U.S. Department 

of Transportation

National Highway 

Traffic Safety 

Administration 

DOT HS 809 541 April 2003 

Technical Report 

SAMPLING ISSUES IN 

REAR-END PRE-CRASH DATA 

COLLECTION 

Published By: 

National Center for Statistics and Analysis 

Advanced Research and Analysis 

This document is available to the public from the National Technical Information Service, Springfield, Virginia, VA 22161
 

This publication is distributed by the U.S. Department of Transportation, National Highway 

Traffic Safety Administration, in the interest of information exchange. The opinions, findings 

and conclusions expressed in this publication are those of the author(s) and not necessarily 

those of the Department of Transportation or the National Highway Traffic Safety 

Administration. The United States Government assumes no liability for its contents or use 

thereof. If trade or manufacturers' names are mentioned, it is only because they are 

considered essential to the object of the publication and should not be construed as an 

endorsement. The United States Government does not endorse products or manufacturers. 
 Technical Report Documentation Page 

1. Report No. 

DOT HS 809 541

2. Government Accession No. 3. Recipient's Catalog No. 

5. Report Date 

April, 2003

4. Title and Subtitle 

Sampling Issues in Rear-End Pre-Crash Data Collection 6. Performing Organization Code 

NPO-121 

7. Author(s) 

Santokh Singh, Ph. D.

8. Performing Organization Report No. 

9. Performing Organization Name and Address 10. Work Unit No. (TRAIS) 

Rainbow Technology Inc. 

17106 Thatcher Court 

Olney, MD 20832

11. Contract or Grant No. 

13. Type of Report and Period Covered 

NHTSA Technical Report 

12. Sponsoring Agency Name and Address 

Mathematical Analysis Division, National Center for Statistics and Analysis 

National Highway Traffic Safety Administration, NPO-121 

U.S. Department of Transportation 

400 Seventh Street, S.W., 

Washington, D.C. 20590

14. Sponsoring Agency Code 

15. Supplementary Notes 

Discussions with Mr. Dennis Utter, Dr. Chou-Lin Chen (Mathematical Analysis Division, NHTSA), and Dr. Mike Goodman (Driver 

Vehicle and Simulation Division, NHTSA) were useful, as were the comments made by them. Proof reading by Mr. Tom Bragan is 

appreciated. 

16. Abstract

A common type of crash that occurs on the roadways is the rear-end crash. Every year a large number of drivers are 

involved in such crashes. In order to develop effective crash countermeasures, it is important to have a better understanding 

of the driving behavior and performance of a driver prior to a rear-end crash. For that purpose, experiments need to be 

conducted in which the drivers can be observed in ‘naturalistic’ settings and data can be collected on the driver-related 

parameters. 

This study discusses some of the sampling issues involved in the process of data collection in the above context. 

Contingency analysis is conducted to suggest criteria for stratifying the target population. A probabilistic approach is used 

for allocating the sample over the strata thus formed. An estimate of the number of vehicles needed to observe a specific 

number of rear-end crashes is obtained. This estimation problem is treated as the ‘discrete waiting-time’ problem. 

Additionally, Binomial probability distribution is used to estimate the number of drivers who would be involved in rear-end 

crashes as a result of deploying a certain number of vehicles. This estimate can be used to assess the potential of a given 

sample for acquiring the desired amount of information. When compared with some of the other methods of allocation 

(equal and proportional), the sample allocation criterion proposed in this study suggested much smaller sample size. Due to 

the random nature of rear-end crashes, the number of vehicles actually required for observing a certain number of rear-end 

crashes is likely to be large, while a smaller number may be deployable due to budgetary restrictions or other operational 

constraints. The sampling strategies are proposed for resolving the issues that may arise in such situation. The approach 

adopted in this study is fairly general and can be used to resolve the sampling issues in similar setups. 

Two databases, the General Estimates System (GES) and the Fatality Analysis Reporting System (FARS), compiled by the 

National Highway Traffic Safety Agency (NHTSA), have been used in this study. 

17. Key Words 

crash involvement propensity, discrete waiting–time, 

equal allocation, inverse sampling, proportional 

allocation, rear-end, stratification, striking, struck

18. Distribution Statement 

Document is available to the public through the National Technical 

Information Service, Springfield, VA 22161 

19. Security Classif. (of this report) 

Unclassified

20. Security Classif. (of this page) 

Unclassified

7 No. of Pages 

27

22. Price 

 Form DOT F 1700.7 (8-72) Reproduction of completed page authorized 

i

Table of Contents 

SECTION/SUBSECTION PAGE 

No. 

EXECUTIVE SUMMARY ii

1. Introduction................................................................................. 1

2. Sampling issues and objectives of the study........................................... 1

3. Data sources and target variables....................................................... 2

4. Stratification and sample allocation criteria. ........................................ 3

4.1. Stratification criteria.................................................................. 3

4.2. Sample allocation criterion.......................................................... 5

5. Probabilistic look at the sampling issues.............................................. 6

5.1. Inverse sampling...................................................................... 7

5.2. Binomial sampling.................................................................... 9

6. Sample designs.............................................................................. 10

6.1. Case 1. Sample designs with the target number, k, of drivers to be involved 

in the rear-end crashes......................................................... 10

6.1.1. Example 1: Sample design with the target number, k=10, in sampling 

from a population stratified by driver’s age............................... 12

6.1.2. Example 2: Sample design with the target number, k=10, in sampling 

from a population stratified by driver’s sex............................... 13

6.2. Case 2. Sample designs with pre-specified sample size, n...................... 14

6.2.1. Example 1: Sample design with pre-specified sample size, n=200, in 

sampling from a population stratified by driver’s age................... 16

6.2.2. Example 2: Sample design with pre-specified sample size, n=200, in 

sampling from a population stratified by driver’s sex.................... 17

7. Conclusions and recommendations.................................................... 18

8. Appendix A. Contingency tables........................................................ 20

9. Appendix B. Analytical details of Crash Involvement Propensity Index ...... 21

10. References................................................................................... 23
 

 National Center for Statistics and Analysis ♦ 400 Seventh St., S.W., Washington, D.C. 20590 ii

EXECUTIVE SUMMARY 

Background and objectives 

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 his/her perception of the complex scenario 

that emerges prior to the crash, but also on the pre-crash driving behavior, response to the 

imminent crash situation and performance in resolving the driving conflicts. Obviously, any 

effort directed towards crash countermeasures must start from data collection that can provide 

information about the driver-related parameters. This would further mean deploying vehicles 

on the roadways, which are equipped with certain recording devices, as well as making the 

voluntary drivers available. Due to the random nature of these crashes, the number of vehicles 

required to deploy for observing a certain number of them involved in such crashes may be 

large. On the other hand, due to budgetary restrictions and operational constraints, the sample 

size actually required may not be permissible. The effort should be, therefore, to make the 

best use of the available resources. 

