A SAMPLING STRATEGY FOR REAR-END PRE-CRASH DATA 

COLLECTION 

Santokh Singh 

NCSA, National Highway Traffic Safety Administration 

400 Seventh Street SW, Washington, DC 20590 

Santokh.Singh@nhtsa.dot.gov 

Abstract 

The involvement of a driver in a rear-end crash and the manner in which his/her vehicle collides with other vehicle(s) 

depends not only on the driver’s perception of the complex scenario that emerges prior to the crash, but also on his/her precrash driving behavior, response to the imminent crash situation, and performance in resolving the driving conflicts. 

Obviously, any effort directed toward crash countermeasures must start from data collection in order to have a better 

understanding of these driver-related parameters in ‘naturalistic’ settings. This would require deploying vehicles on the 

roadways that are equipped with certain devices to record data on such parameters. Due to the random nature of these 

crashes, the number of vehicles actually required to obtain the desired amount of information may be large, thereby making 

data collection an expensive proposition. A sample design is needed that can conserve resources and yet obtains the 

maximum information. 

The present study aims at proposing a sampling strategy for designing an optimal sample. It consists of stratifying the target 

population by driver’s age and allocating the sample over the strata on the basis of driver’s propensity of being in the 

striking/struck role in a rear-end crash. With a specific requirement of observing certain numbers of drivers in these two 

roles, the proposed strategy is compared with other methods of allocation, such as equal and proportional. The sample 

designed through this strategy is found to be much more economical in terms of the number of vehicles that need to be 

equipped and the number of voluntary drivers required. 

1. Introduction 

A common type of crash that occurs on the roadways is the rear-end crash, caused by one vehicle striking the rear of another 

vehicle when both vehicles are in the same traffic lane and are heading in the same direction. These crashes form a significant 

proportion of all crashes and involve a considerable number of drivers every year. Based on the databases, the General 

Estimates System (GES) of the National Automotive Sampling System (NASS) and the Fatality Analysis Reporting System 

(FARS), compiled by the National Highway Traffic Safety Administration (NHTSA), approximately 29.7% of all crashes 

were rear-end crashes in 2001. In terms of drivers crash involvement, of the 191,275,719 licensed drivers in 2001reported by 

Federal Highway Administration (FHWA), approximately 2.1% were involved in rear-end crashes, making up 36.5% of all 

drivers involved in various types of crashes. These figures suggest the necessity of developing crash countermeasures that 

could reduce the occurrence of rear-end collisions. In this regard, it is becoming increasingly apparent that the development 

of any rear-end crash countermeasure would require a better understanding of the driving behavior and performance 

associated with the driver’s response to driving conflict and imminent crash situations. This requires data collection in a 

‘naturalistic’ setting – crash situations as encountered by drivers on the roadways. The vehicles deployed for this type of data 

collection must therefore be equipped with certain devices that could record the data related to the driving behavior and 

performance of a driver prior to a rear-end crash. In the subsequent discussion, these vehicles will be referred to as 

‘experimental vehicles’. The present study is focused on designing an optimal sample for rear-end pre-crash data collection. 

A probabilistic approach is used to formulate a sampling strategy for this purpose. 

2. Objective of the Study 

In order to develop effective rear-end crash countermeasures and accurately estimate their potential benefits, greater 

understanding is needed regarding specific parameters of driving behavior and performance of drivers. These parameters are 

determined by the complex scenario that emerges prior to a rear-end crash. In this context it is important to remember that a 

driver involved in a rear-end crash can be the driver of a striking or struck vehicle, or even of the vehicle that strikes as well 

as struck by another vehicle. Therefore, information on the parameters related to both striking and struck drivers is useful for 

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the development of countermeasures. While data collection is crucial for acquiring this information, an efficient sample 

design is important from the point of view of conserving resources that are required in terms of the experimental vehicles, as 

well as the voluntary drivers that need to be made available to drive these vehicles. With the above objective in mind, an 

important aspect of the data collection process is that the vehicles comprising the sample have to be kept deployed until the

required numbers of drivers have been involved in rear-end striking and struck crashes. In statistical terms, this sampling 

procedure is called ‘inverse sampling’. Since the emergence of scenarios resulting in the occurrence of rear-end crashes is 

random, the inverse sampling may require a large number of vehicles in order to observe a specific number of them involved 

in the rear-end crashes. The objective of the present study is to propose a sampling strategy that is optimal in terms of the 

content of information about the driver-related parameters across the population of drivers. A diligent selection of drivers 

from the target population is crucial for arriving at an optimal sample design. The present study is an attempt in this direction 

and is focused on 

• Choosing a criterion that could be used to stratify the population 

• Choosing a sample allocation criterion 

These criteria will be used in the sampling strategy to design a sample with the pre-specified numbers of drivers that are 

required to be in the striking and struck roles in the rear-end crashes. 

