Traffic Engineering Studies - Module - 2

Statistical Methods In Traffic Safety Analysis – Regression Methods,
Poisson Distribution, Chi- Squared Distribution, Statistical Comparisons- Traffic Management Measures And Their Influence On Accident Prevention.

1. Module - 2
Traffic Engineering Studies
Statistical Methods in Traffic Safety Analysis:
1. Regression Methods
Regression methods are statistical tools that help in finding the relationship between the
number of accidents and various factors that may influence them. The principle is that the
expected number of accidents depends on certain conditions in a predictable way. Engineers
use regression to identify which factors have more impact on accident frequency.
Examples of factors considered:
• Number of two-wheeler vehicles on the road.
• Volume of traffic during night hours.
• Road surface conditions (icy or wet roads).
• Number of pedestrians.
• Pavement width, number of junctions, and road curvature.
Example Explanation: If regression analysis shows that areas with higher pedestrian movement
have more accidents, then engineers may design better crosswalks or signals. Similarly, if
narrow roads show more accidents, widening them can improve safety.
The regression model is of the following form :
Z = a0 + a1 x1 + a2 x2 + a3 x3 + . . . +an xn
Z = accident rate
a1, a2, … an = regression coefficients
x1, x2, ... xn = independent variables
a0 = regression constant.

2. Numerical Example:
1. A city wants to study the effect of traffic volume and pedestrian count on the number of
accidents at a junction. The regression model is given as:
Z = 5+0.01 x1+0.2 x2
Where:
• Z = predicted accident rate per year
• x1 = number of vehicles per day
• x2 = number of pedestrians per day (in 100s)
• a0 = The base accident risk is 5 (constant).
If the junction has 2000 vehicles/day and 300 pedestrians/day, find the expected number of
accidents per year.
Sol:-
Z = 5 + (0.01×2000) + (0.2×3)
Z = 5 + 20 +0.6 = 25.6 Z = 5 + 20 + 0.6 = 25.6Z=5+20+0.6=25.6
Predicted accidents = 26 per year (approx.)
2. Poisson Distribution
Poisson distribution is a statistical method used to predict the probability of rare events. Road
accidents are not daily occurrences for every driver, so they are treated as rare events. Poisson
distribution helps estimate the chances of a certain number of accidents happening over time
or distance.
Example 1: If the probability of a driver having an accident is 1 in 100 per year, and a company
has 500 drivers, Poisson distribution can calculate the likelihood that exactly 4 drivers will
have accidents in a year.
Example 2: Suppose a driver travels 5000 km every year and the chance of accident per
kilometre is very small. Poisson can estimate the probability that this driver will face 0, 1, or 2
accidents in 25 years.
Why it is used: Since accidents are unpredictable and rare, Poisson helps provide a
mathematical basis for safety planning.

3. The mathematical formula for the Poisson distribution is :
P (r) =
�
−�
. �
�

�!

Where:
• P(r) = Probability of exactly r accidents occurring.
• m = Average number of accidents expected. (average number of accidents occurring in
a year at a given location.)
• r = Actual number of accidents we are calculating probability for.
• e = constant, base of natural log.
Different conditions how ‘m’ can be calculated or considered
1. Directly given
Sometimes the problem states: “On average, there are 6 accidents per year.”
Here, m = 6
2. From probability per unit (p) and exposure (M)
When accident probability per km or per unit is given:
m = p × M
p = probability of accident per km (or per unit)
M = distance travelled (or total exposure)
Example: The probability of a vehicle having an accident is 1 accident per 10 million km.
A driver travels 50,000 km in 1 year.
p =
1
1,00,00,000

M = 50,000 km
m = p × M = 0.005
3. From total accidents divided by years (or units)
If we are given accidents over multiple years:
m =
??????���� ??????��??????�����
??????����� �� �����

Example: In a city, accidents in 3 years were 5, 7, and 9.
m =
5+7+9
3
= 7

4. 4. Driver Probability × Number of Drivers
If each driver has a probability ‘p’ of accident in a given year, and there are ‘N’ drivers:
m = p × N
Example 1: Each driver has a 1% chance (0.01) of being in an accident per year.
If there are 800 drivers, what is the expected number of accidents (m) in a year?
m = p × N
= 0.01 x 800
m = 8
Example 2: If 1 in 100 drivers has an accident each year and there are 500 drivers?
m = p × N
= (1/100) x 500
m = 5
Summary:
1. If it says “average accidents per year” → take it directly.
2. If probability per km is given (p is per km) → m = p × M
3. If multiple years are given (p is per driver) → average them.
4. If driver probability is given → m = N × p.
Applying the Poisson distribution formula to determine the probability of a driver
causing an accident, let:
• N = the number of drivers
• M = the kilometres driven by each driver
• P = the probability of having an accident per kilometre travelled
• m = the average rate of occurrence of accidents
The probability number of driver having r accidents
P (r) = ??????
�
−�
. �
�

�!

