Attached 5 DOT HS 811 327

DOT HS 811 327

Sampling and Estimation
Methodologies of CDS

May 2010

DISCLAIMER
This publication is distributed by the U.S. Department of Transportation, National
Highway Traffic Safety Administration (NHTSA), in the interest of information
exchange. The opinions, findings, and conclusions expressed in this publication are
those of the authors and not necessarily those of the Department of Transportation
or the National Highway Traffic Safety Administration. The United States
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ii

TECHNICAL REPORT DOCUMENTATION PAGE
1. Report No.
DOT HS 811 327
4. Title and Subtitle

2. Government Accession No.

Sampling and Estimation Methodologies of CDS

7. Author(s)
Charles Fleming (NCSA)
9. Performing Organization Name and Address
National Highway Traffic Safety Administration
1200 New Jersey Avenue SE
Washington, DC 20590
12. Sponsoring Agency Name and Address
National Center for Statistics and Analysis (NCSA)
National Highway Traffic Safety Administration
1200 New Jersey Avenue SE, Washington, DC 20590

3. Recipients Catalog No.
5. Report Date
May 2010
6. Performing Organization Code
NHTSA/NVS-421
8. Performing Organization Report No.
10. Work Unit No.
11. Contract or Grant No.
13. Type of Report and Period Covered
Technical Report
14. Sponsoring Agency Code

NHTSA

15. Supplementary Notes
Thanks for help and input from Nancy Bondy, Ruby Li, and Donna Glassbrenner.
16. Abstract
Based primarily on two computer programs and expert knowledge, this document describes the method of sampling and
the method of estimation which are used in the Crashworthiness Data System.
17. Key Words
Sampling, estimation, National Automotive Sampling
System, Crashworthiness Data System
19. Security Classification (of this report)
UNCLASSIFIED
Form DOT F 1700.7 (8-72)

18. Distribution Statement:
Document is available to the public from the National
Technical Service www.ntis.gov

20. Security
21. No. of Pages 22. Price
Classification (of
this page)
27
UNCLASSIFIED
Reproduction of completed page authorized

iii

Table of Contents

Disclaimer ........................................................................................................................... ii

Technical Report Documentation Page ............................................................................. iii

1. Introduction:.................................................................................................................... 1

2. Sampling: ........................................................................................................................ 2

2.1 First Stage: Primary Sampling Units ........................................................................ 2

2.2 Second Stage: Police Jurisdiction ............................................................................ 3

2.2.1 Method of Selecting Police Jurisdiction ............................................................... 4

2.3 Final Stage: Police Accident Reports ....................................................................... 7

2.4 Drawing a Sample..................................................................................................... 9

2.4.1 PPS Sampling of PAR's ...................................................................................... 11

2.5 Certainties ............................................................................................................... 11

2.5.1 Sub-divided Police Jurisdictions.......................................................................... 12

3. CDS Weights: ............................................................................................................... 12

3.1. Benchmarking:....................................................................................................... 14

4. Trimming: ..................................................................................................................... 15

Bibliography: .................................................................................................................... 15

Appendix I: List of PSU's ................................................................................................. 18

Appendix II: Non-sampled PJ Counts .............................................................................. 19

Appendix III:Definitions of PAR Strata ........................................................................... 20

iv

List of Figures

Figure 1: cumsum Function ................................................................................................ 6

v

List of Tables

Table 1: Computer Program Variable Names in Analytical Files ..................................... 1

Table 2: PSUGRP ............................................................................................................... 3

Table 3: COLSTRAT.......................................................................................................... 3

Table 4: Calculation of PJWGHT for PSU......................................................................... 4

Table 5: PAR Stratum Weight ............................................................................................ 8

Table 6: Sampling PARS by PPS ..................................................................................... 10

