Before-After Studies Part II

Before-After StudiesPart II

Fall 2015

Before-After StudiesRecap: we need to define the notation that will be used for performing the two tasks at hand.Let:

be the expected number of target crashes of a specific entity in an after period would have been had it not been treated; is what has to be predicted.

be the expected number of target crashes of a specific entity in an after period; is what has to be estimated.

Before-After StudiesThe effect of a treatment is judge by comparing and . The two comparisons we are usually interested are the following:

the ratio of what was the treatment to what it would have been without the treatment; this is defined as the index of effectiveness.

the reduction in the after period of the expected number of target crashes (by kind and severity).

Before-After StudiesThe estimation of the safety of a treatment is done through a 4-step process. This step is done for each entity.

STEP 1: Estimate and predict . There are many ways to estimate or predict these values. Some will be shown in this course.

STEP 2: Estimate and . These estimates depend on the methods chosen. Often, is assumed to be Poisson distributed, thus .

ˆ( )Var ˆ( )Var

ˆ ˆ( )Var

2ˆˆ( )Var

If a statistical model is used: Same as for

Poisson or Poisson-gamma model

Before-After StudiesThe estimation of the safety of a treatment is done through a 4-step process.

STEP 3: Estimate and using and from STEP1 and from STEP 2.

ˆ( )Var

ˆ ˆˆ

Correction factor when less than 500 observations are used

2

ˆˆˆ ˆ ˆ1 { }/Var

Before-After StudiesThe estimation of the safety of a treatment is done through a 4-step process.

STEP 4: Estimate and .ˆ{ }Var ˆ{ }Var

ˆ ˆˆ{ } { } { }Var Var Var

222

2

2

ˆ ˆ{ } { }ˆˆ ˆˆ{ }ˆ{ }1 ˆ

Var V

V

ar

Varar

Before-After StudiesAccounting for change in traffic flow.

Before-After StudiesAdjustment factor for change in traffic flow:

( )( )tf

f Ar f B

d tfr r Note: 1

0f flow F

ˆ L

STEP 1 & STEP 2

Estimates of Coefficients

ˆ{ }Var L

Estimates of Variances

ˆ d tfr r K 2 2 2ˆ ˆ{ } { }d tf tfVar r r K K Var r

Before-After Studies with Traffic Flow Factors

STEP 3 & STEP 4ˆ ˆˆ


222

2

2

ˆ ˆ{ } { }ˆˆ ˆˆ{ }ˆ{ }1 ˆ

Var Var

VarVar

2


Before-After Studies with Traffic Flow Factors

Before-After StudiesEstimation of rtf

( )( )

avgtf

avg

f Ar f B

2 2

22 2

ˆ ˆ{ } { }{ } A avg B avgtf tf

avg avg

c Var A c Var BVar r r

f A f B

CA and CB denote the derivates of “f” with respect to traffic flow Aavg and Bavg.


Using the following equation

2ˆ ˆ{ } { }avg A avgVar f A c Var A

Let CA and CB denote the derivatives of f:

2

1{ } { }n

ii i

YVar Y Var XX

2ˆ ˆ{ } { }avg B avgVar f B c Var B

22 2

ˆ ˆ{ } { }{ }

avg avgtf tf

avg avg

Var f A Var f BVar r r

f A f B


Coefficient of variation (Std Dev / Mean)

Ratio Aavg / Bavg

Before-After StudiesCoefficient of Variation

Factors:

Equation:0.821 7.7 /( ) 1650days AADT


Example: Assume 572 vehicles were counted during a two-hour count for the before period and 637 were counted for the after period on a rural long-distance highway. Now assume that the functional relationship between crashes and flow is given by .Compute and .

0.80f flow F

tfr ˆ{ }tfVar r


0.8 0.8637ˆ 1.114 1.090572tfr

2 2 2 2ˆ{ } 1.114 0.8 0.12 0.12 0.022tfVar r

From Table 8.7

Example: Assume 572 vehicles were counted during a two-hour count for the before period and 637 were counted for the after period on a rural long-distance highway. Now assume that the functional relationship between crashes and flow is given by .Compute and .

