Download Module 6 Statistics Review

STATISTICS REVIEW
Copyright © 2016 STC, UK
Basic Questions
 What to measure?


Number of crashes
Effectiveness of a measure
 Is change due to


Randomness?
Treatment?
Safety Evaluations
 Before and After
 Cross-Sectional
 Hazardous locations
 Multivariate crash models
Terminology
 Population
 Sample
 Descriptive statistics
 Parameter vs. Statistic
Data Descriptors
 Mean

Average number of distribution
 Median

Middle point of a distribution
 Mode

Most frequent number in a
distribution
 Variance
Data Descriptors Example
Consider the following speed
measurements:
45, 46, 51, 45, 48, 39, 46, 52, 43, 44
Mean 45.9 mph Median 45.5 mph
Mode 45.0 mph Variance 14.3 mph
Poisson Distribution
 Discrete variable distribution
 Probability of event occurrence
 Models


Crash data at a single site
Vehicle arrivals at a point
 Mean = Variance
Negative Binomial Distribution
 Discrete variable distribution
 Models

Crash data at several sites
Regression to the Mean (RTM)
How to Address RTM
 More data from similar sites
 More years of data
 Empirical Bayes
Empirical Bayes Method
 Conditional probability
 Number of expected crashes
 Weighted average estimate
based on


Crashes at site
Crashes from other sites (prediction)
Basic Equations
N = N1 w + N2 (1-w)
where:
N: estimated expected crashes
N1: crashes from other similar sites
N2: observed crashes at site
w: weight factor
w = 1/[1+(N1 Y/)φ]
where:
Υ: timeframe in years for crash counts
φ: overdispersion parameter
EB Example
(1/4)
 Road segment





1.8 mi long
ADT 4,000
12 crashes last year
N1 = 0.0224 (ADT0.564)crashes/mi
φ = 2.05/mi
 How many crashes should we
expect next year?
EB Example
(2/4)
 Step 1: Expected number of crashes
N1 = 0.0224(40000.564)(1.8)=4.34 crashes/mi-year
 Step 2: Weight estimation
N1 = 0.0224(40000.564)=2.41 crashes/mi
w=1/[1+(2.41)(1)/2.050]=0.460
 Step 3: Estimated number of crashes
N = 0.460(4.34)+0.540(12)=8.48 crashes/mi-year
𝜎=
1−𝑤 𝑁 =
crashes/mi-year
(1 − 0.460(8.48) = ±2.14
EB Example
(3/4)
 Assuming availability of three
years of data


12, 7 and 8
Same ADT for each year
 How many crashes should we
expect next year?
EB Example
(4/4)
 Step 1: Expected number of crashes
N1 =0.0224(40000.564)(1.8)(3)=13.01 crashes 3 yrs
 Step 2: Weight estimation
w=1/[1+(2.41)(3)/2.050]=0.220 crashes/mi
 Step 3: Estimated number of crashes
N = 0.220(13.01)+0.780(27)=23.92 crashes 3 yrs
𝜎=
1−𝑤 𝑁 =
(1 − 0.220(23.92) = ±4.32
crashes/mi-year 3 yrs
EB Example Summary
 One year data

8.48 ± 2.14 crashes/mi-year
 Three years of data


23.92 ± 4.32 crashes in 3 years
(23.92 ± 4.32)/(3x1.8)= 4.43 ± 0.80
crashes/mi-year
Safety Performance Function
 Relationship of crashes to other
variables
 Basic question

What is the expected number of
crashes?
 Forms



Linear
Non-linear
Multi-variate
Y= a+bx
Y=ea+bx
Y=a0+a1x1+a2x2+…+anxn
Regression Basic Stats
 Form


Y = a+ b X
b slope of line; a intercept
 Basic tests



Is b = 0?
How good is the model?
Is a = 0?
Regression Output
Model Summaryb
Model
1
R
R Square
a
.596
.355
Adjus ted
R Square
.333
Std. Error of
the Es timate
104.27217
R2 35% of observed variation
a. Predictors : (Constant), DRIVEXP
b. Dependent Variable: SCORE
SC=1023.5 +55.848 DR
a
Coefficients
Model
1
(Cons tant)
DRIVEXP
Uns tandardized
Coefficients
B
Std. Error
1023.500
43.620
55.848
13.751
a. Dependent Variable: SCORE
Standardi
zed
Coefficien
ts
Beta
.596
t
23.464
4.061
Sig.
.000
.000
b & a not 0
Approach Steps for Studies
(1/2)
 Identify area of interest
 Collect background information
 Form potential research hypotheses
 Gather/identify data
 Form a single research hypothesis to
be tested with available data
 Perform a descriptive and investigative
analysis of the data
Approach Steps for Studies
 Perform the test(s)
 Compare results to past results
 Develop potential explanations
 Discuss limitations of the findings
 Discuss applicability of findings
 Determine future research
(2/2)