Table 7: Criteria for Sub-dividing Police Jurisdictions .................................................... 12

vi

1 Introduction
The origins of the Crashworthiness Data System can be traced to the National Accident Sampling
System which was implemented for operational use in 1979. Important documents which describe the
design of the original system can be found in [1, 2, 3, 6, 7, 9, 11, 12, 13, 14, 15, 16, 18, 19]. Later, its
name was changed to the National Automotive Sampling System (NASS). In 1988, NASS was divided
into two parts: the General Estimates System (GES) and the Crashworthiness Data System (CDS). The
CDS is one of the crash databases produced by NHTSA. A general description of CDS can be found
in [8].
This report describes the sampling and estimation methodologies of CDS as they are followed in
the operational program. The primary sources of information upon which this report is based are four
documents. They are [4], the two SAS computer programs, PSUWGHT.SAS and CDSWGT.SAS, and
sampling worksheets for manually drawing a sample of police jurisdictions (PJ). The purpose of these
worksheets is to validate the computerized sampling algorithm, in order to make sure that the cases
are selected the same way whether manually or by computer program. This computer program which
is written in DELPHI performs the same task automatically as that of the sampling worksheets. The
output of the DELPHI computer program has been validated by the sampling manager of CDS for
accuracy.
The formulas which appear in this report were inferred from the computer programs, and their
accuracy was confirmed by comparing computations based on these formulas with the output of the
computer programs. To make it easier for the reader to associate the formulas with the logic of the
computer programs, names of the variables which appear in the SAS programs are retained in the
formulas. The correspondence between the names of SAS variables and the variables of the final ana­
lytical files is always maintained except in three cases. The three variables of the computer programs
which are not the same names which appear in the analytical files are listed in Table 1.
Table 1: Computer Program Variable Names in Analytical Files
Variable Name in Computer Program
PSUGRP
RATWGHT
STRATA

Variable Name in Analytical Files
PSUSTRAT
RATWGT
STRATIF

This report is divided into two main parts. The first part describes the method of sampling which is
used in CDS. Afterwards, the second part describes the method of estimation. The same terminology
as well as the definitions of the population of interest and of the stratification which are published in
[10] are followed in this report. Three levels of stratification appear in CDS. The stratification of the
PSU’s constitutes the first level. It is followed by the stratification of the PJ’s, and then the stratification
of the police accident reports (PAR) follows next. To emphasize the importance of the stratification of
the PAR’s, the definitions of the strata are reproduced in Appendix III.

1

2 Sampling
2.1 First Stage: Primary Sampling Units
The collection of primary sampling units (PSU) forms an area frame of the United States. There are
1,195 PSU’s [4] and each one is assigned to one of twelve strata [17] which are defined according to
geographical and demographic characteristics. The definitions of the strata and a detailed account of
their determinations can be found in [17]. From the collection of PSU’s, 24 of them were selected in
1991 such that two PSU’s are selected from each of twelve strata. In a demonstration of the versatility
of the design of CDS to changing conditions, three more PSU’s were added in 2002 for a special
research study until 2008 when the original 24 PSU’s again comprise the sample of PSU’s. For this
special study, three PSU’s were added from 2002 through 2004 to the sampling frame and the sampling
was restricted to include only late model year motor vehicles. From 2005 through 2007, the CDS
continued to include the three extra PSU’s, but they included both late model year and non-late model
year vehicles. The currently used PSU’s for sampling have been used ever since CDS began, although
a re-definition of their sampling weights from the 1979 values did occur in 1991 [4].
The method of calculating the PSU weights is explained in [4], and can be summarized as
1. Counts of fatal and injury crashes were obtained for each of the 1,195 PSU’s.
2. For all PSU’s within each PSU stratum, the number of fatal and injury crashes were summed.
3. The probability of a PSU being selected was determined by dividing the number of fatal and
injury crashes occurring within a PSU by the total number of fatal and injury crashes occurring
in all of the PSU’s within that stratum. The weight of the PSU is simply the inverse of the
probability of the PSU being selected.
That the number of PSU’s should change to accommodate a special study shows the flexibility of
the CDS to changes. For example, in 1996, the Agency had to accommodate a sudden increase in the
number of cases involving deaths and serious injuries which were associated with air bags. In response
to this surge of cases, more PJ’s were added, in order to obtain more crashes involving late model year
vehicles.
Groups of PSU’s (PSUGRP) have been created according to Table 2 for use in the computer pro­
gram for calculating the final published weights. There is a one-to-one correspondence between
PSUGRP and PSUSTRAT, although the numbering scheme is different between them. Appendix I
shows the correspondence between PSUGRP and PSUSTRAT. Nonetheless, the final weights which
are denoted by RATWGHT in the SAS computer programs and by RATWGT in the analytical files
are eventually associated with only a PSU stratum rather than to a specific PSU. Besides having two
PSU’s belonging to a PSU stratum, the strata of the PAR’s are grouped into categories called COL­
STRAT according to the assignments given in Table 3 such that the final published weights, RATWGT,
correspond to a particular PSUSTRAT and to a particular COLSTRAT.

