Accident Analysis and Prevention 54 (2013) 90–98

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Accident Analysis and Prevention

journal homepage: www.elsevier.com/locate/aap

Safety impacts of signal-warning flashers and speed control at

high-speed signalized intersections
Zifeng Wu a , Anuj Sharma a , Fred L. Mannering b,∗ , Shefang Wang a
a
Department of Civil Engineering and Nebraska Transportation Center, University of Nebraska-Lincoln, Lincoln, NE 68583, USA
b
Charles Pankow Professor of Civil Engineering, School of Civil Engineering and the Center for Road Safety, Purdue University, West Lafayette, IN
47907-2051, USA

a r t i c l e i n f o a b s t r a c t

Article history: For many years, to reduce the crash frequency and severity at high-speed signalized intersections, war-
Received 6 December 2012 ning flashers have been used to alert drivers of potential traffic-signal changes. Recently, more aggressive
Received in revised form 8 January 2013 countermeasures at such intersections include a speed-limit reduction in addition to warning flashers.
Accepted 29 January 2013
While such speed-control strategies have the potential to further improve the crash-mitigation effec-
tiveness of warning flashers, a rigorous statistical analysis of crash data from such intersections has not
Keywords:
been undertaken to date. This paper uses 10-year crash data from 28 intersections in Nebraska (all with
Speed limit reduction
intersection approaches having signal-warning flashers; some with no speed-limit reduction, and the
Crash frequency
Crash severity
others with either 5 mi/h or 10 mi/h reduction in speed limit) to estimate a random parameters negative
Nested logit model binomial model of crash frequency and a nested logit model of crash-injury severity. The estimation
Random parameter negative binomial findings show that, while a wide variety of factors significantly influence the frequency and severity of
model crashes, the effect of the 5 mi/h speed-limit reduction is ambiguous—decreasing the frequency of crashes
High-speed signalized intersection on some intersection approaches and increasing it on others, and decreasing some crash-injury sever-
Signal-warning flashers ities and increasing others. In contrast, the 10 mi/h reduction in speed limit unambiguously decreased
both the frequency and injury-severity of crashes. It is speculated that, in the presence of potentially
heterogeneous driver responses to decreased speed limits, the smaller distances covered during reac-
tion time at lower speeds (allowing a higher likelihood of crash avoidance) and the reduced energy of
crashes associated with lower speed limits are not necessarily sufficient to unambiguously decrease the
frequency and severity of crashes when the speed-limit reduction is just 5 mi/h. However, they are suffi-
cient to unambiguously decrease the frequency and severity of crashes when the speed-limit reduction
is 10 mi/h. Based on this research, speed-limit reductions in conjunction with signal-warning flashers
appear to be an effective safety countermeasure, but only clearly so if the speed-limit reduction is at least
10 mi/h.
© 2013 Elsevier Ltd. All rights reserved.

1. Introduction countermeasures that involve speed-limit reductions on intersec-

tion approaches and/or the implementation of warning flashers to
Traffic-safety data indicate that greater than 20% of all traf- provide drivers with additional time to make safer intersection-
fic fatalities in the United States occur at intersections. In 2010 related decisions (Antonucci et al., 2004).
alone, more than 6700 fatalities occurred at intersections in the With regard to speed-limit reductions in general, many stud-
U.S. (National Highway Traffic Safety Administration, 2012). While ies have been conducted to test the effectiveness of changes in the
many factors determine the likelihood of intersection crashes in speed limits due to regulations/laws, variable speed limits, dynamic
general, and fatal crashes in particular, signalized intersections message signs, and special transition zones (Buddemeyer et al.,
with high approach speeds are particularly notorious for generating 2010; Cruzado and Donnell, 2010; Monsere et al., 2005; Parker,
fatal crashes. At such high-speed intersections, studies have shown 1997; Son et al., 2009; Towliat et al., 2006; van den Hoogen and
that the frequency and injury-severity of crashes can be reduced by Smulders, 1994). Findings from these studies suggest that arbitrary
changes in speed limit (changes without a reason that is immedi-
ately obvious to drivers) have little impact upon driver behavior,
and may result in low compliance. However, a speed reduction for
∗ Corresponding author. Tel.: +1 765 496 7913.
certain special cases, such as dangerous curves, or adverse weather
E-mail addresses: ntc-zwu@unl.edu (Z. Wu), asharma@unl.edu (A. Sharma),
flm@purdue.edu, flm@ecn.purdue.edu (F.L. Mannering), sfwang2@gmail.com conditions, has often been shown to lead to a significant reduction
(S. Wang). in operational speeds, even though the magnitude is typically less

0001-4575/$ – see front matter © 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.aap.2013.01.016
 Z. Wu et al. / Accident Analysis and Prevention 54 (2013) 90–98 91