3. Data Sources and Variables for Analysis 

The statistical analysis conducted and the resulting conclusions made in this study are based on the information/data retrieved 

from the following sources: 

• Age distribution of licensed drivers in 2001, reported by the FHWA 

• Drivers involved in rear-end and other crashes in 2001, reported in GES 

• Drivers involved in fatal crashes in 2001, reported in FARS 

While GES obtains its data from a nationally representative probability sample selected from the estimated police reported 

crashes, FARS contains the data only from the files that document all qualifying fatal crashes. For that reason, cases with 

fatal crashes were used from FARS data in lieu of the fatal crashes estimated in GES data. 

The first and foremost task in designing a sample is to select the relevant factors (variables) from a large number of variables 

coded in GES and FARS databases. Keeping in view the fact that our interest is in the driving behavior and driver’s 

performance prior to a rear-end crash, the factor (variable) that most deserves attention is the Manner of collision in the crash; 

we will focus on Rear-end crashes only. The other variables that need to be included in the context of the present study are 

the Occupant role and the Vehicle role; we will consider the occupant’s role as Driver, while two types of Vehicle role will 

be considered; namely Striking and Struck. Vehicle role: Both will be included in Striking as well as in Struck. 

Given a crash, the information about the manner of collision, occupant’s role, and vehicle role can be combined by defining a 

new variable Crash event, which will be used in the subsequent analysis and discussion. 

 1, if the manner of collision is rear-end and the vehicle/driver role is Striking; 





Crash event =  2, if the manner of collision is rear-end and the vehicle/driver role is Struck;







3, if the driver is involved in a crash other than the rear-end or is not involved in any type of crash. 

Last but not least, the perception of the circumstances surrounding a crash as well as the driving behavior and performance of 

the driver prior to a crash, seem to be related to driver’s age. The related variable, coded as Age in GES and FARS databases, 

will be the basis of all analysis done in this study. 

4. Stratification and Sample Allocation Criteria 

The basic aim in any data collection process is to acquire a maximum amount of desired information at the minimum cost 

and effort. Therefore, whether or not there is a restriction on the sample size that can be used in a given situation, achieving 

this aim is easier if the target population is stratified using an appropriate criterion. An efficient criterion to allocate the 

sample over the strata is an additional tool in designing an optimal sample. 

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4.1. Stratification Criterion 

As mentioned earlier, the involvement of a driver in a crash depends on his/her perception of the complex scenario that 

emerges prior to a crash and that the driver’s pre-crash driving behavior and performance play important roles in resolving 

the driving conflicts. This suggests that driver’s age may be one of several factors contributing to the rear-end crash 

involvement of a driver and the role he/she assumes in the crash. This attribute can therefore be considered as one of the 

possible factors for stratification of the target population. Nevertheless, using this factor for this purpose will make sense only 

if there is an evidence of the association between Age and Crash event (the variable that defines a driver’s role in a rear-end 

crash). 

Contingency analysis was performed for testing the association between Age and Crash event. In order to test the 

independence between these categorical variables, the drivers were classified using the following mode: 

A1: Age group 1 (younger than 18) 

A2: Age group 2 (18 to 24) 

A3: Age group 3 (25 to 44) 

A4: Age group 4 (45 to 64) 

A5: Age group 5 (older than 64) 

With this classification in place, the contingency analysis of GES data for the year 2001 was carried out to test the 

hypothesis: there is no association between Age and Crash event. Since the collection of GES data is based on three-stage 

sampling, the statistical software SUDAAN 8.01 was used for this purpose, which takes into account the underlying sampling 

design of the data being used in the analysis. The test procedure yields χ

2

 = 279.9 with 8 degrees of freedom. The 95th 

percentile 9.49 of χ

2

 distribution (with 8 degrees of freedom) being far less than 279.9, the hypothesis of no association is 

discredited, thereby indicating that there is a strong evidence of an association between Age and Crash event. This inference 

provides a strong reason to use driver’s age as the stratification criterion in designing the sample. In the subsequent 

discussion, the classes of drivers defined on the criterion of Age will be referred to as the attribute-based classes. 

4.2. Sample Allocation Criterion 

As mentioned earlier, the basic aim is to design a sample so that a maximum number of drivers involved in rear-end crashes 

may be observed with minimum number of vehicles deployed. The stratification based on age can be effectively used in this 

context, if the sample is designed in such a way that more drivers are included from the strata that consist of drivers who are 

more prone to rear-end crash involvement. Once this is done, the resulting sample would not only increase the likelihood of 

more drivers involved in rear-end crashes and hence yield more data on the driver related parameters, but also provide the 

desired information across the target population. 