2

Table 2: PSUGRP
PSUGRP PSU
1
3,6
2
72,74
3
41,49
4
79,82
5
5,8
6
12,73
7
9,45
8
75,81
9
2,4
10
11,13
11
43,48
12
76,78

Table 3: COLSTRAT
COLSTRAT Strata
AB
A∪B
CJ
C∪J
DK
D∪K
E
E
F
F
G
G
H
H

2.2 Second Stage: Police Jurisdictions
Within each PSU, there are areas called police jurisdictions (PJ). Within a police jurisdiction there is
an administrative center where police accident reports (PAR) are assembled and stored. It should be
noted that the preferred reference to the abbreviation, PAR, is to call it a police crash report. Because
PAR’s may be in an electronic format, typewritten paper copy, or hand written paper copy and since
they may be issued at any time of the day, the logistical demands of drawing a sample of PAR’s require
special consideration.
The method of selecting a police jurisdiction is based on probability proportional to size (PPS)
sampling though exceptions are sometimes made. Even though, according to the original design of
NASS, police jurisdictions are supposed to be re-selected periodically at random, the same police
jurisdictions like the PSU’s have been used since 1995 .
To take into account varying sizes of the PJ’s, police jurisdictions are classified by strata in such a
way so as to create groups of PJ’s which have equal number of fatal and injury crashes within a PSU.
These groups are in essence strata which are determined by the number of instances in which at least
one death, at least one incapacitating injury, or at least one non-incapacitating injury had occurred.
This number is abbreviated by KAB. It serves as the measure of size of a PJ when selecting a police
jurisdiction within one of these PJ strata according to the method of probability proportional to size.
The current CDS sampling uses 1992 KAB data. The number of PJ strata varies according to PSU, and
it depends on the goal to equalize the number of cases across PJ’s within a PJ stratum. The method
which is followed begins by sorting the PJ’s of a PSU by the number of fatal and injury crashes in
descending order of magnitude. Then the PJ’s are grouped into strata of similar size. Usually, the
largest strata contain only one PJ which, in turn, is selected with certainty.

3

2.2.1 Method of Selecting Police Jurisdictions
In explaining the method of selecting a police jurisdiction, P Jil , from P SUi, it is useful to define
certain variables which closely mimic the variables which appear in the computer programs. We will
use concepts which are associated with the method of probability proportional to size sampling in
which the value KAB is the measure of size of a police jurisdiction and only one PJ is selected from a
PJ stratum.

Definition 1. KAB is an abbreviation for killed, incapacitating injury, and non-incapacitating in­
jury as defined by the KABCO [5] system of classifying the severity of injury. Let KABil be the
number of cases in which at least one death or at least one incapacitating injury or at least one nonincapacitating injury occurred in P SUi and in P Jil .
Table 4 presents a list of PJ’s and the associated KAB’s for a hypothetical P SUi. We will refer to
Table 4 when describing the method of selecting PJ’s.
Table 4: Calculation of P JW GHT
PJ
l
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