than the reduction of the posted speed limit. In addition, lowering paper will investigate the safety effects of speed-limit reductions at
the speed limit does not always improve safety because the speed- high-speed signalized intersections with signal-warning flashers,
limit reduction may be too modest for the larger crash-avoidance by considering their effects on crash frequencies and severities.
distances available to drivers and the reduced impact energies of
lower-speed crashes to have a significant effect on the frequency
and severity of crashes. Also, there is the possibility that some 2. Empirical setting
drivers may continue to travel at a speed that they perceive to be
reasonable and safe while others may attempt to comply with the The crash dataset available for this study consists of crash data
reduced posted speed limit—resulting in an increase in speed vari- from 28 intersections in Nebraska, collected over a ten-year period
ance that can completely offset the benefits of the reduced speed from January 1, 2001 to December 31, 2010. As done in previous
limit or in some cases actually result in more dangerous traffic research (for example, Poch and Mannering, 1996), each intersec-
conditions. For example, Boyle and Mannering (2004) found in a tion is broken up into approaches (lane groups at intersections
simulator study that drivers given in-vehicle speed recommen- such as northbound lanes, southbound lanes, eastbound lanes and
dations for adverse weather slowed down substantially relative westbound lanes) meaning that the typical intersection would
to those drivers who were not given such in-vehicle information. generate 4 observations. However, consideration is only given to
However, these in-vehicle-information drivers also sped up when intersection approaches on the primary highways (the higher vol-
the adverse conditions passed, to make up for lost time, caus- ume highways) with signal-warning flashers—which gives a total
ing high variances in speed during and after the hazard. And, in of 56 approaches in the dataset. The crash data were grouped for
other work, Malyshkina and Mannering (2008) found that increas- each approach of the primary highway at each intersection and
ing speed limits on interstate highways by 5 mi/h in Indiana did not 43 of the 56 approaches had no reduction in speed limit (i.e.,
result in an increase in crash-injury severities, partly because of the with 0 mi/h speed limit drop); nine approaches had a 5 mi/h speed
decline in speed variance at the higher speed limit. limit drop; and four approaches had a 10 mi/h speed limit drop.
In contrast to the speed limit reductions, signal-warning flashers Here, the uneven number of approaches for 0 mi/h reduction and
are designed to alert drivers of forthcoming yellow signal indi- 5 mi/h reduction resulted from one intersection having asymmet-
cation at the intersection, giving them more time to adjust their rical signal approach speed; its northbound approach had a 0 mi/h
speed accordingly. There have been a number of research efforts reduction while its southbound approach had a 5 mi/h reduction.
that have studied the effectiveness of these signal-warning flash- The number of crashes occurring in each year is considered
ers. For example, a study by Appiah et al. (2011) concluded that such for each observation so the 56 approaches produce 560 observa-
signal-warning flashers resulted in a 8% reduction in the number of tions because each approach has 10 years of crash data. However,
crashes. In other work, Burnett and Sharma (2011) found that the two intersections had a history of stop-controlled approaches, as
location and timing of signal-warning flashers were key determi- opposed to signalized approaches, within the 10-year study period.
nants in the risk of severe deceleration and/or red-light running at Thus, with these stop-controlled observations removed, there were
high-speed intersections—both of which are fundamental factors 536 observations for the approach-based annual crash-frequency
in determining the frequency and severity of crashes. However, to model.
date, the authors are not aware of any studies that have consid- With regard to the severity of crashes, the dataset includes
ered the joint effects of speed-limit reductions and signal-warning detailed police-reported crash data from 635 crashes that occurred
flashers at high-speed signalized intersections. during the study period. Each crash was documented together with
In terms of the implementation of speed-limit reductions and its crash characteristics, driver characteristics, and location-specific
signal-warning flashers at high-speed intersections, a survey of traffic characteristics including traffic control and traffic flow char-
eight U.S. states (Nebraska, Kansas, Iowa, Missouri, South Dakota, acteristics.
Wyoming, Colorado and California) indicated that they all used The main variables of interest were traffic-control characteris-
signal-warning flashers at high-speed intersections, and that the tics including yellow time, flasher time, and speed-limit reductions,
application of this technology is well supported by guidelines pro- which were studied by defining indicator variables in statistical
vided in the Manual of Uniform Traffic Control Devices (Federal models. The descriptive statistics of the variables found to be sig-
Highway Administration, 2009). In contrast, guidelines for imple- nificant in forthcoming crash frequency and crash severity models
menting speed-limit reductions at high-speed intersections do not are provided in Table 1.
exist, and states may not apply reductions unless there are sig- In Table 1, for the percentage of intersections with sufficient yel-
nificant intersection-related safety concerns, such as a history of low time, yellow time is considered sufficient if the actual yellow
frequent and/or severe-injury crashes or sight-distance restric- time is greater than the suggested yellow time which is calcu-
tions. lated as tr + S85 /(2a + 0.644G), where tr is the standard assumed
The presence of signal-warning flashers further complicates the perception-reaction time (1 s), S85 is the 85th percentile of speed in
issue surrounding the necessity and effectiveness of speed-limit ft/s, a is the standard assumed vehicle deceleration (11.2 ft/s2 ) and
reductions, and can produce a range of possible outcomes. The G is the grade in percent (see Institute of Transportation Engineers,
expected outcome would be that reduced speed limits would be 1985; Mannering and Washburn, 2013). Also, for the percentage of
effective in reducing operating speeds in the presence of signal- intersection approaches with an insufficient signal-warning flasher
warning flashers and thus enhancing overall safety. However, there time, flasher time is considered insufficient if actual flasher time
is the possibility of more complicated effects such as heterogeneous is less than the time required for the drivers driving at signal-
driver compliance with the reduced speed limit. Such heterogene- approach speed limit traveling from the flasher to the stop line
ity may be more likely to occur in the presence of signal-warning
flashers (as drivers may differ greatly in their assessment of the
safety benefits provided by both mitigation measures) and the net
effect of both of these countermeasures may be compromised.1 This case. For example, Wu et al. (2013) showed, by considering vehicle speeds before the
speed-limit reduction and after, that the impact of a 10 mi/h speed-limit reduction
(from 65 mi/h to 55 mi/h) at high-speed intersections with signal-warning flashers
in Nebraska (based on speed data collected from some of the same intersection
1
One key effect here is the possibility that the heterogeneous driver compliance approaches considered in the current paper) reduced mean operating speeds by
will result in an increase in speed variance. However, this does not appear to be the 3.8 mi/h without significantly changing the standard deviation of speeds.
92 Z. Wu et al. / Accident Analysis and Prevention 54 (2013) 90–98