Generally speaking, if the population of drivers is stratified over M strata on a certain criterion, what one needs to look for is 

the likelihood (crash involvement propensity) of a driver belonging to the jth stratum being in one of the two roles in a rearend crash relative to that of the drivers from other M-1 strata. In order to arrive at a suitable measure of the crash involvement 

propensity of drivers belonging to a stratum as compared to other strata, it is important to consider the occurrence of rear-end 

crash-involved drivers from a stratum in the striking/struck role relative to the occurrence of its drivers in the entire 

population of drivers. The important information that one needs in this context is an answer to the question: Given that a 

driver selected at random is from a certain stratum, what is the probability that he/she would be involved in a rear-end crash 

and assume one of the two roles? These probabilities can then be combined into the statistic φj

, called Crash Involvement 

Propensity Index (CIPI) (see details in Appendix A) as proposed in [2] 

C j 

S 2 

φ j =

j , j = 1, 2 , . . . , M , (1) 

M  C 



i ∑ 

 2 

l =1 

S i 

where 

Cj is the number of rear-end crash-involved drivers belonging to the jth stratum (jth subpopulation); 

Sj is the number of drivers in the jth stratum, i.e., the size of the jth subpopulation; Sj > 0, S1 + S2 + ... + SM with NT as the 

size of the population of all drivers; 

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M is the number of disjoint strata that are exhaustive of the target population. 

Note that the numerator in (1) takes into account the likelihood or conditional probability (conditional on stratum) of a driver 

belonging to the jth stratum being involved in a rear-end crash, while the denominator is the normalizing quantity. Obviously, 

the statistic φj

 satisfies the inequality 0 ≤ φj ≤ 1. The statistic CIPI given in equation (1) provides a measure of the 

propensity of drivers belonging to a certain stratum of being in the striking/struck role in the rear-end crashes, relative to that 

of the drivers of other strata. 

It is obvious that more drivers are expected to be involved in rear-end striking (struck) crashes from a stratum in which 

drivers have higher propensity of being in the striking (struck) role in rear-end crashes. This index can therefore be used as 

the constant of proportionality for optimally allocating the sample over the strata. Following this argument, the numbers k1 

(striking) and k2 (struck) can be disbursed over the strata using the relation 

kij = ki φ j , i = 1, 2; j = 1, 2, . . . , M , (2) 

where φj

 is given by (1) and k1j and k2j are, respectively, the required numbers of drivers in the striking and struck roles from 

the jth stratum. Subsequently, using inverse sampling (to be discussed latter in the discussion), the sub-sample size in each 

stratum and hence the total sample size can be determined. 

4.3. Guidelines from the Crash Involvement Propensity 

Before CIPI is used for providing guidelines for designing the sample, i.e., in stratifying the population of drivers, it is 

important to remember that the sample is supposed to consist of voluntary drivers. In that case, due to the anticipated 

operational difficulties, Age group 1 and Age group 5 should preferably be excluded from the sample. Accordingly, the target 

population considered in this study consists of 18 to 64 year-old drivers (Age group 2, Age group 3, and Age group 4). The 

allocation criterion proposed in (1) was calculated for three age groups of drivers as decided earlier, using GES and FARS 

data for the year 2001. The results on the crash involvement propensity (equation (1)) presented in Figure 1 show that Age 

0.105 0.115 

0.780 

0.666 

0.148 

0.186 

0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

1 

Age group 2 Age group 3 Age group 4 

Age group 

Propensity of driver's role 

Striking 

Struck 

Figure 1. Propensity of being in the striking and struck role in a rear-end crashes 

three age groups. 

group 2 (18 to 24) drivers have the highest propensity of being in the striking and struck roles as compared to 25 to 64 yearold drivers, who have much lower propensity of assuming any of these roles. This shows that, when engaged in data 

collection, many more drivers can be expected to have been involved in the rear-end crashes both as striking and struck 

drivers from Age group 2. These results also show that, when involved in a rear-end crash, a 25 to 64 year-old driver is more 

likely to be in the striking role than in the struck, though this tendency is higher among Age group 3 drivers as compared to 

Age group 4 drivers. The tendencies of drivers of different age groups in assuming a role in a rear-end crash that this statistic 

indicates can be exploited in designing a sample. 

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5. Sampling Issue in Rear-End Pre-Crash Data Collection 

Although rear-end crashes form a significant proportion of crashes, their frequency is not so high that a significantly large 

amount of data can be generated by engaging only a small number of experimental units. Being specific about the number of 

observations required on the striking and struck drivers is, therefore, important for the success of any data collection project 

with the current theme. 

Consider the situation where the aim is to obtain information on the driver-related parameters for k1 striking and k2 struck 

drivers. Then the sampling issue involved is to estimate the number, N, of vehicles to be deployed for data collection (i.e., the 

sample size) that is large enough to include at least k1 striking and at least k2 struck drivers. 

6. Probabilistic Formulation of the Sampling Problem 

Like all other road events, the occurrence of rear-end crashes and the role that a driver is likely to assume in such crashes are 

random events. Accordingly, the variable Crash event defined earlier is a discrete random variable. This further means that 

the phenomenon of drivers’ involvement in rear-end crashes and the roles that they assume therein can be treated 

probabilistically. 