PJSTRAT
KABl
85
81
73
67
66
65
53
47
42
35
31
20
15
15
15
14
12
10
4
0

m
1
2
3
4
4
5
5
6
6
7
7
7
7
7
8
8
8
8
8
8

s
1
1
1
1
2
1
2
1
2
1
2
3
4
5
1
2
3
4
5
6

cumsum(KABms )
s≤nm

85
81
73
67
133
65
118
47
89
35
66
86
101
116
15
29
41
51
55
55

RAN Dm

Random Start
n
m
P
KABms

RAN Dm

Select

λm

P JW GHTl

1
1
1
1
0
1
0
0
1
1
0
0
0
0
0
1
0
0
0
0

1
2
3
4

133
67

1
1
1
= 1.99

6

118
65

= 1.82

9
10

89
42
116
35

= 2.12
= 3.31

16

55
14

= 3.93

s=1

1
1
1
.016

85
81
73
.016(133)=2

.504

.504(118)=59

.935

.935(89)=83

.258

.258(116)=30

.368

.368(55)=20

Definition 2. Let l designate a PJ of a PSU, so that P Jil is the lth PJ in the ith PSU, and let m
designate the PJ stratum.

Definition 3. P JST RATim = {P J ∈ P J stratum m of P SUi }, and let nim be the size of P JST RATim.
P JST RATim is a collection of all PJ’s which are assigned to PJ stratum m. For example, for PJ
stratum 8 as given in Table 4 of P SUi , P JST RATi8 = {15, 16, 17, 18, 19, 20}. A catalogue of the
number of PJ strata for each PSU is given in Appendix I.
4

Definition 4. Let RANDim be the random number which is generated by a computer software
package for a U(0, 1) probability distribution for P SUi and P JST RATim .
Every element of a PJ stratum will be assigned the same random number. That is, every PJ stratum
is assigned a unique random number and every element of that stratum is associated with that assigned
random number.
Example 1 presented below illustrates the process of selecting a PJ as well as of making the math­
ematical description of the process more transparent. A key concept which is essential for the process
of selecting an element for a sample when following the method of probability proportional to size is
the cumulative sum.

Definition 5. The cumulative sum of KAB’s up to and including the KAB of the Λth PJ is defined to
be

Λ
cumsum(KABis ) = ' KABis .
s≤Λ
s=1

We see an example of cumsum in the fifth column of Table 4. Let us consider P JST RATi8 . The
cumsum for this PJ stratum is {15, 29, 41, 51, 55, 55}. Each term of the cumsum is the sum of the
previous values of KAB’s including itself. For instance, the third cumsum is: 15+14+12=41. That PJ
which is ultimately selected for the sample will be denoted by a special subscript, λm . More formally,
for the purpose of selecting PJ’s, let the subscript of that PJ of P JST RATim which is selected for
sampling be denoted by λm . Its precise definition is given by

Definition 6. λm = min λ|

λ
'

KABis ≥ RANDim

s=1

n'
im

KABis

s=1

where nim is the number of PJ’s which are contained in P JST RATim . In other words, λm identifies
that PJ of P JST RATim which is selected for the sample; therefore, P Jiλm is that PJ which is selected
from P JST RATim . We will use the cumulative summation to define a step function which lies at the
basis of selecting the PJ’s by means of PPS sampling. In terms of cumulative summations, Definition
6 can be written as
is ) ≥ RANDim
λm = min λ| cumsum(KAB
s≤λ

nim
�

KABis

s=1

Denote the right hand side of the inequality by X, in order to help us to see how λm is determined.
is ) ≥ X
f (X) = min λ| cumsum(KAB
s≤λ

(1)

A graphical depiction of f(X) appears in Figure 1. It is a step function in which the steps occur at each
is )
level, Λ, of cumsum(KAB
where the left end of a level line is empty and the corresponding
s≤Λ
right end is closed. Only the initial four of several levels are shown in Figure 1. For a given X, like
the one drawn in Figure 1, equation (1) says that the chosen element for the sample is determined by
finding all cumsum’s of KAB which either equal X or exceed X and from that list find the PJ with the
minimum cumsum. In reference to Figure 1, the cumsum’s which are bigger than or equal to X are the
3rd , 4th , 5th , 6th , etc. The smallest one is the third cumsum; therefore, of the PJ’s, the third PJ of
5

cumsum(KAB)

cumsum(KABil )

l<= 4

f(X)=cumsum(KABil )

l<=3

cumsum(KABil )