Table 1
Descriptive statistics of crash-related variables.a

Variable Value

Crash-frequency data
Average annual crash frequency on intersection approaches (std. dev.) 1.13 (1.42)
Average percentage of truck volume on intersection approaches (std. dev.) 7.69 (6.12)
Average daily travel in vehicles per lane on intersection approaches (std. dev.) 1851.95 (911.29)
Average percentage of left-turn volume on intersection approaches (std. dev) 12.63 (13.79)
Percentage of intersection approaches with divided medians 83.21
Percentage of intersection approaches with a 5 mi/h reduction in speed limit 16.79
Percentage of intersection approaches with a 10 mi/h reduction in speed limit 7.46
Percentage of intersection approaches with sufficient yellow time (see text for definition). 33.58
Percentage of intersection approaches with an insufficient signal-warning flasher time (see text for definition) 70.15

Crash-severity data
Average daily travel in vehicles per lane on intersection approaches (std. dev.) 2028.43 (921.38)
Average percentage of truck volume on intersection approaches (std. dev.) 7.42 (6.40)
Average percentage of left-turn volume on intersection approaches (std. dev.) 11.14 (11.99)
Percentage of intersection approaches with divided medians 87.87
Percentage of intersection approaches with a 5 mi/h reduction in speed limit 13.86
Percentage of intersection approaches with a 10 mi/h reduction in speed limit 5.98
Percentage of intersection approaches with an insufficient flasher time 26.78
Percentage of intersection approaches with exclusive left turn lanes 94.49
Percentage of crashes classified as out-of-control crashes 5.2
Percentage of crashes classified as angle crashes 60.78
Percentage of crashes classified as head-on crashes 3.94
Percentage of crashes classified as rear-end crashes 30.08
Percentage of crashes classified as property damage only crashes 45.36
Percentage of crashes classified as possible-injury crashes 24.72
Percentage of crashes classified as visible-injury crashes 18.74
Percentage of crashes classified as incapacitating injury crashes 9.92
Percentage of crashes classified as fatality crashes 1.26
a
Variables will have different values in the crash-frequency and crash-severity data because the units of observation are different. In the crash-frequency case the number
of crashes on individual intersection approaches (the unit of observation) is considered, whereas in crash severity each individual crash (the unit of observation) is considered.
Substantial differences in the approach-specific values (such as traffic volume, speed-limit reductions, etc.) between the two data bases reflect the fact that the number of
individual crashes occurring on specific approaches can substantially change values in the injury-severity data since high crash-frequency approaches will be over represented
and low crash-frequency approaches will be under represented (this is in contrast to the crash-frequency where each approach has “equal” representation with one value
per approach per year).

(time required is the distance to the stop line in feet divided by the As is well known in the literature (Lord and Mannering, 2010),
speed limit of the approach in ft/s). a Poisson model may not always be appropriate because the
Poisson distribution restricts the mean and variance to be equal
(E[ni ] = VAR[ni ]). Crash-frequency data are typically overdispersed
3. Methodology
(E[ni ]  VAR[ni ]) so estimation with a Poisson model will result
biased parameter estimates. To account for this possibility, the
To assess the safety impacts of signal-warning flashers and
negative binomial model is often used. This model is derived by
speed control at high-speed signalized intersections, consideration
rewriting,
will be given to the frequency of crashes and then to the severity
of crashes once a crash has occurred. Turning first to the analy- i = EXP(ˇXi + εi ) (3)
sis of crash frequency, count-data modeling techniques have been
shown to be an appropriate methodological approach because the where EXP(εi ) is a Gamma-distributed error term with mean 1 and
number of crashes assigned to an intersection approach is a non- variance ˛2 . The addition of this term allows the variance to dif-
negative integer (see Lord and Mannering, 2010). These, count data fer from the mean with VAR[ni ] = E[ni ][1 + ˛E[ni ]] = E[ni ] + ˛E[ni ]2 .
are generally modeled with a Poisson regression or its derivatives The negative binomial probability density function is (Washington
which include the negative binomial and zero-inflated models (see et al., 2011):
Shankar et al., 1997; Lee and Mannering, 2002; Lord and Mannering,  1/˛   ni
1/˛  (1/˛) + ni i
2010; Washington et al., 2011). For the basic Poisson model, the P(ni ) = (4)
(1/˛) + i  (1/˛)ni ! (1/˛) + i
probability P(ni ) of intersection approach i having ni crashes per
year is, where  (.) is a gamma function. Note that the Poisson regres-
n sion is a limiting model of the negative binomial regression as
EXP(−i )i i ˛ approaches zero. Thus, if ˛ (often referred to as the dispersion
P(ni ) = (1)
ni ! parameter) is significantly different from zero, the negative bino-
mial is appropriate and if it is not, the Poisson model is appropriate
where i is the Poisson parameter for intersection approach i, which
(Washington et al., 2011).
is intersection approach i’s expected number of crashes, E[ni ]. Pois-
Random parameters can be introduced to account for possible
son regression specifies the Poisson parameter i (the expected
heterogeneity (unobserved factors that may vary across intersec-
number of accidents) as a function of explanatory variables by using
tions). In this case the model is structured so that each of the 28
the function,
intersections (each of which have two approaches) can have their
i = EXP(ˇXi ) (2) own ˇ (note that, with ten years of data and typically two of the
four intersection approaches having the signal-warning flashers,
where Xi is a vector of explanatory variables and ␤ is a vector of a typical intersection generates 20 observations). This is in con-
estimable parameters (Washington et al., 2011). trast to the traditional random parameters approach where each
 Z. Wu et al. / Accident Analysis and Prevention 54 (2013) 90–98 93