6.1. Categorization of Drivers in the Target Population 

Consider the target population of drivers (i.e., 18 to 64 year old drivers). Based on the variable Crash event defined in 

Section 3, the drivers in this population can be categorized into the following categories: 

C1: drivers involved in the rear-end crashes with their vehicles in the striking role; 

C2: drivers involved in the rear-end crashes with their vehicles in the struck role; 

C3: drivers involved in crashes other than the rear-end crashes or not involved in any crash. 

This event-based categorization of drivers classified into age groups A2, A3, and A4 is shown in Figure 2. Note that the areas 

demarked in Figure 2 are merely representative of the class/category of drivers and not of their actual sizes. In the 

Figure 2. Categorization of drivers (classified by Age), based on Crash event. 

subsequent analysis, the age-based classes A2, A3, and A4 will be referred to also as subpopulations. Having done this 

categorization of each subpopulation, the task is to determine the sub-sample sizes. 

For that purpose, define events E1, E2, and E3 as 

E1: driver is involved in a rear-end crash and is the driver of the striking vehicle; 

E2: driver is involved in a rear-end crash and is the driver of the struck vehicle; 

E3: driver is involved in a crash other than the rear-end crash or is not involved in any type of crash. 

The definitions of these events suggest that each driver in an attribute-based class can be categorized in one of the three 

categories C1, C2, and C3, depending on the occurrence of the events E1, E2, and E3, respectively. For instance, if a driver is 

involved in a rear-end crash and is the driver of the striking vehicle, then he/she will be considered as belonging to C1. 

Similarly, if a driver is involved in a head-on collision or is not involved in any type of crash, then he/she will be considered 

as belonging to C3. This categorization of drivers will be referred to as the event-based categorization. 

Page 115 
 

 

 

 

 

 

 

Due to the uncertainty inherent with the road events, each of the event-based categories of subpopulation can be associated 

with certain probability. Specifically, let 

p1 = P{E1}, i.e., the probability that a driver belongs to category C1 of the population; 

p2 = P{E2}, i.e., the probability that a driver belongs to category C2 of the population; 

p3 = P{E3}, i.e., the probability that a driver belongs to category C3 of the population ( p = 1 − p − p ). 3 1 2 

6.2. Inverse Sampling 

Recall that the objective of this study is to design a sample to collect data on at least k1 drivers involved in rear-end crashes 

assuming the role of striking drivers and k2 involved in such crashes assuming the role of struck drivers. This further means 

that one needs to continue collecting data on the pre-crash driving behavior and performance of drivers until at least the 

specified numbers of drivers k1 and k2, respectively, have fallen in the categories C1 and C2. This immediately suggests that 

the process of data collection can be thought of as throwing balls successively into three boxes, until at least k1 balls of C1-

type type are obtained in the first box and at least k2 balls of C2-type are obtained in the second box. Since the outcome of a 

throw resulting in a ball that goes into the third box does not affect termination of this process, no attention is paid to what 

happens to that box. Thus, the sample size in this experimentation (throwing balls into three boxes) is nothing but the number 

of trials one has to wait through in order to achieve the objective. Correspondingly, speaking in terms of drivers/vehicles, the 

sample size in the present context is the number of drivers/vehicles one has to wait through in order to collect data on at least 

k1 drivers in the striking role and at least k2 drivers in struck role. In statistical terms, this process is called ‘inverse sampling’ 

and is described in the flow chart presented in Figure 3, where Ns

 is the number of striking drivers, and Nst the number of 

struck drivers. 

YES 

NO 

NO 

YES 

YES 

Sample size = Road events 

Target numbers k1 (striking) and k2 (struck) 

Road events = 0 

Road events = Road events + 1 

STOP 

Is crash C1- type? 

Nst=Nst+1 

Ns = Ns+1 

Is Ns = k1 and 

Nst = k2 ? 

NO 

Is crash C2- type? 

Figure 3. Flow chart describing inverse sampling. 

A typical sequence of the drivers involved in all types of road events, before 3 of them are involved in C1-type of crashes and 

2 in C2-type, would look like shown in Figure 4. 

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Driver/Vehicle 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 

Crash Event C3 C2 C3 C3 C3 C1 C3 C3 C2 C3 C1 C2 C2 C3 C1 

Run of 2 4 3 2 1 1 2

Figure 4. Sequence of road events before at least 3 drivers are involved in rear-end striking and 2 in rear-end struck crashes. 

It would not be out of context to mention at this point that the total number of drivers, 15 in this instance, is subject to 

randomness due to uncertainty involved in road events. 

6.3. Probability Distribution of the Number of Striking and Struck Drivers Involved in Rear-End Crashes 

It is now the time to put all the information together which we gained about rear-end crashes. First of all, it is clear that as the 

experimentation (data collection) goes on, the vehicles deployed on the road and hence the drivers engaged would be 

involved in crashes; the probability being p1 that a driver involved in a rear-end crash would be the Striking driver and p2 the 

probability that a driver involved in a rear-end crash would be the driver who was Struck. Any other type of road event, not 

covered by the above two crash scenarios, occurs with probability p3 (=1 - p1 - p2). It is important to note that the termination 

of data collection depends only on the number of drivers assuming striking and struck roles that have happened up to and 

including a rear-end crash and not on the occurrence of the other type of road event. However, the total number of drivers we 

will have to wait through (length of the sequence in Fig. 4) to terminate data collection depends on how often C3 occurs. 