l<= 2

cumsum(KABil )

l<= 1

X0

X1

X2 X X3

X4

Figure 1: cumsum Function

the PJ stratum is selected. As X changes, so might f(X), and since a different RANDim will produce
a different X, we might select a different PJ, hence we see the origins of the random sampling of the
PJ’s.
Definition 6 describes the basic idea of selecting a PJ:
1. List all PJ’s of a PSU in descending order according to KAB’s.
2. Define the boundaries of the PJ strata such that the sum of the KAB’s are approximately the same
across the strata and such that it is feasible for the researchers to accomplish their assignments.
3. Order the PJ’s according to their KAB’s within a PJ stratum.
4. Determine those PJ’s which will be selected with probability 1.
5. Given a PJ stratum, list all cumsum’s of KAB’s which equal or exceed a random start.
6. Pick that PJ which corresponds to the smallest cumsum of that list.
Referring to Table 4 and to P JST RATi8 , we see that RANDi8 = .368. The random start is
.368(55) = 20. All PJ’s with cumsum’s exceeding 20 are {16, 17, 18, 19, 20}. The PJ with the smallest
cumsum is P Ji16 . Because it is the second PJ in P JST RATi8 , λ8 = 16. This second PJ is the one
selected for the sample.
Another example to consider is P ST RATi6. The cumsum is {47, 89}. The random number which
is produced from a computer program is .935. The random start is .935(89)=83. The cumsum which
6

exceeds 83 is {89}. The smallest cumsum is 89, and it corresponds to P Ji9 which is the second PJ in
P ST RATi6 , hence, λ6 = 9.
In summary, P Jiλm is the selected police jurisdiction of P JST RATil when it occurs that
λm = min λ|

λ
�

KABis ≥ RANDim

s=1

nim
�

KABis

s=1

The probability of the event of selecting P Jiλm is
KABiλm
P (P Jiλm ) = n'
im
KABis
s=1

Its reciprocal is called P JW GHTiλm , and it is given by

P JW GHTiλm =

n'
im

KABis

s=1

KABiλm

(2)

Example 1. The example given in Table 4 illustrates the method which is employed in CDS to select
a police jurisdiction.
In this example, we notice that in PJ strata 1, 2, and 3 there is only one PJ. If a total KAB count of
a PJ exceeds 70, then it is selected with certainty.
The selection of an element for a sample based on PPS sampling depends on the order of the PJ’s.
A different order of the PJ’s will produce different sets of cumsum’s. When selecting PJ’s, the PJ’s are
ordered with respect to KAB in descending order such that the PJ with the largest KAB is listed first.

2.3 Final Stage: Police Crash Reports
The statistical enumerators who are called researchers enter the PAR’s from the sampled PJ’s into a
computer program for case selection. The researchers conduct vehicle and scene inspections within
four days of the case being selected. They conduct interviews, draw the scene diagram, code the
case, and obtain medical records. When determining the number of PJ’s to be selected in a PSU,
consideration is given to the area of the PJ, the number of jurisdictions which investigate motor vehicle
traffic crashes, the number of those which involve an injury or death, the estimated number of cases
which should be drawn from a PSU, and the distance between PJ’s.
Sampling of PAR’s is performed weekly in all 171 police jurisdictions which are located in 24
PSU’s. A researcher, who is a trained statistical enumerator, once a week examines the PAR’s which
apply to the police jurisdiction to which he is assigned. The researcher will classify each PAR as
to stratum, only if the PAR qualifies for admission into the sampling frame. Elements of the CDS
sampling frame of any police jurisdiction must comply with the same criteria for admission to the
sampling frame as the criteria which defines the population [10]. The weekly drawing of a PAR from
a police jurisdiction sampling frame is the last stage of sampling.
7