observation (in this case each year/intersection-approach combi- Kweon, 2002; Abdel-Aty, 2003; Yamamoto and Shankar, 2004;
nation) would get their own ˇ. The advantage of having a single Eluru et al., 2008; Savolainen and Mannering, 2007; Milton et al.,
parameter for the approaches in the same intersection (as opposed 2008; Malyshkina and Mannering, 2009; Christoforou et al., 2010;
to allowing each approach to have its own parameter) is that the Kim et al., 2010; Anastasopoulos and Mannering, 2011; Morgan
model takes into account additional information (the fact that the and Mannering, 2011; Ye and Lord, 2011; Patil et al., 2012; Xiong
approaches are from the same intersection and thus are likely to and Mannering, 2013). A review of crash-injury severity models
share many of the same unobserved effects). This additional infor- and methodological approaches can be found in Savolainen et al.
mation is traded off against the restriction being placed on the (2011). Studies have shown that the choice of one methodological
parameters (that they are constrained to be the same for each approach over another is often data dependent, although the para-
intersection approach in a given intersection). Subsequent model metric restrictions of the ordered probability models can preclude
estimations clearly show that constraining the approach parame- them as a feasible alternative (Savolainen et al., 2011).3
ters to be the same for each intersection is statistically justified.2 After extensive consideration of the standard multinomial logit,
To develop such a random-parameters model, individual estimable mixed logit and nested logit (Savolainen et al., 2011), the nested
parameters are written as (see Greene, 2007; Anastasopoulos and logit model provided the best overall statistical fit in current study.4
Mannering, 2009; Washington et al., 2011), The nested logit model is a generalization of the standard multi-
nomial logit model that overcomes the restriction that requires
ˇj = ˇ + ϕj (5)
the assumption that the error terms are independently distributed
where ϕj is a randomly distributed term for each intersection j, across injury outcomes. As shown in past work, this independence
and it can take on a wide variety of distributions such as the nor- may not always be the case if some crash-injury severity levels
mal, log-normal, logistic, Weibull, Erlang, and so on. Given Eq. (5), share unobserved effects (Savolainen and Mannering, 2007). For
the Poisson parameter i becomes i |␸j = EXP(␤Xi + εi ) in the neg- example, with the five injury categories that we will consider in
ative binomial model with the corresponding probabilities P(ni |␸j ) this paper (no injury, possible injury, visible injury, incapacitating
(see Eq. (4)). The log-likelihood function for the random parameters injury and fatality),5 it is possible that adjacent injury-severity cat-
negative binomial in this case can be written as, egories may share unobserved effects that relate to lower-impact
 collisions, thus violating the assumption that the error terms are
LL = ln g(ϕj )P(ni |ϕj )dϕj (6) independently distributed across outcomes, an assumption needed
∀i ϕj for the derivation of the standard multinomial logit model (see
McFadden, 1981). The nested logit model deals with possible corre-
where g(.) is the probability density function of the ϕj . lation of unobserved effects among discrete outcomes by grouping
Because maximum likelihood estimation of the random- outcomes that share unobserved effects into conditional nests.
parameters negative binomial models is computationally cumber- The outcome probabilities are determined by differences in the
some (due to the required numerical integration of the negative functions determining these probabilities with shared unobserved
binomial function over the distribution of the random param- effects canceling out in each nest. The nested logit model has the
eters), a simulation-based maximum likelihood method is used following structure for crash n resulting in injury outcome i (see
(the estimated parameters are those that maximize the simulated McFadden, 1981; Washington et al., 2011):
log-likelihood function while allowing for the possibility that the
variance of ϕj for intersection-level parameters is significantly EXP[ˇj|i Xjn ]
greater than zero). The most popular simulation approach uses Pn (j|i) =
(7)
∀J EXP[ˇJ|i XJn ]
Halton draws, which has been shown to provide a more efficient
distribution of draws for numerical integration than purely random