5. Numerical Example:
1. The accident records for three uncontrolled consecutive years at an uncontrolled junction
indicate the following number of accidents :
Year No. of Accidents
1972 3
1973 6
1974 9
Calculate the probability of 4 accidents occurring at the site.
Sol:-
Step 1: Average accidents per year:
m =
3+6+9
3
= 6
Step 2: Probability of accidents for r = 4:
P (r) =
??????
−??????
. ??????
??????

??????!

P (4) =
??????
−6
. 6
4

4!
=
0.002478 X 1296
24
= 0.1339
2. A fleet has 2,000 drivers. Each driver’s chance of having at least one accident this year is
0.2% (p=0.002). What is the probability that at least 3 drivers have an accident this year?
Step 1: Average number of accidents
m = n x p = 2000 x 0.002
m = 4
Step 2: Probability of at least 3 accidents in year – (Using the complement rule)
P (r = at least 3) = 1− [ P (r = 0) +P (r = 1) + P (r = 2) ]
= 1 – [
??????
−4
?????? 4
0

0!
+
??????
−4
?????? 4
1

1!
+
??????
−4
?????? 4
2

2!
]
P (r = at least 3) = 0.761

6. 3. It is observed that on an average a vehicle driver drives 5000 km during the course of a
year. The probability of having an accident is 100 per 200 million vehicle-kilometers. What
is the probability of a driver having at least two accidents during his driving career
extending to 25 years?
Sol:-
• A driver drives 5000 km per year.
• Probability of accident = 100 per 200 million km.
• Driving career = 25 years.
• Find probability of at least 2 accidents.
Step 1: Find accident probability per km
p =
100
200,000,000
= 0.0000005
Step 2: Total distance in 25 years
M = 5000×25
M = 125,000  km
Step 3: Average number of accidents expected
m = p×M
= 0.0000005×125,000
m = 0.0625
So, on average, the driver will face 0.0625 accidents in 25 years (less than 1).
Step 4: Probability of at least 2 accidents – (Using the complement rule)
P (r = at least 2) = 1− [ P (r = 0) + P (r = 1) ]
= 1 – [
??????
−0.0625
. 0.0625
0

0!
+
??????
−0.0625
. 0.0625
1

1!
]
= 1 – [ ??????
−0.0625
+ { ??????
−0.0625
x 0.0625 } ]
= 0.001874
4. It has been found that on an average 1 in 100 drivers in a bus company are involved in an
accident every year. If there are 500 drivers in the company, what is the probability that
there are exactly 4 drivers who are involved in an accident during a year?

7. Sol:-
Step 1: Find accident probability per driver
p =
1
100
= 0.01
Number of drivers : n = 500
Step 2: Average number of accidents expected
m = np = 500 x 0.01
m = 5
Step 3: Probability of accident for 4 drivers involved in accident : r = 4:
P (r) =
??????
−??????
. ??????
??????

??????!

P (4) =
??????
−5
. 5
4

4!
=
0.0183 X 625
24
= 0.48
5. A company has 500 drivers. The accident rate is 2 per 100,000,000 km. Each driver travels
20,000 km in one year. In that one year, how many drivers are expected to have exactly 1
accident?
Sol:-
Step 1: Find accident probability per km
p =
2
100,000,000
= 2 x 10
-8
Step 2: Average number of accidents expected
m = p x distance = 2 x 10
-8
x 20,000
m = 0.0004
Step 3: Probability of drivers involved in exactly 1 accident : r = 1, N = 500
P (r) = N
??????
−??????
. ??????
??????

??????!

P (1) = 500 ??????
??????
−0.0004
. 0.0004
1

1!
= 0.20 driver

8. 3. Chi-Square Test
The Chi-Square test is used to check whether changes in accident numbers are due to actual
improvements or just random chance. It is often used to evaluate the effectiveness of new safety
measures (improvements).
Example: A junction had 20 accidents per year. After traffic signals were installed, accidents
reduced to 8 per year. But traffic volume in the city also grew by 10%. The Chi-Square test
helps to decide whether the reduction was really due to the signals or just due to normal
variations in traffic.
How it works: It compares 'before' and 'after' data, while also considering other influencing
factors like increase in traffic. If the test result shows a significant difference, it proves the
safety measure is effective.
The value of Chi-squared is
??????
2
=
(�−�??????)
2

(�+�)??????

Where:
a = number of accidents after improvements
b = number of accidents before improvements
C = control ratio (accounts for growth in traffic/weather)
Assuming a 5% level of significance, from below figure, we find the value x
2
to be 3.841 with
one degree of freedom. If x² > 3.841, indicates that any improvement has significant impact /
change in road safety. If x² < 3.841, indicates that any improvement has no significant impact
/ change in road safety.
Simple Analogy: It is like testing a new medicine. If patients recover faster with the new
medicine compared to without it, the Chi-Square test confirms the medicine actually works,
rather than the recovery being due to luck.