The objective of this study is to propose sampling strategies by which the maximum amount 

of information, both in terms of the number of rear-end crashes and coverage of the target 

population, could be obtained by deploying the minimum number of vehicles. Toward that 

end, sample stratification criteria are proposed. In addition, two estimates are proposed: (i) the 

number of vehicles that would be involved in rear-end crashes out of a certain number 

deployed, and (ii) the least number of vehicles that would be required for observing a specific 

number of vehicles involved in rear-end crashes. These estimates are required for 

implementation of the approach proposed in this study to resolve the sampling issues. 

Data and methodology 

Two 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) are used in the analysis. 

The contingency analysis is used for proposing criteria that can be effectively used for 

stratifying the population of drivers for rear-end pre-crash data collection. Based on the 

likelihood ratio, a statistic is proposed for optimally allocating the sample size over the strata. 

The problem of estimating the number of vehicles/drivers required (sample size) until a given 

number of them is involved in rear-end crashes is treated as the ‘discrete waiting-time 

problem’. The sampling in this case is called ‘inverse sampling’. In addition, Binomial 

probability distribution is used for estimating the number of vehicles/drivers that would be 

involved in rear-end crashes out of a certain number engaged in data collection. 

Conclusions and recommendations 

The contingency analysis of GES and FARS data for the year 2000 showed that the driver 

attributes age and sex were associated with the driver’s involvement in a rear-end crash. 
 

 National Center for Statistics and Analysis ♦ 400 Seventh St., S.W., Washington, D.C. 20590 iii

Accordingly, these two factors were considered as appropriate criteria for stratifying the 

population of drivers. 

The statistic called Crash Involvement Propensity Index (CIPI) proved to be an efficient tool 

in optimally allocating the sample size over the strata, by making greater provision in the 

sample for the strata whose drivers were more prone to rear-end crash involvement. CIPI also 

provided a useful guideline for disbursing the target number over the strata, thus specifying an 

appropriate target number for each stratum. 

CIPI-based sample allocation was compared for its efficacy with some of the other possible 

allocations: proportional and equal. The estimates obtained using different allocation methods 

showed that CIPI-based allocation was most efficient in both optimally disbursing the target 

number of drivers to be involved in rear-end crashes over the strata and allocating the sample 

when the sample size is fixed in advance. 

Estimates of the numbers of vehicles that would be involved in rear-end crashes, as a result of 

deploying a certain number, showed that the sub-sample sizes corresponding to some of the 

strata would not generate sufficient amount of information in terms of the number of crashinvolved drivers. This shortcoming was overcome by using the proposed sampling strategy. In 

this way, optimal sample could be designed that would not only be well representative of the 

drivers across the target population, but also be likely to produce sufficient information in 

terms of the drivers involvement in rear-end crashes. In yet another situation, the target 

numbers of rear-end crash-involved drivers from certain strata, as suggested by an allocation, 

may not be satisfactory. The sampling strategy proposed for handling such situation resulted 

in the smallest possible sample size. 

The approach adopted in this study is general and can be used for resolving similar sampling 

issues involved in other data collection processes with similar setups. 

 

 National Center for Statistics and Analysis ♦ 400 Seventh St., S.W., Washington, D.C. 20590 1

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 

2000. In terms of drivers crash involvement, of the 190,625,023 licensed drivers in 2000, 

reported by Federal Highway Administration (FHWA), approximately 2.2% were involved in 

rear-end crashes, making up 36% of all drivers involved in various types of crashes. These 

figures suggest the necessity of developing crash countermeasures that could prevent 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 the drivers in their own cars, driving unobserved on the roadways. The 

vehicles deployed for this type of data collection must therefore be equipped with certain 

devices that could record the parameters 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 the sampling issues involved in 

rear-end pre-crash data collection. A probabilistic approach is used to formulate sampling 

strategies that can be used to resolve these issues. The sampling issues and specific objectives 

of the study are described in Section 2. Section 3 contains a brief introduction to the databases 

used in this study, as well as the rationale used in selecting the variables for statistical 

analysis. Some possible criteria for sample stratification and a statistic for sample allocation 

over the strata are proposed in Section 4. Section 5 is devoted to probabilistic formulation of 

the sampling issues. This section also provides analytical details of the estimates required for 

the sampling strategies proposed in this study. The implementation of these strategies is 

demonstrated through examples in Section 6, followed by discussion of the results and 

recommendations in Section 7. The contingency tables and the analytical details of the 

statistic Crash Involvement Propensity Index (CIPI) [2] are included in the Appendices 

(Section 8 and Section 9, respectively). The references used in this study are listed in Section 

10.

2. Sampling issues and objectives of the study

In real-life driving conditions, the parameters related to the driving behavior and performance 

of a driver are determined by the complex scenario that emerges prior to a rear-end crash. 

While data collection is crucial for acquiring information on the driver-related parameters, 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, two aspects of the 

sample design need to be considered. If the aim is to collect data on a specific number of rear-
 

 National Center for Statistics and Analysis ♦ 400 Seventh St., S.W., Washington, D.C. 20590 2

end crashes, then the vehicles comprising the sample have to be kept deployed until the 

required number of these have been involved in rear-end 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. On the other hand, due to budgetary restrictions on the number of experimental 

vehicles or other operational constraints, such as the availability of voluntary drivers, only a 

fixed number of vehicles may be available for the experimentation (data collection). In other 

words, the sample size is fixed in advance. One must then look at the second aspect of the 

sample design. In this case, it is important to estimate the number of rear-end crashes that 

would occur if the pre-specified number of experimental vehicles were deployed for data 

collection. This estimate can help in assessing the amount of information that a sample of 

given size would generate. If the given sample size is not large enough to produce a sufficient 

amount of data on the rear-end crashes, some sampling strategy needs to be used by which 

one could make the best use of the available number of experimental resources. 

The objective of the present study is to propose sampling strategies that are optimal both in 

terms of the number of rear-end crashes and the content of information about the driverrelated parameters across the population of drivers. In this context, it is important to keep in 

mind that an experimental unit in the current data collection process is comprised of a vehicle 

and a voluntary driver. A diligent selection of drivers from the target population is crucial for 

arriving at an optimal sample design. One of the ways to do this is to stratify the population 

using an appropriate criterion and select drivers from the strata proportional to the crash 

involvement propensity of drivers in each stratum. 

In the subsequent sections, for the purpose of rear-end pre-crash data collection, we will 

i. propose efficient criteria for stratification of the target population of drivers, 

ii. propose a criterion that can be used to optimally allocate the sample over the strata, 

iii. estimate the sample size that is large enough to be able to observe a specific number 

of drivers who would be involved in the rear-end crashes, 

iv. estimate the number of drivers who are likely to be involved in the rear-end crashes as 

a result of a specific number engaged in data collection. 

3. Data sources and target variables 

The statistical analysis conducted and the resulting conclusions made in this study are based 

on the information/data retrieved from the following sources: 

1. Age/Sex distribution of licensed drivers for 2000, reported by the FHWA 

2. Drivers involved in rear-end and other crashes in 2000, reported in GES 

3. Drivers involved in fatal crashes in 2000, reported in FARS 
 

 National Center for Statistics and Analysis ♦ 400 Seventh St., S.W., Washington, D.C. 20590 3

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. 