LSin = LN[ exp(ˇJ|i XJn )] (8)
draws (see Greene, 2007). ∀J
Finally, to assess the impact of specific variables on the mean
EXP[ˇi Xin + i LSin ]
number of crashes, marginal effects are computed (see Washington Pn (i) =
(9)
et al., 2011). Marginal effects are computed for each observation ∀I EXP[ˇI XIn + I LSIn ]
and then averaged across all observations. The marginal effects give
where Pn (i) is the unconditional probability of crash n having
the effect that a one-unit change in x has on the expected number
injury outcome i, X’s are vectors of measurable characteristics that
of crashes at each approach, i .
determine the probability of injury outcomes, ␤’s are vectors of
With regard to the injury-severity of crashes given that a crash
estimable parameters, and Pn (j|i) is the probability of crash n having
has occurred, discrete outcome models have been widely used. In
injury severity j conditioned on the injury severity being in injury-
this study, possible injury outcomes (the police-reported injury sta-
severity category i, J is the conditional set of outcomes (conditioned
tus of the most severely injured vehicle occupant in the crash)
include: no injury, possible injury, visible injury, incapacitating
injury, and fatality. To address this type of discrete outcome data,
over the years researchers have used a variety of methodologi- 3
As pointed out in Savolainen et al. (2011), ordered probability models are partic-
cal approaches including ordered probability models, multinomial ularly susceptible to under-reporting of less severe crashes and such models place
an often unrealistic restriction on the effect variables can have on crash-injury out-
logit models, nested logit models, mixed (random parameters) logit
comes. This is because traditional ordered probability models cannot allow a variable
models, dual-state multinomial logit models and finite-mixture to simultaneously decrease (or simultaneously increase) the probability of the low-
random-parameter models (Shankar et al., 1996; Duncan et al., est and highest severity levels (it should be noted that some recent work by Eluru
1998; Chang and Mannering, 1999; Khattak, 2001; Kockelman and et al. (2008) develops a generalized ordered probability model that relaxes the vari-
able restriction of standard ordered probability models). See Savolainen et al. (2011),
for further discussion of this point.
4
The mixed logit model did not produce any statistically significant random
2
The model estimation that constrained the parameters of approaches in the parameters at the 95% confidence level (only one parameter was found to be signif-
same intersection to be identical had a log-likelihood at convergence (LL(␤intersection )) icant even at the 90% confidence level). As will be shown, the standard multinomial
of −732.05 whereas the model that allowed all approaches to have their own param- logit could be statistically rejected relative to the nested logit model.
eter converged at (␤approach ) −760.67. The substantially higher value of LL(␤intersection ), 5
These severity levels follow the traditional “KABCO” scale: fatal injury or killed
clearly suggests that constraining the parameters of approaches on the same inter- (K), incapacitating injury (A), non-incapacitating (B), possible injury (C), and prop-
section to be identical provides a superior statistical fit. erty damage only (O).
94 Z. Wu et al. / Accident Analysis and Prevention 54 (2013) 90–98

in the model (−1104.04). The random parameter model esti-

mated is also superior to the simple fixed parameter model which
produced a much lower log-likelihood at convergence of −797.77.
Finally, the statistical significance of the dispersion parameter, ˛,
shows that it significantly different from zero and that the nega-
tive binomial model is appropriate relative to the simple Poisson
model.
Turning to specific parameter estimates, higher truck percent-
No-injury Lower intermediate injury Incapacitating Fatal Injury ages produce a positive parameter indicating that an increase
injury in truck percentages increases the frequency of crashes. This is
expected given that the poorer braking performance of trucks can
be expected to be problematic at high-speed intersections. The
marginal effects in Table 2 show that a 1% increase in truck percent-
age increases the mean number of crashes per year on the approach
by 0.0142.
Also, as expected, increases in the traffic volume per lane
Possible Visible increase the frequency of crashes on intersection approaches. Here,
injury injury marginal effects show that an increase in average traffic volume of
1000 vehicles per day per lane will increase the expected number
Fig. 1. Nested logit structure of the crash-injury severity model.
of crashes by 0.24 per year (see Table 2). As this number indicates,
any substantial increase in volume can be a real safety concern.
on i), I is the unconditional set of outcome categories, LSin is the Intersection approaches with divided medians were found to
inclusive value (logsum), and i is an estimable parameter. have higher crash frequencies with marginal effects showing that
For an example of a nested structure, consider a model that has a divided-median intersection approach has a 0.81 higher annual
correlation of unobserved effects among the intermediate injury crash frequency relative to undivided median approaches. Here,
outcomes of possible injury and visible injury. In this case, in the median space between opposing approaches is likely causing a
Eq. (9), the outcome categories i would include no injury, inca- problem by increasing the time required for left-turning vehicles,
pacitating injury, fatal injury, and a “lower intermediate injury” and vehicles traveling through the intersection on the minor road,
category (which would determine the unconditional probability of to cross the approach lanes and median to clear the intersection.
the crash resulting in a possible- or visible-injury outcome). The Sight distance may also be an issue with divided medians in some
lower-intermediate-injury category (Pn (i) in Eq. (9)) would include cases.
a LSin as the inclusive value (logsum) which would be the denom- The estimated parameter for the insufficient signal-warning
inator from the binary logit model estimated in Eq. (7) (as shown flasher-time indicator (see earlier definition) was found to be a
in Eq. (8)) with possible outcomes of possible injury and visible normally distributed random parameter with a slightly positive
injury conditioned on the fact that the crash resulted in a lower- but insignificant effect on average (the parameter mean). However,
intermediate-injury category (that is, possible injury and visible this parameter estimate did have a highly statistically significant
injury). Visually this model structure is shown in Fig. 1. standard deviation. Given the estimated standard deviation, the
Estimation of a nested model logit model is readily undertaken mean, and the normal distribution of parameters, we find that the
using a full information maximum likelihood approach that ensures presence of insufficient flasher-time increases crash frequencies
that variance-covariance matrices are properly estimated (this is at 57% of intersections and decreases crash frequencies at 43% of
in contrast to older sequential maximum likelihood estimation intersections. The variation in this parameter about zero suggests
approach which underestimated the variance–covariance matri- that the influence of insufficient signal-warning flasher times varies
ces resulting in an over estimation of the t-statistics of parameter considerably among intersection approaches and this may be due
estimates, see Greene, 2007 for additional details). to, among other factors, how local drivers react to flashers. Because
In comparing nested and un-nested logit models, it is impor- a large percentage of drivers on the intersection approaches are
tant to note that if the estimated value of i is not significantly likely regular users, this finding may be picking up site-specific
different from 1, the assumed shared unobserved effects in the anomalies among intersections or the possibility that the driver
lower-nest are not significant and the nested model reduces to a populations adjust to minor variations in signal-warning flashing
simple multinomial logit model (see Eqs. (8) and (9) with i ’s = 1). times in different ways and this would explain the plus/minus vari-
As was the case for the random-parameters negative bino- ation in this parameter estimate.
mial model, to assess the impact of specific variables on the The sufficient yellow-time indicator (see earlier definition) also
crash-severity probabilities, marginal effects are computed for resulted in a normally distributed random parameter with a statis-
each observation and then averaged across all observations (see tically insignificant mean and a significant standard deviation. In
Washington et al., 2011). Here, the marginal effects give the impact this case, intersections with sufficient yellow times had reduced
that a one-unit change in an explanatory variable, xi , has on the crash frequencies 55% of the time and increased crash frequen-
probability of crash injury-severity outcome i. cies 45% of the time. Once again this heterogeneous effect across
intersections may be the result of site-specific anomalies and/or
4. Estimation results: crash frequency adaptive driver behavior.
The percentage of total approach traffic making left turns also
The parameter estimation results and the corresponding aver- produced a normally distributed random parameter with a negative
age marginal effects are shown in Table 2. Table 2 shows that the mean (although statistically insignificant from zero). The distribu-
random-parameters negative binomial model estimation includes tion of parameters is such that higher left-turn percentages have
5 significant fixed parameters and 4 significant random parameters. a negative effect on crash frequencies at 59% of the intersections
Overall model fit is quite good as indicated by the log-likelihood at and a positive effect on crash frequencies at 41% of the intersec-
convergence (−732.05) which shows a very substantial improve- tions. It is again speculated that this variation is likely the result of
ment relative to the log-likelihood with only the constant included site-specific anomalies and diver adaptation.
 Z. Wu et al. / Accident Analysis and Prevention 54 (2013) 90–98 95

