### Crash Modification Factors & Functions

```Spring 2013
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Crash modification factors (CMFs) are
becoming increasing popular:
◦ Simple multiplication factor
◦ Used for estimating safety improvement
programs (countermeasures)
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Also known as Crash Reduction Factors
(more positive)
Many states have developed CMFs
NCHRP Project 17-25
HSM and SafetyAnalyst will include
numerous CMFs
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http://www.cmfclearinghouse.org/
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http://safety.fhwa.dot.gov/tools/crf/resour
ces/fhwasa10032/
1960s
1970s
Highway Safety Act of 1966 required
DOTs to implement safety
improvement programs (SIP)
NCHRP Project 17-2A proposed a
model methodology for programming
safety improvements.
Report 162 recommended using a
simple before-after study for
developing CMFs (called CRFs at the
time)
1990s - now
Numerous research projects
conducted on the development of
CMFs
General Framework
Crash Modification Factor
A CMF is a multiplicative factor used to compute the
expected number of crashes after implementing a given
countermeasure at a specific site. The CMF is multiplied by
the expected crash frequency without treatment. A CMF
greater than 1.0 indicates an expected increase in crashes,
while a value less than 1.0 indicates an expected reduction
in crashes after implementation of a given countermeasure.
For example, a CMF of 0.8 indicates an expected safety
benefit; specifically, a 20 percent expected reduction in
crashes. A CMF of 1.2 indicates an expected degradation in
safety; specifically, a 20 percent expected increase in
crashes.
General Framework
Crash Modification Factor
Example
The CMF for installing a traffic signal at a rural stopcontrolled intersection is 0.23 for angle crashes (Harkey et
al., 2008). If a specific stop-controlled intersection is to be
converted to a signalized intersection and the expected
number of crashes at this intersection is 6.24 angle crashes
per year, the expected crash frequency after signalization
would be equal to 6.24*0.23 = 1.44 angle crashes per year.
Stated in terms of the expected change in crashes, the CMF
indicates a 77 percent (i.e., 100*(1 – 0.23)) expected
reduction in angle crashes after the installation of a traffic
signal.
General Framework
Crash Modification Function
A CMFunction is a formula used to compute the CMF for a specific site
based on its characteristics. It is not always reasonable to assume a
uniform safety effect for all sites with different characteristics (e.g.,
safety benefits may be greater for sites with high traffic volumes). A
countermeasure may also have several levels or potential values (e.g.,
improving intersection skew angle, or widening a shoulder). A crash
modification function allows the CMF to change over the range of a
variable or combination of variables.
Where possible, it is preferable to develop CMFunctions as opposed to a
single CMF value since safety effectiveness most likely varies based on
site characteristics. In practice, however, this is often difficult since more
data are required to detect such differences.
General Framework
Crash Modification Function
General Framework
Standard Error
The standard error is the standard deviation of a sample mean. The
standard error provides a measure of certainty (or uncertainty) in the
CMF. A relatively small standard error, with respect to the magnitude of
the CMF estimate, indicates greater certainty in the estimate of the CMF,
while a relatively large standard error indicates less certainty in the
estimate of the CMF. The standard error is used in the calculation of the
confidence interval. In some cases, the variance of the CMF may be
reported instead of the standard error. The standard error is simply the
square root of the variance as shown in equation below.
Standard Error = Variance
General Framework
Confidence Interval
A confidence interval is another measure of the certainty of a CMF. A
CMF is simply an estimate of the actual safety effect of a
countermeasure based on observations from a sample of sites. The
confidence interval provides a range of potential values of the CMF
based on the standard error. As the width of the confidence interval
increases, there is less certainty in the estimate of the CMF. If the
confidence interval does not include 1.0, it can be stated that the CMF is
significant at the given confidence level. If, however, the value of 1.0
falls within the confidence interval (i.e., the CMF could be greater than
or less than 1.0), it can be stated that the CMF is insignificant at that
confidence level. It is important to note insignificant CMFs because the
treatment could potentially result in 1) a reduction in crashes, 2) no
change, or 3) an increase in crashes. These CMFs should be used with
caution (AASHTO, 2010).