These facts provide sufficient reason to treat the sample size problem as a discrete waiting-time problem [3]. 

Define the event E as

 E = {at least k1 drivers are involved in C1 -type of crashes and at least k 2

 drivers in C2 -type} 

It is easy to see that the event E can happen in one of the two ways; namely when a crash occurs that ends with exactly k1 

drivers involved in C1-type of crashes and at least k2 drivers in C2-type, or when a crash occurs that ends with exactly k2 

drivers involved in C2-type of crashes and at least k1 drivers in C1-type. Although the happening of the event E is decided by 

the occurrence of C1 or C2-type of crash event, as mentioned earlier, the occurrence of all other road events included in C3-

type decides the length of the sequence of road events (all types) that would be required to have happened, before the event E 

happens. One such sequence is shown in Figure 4. Continuing the argument, it can be concluded that the number of drivers, 

N, required for the happening of E is a random variable that assumes values 

k1 + k 2, k1 + k 2 + 1, k1 + k 2 + 2, . . . . . . . . 

and follows a negative multinomial distribution (see Appendix B). Accordingly, the probability that n vehicles/drivers will 

be required for the event E to happen is given by (Appendix B). 

n −1  (n−k )!

i

i rl 3 P{N = n } = ∑

2 

  pi

k ∑ pl p3 

r , (3) 

i=1 

ki −1 rl ! r3

! 

where the summation ∑ is taken over rl and r3(l ≠ i) , such that, r ≥ k and r + r = n − k .

l l l 3 i 

7. Estimation of Sample Size 

Once the probability distribution of N has been established, the sample size is nothing but the expected value of N. Using the 

probability distribution given by (3), the expected number of vehicles required in order to compose a sample that consists of 

at least k1 drivers involved in C1-type of crashes and at least k2 drivers in C2-type is given by (Appendix B) 

2  ∞

 n −1  ki 

(n − ki )! r r3 



E N = ∑  ∑ n 

k −1

 pi ∑ pl

l p3  (4) 

i =1 n=k1 +k2  i  rl ! r3

! 

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where the summation ∑ is taken over rl and r3(l ≠ i) , such that, rl ≥ kl and rl + r3 = n − ki . For small values of p1

, p2, 

p3, the convergence of the sum in (4) is bound to be slow. In order to overcome this computational difficulty a simplified 

form of EN can be derived (see Appendix B) 

 k   p  k   p 

ˆ 1 1 1 2 N =  1− I



k1 +1, k2

; 

 + 1− I



k2 +1,k2

; 

  , (5)  p 1  p + p  p  p + p    1 2  2   1 2    

where I (l, m; a) is the Incomplete beta function with l, m ≥ 0 and 0 ≤ a ≤1 (Appendix B). 

8. Example: Sample Design for Rear-End Pre-Crash Data Collection with the Objective of 10 Striking and 10 

Struck Drivers 

In this section, the proposed sampling strategy is implemented and compared with other strategies (equal and proportional 

allocation). GES and FARS data for the year 2001 were used to estimate the sample size required for observing k1 (=10) 

drivers in the striking role and k2 (=10) drivers in the struck role in the rear-end crashes. As suggested by the contingency 

analysis, age of the driver was used as the stratification criterion. However, as mentioned earlier, due to the anticipated 

operational difficulties, Age group 1 and Age group 5 were excluded in the following analysis. As a first step, the target 

numbers k1 and k2 were partitioned into 3 numbers each, ki1, ki2, ki3 + ki1 + ki2 = ki

, i= 1, 2,with k1j representing the number of 

striking drivers and k2j representing the number of struck drivers from stratum j. This was done in three ways: (i) kij’s 

determined by the propensity statistic given by (1), (ii) equal kij’s, and (iii) kij’s proportional to strata sizes. 

The results for case (i) are presented in Table 1. These results show that, if CIPI is used as the sample allocation criterion, 

then, of the 10 striking drivers involved in rear-end crashes, 8 would be from Age group 2, and 1 each from Age group 3 and 

Age group 4. Similarly, of the 10 struck drivers involved in rear-end crashes, 7 would be from Age group 2, 1 from Age 

group 3, and 2 from Age group 4. Using the estimate of the sample size from (5) for each group of the target population, the 

respective sub-sample sizes for the three age groups were found to be 410, 138, and 175. Thus, with age as the stratification 

criterion and CIPI as the allocation criterion, the expected total number of vehicles/drivers required for data collection adds 

up to 723. 

 Table 1. Sample design for observing 10 drivers in the striking role and 10 in the struck role, in rear-end crashes 

(stratification by age, allocation by crash involvement propensity). 