9. Fig: Critical Values for Chi-Square Distribution

10. Numerical Example:
1. (Comparison - Before and After)
In an ordinary square junction of two roads there were 20 accidents in a year. After provision
of traffic signals, the number of accidents dropped down to 8 per year. In the sector of the city
where this junction is situated, the general trend observed was that number of accidents
increased at a rate of 10 per cent during the period covered by the above two observations.
Test whether the improvement in junction design has a significant effect at 5% significance
level.
Sol:-
After signals : a = 8 accidents
Before signals : b = 20 accidents
Accident increase by 10% i.e, 110%
C =
110
100
= 1.1
??????
2
=
(&#3627408462;−&#3627408463;??????)
2

(&#3627408462;+&#3627408463;)??????
=
(8 − 20 ?????? 1.1)
2

(8 +20)1.1
=
196
30.8

&#3627408537;
??????
= 6.36
• From Chi-Square table: ??????
2
for 5% significance level and 1 degree of freedom is 3.841.
• Since 6.36 > 3.841, the improvement in junction design has a significant effect.
2. (Comparison – Year to Year)
The accident data pertaining to a metropolitan city for the year 1965 and 1970 are given below
:.
Year 1965 1970
Accidents 300 400
Vehicle-kilometre of travel 250 million 300 million
Test whether there is any significant increase in the accident rates in 2 years.
Sol:-
After signals : a = 400 accidents
Before signals : b = 300 accidents
Here, the traffic has increased from 250 to 300 million vehicle kilometres, and thus the
control rate C is

11. C =
300
250
= 1.2
??????
2
=
(&#3627408462;−&#3627408463;??????)
2

(&#3627408462;+&#3627408463;)??????
=
(400 − 360 ?????? 1.2)
2

(400+360)1.2
=
1600
840

&#3627408537;
??????
= 1.90
• From Chi-Square table : ??????
2
for 5% significance level and 1 degree of freedom is 3.841.
• Since 1.90 < 3.841, the increase in the accident rates in 2 years is not significant.

3. (Comparison – Junction to Junction)
In a year, Junction A had 25 accidents with 15 million vehicle-km of traffic. Junction B had
30 accidents with 25 million vehicle-km of traffic. Which junction is more dangerous?
Sol:-
Compute accident rate per million vehicle-km.
• Junction A:
RateA= 25 / 15 = 1.67 accidents per million km
• Junction B:
RateB = 30 / 25 =1.20 accidents per million km
Even though total accidents are higher at B (30), the rate is higher at A (1.67 > 1.20).
Advantages and Limitations of Statistical Analysis Methods in Accident Studies
1. Poisson Distribution
Advantages
• Suitable for rare and random events such as road accidents.
• Simple to apply and requires only the average number of crashes (m).
• Helps identify whether accident occurrence at a location is within normal variation or
unusually high.
• Useful for black-spot analysis and preliminary screening of accident-prone sites.
Limitations
• Assumes that the average accident rate is constant, which may not be true in real traffic.
• Ignores the influence of factors like road width, traffic flow, or weather.
• Not reliable for high-frequency crash sites (variance not equal to mean).
• Cannot explain why crashes occur, only the probability of occurrence.

12. 2. Regression Analysis
Advantages
• Relates accident frequency with multiple influencing factors such as traffic volume,
curvature, and pavement width.
• Useful for predicting future crash rates and identifying significant causes.
• Helps quantify the effect of each variable on accident frequency.
• Supports data-driven decision-making for safety improvement and design changes.
Limitations
• Requires a large and reliable data set for accurate results.
• Sensitive to errors in data; poor correlation leads to misleading results.
• Assumes a linear relationship between variables unless modified.
• Does not capture the randomness of crash occurrence (handled better by Poisson).
3. Chi-Square Test
Advantages
• Simple and effective for before–after studies of safety measures.
• Tests whether a change in accident frequency is statistically significant.
• Works well for categorical or frequency data (for example, with treatment or without
treatment).
• Useful for evaluating countermeasure effectiveness and comparing locations.
Limitations
• Only tells if a difference is significant, not how large or why it occurred.
• Needs a minimum sample size (expected frequency of at least 5).
• Cannot handle continuous data or predict accident rates.
• Sensitive to errors or bias in data collection.