Keeping in mind that our interest is in the driving behavior and performance of the driver 

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

collision in a crash. Of the several manners of collision coded in these databases, we will 

focus on Rear-end collision. Since the manner of collision of a vehicle amounts to the 

involvement of its driver in a certain type of crash, and our interest is in the rear-end crashes, 

for the subsequent analysis, we define a new variable Crash event as 











=

Other , if the manner of collision is other than the rear - end

Rear - end, if the manner of collision is rear - end

Crash event

Last but not least, the perception of the circumstances surrounding a crash as well as the 

driving behavior and performance of a driver prior to a crash seem to be related to driver 

attributes age and sex. Accordingly, our focus in this study will be on two more variables Age 

and Sex. 

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, it can help a great deal in achieving this 

aim if the target population is first stratified using an appropriate criterion and then an 

efficient criterion is used to optimally allocate the sample over the strata thus formed. 

4.1. Stratification criteria

While the involvement of a driver in a crash depends on his/her perception of the complex 

scenario that emerges prior to a crash, a driver’s pre-crash driving behavior and performance 

plays an important role in resolving the driving conflicts. This suggests that the driver 

attributes age and sex may be two of several factors contributing to the rear-end crash 

involvement of a driver. These attributes can, therefore, be considered as possible factors for 

stratification of the target population. Nevertheless, using these factors for this purpose will 

make sense only if there is an evidence of the association between Age/Sex and Crash event. 

Contingency analysis was performed for testing the association between driver’s Age and 

Crash event. This is one of the useful techniques that can be used for analyzing the data that 

can be meaningfully classified in a contingency table, such as Table A.1 (Appendix A). In 

order to test the independence between driver attribute age and crash event, the drivers were 

classified using the following mode. 
 

 National Center for Statistics and Analysis ♦ 400 Seventh St., S.W., Washington, D.C. 20590 4

• Classification of drivers based on age: 

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 criterion of classification in place, the contingency analysis of GES data for the year 

2000 was carried out to test the hypothesis: there is no association between driver’s 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 results are presented in the 

contingency table (Table A.1, Appendix A.) that yield 2 χ = 159.2 with 4 degrees of freedom. 

The 95th percentile 9.49 of 2 χ distribution (with 4 degrees of freedom) being far less than 

159.2, the hypothesis of no association is discredited, thereby indicating that there is a strong 

evidence of an association between driver’s Age and Crash event. 

Based on GES and FARS data for the year 2000, it can be seen that among drivers older than 

17, 18 to 24 year old drivers (considered as young drivers in this study) have the highest rate 

of involvement in rear-end crashes (1,964 per 100,000 young drivers). The drivers from this 

age group can, therefore, provide more data on the driver-related parameters as compared to 

the drivers belonging to other age groups. However, before a sample is designed from this age 

group, it is worth investigating if the attribute sex should be used for stratifying the population 

of young drivers. Accordingly, the young drivers were classified using the following mode. 

• Classification of young drivers based on Sex: 

Ym: Young male (18 to 24 male drivers) 

Yf : Young female (18 to 24 female drivers) 

The contingency analysis was carried out to test the hypothesis: there is no association 

between Sex of the young driver and Crash event. The results are presented in the contingency 

table, Table A.2, (Appendix A) that yield 2 χ = 6.96 with 1 degree of freedom. Since the 95th 

percentile 3.84 of 2 χ distribution with 1 degree of freedom is less than 6.96, we would reject 

the hypothesis of independence between Sex of the young driver and Crash event and 

conclude that the young driver’s sex has some bearing on his/her involvement in a rear-end 

crash. 

The statistical evidence of the association between Crash event, on the one hand, and driver’s 

Age or young driver’s Sex on the other (as shown through the above analysis) gives a strong 

 reason to use these driver attributes as the stratification criteria in designing a sample for 

rear-end pre-crash data collection. 
 

 National Center for Statistics and Analysis ♦ 400 Seventh St., S.W., Washington, D.C. 20590 5

In the subsequent discussion, the classes of drivers defined on the criteria of Age and Sex will 

be referred to as the attribute-based classes. 

4.2. Sample allocation criterion 

The stratification based on age/sex can be effectively used in the present 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 ith stratum being involved in a rear-end 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 in a stratum 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? These probabilities can then be combined into the statistic φi , called Crash 

Involvement Propensity Index (CIPI) [2], given by (for analytical details refer to Appendix B) 

 , i 1, 2, . . . , M ,

S

C

S

C

M

j 1

2

j

j

2

i

i

i =















 =

∑

=

φ (1) 

where 

Ci is the number of rear-end crash-involved drivers belonging to the ith stratum (ith 

subpopulation), 

Si is the number of drivers in the ith stratum, i.e., the size of the ith subpopulation; Si > 0 , 

M NT S1 + S2 + . . .+ S = (size of the population of all drivers), 

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

Note that the numerator in (1) takes into account the likelihood or conditional probability 

(conditional on stratum) of a driver belonging to the ith stratum being involved in a rear-end 

crash (Appendix B), while the denominator is the normalizing quantity. Obviously, the index 

φ i satisfies the inequality 0 ≤ φ i ≤ 1 . The statistic CIPI given in equation (1) provides a 

measure of the propensity of drivers belonging to a certain stratum of being involved in the 
 

 National Center for Statistics and Analysis ♦ 400 Seventh St., S.W., Washington, D.C. 20590 6

rear-end crashes, relative to that of the drivers of other strata. 

Used as the constant of proportionality for allocating the sample over the strata, this index can 

split the sample size n* in such a way that larger sub-sample sizes are assigned to the strata 

that consist of drivers with higher rear-end crash involvement propensity. Specifically, the 

strata sample sizes can be computed from the relation 

 , i 1, 2, . . . , M

S

C

S

C

n n

M

j 1

2

j

j

2

i

i

* i =



















































 =

∑

=

 (2) 

where ni is the sample size for the ith stratum and n* is the total sample size. 

Obviously, * M

i 1 i ∑ n = n = . 

5. Probabilistic look at the sampling issues

As mentioned earlier, there are mainly two sampling issues involved in designing a sample for 

the rear-end pre-crash data collection: (i) to estimate the sample size required for observing a 

target number of drivers involved in rear-end crashes, and (ii) to utilize a sample of prespecified size for obtaining the maximum possible information in terms of the rear-end crash 

involvement. Before methods can be developed to resolve these issues, it is important to 

remember that a driver’s involvement in a rear-end crash is one of the several road events that 

are random. This allows us to formulate the current sampling problem, probabilistically. 

