The Empirical Bayes Method for Safety Estimation

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

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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.ctre.iastate.edu/educweb/CE552/docs/Bay
  es_tutor_hauer.pdf

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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?

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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

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Common Approach Use Last 3 Years of Crash Data
2008 2009 2010
Observed Crashes 30 19 21
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More Data Gives a Different Result
2001 2002 2003 2004 2005 2006 2007 2008 2009 2010
Observed Crashes 22 23 16 16 9 14 17 30 19 21
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RTM Example with Average Observed Crashes
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True Safety Impact of a Measure
3-year average before (Xa)
Long-term average (m)
Observed safety effect
True safety effect
3-year average after
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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

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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
or 5.7 of the mean
However, if there are relatively few crashes per
time period (say, 1 crash per 10 years) the
estimate varies greatly
or 180 of the mean!
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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

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

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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?

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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

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Empirical Bayes

• Increases precision
• Reduced RTM bias
• Uses information from the site, plus
• Information from other, similar sites

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Safety Performance Functions

• SPF Mathematical relationship between crash
  frequency per unit of time (and road length) and
  traffic volumes (AADT)

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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

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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
  f
• variance ?1?/(fL)
• Where
• µaverage crashes/km-yr (or /yr for
  intersections)
• ?µYL (or µY for intersections) number of
  crashes/time
• festimated by the regression (units must be
  complementary with L, for intersections, L is
  taken as one)

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SPF Example

• Regression model for total crashes at rural
  4-leg intersections with minor-road STOP control

Np exp(-8.69 0.65 lnADT1 0.47 lnADT2)
where Np Predicted number of
intersection-related crashes per year within
250 ft of intersection ADT1 Major-road
traffic flow (veh/day) ADT2 Minor-road
traffic flow (veh/day)
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Calculating the Long-Term Average Expected Crash
Frequency

• The estimate of expected crash frequency
• Ne w (Np) (1 w) (No)
• Weight (w 0ltwlt1) is calculated from the
  overdispersion parameter

Predicted Accident Frequency
Observed Accident Frequency
Expected Accident Frequency
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Weight (w) Used in EB Computations

• w 1 / ( 1 k Np)
• w weight
• k overdispersion parameter for the
• SPF
• Np predicted accident frequency for site

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Graphical Representation of the EB Method
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Predicting Future Safety Levels from Past Safety
Performance

• Ne(future) Ne(past) x (Np(future) / Np(past))
• Ne expected accident frequency
• Np predicted accident frequency

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Predicting Future Safety Levels from Past Safety
Performance

• The Np(future)/Np(past) ratio can reflect changes
  in
• Traffic volume
• Countermeasures (based on CMFs)

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CMFsHow 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

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CMFsHow to Use Them

• Expected crash frequencies and CMFs can be
  multiplied together

Ne(with) Ne(without) CMF Crashes Reduced
Ne(without) - Ne(with)
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CMFsSingle Factor

• CMF for shoulder rumble strips
• Rural freeways (CMFTOT 0.79)

Ne(with) Ne(without) x 0.79

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

• CMFs can be multiplied together if their effects
  are independent
• Ne(with) Ne(without) CMF1 CMF2

Are countermeasure effects independent?
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EB applications

• HSM
• IHSDM
• Safety Analyst

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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

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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

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• Questions?