Table 2
Model estimation results for random parameters negative binomial model of intersection crash frequency (all random parameters are normally distributed).

Variable Parameter t-Stat. Average

estimate marginal effect

Constant −1.91 −7.74

Truck percentage 0.0193 2.07 0.0142
Average daily travel per lane (in thousands of vehicles) 0.33 5.51 0.24
Divided median indicator (1 if intersection approach has a divided median, 0 1.11 7.24 0.81
otherwise)
Insufficient signal-warning flasher-time indicator (1 if the actual signal-warning 0.14 (0.80) 1.31 (10.99) 0.10
flasher time is less than the time required for the drivers driving at signal-approach
speed limit traveling from the flasher to the stop line, 0 otherwise) (standard
deviation of parameter distribution)
Sufficient yellow time indicator (1 if the actual yellow time is greater than the −0.09 (0.70) −0.77 (6.86) −0.06
suggested yellow time, 0 otherwise; see text for definition) (standard deviation of
parameter distribution)
Percentage of approach traffic making left turns (standard deviation of parameter −0.0051(0.0231) −1.20 (6.67) −0.0037
distribution)
5 mi/h speed-limit reduction indicator (1 if speed limit is reduced by 5 mi/h, 0 −0.32 (0.72) −2.24 (4.91) −0.23
otherwise) (standard deviation of parameter distribution)
10 mi/h speed-limit reduction indicator (1 if speed limit is reduced by 10 mi/h, 0 −0.47 −2.00 −0.34
otherwise)
Dispersion parameter, ˛ 8.19 1.98

Number of observations 536

Log-likelihood with constant only −1104.04
Log-likelihood at convergence −732.05

Turning now to the specific variables of interest, the effect of 5 mi/h speed-limit reduction reduces crash frequencies at 67% of
various reductions in speed limit in the presence of signal-warning intersections and increases them at 33% of intersections. Here,
flashers, we find that a 5 mi/h reduction results in a normally dis- among potentially other factors relating to site-specific conditions,
tributed random parameter with a statistically significant mean there is the possibility that the 5 mi/h speed-limit reduction is sim-
of −0.32 and a standard deviation of 0.72. This suggests that the ply not sufficiently large enough to unambiguously decrease the

Table 3
Nested logit model for crash severity at high speed signalized intersections. Severity levels (see Fig. 1): NI = no injury (upper nest); PI = possible injury (lower nest), VI = visible
injury (lower nest), INI = incapacitating injury (upper nest), F = fatality (upper nest), and LII = lower intermediate injury (upper nest).

Severity level Variable Parameter estimate t-Stat.