0.80f flow F

tfr ˆ{ }tfVar r

Before-After StudiesContinuing with the previous example. Now, assume that a road section has been resurfaced. In the two-year ‘before’ period, 30 wet-pavement crashes were recorded on this section. In the two-year ‘after’ period, 40 wet-pavement crashes were reported. As before, 572 vehicles were counted during a two-hour count for the before period and 637 were counted for the after period. The function relationship is still the same: . In addition, there were 50 wet-pavement days for the before period and 40 wet-pavement days for the after period. Estimate , and the standard deviation of these estimates.

0.80f flow F

Before-After Studies

ˆ 40

STEP 1: Estimate and .

0.81.114 1.090tfr

ˆ 0.8 1.090 30 26.16

40 / 50 0.8dr


ˆ{ } 40Var


2 2 240ˆ{ } 1.090 30 30 0.02250ˆ{ } 35.4

Var

Var


ˆ 26.16 40 13.84


2

4026.16ˆ

35.41 26.16ˆ 1.45


ˆ{ } 35.4 40 75.4Var


22

2

2

35.411.45 40 26.16ˆ{ }35.41 26.16

ˆ{ } 0.144

Var

Var

Premise: the safety of a site is estimated using two sources of information:◦ 1) information obtained from sites that have the

same characteristics (reference population)◦ 2) information obtained from the actual site

where the EB method is being applied Reference population

◦ Method of moments◦ Statistical model

Empirical Bayes Method

Formulation:Empirical Bayes Model

ˆ (1 )EB y

1ˆ

1

ˆ

where

Dispersion parameter

Mean

Note: we use previously.

Formulation of the variance:2ˆˆ{ }Var

ˆ ˆ{ } 1EB EBVar

Empirical Bayes Model

The EB Variance

If the estimate of and is available, one can estimate the coefficients and from the gamma distribution (two-parameter).


ˆ{ }Var

1ef

{ }Var

2

{ }Var

It can be shown that by using the Bayes theorem, we can incorporate the crashes occurring on the given site to develop a new gamma function:

1 (1 )1 y y

EB

ef

y

2{ }

1EB

yVar


1EBy

Empirical Bayes ModelEstimating and . ˆ{ }Var

Method of Moments

1

/n

i

y K n n

K(n) = the number of crashes on each entityn= the number of entities

22

1

/n

i

s y y K n n

Sample mean

Sample variance


Method of Moments

ˆ y

2ˆ{ }Var s y

Estimated mean

Estimated variance


Statistical Model

ˆ exp xβ2ˆˆ{ }Var

Estimated mean

Estimated variance

Empirical Bayes ModelBefore-After Study using the EB model

STEP 1: Develop statistical models.Using data from the control group, develop one or several statistical models.

From the model(s), estimate the dispersion parameter .

ˆ exp xβ


STEP 2: Estimate and for the before period.

ˆEB ˆ{ }EBVar

ˆ

ˆ

bEB

b

y

t

= crash count during the period “t” years (labeled as tb)

= expected annual number of crashes for the before period

ˆEB

by

ˆ ˆ{ } 1EB EBVar


STEP 3: Estimate . tfr

( )( )tf

f Ar f B

ˆ expaf A xβ

ˆ expbf B xβ

For each site, use the characteristics for the after period

For each site, use the characteristics for the before period

ˆ ˆtf a EBr t


STEP 4: Estimate the number of collision for the after period.

= the number of years for the after periodat

STEP 5: Estimate . (same as before)

STEP 6: Estimate and . ˆ( )Var ˆ( )Var

2ˆ

ˆ( )

ˆ

EB tf a

b

r tVar

t


ˆ ˆ( )Var

STEP 7: Estimate and using the output from STEP 4, STEP 5 and STEP 6.

ˆ ˆˆ

Before-After Study using the EB model


2




222

2

2

ˆ ˆ{ } { }ˆˆ ˆˆ{ }ˆ{ }1 ˆ

Var Var

VarVar

Before-After Study using the EB model


Before-After Studies Part II

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