13. Statistical Comparison in Traffic Engineering
Statistical comparison means using numbers and simple tests to check if two sets of traffic data
are really different or not, or if the difference is just by chance.
Common Methods
1. Descriptive Statistics
• Measures mean, median, and standard deviation.
• Example: Studies the average speed, variation in traffic flow.
2. t-Test
• Compares two averages.
• Example: Compares speed before vs. after adding a speed hump.
3. Chi-Square Test
• Analyzing accident frequency by vehicle type or location.
• Example: Types of accidents at Junction A vs. Junction B.
4. ANOVA (Analysis of Variance)
• Compares averages across more than two groups.
• Example: Delay (waiting/travel time) at three different intersections.
5. Regression Analysis
• Finds relationship between factors.
• Example: How accident frequency depends on traffic volume, road width, or
weather.
6. Poisson Models
• Used for rare events (like crashes).
• Example: Predicting the number of accidents in a year on a road.
Applications in Traffic Engineering
1. Before-and-After Studies
• Example: Comparing accident severity before and after installing a pedestrian
overpass.
2. Comparative Evaluation
• Example: Which intersection design (roundabout vs. signalized) reduces delays
more effectively?
3. Safety Impact Analysis
• Example: Testing if helmet enforcement significantly reduces motorcycle fatalities.

14. 4. Policy Evaluation
• Example: Speed limit reduction effectiveness.
Traffic Management Measures and Their Influence on Accident Prevention
• The fundamental approach in traffic management measures is to retain as much as
possible existing pattern of streets, but to alter the pattern of traffic movement on these,
so that the most efficient use is made of the system.
• In doing so, minor alterations to traffic lanes, islands, and curbs are inevitable under
part of management measures.
• The general aim is to reorient the traffic pattern on existing streets so that the conflict
between the vehicles and the pedestrians is reduced.
Traffic Management Measures and Their Influence on Accident Prevention are as below;
a) Travel Demand Management (TDM)
Restricts the use of private vehicles and encourages public transport, car-pooling, park-
and-ride schemes, and road pricing.
Influence :
• Reduces the number of vehicles on roads, thereby cutting down congestion and
potential conflicts that cause accidents.
b) One - way Streets
Narrow streets are common in the congested parts of cities, particularly old cities. Bi-
directional traffic in narrow streets is difficult, particularly under Indian conditions.
Under this condition, traffic is permitted in only one direction in the street.
Influence:
• Reduces conflict points at intersections.
• Eliminates head-on collisions.
• Improves flow speed.
• Improvement in parking facility.
c) Prohibited Right Turn
At an intersection, especially where a minor road meets a major road, left-turning traffic
is easy to manage, whereas right-turning poses traffic problems of conflict and loss of
capacity. Traffic intending to turn right is made to go to the most favorable intersection.

15. Another way of dealing with it is to introduce an early cut-off or late start arrangement
of traffic signals.
Influence:
• Prevents dangerous crossing/turning conflicts at intersections, reduces delays,
and lowers crash chances.
d) Tidal Flow Operation
Traffic is predominantly towards the city centre in the morning, the reverse is true in
the evening. In peak hours, more lanes are given to the heavy traffic direction and fewer
to the lighter side.
Example: In a 6-lane road, 4 lanes for the busy direction, 2 lanes for the other.
Influence:
• Reduces congestion, avoids unsafe overtaking, and ensures orderly flow.
e) Exclusive Bus Lanes
Separate lanes marked only for buses, and restricted for other vehicles. This is possible
only in situations where the carriageway is of adequate width and a lane can be easily
spared for buses.
Influence:
• Makes buses safe from mixed traffic and promotes public transport.
f) Closing of Side – Streets
Main Street may have a number of side streets where the traffic may be very light. In
such situations, it may be possible to close some of these side streets without adversely
affecting the traffic, and yet with a number of benefits.
Influence:
• Since the interference from the traffic from the side streets is eliminated, the
speed increases and the journey time reduces.
• The accidents get reduced the closed side streets can be utilised for parking of
vehicles.
g) Pedestrianisation
Certain busy city areas (markets, shopping streets) are closed to vehicles and kept only
for pedestrians.

16. Influence:
• Prevents pedestrian–vehicle conflicts, lowers pedestrian accidents, and
improves environmental safety.
How can their effectiveness be quantified statistically?
Traffic management measures help reduce congestion and improving safety.
Their effectiveness should be quantified using statistical data to ensure that the observed safety
improvements are real and measurable.
The following methods are commonly used for this evaluation:
Before–After Crash Studies
• Compare the number of accidents before and after a safety measure is applied.
• If the number of crashes reduces after the change, it shows the measure is working.
Chi-Square Test
• Used to check if the reduction in crashes is meaningful or just random.
• A higher Chi-square value means the measure has a real effect on reducing accidents.
Regression Method
• A simple mathematical approach to predict expected accidents based on road and traffic
conditions.
• If the actual accidents are less than the predicted number, the measure is considered
successful.
Poisson Method
• Used when accidents are random and occur rarely.
• It helps calculate the probability of accidents happening after applying a safety measure.
• If the chance of accidents becomes much lower, the safety improvement is confirmed.