Lower nest
PI Rear-end crash indicator (1 if the crash was a rear-end crash, 0 otherwise) 2.12 4.09
Left-turn lane indicator (1 if left-turn lane is present on the intersection approach, 0 otherwise) 1.20 2.01

VI Constant 1.79 2.04

5 mi/h speed-limit reduction indicator (1 if speed limit is reduced by 5 mi/h, 0 otherwise) −0.96 −2.10
Truck percentage −0.060 −2.02
Average daily travel per lane (in thousands of vehicles) 0.44 2.47
At-fault driver-age indicator (1 if the at-fault driver was more than 60 years old, 0 otherwise) 0.77 2.21
At-fault male-driver indicator (1 if the at-fault driver was male, 0 otherwise) −0.52 −1.90
Angle crash indicator (1 if the crash was an angle crash, 0 otherwise) −0.85 −1.77

Upper nest
NI Constant 1.07 2.34
Head-on indicator (1 if the crash was head-on crash, 0 otherwise) −1.19 −2.46
Divided median indicator (1 if intersection approach has a divided median, 0 otherwise) −2.10 −2.80
Sufficient yellow time indicator (1 if the actual yellow time is greater than the suggested yellow 0.67 3.56
time 0 otherwise; see Table 1 for definition)
10 mi/h speed-limit reduction indicator (1 if speed limit is reduced by 10 mi/h, 0 otherwise) 0.85 2.38
Multiple-vehicle indicator (1 if crash involved more than two vehicles, 0 otherwise) −1.14 −3.24

LII Percentage of approach traffic making left turns −0.0125 −1.70

Divided median indicator (1 if intersection approach has a divided median, 0 otherwise) −1.23 −1.63
Inclusive value (logsum) 0.24 −5.85a

InI Constant −3.28 −3.63

At-fault driver drinking indicator (1 if the at-fault driver had been drinking, 0 otherwise) 1.91 3.27
Angle crash indicator (1 if the crash was an angle crash, 0 otherwise) 1.34 3.50

F Constant −2.02 −1.61

Average daily travel per lane (in thousands of vehicles) −1.49 −2.23
At-fault driver drinking indicator (1 if the at-fault driver had been drinking, 0 otherwise) 2.06 1.80

Number of observations 635

Log-likelihood at zero, LL(0) −1071.61
Log-likelihood at convergence, LL(ˇ) −753.51
McFadden 2 (1 − LL(ˇ)/LL(0)) 0.30
a
As opposed to all other t-statistics which are computed as ˇ − 0 (since we are interested in whether the parameter is significantly different from zero) divided by the
standard error, the inclusive value t-statistic is computed as ˇ − 1 divided by the standard error, since the statistical difference from 1 indicates whether the nested structure
is valid as opposed to a traditional multinomial logit.
96 Z. Wu et al. / Accident Analysis and Prevention 54 (2013) 90–98

Table 4
Average marginal effects of the nested logit model for crash severity at high speed signalized intersections. Severity levels (see Fig. 1): NI = no injury (upper nest); PI = possible
injury (lower nest), VI = visible injury (lower nest), INI = incapacitating injury (upper nest), F = fatality (upper nest), and LII = lower intermediate injury (upper nest).

Variable NI PI VI INI F LII

Traffic-flow characteristics
Truck percentage −0.0052 −0.000724
Average daily travel per lane (in thousands of vehicles) 0.0383 0.0181 0.0053

Traffic-control characteristics
5 mi/h speed-limit reduction indicator (1 if speed limit is reduced by 5 mi/h, 0 otherwise) −0.0831 −0.0116
10 mi/h speed-limit reduction indicator (1 if speed limit is reduced by 10 mi/h, 0 otherwise) 0.196
Sufficient yellow time indicator (1 if the actual yellow time is greater than the suggested yellow 0.154
time 0 otherwise; see text for definition)
Divided median indicator (1 if intersection approach has a divided median, 0 otherwise) −0.486
Left-turn lane indicator (1 if left-turn lane is present on the intersection approach, 0 otherwise) 0.104 0.024

Driver characteristics
At-fault driver-age indicator (1 if the at-fault driver was more than 60 years old, 0 otherwise) 0.067 0.0093
At-fault male-driver indicator (1 if the at-fault driver was male, 0 otherwise) −0.045 −0.0063
At-fault driver drinking indicator (1 if the at-fault driver had been drinking, 0 otherwise) 0.160 0.025

Crash characteristics
Angle crash indicator (1 if the crash was an angle crash, 0 otherwise) −0.073 0.112 −0.010
Head-on indicator (1 if the crash was head-on crash, 0 otherwise) −0.276
Rear-end crash indicator (1 if the crash was a rear-end crash, 0 otherwise) 0.184 0.043
Multiple-vehicle indicator (1 if crash involved more than two vehicles, 0 otherwise) −0.265

frequency of crashes. That is, in the presence of potentially hetero- lower nest of lower-intermediate injuries (possible injury and vis-
geneous driver responses to decreased speed limits, the benefits ible injury) as depicted in Fig. 1.8 As shown in Table 3, the inclusive
drivers accrue from lower speeds (smaller distances covered during value (logsum) of the lower nest produced a parameter estimate of
reaction time, which allow a higher likelihood of crash avoid- 0.24 with a standard error of 0.13 which gives a t-statistic of −5.85
ance), at the 5 mi/h speed-limit reduction level, are not necessarily ([ˇ − 1]/s.e.) showing that the logsum’s parameter estimate is sig-
sufficient to unambiguously decrease the frequency of crashes.6 nificantly different from one, thus validating the form of the nested
However, this ambiguity seems to be resolved at the 10 mi/h speed logit relative to the standard multinomial logit and indicating the
limit reduction level. For the 10 mi/h speed-limit reduction indi- presence of shared unobserved effects between possible and visible
cator, the parameter is fixed and negative indicating a decrease in injury-severity categories.9
approach crash frequencies. In fact, the marginal effects in Table 3 As Tables 3 and 4 indicate, all parameter estimates are of plau-
show that this decrease is reasonably large with 0.34 fewer crashes sible sign and magnitude (as reflected in the computed marginal
per year for approaches that had a 10 mi/h reduction in speed limits effects). Turning specifically to the variables of interest (the
combined with signal-warning flashers (given that the mean num- speed-limit reduction indicators), the 5 mi/h speed limit reduction
ber of crashes at all intersection approaches is 1.13 crashes per year, indicator was only found to be significant in the visible-injury out-
0.34 crashes per year constitutes a significant safety improvement). come. Marginal effects in Table 4 show that a 5 mi/h speed-limit
This is an important finding in that it clearly shows that speed limit reduction reduces the probability of visible injury by 0.0831. This
reductions of at least 10 mi/h are needed to have an unambiguously implies that the probability of other injury categories (no injury,
positive effect on safety.7 possible injury, incapacitating injury, and fatality) all increase in
the presence of a 5 mi/h speed-limit reduction.10 As such, the net
5. Estimation results: injury severity effect of a 5 mi/h speed-limit reduction on crash severity is ambigu-
ous because it reduces the probability of visible injury, but increases
Table 3 shows the nested logit model estimation results and the the probability of other less severe and more severe crash-injury
corresponding marginal effects are presented in Table 4. After mul- outcomes.
tiple trials, the appropriate nested logit model formulation had a In contrast, the effect of the 10 mi/h reduction in speed limit
(whose indicator variable was found to be only significant in the