The list of PAR’s which is constructed each week at a police jurisdiction is divided into ten strata.
These ten strata are: A, B, C, D, E, F, G, H, J, and K. The definitions of them are given in Appendix
III. The last two strata were added to the original strata in 1991.
In the late 1980’s, the Agency required that the number of serious injury crashes which are selected
for the sample be increased. A group in the National Center for Statistics and Analysis (NCSA) con­
sidered many options to achieve the desired sampling size. Each option was analyzed by such factors
as the number of potential cases by severity of injury both in terms of weighted and unweighted values
and the feasibility and ease of implementation. It was determined that adding two strata, J and K, des­
ignated for hospitalization would significantly increase the number of cases of serious injuries while
maintaining the weighting requirement. A pilot study was conducted for three months in 1990 to dis­
cover whether or not there is sufficient information contained on a PAR to determine if hospitalization
had occurred for a case. By adding the two strata, the number of cases of serious injuries which could
be available for sampling increased by 40%. In 1991, the two strata were added such that stratum J
corresponds to late model year vehicles and stratum K corresponds to non-late model year vehicles.
In every one of the 171 police jurisdictions, a weekly sampling of PAR’s is performed when a
contractor by means of a computer program draws a sample of PAR’s once all cases have been entered
by a researcher. Afterwards, elements of the sample are transmitted to the researcher. This computer
program is written in DELPHI for use in an Oracle database. Sampling worksheets upon which the
DELPHI program is based were used when preparing this report in lieu of the DELPHI program to
infer the sampling method of the PAR’s.
The sizes of the samples depend on the availability of the researchers, consequently, the eventual
sampling size is not always an optimum sampling size for achieving a prescribed precision in the
estimates. Instead, the sizes of the samples of PAR’s regardless of PSU will vary from week to week.
In practice, the sampling of the PAR’s is performed independently of the police jurisdiction since the
operational sampling frame is formed by taking the union of the sampling frames of all sampled police
jurisdictions within a PSU. It is from this combined sampling frame that the sample is drawn based
on PPS sampling in which the measure of size of a PAR is the product of stratum weight by the size
of the police jurisdiction with respect to its PJWGHT which is calculated by the method described in
Section 2.2.1.
The stratum weights which are assigned to the PAR strata were created, in order to increase the
number of cases of severe injuries in the sample. The weighting factors reflect the importance of these
cases. The determination of the stratum weights was originally done in 1988 to produce a sample from
which the proportion of late model year motor vehicle crashes to non-late model year motor vehicle
crashes was 3 to 2. This ratio was changed to 7 to 3 when strata J and K were added.
The stratum weights which are called STRTWGHT are given in Table 5. In Table 5, PAR strata
Table 5: PAR Stratum Weight
PAR Stratum
A
B
C
D E F G H
J
K
STRTWGHT
400 400 175 25 7 3 2 1 400 300
Late Model Year
X
X
X
X
X
Non-late Model Year X
X
X
X
X

8

are designated Late Model Year and Non-late Model Year. Late model vehicles include the model
year of the current year and the last three model years and the up-coming model year. Non-late Model
vehicles include vehicles of a model year older than four years. [10]

2.4 Drawing a Sample
A discussion of the sampling of PAR’s which are taken from the list of qualifying PAR’s will begin
with some definitions.
1. Let CONT DAT Ej be the j th date on which a researcher examines P ARijkln
of P Jil where n identifies the nth PAR which is inspected on CONT DAT Ej .

Definition 7.

2. Let ST RT W GHTk be a stratum weight as given in Table 5 which is assigned to PAR stratum k.
3. Let P ARST RTijk be the set of PAR’s which belong to PAR stratum k of P SUi and which were
examined on CONT DAT Ej . The subscript i identifies the PSU, k identifies the PAR stratum,
and j identifies the CONTDATE.
4. Recall from Definition 2 that the subscript, l, identifies a PJ.
Upon being examined by a researcher, a PAR will not only be assigned to a PAR stratum according
to the definitions appearing in Appendix III, but it is assigned by the Oracle database to a sequence
number (SEQNUM) in sequential order as the PAR’s are examined. Therefore, each qualifying PAR
when it is listed in a sampling frame is assigned two identification numbers: the SEQNUM which is
based on the order of discovery and a sampling frame identification number which is based on the time
of the collision. This sampling frame identification number will be called the sequence number. The
combination of these two identification numbers serves the purpose of ordering the PAR’s within a
stratum for PPS sampling. The process of sampling PAR’s begins with their ordering. The PAR’s are
first sorted alphabetically by PAR strata and, then, within a PAR stratum by the sequence numbers in
ascending order.