For that purpose, define the events E1 and E2 as 

E1: Driver is involved in a rear-end crash, 

E2: 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 two categories C1 and C2, depending on the occurrence of the events 

E1 and E2, respectively. For instance, if a driver is involved in a rear-end crash, 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 C2. This 

categorization of drivers will be referred to as the event-based categorization and is used for 

categorizing each age- and sex-based class of drivers defined in Section 4. The resulting 

categorization is shown in Figure 1, where Figure 1(a) shows the event-based categorization 
 

 National Center for Statistics and Analysis ♦ 400 Seventh St., S.W., Washington, D.C. 20590 7

 Figure1. Categorization of drivers (classified by Age and Sex), based on Crash event: Rear 

 end crash involvement or otherwise. 

of drivers classified into age groups A1, A2, A3, A4, and A5, and Figure 1(b) shows the eventbased categorization of young male (Ym) and young female (Yf) drivers. Note that the areas 

demarked in Figure 1 are merely representative of the class/category of drivers and not of 

their actual sizes. In the subsequent analysis, the age- and sex-based classes A1, A2, A3, A4, A5,

Ym, Yf, will be referred to also as subpopulations. 

5.1. Inverse sampling

Consider, first, the situation where the objective is to determine the number of vehicles that 

need to be included in the fleet of experimental vehicles so that at least k of the drivers would 

be involved in rear-end crashes. Keeping in view the fact that the drivers in each 

subpopulation are further subdivided into the event-based categories C1 and C2, the sampling 

process can be thought of as drawing objects (drivers) from a box (class) containing N objects 

of two types (C1 and C2) until the target number (k) of objects of one type (drivers from C1) 

are included in the sample. 

Due to the uncertainty inherent in the road events, each of the event-based categories of a 

subpopulation can be associated with a certain probability of occurrence in the object drawing 

process, described above. Specifically, let 

p be the probability that a driver is involved in a rear-end crash, i.e., the probability that 

the driver belongs to C1, 

q be the probability that a driver is involved in a crash other than the rear-end crash or is 

not involved in any type of crash, i.e., the probability that the driver belongs to C2 

( q=1− p ). 

(b) Event-based categorization of young 

drivers (classified by Sex) (a) Event-based categorization of drivers 

(classified by Age) 
 

 National Center for Statistics and Analysis ♦ 400 Seventh St., S.W., Washington, D.C. 20590 8

Equivalent to the box problem, as described above, the data collection in the present case has 

to continue until k drivers are involved in rear-end crashes; the probability being p that a 

driver would be involved in a rear-end crash. This has repercussions in that the termination of 

data collection depends on the number (k) of drivers involved in rear-end crashes that have 

occurred up to and including a crash and not on any other road event. These facts provide 

sufficient reason to treat the current sample size problem as a discrete waiting-time problem. 

For that purpose, we define the event E as 

E = {exactly k drivers are involved in rear-end crashes, i.e., belong to C1 }. 

Obviously, the event E can happen only when a crash occurs that ends with exactly k drivers 

involved in rear-end crashes. A typical sequence of the drivers involved in all types of road 

events, before 3 of them, for example, are involved in rear-end crashes, would look like the 

one shown in Figure 2. Recalling the definition of the events E1 and E2 , it is easy to see that 

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

Crash/Event type Ŕ Ŕ Ŕ Ŕ ® Ŕ Ŕ Ŕ Ŕ Ŕ ® Ŕ Ŕ Ŕ ®

® driver involved in rear-end crash 

 Ŕ driver involved in other than rear-end 

 crash or not involved in any crash

Figure 2. Sequence of crash events required to wait through for the involvement of k = 3 

 drivers in rear-end crashes. 

while the occurrence of the event E is governed by the number of occurrences of E1 , the 

occurrence of E2 decides the length of the sequence of road events that would be required to 

have happened before the event E happens. It follows that the number of drivers (vehicles) 

required for the occurrence of E is a random variable that assumes values: 

k , k + 1, k + 2, . . . , Nq; 1 ≤ k ≤ Np , where N is the total number of drivers in a 

population/subpopulation. 

If X is the number of drivers one has to wait through before k of them are involved in rearend crashes, then the probability distribution of X can be described by the Hypergeometric 

waiting-time distribution, defined as [1] 

Run of Ŕ
 

 National Center for Statistics and Analysis ♦ 400 Seventh St., S.W., Washington, D.C. 20590 9

 , x k , k 1,. . .,Nq; 1 k Np

Np

N

Np k

N x

k 1

x 1

Pr ob ( X x ) = + ≤ ≤





 









 





−

− 



 





−

−

= = . (3) 

Using the probability distribution from equation (3), the expected number of drivers required 

in order to have at least k of them from C1 is given by 

∑

= 



 









 





−

− 



 





−

−

=

Np

x k

Np

N

Np k

N x

k 1

x 1

nˆ ( N, k , p ) x

which simplifies to 

 Np 1

k( N 1 )

nˆ ( N , k , p )

+

+ = . (4) 

and the variance of the sample size is given by 

 ( 1)

( 1)

( 1)

1

(2 1)( 1)

( 2)( 1)

( 1) ( 1) ( , ) 2

2 2

− +

+

+ − +

+ +

+

+ +

− + = k k

Np

k N

Np

k k N

Np Np

Nq Nq k k V k p (5) 

that can be used to calculate the confidence interval for the number of drivers required in 

order to observe at leas k of them involved in rear-end crashes. 

5.2. Binomial sampling

Consider now the situation when the sample size n* is fixed in advance and the requirement 

is to find the expected number k

ˆ of drivers who would be involved in rear-end crashes, i.e., 

would be in C1. Obviously, in this case, during the experimentation, each of the n* drivers 

would fall either in C1 or in C2 with p as the probability of a driver being in C1. This 

phenomenon can, therefore, be described by Binomial probability distribution with 

parameters n* and p. Accordingly, the expected number of drivers who would be involved in 

rear-end crashes in a sample of size n* is given by [1] 

k n p n p * * ( , ) ˆ = . (6) 

with the variance 

 V n p n pq * * ( , ) = (7) 

 

 that can be used to calculate the confidence interval for the number of drivers involved in 

rear-end crashes out of a sample of size n* . 
 

 National Center for Statistics and Analysis ♦ 400 Seventh St., S.W., Washington, D.C. 20590 10

6. Sample designs 

Whether or not the sample size is fixed in advance, our main concern in pre-crash data 

collection is to obtain maximum information on the driver-related parameters in rear-end 

crashes at the minimum cost. Based on the previous analysis, the most important information 

that can be utilized to achieve this objective is the differential that exists in terms of the crash 

involvement propensity of drivers belonging to different age- or sex-based classes of drivers. 