Stratum 

of drivers 

( j ) 

REAR-END CRASHES: 

STRIKING DRIVERS 

REAR-END CRASHES: 

STRUCK DRIVERS 

Stratum 

sample size 

( n j ) 

Constant of 

Proportionality 

( α j ) 

Number of 

Striking drivers 

( k = α .k )

1 j j 1 

Probability 

(Striking) 

( p1 ) 

Constant of 

Proportionality 

( β j ) 

Number of 

Struck drivers 

( k = β .k )

2 j j 2 

Probability 

(Struck) 

( p2 ) 

1. Age group 2 0.780278 8 0.026936 0.665939 7 0.017725 410 

2. Age group 3 0.104673 1 0.012703 0.148212 1 0.013868 138 

3. Age group 4 0.11505 1 0.010875 0.1858484 2 0.013545 175 

Total sample size required for 10 drivers in striking role and 10 drivers in struck role 723 

Source: National Center for Statistics and Analysis, NHTSA, GES and FARS 2001, FHWA 

For the purpose of comparison, two other methods of allocation were also considered: equal and proportional. The results 

presented in Table 2 show that with the numbers ki1 = 4, ki2 = 3, and ki2 = 3, i = 1, 2, larger number of drivers, 297 and 329, 

would be required, respectively, from Age group 3 and Age group 4. This is obviously due to the fact that Age group 3 and 

Age group 4 drivers have a lower propensity of being in the striking/struck role as compared with Age group 2 drivers and 

yet the samples selected from them are supposed to produce almost the same number in these two roles. This in turn raises 

the requirement of total number of vehicles/drivers in the sample to a larger number (873) as compared to CIPI-based 

allocation that requires 723 drivers with the same outcome. 

Page 118 
 

 

 

 

 

 

 

 

 

 

 

Table 2. Sample design for observing 10 drivers in the striking role and 10 in the struck role in rear-end crashes 

(stratification by age, equal allocation). 

Stratum 

of Drivers 

( j ) 

REAR-END CRASHES: 

STRIKING DRIVERS 

REAR-END CRASHES: 

STRUCK DRIVERS 

Stratum 

Sample Size 

( n j ) 

Constant of 

Proportionality 

( α j ) 

Number of 

Striking Drivers 

( k = α .k )

1 j j 1 

Probability 

(Striking) 

( p1 ) 

Constant of 

Proportionality 

( β j ) 

Number of 

Struck Drivers 

( k = β .k )

2 j j 2 

Probability 

(Struck) 

( p2 ) 

1. Age group 2 0.4 4 0.026936 0.4 4 0.017725 247 

2. Age group 3 0.3 3 0.012703 0.3 3 0.013868 297 

3. Age group 4 0.3 3 0.010875 0.3 3 0.013545 329 

Total sample size required for 10 drivers in striking role and 10 drivers in struck role 873 

Source: National Center for Statistics and Analysis, NHTSA, GES and FARS 2001, FHWA 

The target numbers, 10 of rear-end crash-involved drivers in striking role and 10 in the struck role, were disbursed also using 

proportional allocation (i.e., proportional to strata sizes). The results presented in Table 3 show that with this allocation, from 

Age group 2 only 1 driver is expected to be in the striking role and 1 in the struck. This would require 94 drivers from this 

age group to be included in the sample. In order to observe 4 striking and 4 struck drivers, 458 drivers need to be included 

from Age group 3. Similarly, in order to observe 5 striking and 4 struck drivers, 405 drivers need to be included from Age 

group 4. Thus, with proportional allocation, 957 vehicles/drivers would be required for the same target numbers of striking 

and struck drivers. This number is larger than the one suggested by equal allocation (873) and much larger as compared to 

sample size (723) suggested by CIPI-based allocation.

 Table 3. Sample design for observing 10 drivers in the striking role and 10 in the struck role in rear-end crashes 

(stratification by age, proportional allocation). 

Stratum 

of Drivers 

( j ) 

REAR-END CRASHES: 

STRIKING DRIVERS 

REAR-END CRASHES: 

STRUCK DRIVERS 

Stratum 

Sample Size 

( n j ) 

Constant of 

Proportionality 

( α j ) 

Number of 

Striking Drivers 

( k = α .k )

1 j j 1 

Probability 

(Striking) 

( p1 ) 

Constant of 

Proportionality 

( β j ) 

Number of 

Struck Drivers 

( k = β .k )

2 j j 2 

Probability 

(Struck) 

( p2 ) 

1. Age group 2 0.137863 1 0.026936 0.137863 1 0.017725 94 

2. Age group 3 0.484651 5 0.012703 0.484651 5 0.013868 458 

3. Age group 4 0.377486 4 0.010875 0.377486 4 0.013545 405 

Total sample size required for 10 drivers in striking role and 10 drivers in struck role 957 

Source: National Center for Statistics and Analysis, NHTSA, GES and FARS 2001, FHWA 

9. Conclusions and Recommendations 

The contingency analysis of GES and FARS data for the year 2001 provided strong evidence of the association between 

driver age and the crash involvement of drivers (rear-end or otherwise). Driver’s age can therefore be used as a criterion for 

stratifying the population so that different age groups can have appropriate representation in the sample. 