General Framework
Confidence Interval
A confidence interval is calculated by multiplying the standard error by a
factor (i.e., the cumulative probability) and adding and subtracting the
resulting value from the CMF estimate. The equation below is used for
calculating the confidence interval.
Confidence Interval = CMF ± (Cumulative Probability * Standard Error)
Confidence Interval
Cumulative Probability
99%
2.576
95%
1.960
90%
1.645
General Framework
Crash Modification Factor
Example
The CMF for a given countermeasure is 0.761 with a standard error of 0.168. An
engineer would like to calculate the 95 percent confidence interval for this CMF.
The first step is to determine the appropriate cumulative probability factor from
Table 1, given the desired confidence interval. The factor for a 95 percent
confidence interval is 1.960. The 95 percent confidence interval is then
calculated by adding and subtracting 1.96 times the standard error of 0.168
from the CMF estimate of 0.761.
95% Confidence Interval = 0.761 ± 1.960(0.168)
This gives a confidence interval of 0.432 to 1.090. Note the value of 1.0 lies
within the confidence interval. As such, it cannot be stated with 95 percent
confidence that the true value of the CMF is not 1.0 (i.e., it cannot be stated
with 95 percent confidence that the treatment had any effect).
General Framework
Crash Modification Factor
CMFs can be used by several groups of transportation
professionals for various reasons. The primary user groups
include highway safety engineers, traffic engineers, highway
designers, transportation planners, transportation
researchers, and managers and administrators. CMFs can be
used to:
• Estimate the safety effects of various countermeasures.
• Compare safety benefits among various alternatives and
locations.
• Identify cost-effective strategies and locations in terms of
crash effects.
• Check reasonableness of evaluations (i.e., compare new
analyses with existing CMFs).
• Check validity of assumptions in cost-benefit analyses.
General Framework
CMF is defined as:
Nw
CMF 
Nw/ o
N w = expected number of crashes with a treatment
or improvement.
N w / o = expected number of crashes without a treatment
or improvement.
Note: a number above one implies that an increase in
expected number of crashes is observed between sites
with treatment and without treatment
General Framework
Application for single improvement:
Na  CMF  Nb
Nb = expected number of crashes before a treatment
or improvement is implemented. (note: also called
base condition)
N a = expected number of crashes after a treatment
or improvement is implemented.
Note:
N  Nw/ o  CMF 1
Application for multiple improvements:
CMFc  CMF1  CMF2  CMF3 
 CMFn
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Simple/Naive Before-After Study
Before-After Study with Control Group
Before-After Study with EB method
Cross-sectional study
Case-control study
Cohort study
Meta-Analysis
Expert Panel
Coefficients of regression models
EXAMPLE
Source: Harwood et al. 2000 FHWA Report-RD-99-207
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How were units sampled for the study?
Do the data collected in the study refer directly to
the outcome of interest or to aggregated data?
Was crash or injury severity specified?
Were study results tested for statistical significance
or their statistical uncertainty otherwise estimated?
Did the study use appropriate techniques for
statistical analysis?
Can the causal direction between treatment and
effect be determined?
How well did the study control for confounding
factors?
Did the study have a clearly defined target group,
and were effects found in the target group only?
Are study results explicable in terms of wellestablished theory?
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Evaluation of Design Safety
Evaluation of Design Exceptions
Evaluation of Design Consistency
APPLICATION
Highway Design
A 3-mile two-lane rural highway segment linking two cities
is being designed. This segment contains one tangent
section without any vertical curves. The lane and shoulder
widths for the initial proposed design are 12 ft and 5 ft
respectively. The traffic volume is estimated at 5,000 vpd.
The designer has also identified an alternative design,
which consists of having 10-ft lanes with 7-ft shoulders. All
other conditions are the same for both designs. Determine
the change in safety between the initial and alternative
designs.