The attribute-based classes will henceforth be referred to as strata. The most efficient way of 

utilizing this differential is to allocate more drivers (out of the pre-specified sample size) to 

those strata that have a higher crash involvement propensity. Similarly, more drivers are 

expected (out of the target number required to be involved in rear-end crashes) from those 

strata that have higher crash involvement propensity. In either case, some of the strata may not 

generate as much information, in terms of the number of drivers involved in rear-end crashes, 

as is required in order to arrive at useful conclusions. This may happen because some of the 

strata may ask for larger numbers of vehicles to be deployed than the corresponding numbers 

that can actually be allocated due to the over all restriction on the sample size. However, an 

efficient sampling strategy can help a lot in making up the discrepancy. In the subsequent 

sections, the sampling strategies are proposed for this purpose and are demonstrated through 

examples. Two situations are considered: (i) when the number of drivers that must be involved 

in rear-end crashes, before the data collection is stopped, is pre-specified and the aim is to 

obtain sufficient information on crash-involved drivers from each attribute-based stratum, and 

(ii) when the sample size is pre-specified and the aim is to allocate drivers over the attributebased strata so that each sub-sample can produce sufficient information in terms of the rearend crash involvement of drivers. While situation (i) can be handled by inverse sampling, both 

binomial and inverse sampling are required for resolving the sampling issues arising in 

situation (ii). The following two examples not only demonstrate the implementation of the 

proposed sampling strategies, but also compare CIPI-based allocation with some of the other 

possible allocations: equal and proportional. 

6.1. Case 1. Sample designs with the target number, k, of drivers to be 

involved in the rear-end crashes 

Consider the situation where the data on pre-crash driver-related parameters need to be 

collected on k drivers involved in rear-end crashes and there is no restriction on the number of 

vehicles/drivers needed for achieving this objective. The proposed approach to arrive at an 

optimal sample design in this situation is demonstrated through examples in the following 

sections. 

GES and FARS data for the year 2000 were used to estimate the sample size required for 

observing k (=10) drivers involved in rear-end crashes. Due to the anticipated operational 

difficulties, Age group 1 and Age group 5 were excluded. Specifically, two target populations 

were considered: 18 to 64 year old drivers (Age group 2, Age group 3, and Age group 4), and 

18 to 24 year old (young) drivers. As suggested by the contingency analysis in Section 4.1, 

two stratification criteria were used: age of the driver in the first population and driver’s sex 

in the second. The target number k of drivers required to be involved in rear-end crashes was 
 

 National Center for Statistics and Analysis ♦ 400 Seventh St., S.W., Washington, D.C. 20590 11

partitioned into M numbers, k1, k2,...,kM ( k k M

i ∑ i = =1 ), with ki representing the target number 

of rear-end crash-involved drivers from stratum i. This was done in four ways: (i) ki’s 

determined by CIPI, (ii) equal ki’s, (iii) ki’s proportional to strata sizes, and (iv) ki’s adjusted 

using the sampling strategy. The steps required to arrive at an optimum sample size through 

the sampling strategy are described in Figure 3. The strata sample sizes and hence the total 

sample size estimated by CIPI and the sampling strategy use inferences made in the previous 

sections as well as the estimates obtained there. 

Figure 3. Sampling strategy when the target number, k, of drivers 

 involved in rear-end crashes is to be observed. 

The results, based on one-year crash statistics, are presented in Table 1 (for age-based 

stratification, M=3) and Table 2 (for sex-based stratification, M=2). The last three columns 

(columns 5, 6, and 7) in these tables, respectively, represent the number of drivers likely to be 

involved in rear-end crashes from each stratum (ki), the required stratum sample size (ni) and 

its 95% confidence interval. 

1. Stratify the target population over M

classes (strata), based on an appropriate

criterion (e.g., Age or Sex ). 

4. Use inverse sampling (equation (4))

to estimate the sub-sample size ni

for each stratum, i = 1,2,...,M . 

5. Add stratum sample sizes ni to obtain

total sample size n , ∑ = = M

i 1 i n n . 

2. Use CIP (equation (1)) to allocate k (the least

number of drivers on which rear-end pre-crash

information is required) over the strata, thus

computing k k , i 1,2,..,M , i =φi = ∑ = = M

i 1 i k k . 

3. Re-assign ki if for certain strata the

representation in the target number k is not

sufficient (as assessed through CIPI in Step 2). 

6. Adjust ki ’s if certain ni’s or n is large

and go to Step 4, else exit with the

current strata sample sizes. 
 

 National Center for Statistics and Analysis ♦ 400 Seventh St., S.W., Washington, D.C. 20590 12

6.1.1. Example 1: Sample design with the target number, k=10, in sampling 

from a population stratified by driver’s age 

In this example, the stratification of 18 to 64 year old drivers was done on the criterion of age. 

The estimated sample sizes and the corresponding 95% confidence intervals are presented in 

Table 1. These results show that, if CIPI is used as the sample allocation 

Table 1. Sample designs for observing the target number, k=10, of drivers involved in rearend crashes, using CIPI, equal and proportional allocations and the sampling strategy in 

sampling from a stratified (by age) population of drivers 

Allocation 

criterion 

Stratum 

of drivers 

 ( i )

Probability 

rear-end crash

involvement 

( pi )

Constant of 

proportionality

(αi )

Number of 

drivers involved 

in rear-end crashes 

( k k i i = α . )

Stratum 

sample size 

( ni ) 

95% Confidence 

interval 

[ i LOWER iUPPER n , n ]

1. Age group 2 0.03742 0.74949 8 213 [68, 359] 

2. Age group 3 0.02393 0.13294 1 42 [2, 123] 

3. Age group 4 0.01575 0.11758 1 64 [1, 187] 

CIPI 

Total 10 319 

1. Age group 2 0.03742 0.33333 3 + 1 107 [4, 210] 

2. Age group 3 0.02393 0.33333 3 125 [3, 266] 

3. Age group 4 0.01575 0.33333 3 191 [3, 404] 

EQUAL 

Total 10 423 

1. Age group 2 0.03742 0.13721 1 37 [0, 97] 

2. Age group 3 0.02393 0.49472 5 207 [3, 387] 

3. Age group 4 0.01575 0.36807 4 233 [3,471] 

PROPORTIONAL 

Total 10 477 

1. Age group 2 0.03742 n 5 134 [19, 249] 

2. Age group 3 0.02393 n 3 125 [2, 265] 

3. Age group 4 0.01575 n 2 127 [1,302] 

SAMPLING 

STRATEGY 

Total 10 386 

Source: National Center for Statistics and Analysis, NHTSA, GES and FARS 2000, FHWA 

n not required 

criterion, then, of the 10 drivers required to be involved in rear-end crashes, 8 would be from 

Age group 2, requiring 213 drivers, and 1 each from Age group 3 and Age group 4, requiring, 

respectively, 42 and 64 drivers to participate. Thus, with age as the stratification criterion and 

CIPI as the allocation criterion, the total number of vehicles/drivers required for data 

collection adds up to 319. 

For the purpose of comparison, two other methods of allocation were also considered: equal 

and proportional. The results presented in Table 1 show that with the target numbers k1 = 4, k2
 

 National Center for Statistics and Analysis ♦ 400 Seventh St., S.W., Washington, D.C. 20590 13

= 3, and k3 = 3 for the three strata, larger number of drivers, 125 and 191, 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 have lower crash involvement propensity as compared to Age group 

2 and yet the samples selected from them are supposed to produce almost the same number 

involved in the rear-end crashes. This in turn raises the requirement of total number of 

vehicles/drivers in the sample to a larger number (423) as compared to CIPI-based allocation 

that requires 319 drivers. 