As demonstrated through examples, the statistic CIPI can provide a useful guideline to optimally allocate the sample over the 

strata by making greater provision in the sample for the strata that are more prone to rear-end crash involvement. Based on 

this statistic, it was found that 18 to 24 year-old drivers were most prone to rear-end crash involvement. In order to produce 

maximum amount of data, this age group should therefore contribute most to the sample. 

The CIPI-based sample allocation was compared with some other possible methods of allocation, equal number of drivers 

Page 119 
 

 

 

 

 

 

from each stratum and strata sample sizes proportional to the strata sizes. The example demonstrated that due to the 

differential that exists among the strata with respect to the crash involvement propensity, for the same target number of crashinvolved drivers, both equal and proportional allocations resulted in larger strata sample sizes and hence larger total sample 

size as compared to the one suggested by CIPI-based allocation. 

The proposed sampling strategy is neither data dependent nor population dependent. In fact, the approach used in this study is 

general and can be used for designing an optimal sample for data collection in similar setups. The sample allocation criterion 

proposed in this study can also be used when the total sample size is fixed in advance and the requirement is merely to 

allocate the sample size over the strata in order to get the best out of the restricted sample size. In that case the fixed sample 

size needs to be disbursed using CIPI. The expected number of drivers involved in rear-end striking and struck cashes in each 

age group can then be obtained using the binomial distribution. 

10. References 

[1] Wilks, S. S., Mathematical Statistics, John Wiley and Sons, New York (1962). 

[2] Singh, Santokh, ‘Driver Attributes and Rear-End Crash Involvement Propensity’, (NHTSA Technical report No. DOT 

HS 809 540, March 2003. 

[3] McCarthy, P. J., ‘Approximate Solutions for Means and Variances in a Certain Class of Box Problems’, Annals of 

Mathematical Statistics, Vol. 18 (3), 1947. 

Page 120 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

11. Appendix A. Analytical Details of Crash Involvement Propensity Index 

11.1 Crash Involvement Propensity Index 

This appendix supplies the analytical details of the statistic CIPI, used in Section 4.2. 

Consider a situation in which, based on a certain criterion, the drivers are divided into M subpopulations and our interest is in 

comparing these subpopulations with respect to their propensity of being involved in a rear-end crash. In order to develop a 

reasonable measure of the crash involvement propensity of a driver belonging to a subpopulation as compared to other 

subpopulations, it is important to consider the occurrence of rear-end crash-involved drivers in this subpopulation relative to 

the occurrence of its drivers in the entire population of drivers. The important information that one needs in this context is an 

answer to the question: Given that a driver selected at random is from a certain subpopulation, what is the probability that 

he/she would be involved in a rear-end crash? In other words, what one needs to look for is the likelihood of a driver of each 

subpopulation being involved in a rear-end crash. 

For this purpose, we consider the space Ω of all drivers belonging to a subpopulation and the subspace Ω C of those drivers 

of this subpopulation who are involved in rear-end crashes. 

Let 

L(Ω) be the probability that a driver selected at random belongs to the subpopulation Ω , 

L (Ω C ) be the probability that a driver selected at random is involved in a rear-end crash, given that he/she belongs to the 

subpopulations Ω . 

Let N be the number of drivers in the entire population of drivers that has been divided into M subpopulations, based on a 

certain criterion, Si the number of drivers in the subpopulation i, and Ci

 the number of drivers who are involved in rear-end 

crashes from this subpopulation, i= 1, 2,...,M. Then the crash involvement propensity of drivers belonging to subpopulation i 

can be defined as 

(i)

L(Ω )

λi =

C 

, i = 1, 2, . . . , M , (A.1) (i)

L(Ω ) 

where 

( i ) C i L ( Ω C ) = , S i > 0

S i 

and 

( i ) S i L ( Ω ) = ,

N 

so that λi defined in (A.1) becomes 

 Ci 

λi = N 



2 

, i = 1, 2, . . . , M . (A.2) 

S  i 

Note that λi in (A.2) is the conditional probability of a driver being involved in a rear-end crash, given that he/she belongs to 

that i-th subpopulation. In order to compare the crash involvement propensity of mutually disjoint subpopulations A1, 

A2,...,AM into which the population of all licensed drivers is divided, these probabilities can be combined to define the Crash 

Involvement Propensity Index (CIPI) 

λi

φi = M , i = 1, 2, . . . , M . 

∑λ j

j=1 

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Using λi

 from (A.2), the CIPI can be derived in the usable form 

Ci 

S 2 

φ =

i 

, i = 1, 2, . . . , M . (A.3) i

M  C 

∑ 2 

j

    j =1 S

 j 

12. Appendix B. Analytical Details of the Probability Distribution and Expected Value of N 

12.1. Probability Distribution of N 

This appendix supplies the analytical details of the probability distribution of the random variable N, referred to in Section 

6.3. 