APPLICATION
Step 1: Estimate the Expected Number of Crashes for the
Base Conditions
Nbase  0.0002244  ADT  L
Nbase  0.0002244  5,000  3
Nbase  3.37crashes / year
Statistical model taken from upcoming Highway Safety
Manual: 12-ft lanes and 6-ft shoulder widths
APPLICATION
Step 2: Adjust Base Conditions to Reflect Existing Design
N w / o  ( AMFshoulderwidth )  Nbase
N w / o  ( AMFshoulderwidth )  3.37
N w / o  1.075  3.37
N w / o  3.62crashes / year
AMF taken from figure shown earlier.
APPLICATION
Step 3: Specify Design Change and Identify Appropriate
CMFs
CMFlanewidth  1.30
CMFshoulderwidth  0.935
CMFc  CMFlanewidth  CMFshoulderwidth
CMFc  1.30  0.935
CMFc  1.216
APPLICATION
Step 4: Compute Safety Change
 1.216 
N  3.62 
 1
 1.075 
N  0.47crashes / year
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Relationship between CMFs and base
conditions
◦ CMF is used to adjust for change in condition A to B
◦ The assumption is that B is not represented in the
base model parameter
◦ Difficult to determine whether the condition B was
included in the original statistical model
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Observed Crash Counts (EB)
◦ Not valid for a substantial change in highway
design (between different alternatives)
◦ Cannot be used for design exceptions
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Combination of Improvements
◦ CMFs are assumed to be independent
◦ In reality, highway improvements from which the
CMFs are estimated include several design
elements changes simultaneously.
◦ In addition, CMFs are probably not independent
(shoulder width versus lane width)
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Constant versus function
◦ Current CMFs are independent of exposure
◦ Not true, if we assume a non-linear relationship
between safety and traffic flow
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Crash Severity and Crash Type
◦ Certain improvements can also influence crash
severity or types.
◦ Current CMFs do not consider changes in severity
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Crash Migration
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Statistical Inferences
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Expert Panels/Meta-Analysis
◦ Def: Interventions influencing crashes at adjacent
sites (note: could also be for different time
periods or crash type)
◦ CMFs do not factor in crash migration
◦ Most research do not provide confidence intervals
◦ High uncertainty/high reduction versus low
uncertainty/low reduction
◦ Now, researchers have started to provide
◦ Subjective estimation (panel)
◦ Panel does not know characteristics of data (both)
◦ No confidence intervals available (panel)
Calculating the variance for the product of SPFs + CMFs
Based on the theory of Independent Random Variables
z  x1 x2 x3 
where,
z
= the product of independent random variables; and,
x' s
= random variable taken from any kind of distribution.
It should be pointed out that the mean and variance estimates are defined as
2
E  x   and E  x       (second central moment), respectively.
See Lord, D. (2008) Methodology for Estimating the Variance and Confidence
Intervals of the Estimate of the Product of Baseline Models and AMFs. Accident
Analysis & Prevention, Vol. 40, No. 3, pp. 1013-1017.
Mean of a Product
z  x1 x2 x3 
Ez  Ex1 Ex2 Ex3 
z  x1x 2x3
Variance of a Product
z x x x 
2
2
1
2 2
2 3
     
E z 2  E x12 E x22 E x32 
2
2
2
2
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 z z   x1 x1  x 2 x 2  x3  x3 
Note:
n
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E  x   E   x       
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n
for
n2
2
2
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E  x   E  x        
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2
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E  x      E 2  x     E  2      2
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Comparison:
Full Model:
1
 2 x2 3 x3 4 x4 
  0 LF e
Baseline Model:
   0 LF 
1
    CMF1  CMF2  CMF3
SCENARIO 1 − 12-ft lanes, 8-ft shoulders, 4 curves/mile
SCENARIO 2 − 12-ft lanes, 12-ft shoulder width, and 1
curve/mile
Comparison – Scenario 1 (KAB):
Comparison – Scenario 1 (KABCO):
Comparison – Scenario 2 (KAB):
Comparison – Scenario 2 (KABCO):
Comparison
Lord, D., P.-F. Kuo, and S.R. Geedipally (2010) Comparing the Application of
the Product of Baseline Models and Accident Modification Factors and Models
with Covariates: Predicted Mean Values and Variance. Transportation
Research Record 2147, pp. 113-122.
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