The target number (k = 10) of crash-involved drivers was also allocated using proportional 

allocation (i.e., proportional to strata sizes). The results presented in Table 1 show that with 

this allocation, only 1 driver is expected to be involved in a rear-end crash out of 37 selected 

from Age group 2. Age group 3 and Age group 4 need to contribute, respectively, 207 and 

233 drivers to the sample in order to observe, respectively, 5 and 4 drivers involved in rearend crashes. Thus, with proportional allocation, 477 vehicles/drivers would be required in 

order to observe 10 drivers involved in rear-end crashes. This number is larger than the one 

suggested by equal allocation (423) and much larger as compared to 319 suggested by CIPIbased allocation. 

In certain situations, based on a sample allocation, the contributions of some strata to the 

target number k may not be satisfactory and ki’s need to be adjusted. In such situations, it is 

advisable to use a sampling strategy rather than determining ki’s arbitrarily. Assume for the 

time being, that in the present case, this is the situation. Using the sampling strategy proposed 

in Figure 3, optimal values of k1, k2, and k3 were determined as 5, 3, and 2, respectively. The 

estimates of the corresponding sample sizes required from Age group 2, Age group 3 and Age 

group 4 were obtained as 134, 125, and 127, respectively. After adjustment of ki’s, the total 

sample size requirement for observing 10 drivers involved in rear-end crashes is much larger 

(386) as compared to CIPI-based (319). This is obviously due to the requirement of higher 

rear-end crash involvement of drivers from the strata which, otherwise, have low crash 

involvement propensity. 

The above results show that of the three sample allocations used in this example, the one 

based on CIPI requires the smallest number (319) of vehicles/drivers to be engaged in data 

collection. The proposed sampling strategy increases the likely contributions of Age group 3 

and Age group 4 from 1 each (as suggested by CIPI) to, respectively, 3 and 2 to the 

subpopulation of rear-end crash-involved drivers. This, however, increases the total sample 

size requirement from 319 (CIPI-based allocation) to 386 which is still less than the sample 

size based on equal (423) and proportional (477) allocations. 

6.1.2. Example 2: Sample design with the target number, k=10, in sampling 

from a population stratified by driver’s sex 

It can be seen in Table 1 (previous section) that among the three age groups, Age group 2 has 

the highest crash involvement propensity, 0.74949, used as constant of proportionality in 

CIPI-based allocation. This suggests that if the voluntary drivers are selected from only Age 

group 2, it may be economical in terms of the number of vehicles/drivers required for data 

collection. Accordingly, the sub-population of young drivers was considered as the target 
 

 National Center for Statistics and Analysis ♦ 400 Seventh St., S.W., Washington, D.C. 20590 14

population and stratified by driver’s sex. In order to design sample, the target number k (=10) 

of young drivers required to be involved in rear-end crashes was split over the two strata 

(male and female drivers), using three methods of allocation: CIPI, equal, and proportional. 

The results are presented in Table 2.

Table 2. Sample designs for observing the target number, k=10, of drivers involved in rearend crashes, using CIPI, equal and proportional allocations in sampling from a stratified (by 

sex) population of young drivers 

Allocation 

criterion 

Stratum 

of drivers 

 ( i ) 

Probability 

rear-end crash

involvement 

( pi )

Constant of 

proportionality

(αi ) 

Number of 

drivers involved 

in rear-end crashes 

( k k i i = α . )

Stratum 

sample size 

( ni ) 

95% Confidence 

interval 

[ i LOWER iUPPER n , n ]

1. Young male 0.04199 0.55045 6 143 [31, 255] 

CIPI 2. Young female 0.03260 0.44955 4 123 [4, 241] 

Total 10 266 

1. Young male 0.04199 0.50000 5 119 [17,221] 

2. Young female 0.03260 0.50000 5 153 [21, 286] EQUAL 

Total 10 272 

1. Young male 0.04199 0.51268 5 119 [17,221] 

PROPORTIONAL 2. Young female 0.03260 0.48732 5 153 [21, 286] 

Total 10 272 

Source: National Center for Statistics and Analysis, NHTSA, GES and FARS 2000, FHWA 

The CIPI-based allocation suggests that in order to observe 10 young drivers (6 male and 4 

female) involved in rear-end crashes, the sample needs to be comprised of 143 young male 

and 123 young female drivers. On the other hand, if equal number (5) of young male and 

young female drivers involved in rear-end crashes are to be observed, then 119 drivers will 

have to be selected from young male drivers and much larger (153) from young female 

drivers. The proportional allocation results in the same estimates of the strata sample sizes as 

the equal allocation. The results show that the allocation based on CIPI requires the smallest 

sample size for observing the target number (10) of drivers involved in rear-end crashes. 

6.2. Case 2. Sample designs with pre-specified sample size, n

Consider the situation where the sample size (n), rather than the number of drivers required to 

be involved in rear-end crashes (k), is pre-specified and the interest is in estimating the 

expected number k

ˆ of drivers who would be involved in rear-end crashes out of n. The 

proposed approach to resolve the sampling issues in this situation is demonstrated through 

examples in the following sections. GES and FARS data for the year 2000 were used to 

estimate different quantities required in estimating the sample sizes ni and hence n (=∑ =

M

i 1ni ). 

As in the previous example, the population of 18 to 64 year old drivers was stratified by 
 

 National Center for Statistics and Analysis ♦ 400 Seventh St., S.W., Washington, D.C. 20590 15

driver’s age and that of 18 to 24 old drivers by driver’s sex. The results were obtained using 

four methods of sample allocation: (i) ni’s determined by CIPI, (ii) equal ni’s, (iii) ni’s 

proportional to strata sizes, and (iv) ni’s adjusted using sampling strategy. The steps required 

for adjusting ni’s based on the sampling strategy are shown in Figure 4. The estimates of the 

Figure 4. Sampling strategy with restriction on the sample size

strata sample sizes based on CIPI and the sampling strategy use inferences made in the 

previous sections as well as the estimates obtained there. While using the sampling strategy, it 

is assumed that some adjustment in the number of available experimental vehicles/drivers is 

permissible. 

The results using the four methods of allocation mentioned above are presented in Table 3 

(age-based stratification) and Table 4 (sex-based stratification). The last three columns 

(columns 5, 6, and 7) in these tables, respectively, represent the strata sample sizes (ni) out of 

6. Compute total sample 

size: n, ∑ = = M

i n 1ni .

7. Assess if the revised sample size n can be

accommodated in the budget provided.

8. Repeat steps 4 through 7, if the

revised sample size becomes too large

to be permissible, else exit with the

current sample size and its allocation.

1. Stratify driver population over M classes (strata),

based on an appropriate criterion (e.g., Age or Sex ).

2. Allocate pre-specified sample size n over the

strata formed in step 1, using CIP (equation

(2), thus calculating ni , i = 1,2,...,M .

3. Estimate the number, i k , of drivers likely to be

involved in rear-end striking crashes out of the

allocated strata sample sizes, ni, using equation (6).