Consider the situation in which we are interested in the occurrence of crashes that involve at least k1 drivers in C1-type crash 

events (that occur with probability p1) and at least k2 drivers in C2-type crash vents (that occur with probability p2), while all 

other type of crash events (forming one category) occur with non-zero probability, say, p3 (=1 - p1 - p2).. The sampling 

problem in this case can be visualized as throwing balls into three boxes one by one (each box representing a category of 

drivers C1, C2, or C3) until at least k1 balls are obtained in the first box and at least k2 balls are obtained in the second box. No 

attention is paid to what happens to the third box, as the number of balls in this box is not decisive of the termination of 

experiment (throwing balls in the boxes). The number of balls required or the number of trials, N, one has to wait through in 

order to accomplish this task is a random variable which has a Discrete waiting-time distribution. Accordingly, the 

probability that at the n -th throw k1 balls will be obtained in the first box, before k 2

 balls are obtained in the second box 

without paying attention to what happens to the third box is immediately seen to be 

n −1  k (n − k )! r r 1 1 2 3

  p1 ∑1 p2 p3 , (B.1) 

k 1 −1  r ! r ! 2 3 

where the summation ∑1 is taken over r2 and r3 , such that r ≥ k and r + r = n − k .

2 2 2 3 1 

Similarly, the probability that at the n -th throw k2 balls will be obtained in the second box, before k1 balls are obtained in the 

first box, without paying attention to what happens to the third box is immediately seen to be 

n −1  k (n − k )! r r 2 2 1 3

  p2 ∑2 p1 p3 , (B.2) 

k 2 −1  r ! r ! 1 3 

where the summation ∑2 is taken over r1 and r3 , such that r ≥ k and r + r = n − k .

1 1 1 3 2 

Combining (B.1) and (B.2), the probability that n throws will result into k1 balls of C1-type in the first box and at least k2 

balls of C2-type in the second box is given by 

n −1  k (n − k1 )! r r n −1  k (n − k 2 )! r r 1 2 3 2 1 3 P(N = n) =  p1 ∑1 p2 p3 +   p2 ∑2 p1 p3 (B.3) 

k1 −1 r ! r ! 

k 2 −1 r ! r !

2 3 1 3 

where the summation ∑1 is taken over r2 and r3 , such that r2 ≥ k2 and r2 + r3 = n − k1 , and the summation ∑2 is taken 

over r1

 and r3 , such that r1 ≥ k1 and r1 + r3 = n − k 2. 

12.2. Expected Value of N 

This appendix supplies the analytical details of the expected value of the random variable N, used in Section 7. 

Using (B.3), the expected number of trials required for the occurrence of at least k1 balls of C1-type and k2 balls of C2-type in 

Page 122 
 

 

 

 

 

 

the sample drawn from a population that is categorized into three categories is given by 

1 n k !

1

( − ) −1

∞

∑ 1 













1 







n n k1 k

E(N) r r 2 3 r r 2 ∑ 1 3 p (B.4) = 1 1 2 3 2 2 3 p p n p p1 p

k k ! − 3

! ! 3

r !

2 r r1 k r 1+k n= 2 2







∑ +







n k2

! 

n 

1 

The convergence of the sums involved in (B.4) depends on the magnitude of the probabilities, p1 and p2; the smaller the 

magnitude of these quantities, the longer it will take for the summation to converge. However, the problem can be reduced by 

one dimension if we consider only those balls that go into first two boxes, that is, by considering the first two boxes as one 

unit. Thus, instead of counting the number of balls going into the first two boxes separately, only the number of balls going in 

)

the single unit are counted; the probability being p1 + p2 of balls going into this unit. With this formulation, it is obvious that 

−

the number of balls necessary to obtain k1 balls in the first box and k2 balls in the second box of this unit is a random variable 

(

X which takes on values k1 + k2, k1 + k2 + 1, k1 + k2 + 2, ......... ∞ and follows a negative multinomial distribution. 

The expected number of trials for the occurrence of the desired event is given by 

−

−







E [N E ] ( ), ( )]] [E [ X p p p 1 2 3 ∞

 

E k 

























, 















p p 1 2 E k ( p1+ p 2  ), ( )











= p 1 2 3

p1 + p p p 2 1 2

This can be further simplified as 







1

1 21 

k is the value of the incomplete beta function which for the specified parameters l, m 



















k k p p 1 2 E [N I ] − (k k I 1, , , − (k k +1, (B.5) 1 2 2 2

p p p p p p 1 1 2 2 1 2 + +

I (k where 1 + 1, ; p ) , and a is defined 2 1 

as 













Γ(l ) a + m l−1 1 −

1 and Γ(m) is the value of the gamma function evaluated at m . 

I(l, m; a dx ) (1− ) (B.6) m

∫

x x ,

Γ Γ ( ) ( l m) 0 

where l, m ≥ 0, 0 ≤ a ≤

∞

= +

+

= + +

=

Page 123