6
The possible increase in speed variance caused by the reduction in speed limit
8
could also be playing a role here in that the crash-avoidance benefits caused by lower This is in contrast to the earlier work of Savolainen and Mannering (2007) which,
speeds are being partially offset by increasing speed variances (making crashes more in their analysis of motorcycle-rider injuries, found the lowest injury-severity cate-
likely). However, as discussed in footnote 1, increasing speed variance is not statis- gories shared unobserved effects as opposed to the intermediate categories. This and
tically supported for the intersections in our sample for which speed data were other research suggests appropriate nesting structures tend to be quite data-specific
collected (Wu et al., 2013). in the case of injury-severity analyses.
7 9
There is the possibility that speed-limit reductions are more likely to be used Recall an inclusive value that is not significantly different from one indicates that
at intersection approaches with high crash frequencies. If this is the case, in the the model reduces to the standard multinomial logit model. It is also noteworthy
presence of omitted variables and unobserved heterogeneity, the parameter esti- that the inclusive value parameter is between zero and one, which is the range
mates of the speed-limit reduction indicators will be estimated with a upward bias needed for model validity (McFadden, 1981).
10
with regard to frequencies because the speed-limit indicators will be picking up Note that the fact that the 5 mi/h speed-limit reduction indicator was found
unobserved factors that make these approaches more likely to have high crash fre- to be significant only for the visible-injury outcome (an intermediate severity out-
quencies. Our review of speed-limit placement policies, rich model specification, come) is a further indication that an ordered probability model of crash-severity
and significant negative parameter estimates for speed-limit reduction indicators outcomes is not appropriate for these data. This is because ordered probability
suggest that the impact of this potentially non-random implementation of speed- model structures (such as the standard ordered probit model) do not allow for the
limit reductions is likely to be minimal. However, in the worst case, our findings possibility of variables influencing only intermediate outcomes. That is, they do not
can be considered as a lower bound of the effectiveness of speed-limit reductions. allow for the possibility that a variable can simultaneously decrease or simulta-
Please see Carson and Mannering (2001) for a discussion of the non-random imple- neously increase the extreme outcomes as is the case here—where the 5 mi/h speed
mentation of safety countermeasures with regard to the placement of ice-warning reduction indicator simultaneously increases the probability of no injury and fatality
signs in Washington State. crashes.
 Z. Wu et al. / Accident Analysis and Prevention 54 (2013) 90–98 97

no-injury outcome) has an unambiguous effect in that it increases flashers are an effective safety countermeasure, but only clearly so
the probability of a no-injury crash by a substantial 0.196 (as shown if the speed-limit reduction is 10 mi/h. As a final point, it should
in Table 4) and thus simultaneously decreases the probability of all be noted that the data used in this study only included speed-limit
of the more severe injury outcomes (visible injury, possible injury, reductions of 5 mi/h (from 60 to 55 mi/h and from 55 to 50 mi/h)
incapacitating injury, and fatality).11 and 10 mi/h (from 65 to 55 mi/h). A fruitful area for further research
These injury-severity findings corroborate the findings in the would be to consider the effect of different base speed limits and
crash-frequency model where it was found that the effect of a speed-limit reductions.
5 mi/h speed-limit reduction was also ambiguous—reducing crash
frequencies on 67% of the intersection approaches while increasing
Acknowledgements
crash frequencies on 33% of the intersection approaches.
Again, it is speculated that, in the presence of potentially het-
This work was supported by the Nebraska Department of Roads,
erogeneous driver responses to decreased speed limits, the larger
for which the authors are very thankful. The contents of this paper
distances covered during reaction time at lower speeds (allowing
reflect the views of the authors, who are responsible for the facts
a higher likelihood of crash avoidance) and the reduced energy of
and the accuracy of the data presented herein, and do not nec-
crashes associated with lower speed limits (and the lower speeds
essarily reflect the official views or policies of the sponsoring
at impact due to the additional reaction-time distance provided)
organizations, nor do the contents constitute a standard, specifi-
are not necessarily sufficient to unambiguously decrease the fre-
cation, or regulation.
quency and severity of crashes when the speed-limit reduction is
just 5 mi/h. However, they are sufficient to unambiguously decrease
the frequency and severity of crashes when the speed-limit reduc- References
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