4. Reset the number of drivers that are required to be involved in rearend striking crashes, ki, for the strata in which the representation is

not sufficient (using estimates obtained in step 3). 

5. Use inverse sampling (equation (4)) to estimate

the sample sizes, ni, required for the strata in

which adjustments were done in step 4.
 

 National Center for Statistics and Analysis ♦ 400 Seventh St., S.W., Washington, D.C. 20590 16

the pre-specified sample size n, estimates of the corresponding numbers of drivers likely to be 

involved in rear-end crashes (ki) and their 95% confidence intervals. 

6.2.1. Example 1: Sample design with pre-specified sample size, n=200, in 

sampling from a population stratified by driver’s age 

Consider the situation where the population of 18 to 64 old drivers is partitioned on the 

criterion of age and the interest is in allocating n (=200) drivers in order to observe maximum 

possible number of them involved in rear-end crashes. As mentioned earlier, four methods of 

allocation were used. The results presented in Table 3 show that CIPI-based, equal, and 

proportional allocations would, respectively, result in 7, 6, and 4 drivers involved in rear-end 

crashes, out of the pre-specified sample of 200 drivers. On the other hand, the allocation 

based on the sampling strategy would require 7.5% increase in the total sample size in order 

to increase the representations of Age group 3 and Age group 4 among the crash-involved 

drivers. Specifically, with the sampling strategy, a sample of 215 drivers would be required in 

order to observe 7 drivers involved in rear-end crashes. 

The sample allocation based on CIPI suggests that the selection of 150 drivers from Age 

group 2, 27 from Age group 3 would, respectively, result in 6 and 1 driver(s) involved in rearend crashes. The point estimate of k3 corresponding to Age group 4 in this allocation indicates 

that no driver would be involved in rear-end crash from this stratum. However, to be 

optimistic, as the upper confidence limit of the estimate of k3 suggests, the selection of 23 

drivers from this age group might result in the involvement of 1 driver in a rear-end crash. 

The equal allocation of 200 drivers suggests that of the 6 drivers likely to be involved in rearend crashes, 3 would be from Age group 2, 2 from Age group 3, and 1 from Age group 4. 

The proportional allocation, on the other hand, suggests a completely different sample design. 

The selection of 27, 99, and 74 drivers, respectively, from Age group 2, Age group 3, and Age 

group 4 is likely to result in 1, 2, and 1 driver(s) involved in rear-end crashes, respectively. 

This example demonstrates that of the three allocations: CIPI-based, equal, and proportional, 

CIPI-based allocation is likely to generate data on the maximum number (7) of crash-involved 

drivers. The equal allocation can be considered as the second best choice that is likely to 

result in 6 crash-involved drivers. Due to poor performance (with the likelihood of only 4 

drivers involved in rear-end crashes), the proportional allocation cannot be recommended in 

this case. In fact, the strata sizes resulting from this allocation are too heterogeneous and are 

not commensurate with the corresponding crash involvement propensities. 

In a given situation, for certain reasons, the representation of some strata in the sample may 

not be satisfactory, thus requiring a revision of the sample allocation. The sampling strategy 

proposed in Figure 4 can help a great deal in resolving this issue. Using this strategy, the 

strata sample sizes were estimated as 100, 65, and 50 drivers, respectively, from Age group 2, 

Age group 3, and Age group 4. As compared to CIPI-based allocation, the revised allocation 

requires significant increase in the sample sizes from Age group 3 and Age group 4. This, 

however, is adjusted to a large degree by a decrease in the sample size from Age group 2, so 

as to keep the total sample size to its minimum possible. The results show that the revised 
 

 National Center for Statistics and Analysis ♦ 400 Seventh St., S.W., Washington, D.C. 20590 17

allocation would result in 4, 2, and 1 driver(s) involved in rear-end crashes, respectively, from 

Age group 2, Age group 3, and Age group 4. 

Table 3. Sample designs with pre-specified sample size, n=200, using CIPI, equal and 

proportional allocations and the sampling strategy in sampling from a stratified (by age) 

population of drivers 

Allocation 

criterion 

Stratum 

of drivers 

 ( i )

Probability 

rear-end crash 

involvement 

( pi )

Constant of 

proportionality

(αi )

Allocated 

stratum 

sample size 

 ( ni =αi ⋅ n )

Drivers likely to 

be involved in 

rear-end crashes 

 ( i k )

95% Confidence 

interval 

[ iLOWER iUPPER k , k ]

1. Age group 2 0.03742 0.74949 150 6 [1, 11] 

2. Age group 3 0.02393 0.13294 27 1 [0, 3] 

3. Age group 4 0.01575 0.11758 23 0 [0, 1] 

CIPI 

Total 200 7 

1. Age group 2 0.03742 0.33333 67 3 [0, 6] 

2. Age group 3 0.02393 0.33333 67 2 [0, 4] 

3. Age group 4 0.01575 0.33333 66 1 [0, 3] 

EQUAL 

Total 200 6 

1. Age group 2 0.03742 0.13721 27 1 [0, 3] 

2. Age group 3 0.02393 0.49472 99 2 [0, 5] 

3. Age group 4 0.01575 0.36807 74 1 [0, 3] 

PROPORTIONAL 

Total 200 4 

1. Age group 2 0.03742 n 100 4 [0, 8] 

2. Age group 3 0.02393 n 65 2 [0, 4] 

3. Age group 4 0.01575 n 50 1 [0, 3 

SAMPLING 

STRATEGY 

Total 215 7 

Source: National Center for Statistics and Analysis, NHTSA, GES and FARS 2000, FHWA n not required 

6.2.2. Example 2: Sample design with pre-specified sample size, n=200, in 

sampling from a population stratified by driver’s sex 

The pre-specified sample size n (=200) was allocated over two sex-based strata of young 

drivers by using three methods of allocation: CIPI, equal, and proportional to the strata sizes. 

The results are presented in Table 4. These results show that the sample sizes assigned on the 

basis of CIPI, equal, and proportional allocations would result, respectively, in 8, 7, and 7 

drivers involved in rear-end crashes. 
 

