM bias  Uses information from the site, plus …  Information from other, similar sites

3-17 Safety Performance Functions SPF = Mathematical relationship between crash frequency per unit of time (and road length) and traffic volumes (AADT)

3-18 How Are SPFs Derived?  SPFs are developed using negative binomial regression analysis  SPFs are based on several years of crash data  SPFs are specific to a given reference group of sites and severity level  Different road types = different SPFs  Different severity levels = different SPFs

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 = 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)

3-20 Regression model for total crashes at rural 4-leg intersections with minor-road STOP control SPF Example where: N p = Predicted number of intersection-related crashes per year within 250 ft of intersection ADT 1 = Major-road traffic flow (veh/day) ADT 2 = Minor-road traffic flow (veh/day) N p = exp( lnADT lnADT 2 )

3-21 Calculating the Long-Term Average Expected Crash Frequency The estimate of expected crash frequency: N e = w (N p ) + (1 – w) (N o ) Weight (w; 0<w<1) is calculated from the overdispersion parameter Expected Accident Frequency Predicted Accident Frequency Observed Accident Frequency

Weight (w) Used in EB Computations w = 1 / ( 1 + k N p ) w = weight k = overdispersion parameter for the SPF N p = predicted accident frequency for site 3-22

3-23 Graphical Representation of the EB Method

Predicting Future Safety Levels from Past Safety Performance N e(future) = N e(past) x (N p(future) / N p(past) ) N e = expected accident frequency N p = predicted accident frequency 3-24

Predicting Future Safety Levels from Past Safety Performance 3-25  The N p(future) /N p(past) ratio can reflect changes in:  Traffic volume  Countermeasures (based on CMFs)

CMFs—How to Use Them  CMFs are expressed as a decimal factor:  CMF of 0.80 indicates a 20% crash reduction  CMF of 1.20 indicates a 20% crash increase

CMFs—How to Use Them  Expected crash frequencies and CMFs can be multiplied together: N e(with) = N e(without) CMF Crashes Reduced = N e(without) - N e(with)

3-28 CMFs—Single Factor  CMF for shoulder rumble strips  Rural freeways (CMF TOT = 0.79) N e(with) = N e(without) x 0.79

3-29 CMF Functions CMFs for Lane Width (two-lane rural roads) (Harwood et al., 2000)

CMFs for Combined Countermeasures  CMFs can be multiplied together if their effects are independent: N e(with) = N e(without) CMF 1 CMF 2 Are countermeasure effects independent?

EB applications  HSM  IHSDM  Safety Analyst

EB applications HSM Part C  Estimate long-term expected crash frequency for a location under current conditions  Estimate long-term expected crash frequency for a location under future conditions  Estimate long-term expected crash frequency for a location under future conditions with one or more countermeasures in place HSM Part B  Evaluate countermeasure effectiveness using before and after data

EB applications Site-Specific EB Method  Based on equations in this presentation Project-Level EB Method  If project is made up of components with different SPFs, then there is no single value of k, the overdispersion parameter EB Before-After Effectiveness Evaluation  See Chapter 9 in HSM Part B

Questions?