 National Center for Statistics and Analysis ♦ 400 Seventh St., S.W., Washington, D.C. 20590 18

Table 4. Sample designs with pre-specified sample size, n, using CIPI, equal and proportional 

allocations in sampling from a stratified (by sex) population of young drivers 

Allocation 

criterion 

Stratum 

of drivers 

( i ) 

Probability 

rear-end crash

involvement 

( pi )

Constant of 

proportionality

(αi ) 

Allocated 

stratum 

sample size

( ni =αi ⋅ n )

Drivers 

involved in 

rear-end crashes 

( i k )

95% Confidence 

interval 

[ i LOWER iUPPER k , k ]

1. Young male 0.04199 0.55045 110 5 [1, 9] 

2. Young female 0.03260 0.44955 90 3 [0, 6] CIPI 

Total 200 8 

1. Young male 0.04199 0.50000 100 4 [0, 8] 

EQUAL 2. Young female 0.03260 0.50000 100 3 [0, 6] 

Total 200 7 

1. Young male 0.04199 0.51268 110 4 [0, 8] 

PROPORTIONAL 2. Young female 0.03260 0.48732 90 3 [0, 6] 

Total 200 7 

Source: National Center for Statistics and Analysis, NHTSA, GES and FARS 2000, FHWA 

The CIPI-based allocation suggests the selection of 110 young male and 90 young female 

drivers in the sample that is likely to result in the involvement of 5 young male and 3 young 

female drivers in the rear-end crashes. The equal allocation suggests that a sample comprised 

of 100 young male and 100 young female drivers is likely to result in 4 young male and 3 

young female drivers. Although the proportional allocation also suggests the involvement of 4 

young male and 3 young female drivers in rear-end crashes, the required sample sizes are 

different. This allocation suggests the selection of 103 young male and 97 young female 

drivers to compose a sample of 200 young drivers. 

This example demonstrates that highest number (8) of drivers are likely to be involved in rearend crashes, if the sample of pre-specified size (200) is selected from the population of young 

drivers and CIPI is used as allocation criterion. The equal and proportional allocations are the 

other possible choices with slightly different sample allocation, but with the likelihood of 7 

drivers involved in rear-end crashes. 

7. Conclusions and recommendations 

The contingency analysis of GES and FARS data for the year 2000 provided strong evidence 

of the association between driver attributes, age and sex, and the crash involvement of drivers 

(rear-end or otherwise). These factors can, therefore, be used for stratifying the population in 

order to have appropriate representation in the sample from age- or sex-based classes of 

drivers for data collection on rear-end crashes. 
 

 National Center for Statistics and Analysis ♦ 400 Seventh St., S.W., Washington, D.C. 20590 19

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. Thus, when age 

was used as the stratification criterion, out of 319 drivers required for 10 of them to be 

involved in rear-end crashes, the largest number (n1 = 213) was allocated to this stratum. 

Similarly, among young drivers, male drivers were found to be more prone to rear-end crash 

involvement. Thus, when sex was used as the criterion for stratifying only the young drivers, 

a larger number (143) was allocated to the stratum of young male drivers as compared to 123 

allocated to the stratum of young female drivers. When the sample size was assumed fixed (n 

=200) in advance and the population was stratified by age, the largest number, 150, was 

recommended from the stratum of young male drivers, with the likelihood of 6 drivers being 

involved in rear-end crashes. 

The CIPI-based sample allocation in both modes of stratification, driver’s age and sex, was 

compared with some other possible methods of allocation, equal number of drivers from each 

stratum and strata sample sizes proportional to the strata sizes. The examples illustrated that 

due to the differential that exists among the strata with respect to the crash involvement 

propensity, for the same target number of crash-involved 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. Similarly, in the case of prespecified sample size, both equal and proportional allocation, indicate the likelihood of a 

smaller number of crash-involved drivers as compared to CIPI-based sample allocation. 

So far as the usefulness of the proposed sampling strategies is concerned, two types of 

situations can be handled: (i) when some strata are not well represented in the target number 

of rear-end crash-involved drivers, and (ii) when some strata sample sizes (out of prespecified sample size) do not promise a satisfactory number of crash-involved drivers. 

The proposed sampling strategies are neither data dependent nor population dependent. In 

fact, the approach used in this study is fairly general and can be used for resolving similar 

sampling issues involved in data collection in similar setups. 
 

 National Center for Statistics and Analysis ♦ 400 Seventh St., S.W., Washington, D.C. 20590 20

8. Appendix A. Contingency tables

Table A.1. This table shows sample size and weighted size for each class (age group), that 

were computed by SUDDAN while carrying out the contingency analysis for testing the 

association between Age and Crash event (Section 4.1). 

 Table A. 1. Contingency table: Driver’s Age vs. Crash event 

Crash event 

Driver’s age Statistics 

Rear-end Other 

Total 

Age group 1 Sample size 1,708 4,097 5,805 

(< 18) Weighted size 234,832 463,310 698,142 

Age group 2 Sample size 6,243 13,731 19,974 

(18 to 24) Weighted size 791,242 1,460,653 2,251,895 

Age group 3 Sample size 15,162 27,471 42,633 

(25 to 44) Weighted size 1,795,324 2,744,902 4,540,226 

Age group 4 Sample size 7,852 14,325 22,177 

(45 to 64) Weighted size 893,748 1,458,380 2,352,128 

Age group 5 Sample size 3,165 8,514 11,679 

(>64) Weighted size 422,690 1,052,585 1,475,275 

Sample size 34,130 68,138 102,268 

Total

Weighted size 4,137,836 7,179,830 11,317,666 

Source: National Center for Statistics and Analysis, NHTSA, GES and FARS 2000 

Table A.2. This table shows sample size and weighted size for each class (Sex of young 

driver), that were computed by SUDDAN while carrying out the contingency analysis for 

testing the association between Sex of the young driver and Crash event (Section 4.1).

Table A.2. Contingency table: Young driver’s Sex vs. Crash event 

Crash event 

Driver Age/ Sex Statistics 

Rear-end Other 

Total 

Young (18 to 24 ) Sample size 3,701 8,541 12,242 

Male Weighted size 453,724 878,006 1,331,730 

Young (18 to 24 ) Sample size 2,542 5,186 7,728 

Female Weighted size 337,517 582,519 920,036 

Sample size 6,243 13,727 19,970 

Total 

Weighted size 791,241 1,460,525 2,251,766 

 Source: National Center for Statistics and Analysis, NHTSA, GES and FARS2000
 

 National Center for Statistics and Analysis ♦ 400 Seventh St., S.W., Washington, D.C. 20590 21

9. Appendix B. Analytical details of Crash Involvement Propensity Index 

B.1 Crash Involvement Propensity Index 

This appendix supplies the analytical details of the statistic Crash Involvement Propensity

Index (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 1, 2 , . . . , M

L( )

L( )

( i )

( i )

i C = = Ω

Ω

λ , (B.1) 

where 

 

i

i i

C S

C L ( Ω ) = ( ) , S 0 i > 

and 

N

S L i i ( Ω ) = ( ) , 
 

 National Center for Statistics and Analysis ♦ 400 Seventh St., S.W., Washington, D.C. 20590 22

so that λi defined in (B.1) becomes 

 , i 1, 2 , . . . , M

S

C N 2

i

i

i = 















λ = . (B.2) 

Note that λi in (B.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) 

 , i 1 , 2 , . . . , M

M

j 1

j

i

i = =

∑

=

λ

λ φ . 

Using λi from (B.2), the CIPI can be derived in the usable form 

 , i 1, 2 , . . . , M

S

C

S

C

M

j 1

2

j

j

2

i

i

i =















 =

∑

=

φ . (B.3) 
 

 National Center for Statistics and Analysis ♦ 400 Seventh St., S.W., Washington, D.C. 20590 23

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. 


DOT HS 809 541

April 2003