Definition 8.

1. Let SEQUENCENUMBER ijn = ACCT IMMMijn ∗ 1000000 +
ACCT IMHHijn ∗ 10000 + SEQNUMijn be the sampling frame ordering number where
ACCTIMMM and ACCTIMHH correspond to the minute and hour of the day on which the col­
lision was reported to have occurred.

For example, suppose a collision was reported to have occurred on 1 July at 1032, then ACC­
TIMHH=10 and ACCTIMMM=32. Suppose that the submitted PAR was the 38th one which was
examined by the researcher, then SEQUENCENUMBER ijn = 32100038.
Once elements of the sampling frame are ordered by PAR stratum and then within a PAR stratum
with respect to SEQUENCENUMBER ijn , PAR’s are selected using PPS sampling as illustrated
in Example 2.

Example 2. In Table 6, we see that the PAR’s are ordered first according to PAR stratum and then
according to the SEQUENCENUMBER within a given PAR stratum. For each PAR, a PARWGHT is
9

calculated by PARWGHT=STRTWGHT*PJWGHT. We see that the ordering of the PAR’s is performed
independently of the PJ’s. The entries in Table 6 correspond to one PSU and to only one CONTDATE.
Suppose that it has been decided that three PAR’s are to be selected for investigation, then the
sampling interval will be 98.90
= 32.96667. To begin the process, we need a random number to
3
determine the first element of the sample. Suppose .308 is that random number which a computer
program generated, then the starting weight for sampling will be: .308*32.96667=10.15373. The first
element to be selected for the sample, therefore, will be 32100038. The next element to be selected will
be the one for which its cumsum is the smallest one which does exceed 10.15373+32.96667=43.1204.
Thus 35170045 is the second element of the sample. Likewise, for the third and last element of the
sample, 29070044 identifies the PAR with the smallest cumsum which exceeds 10.15373 + 32.96667 +
32.96667 = 76.08707.
Table 6: Sampling PAR’s by PPS
SEQUENCENUMBER
32100038
48090042
1030004
13140058
35100026
35170045
57070059
3000018
16070012
30020050
45120021
55140057
4130046
5000054
5070030
6080031
14090019
16160051
20120022
22170014
25180052
28120055
29070044
29140029
30230056
35100010
35220033
40110037
41080053
49230048
50150007
55180032
57070006
59040001

SEQNUM
38
42
4
58
26
45
59
18
12
50
21
57
46
54
30
31
19
51
22
14
52
55
44
29
56
10
33
37
53
48
7
32
6
1

PJ
4
6
9
2
1
6
2
3
9
2
1
2
6
2
10
10
3
2
1
3
2
2
6
1
2
9
10
4
2
6
9
10
9
9

PJ Stratum
4
5
6
2
1
5
2
3
6
2
1
2
6
2
7
7
3
2
1
3
2
2
5
1
2
6
7
4
2
5
6
7
6
6

PAR Stratum
E
E
F
F
F
F
F
G
G
G
G
G
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H
H

10

STRTWGHT
7
7
3
3
3
3
3
2
2
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1

PJWGHT
1.98507
1.81538
2.11905
1.00000
1.00000
1.81538
1.00000
1.00000
2.11905
1.00000
1.00000
1.00000
1.81538
1.00000
3.31000
3.31000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.81538
1.00000
1.00000
2.11905
3.31000
1.98507
1.00000
1.81538
2.11905
3.31000
2.11905
2.11905

PARWGHT
13.93
12.74
6.36
3.00
3.00
5.46
3.00
2.00
4.24
2.00
2.00
2.00
1.82
1.00
3.31
3.31
1.00
1.00
1.00
1.00
1.00
1.00
1.82
1.00
1.00
2.12
3.31
1.99
1.00
1.82
2.12
3.31
2.12
2.12

cumsum
13.93
26.67
33.03
36.03
39.03
44.49
47.49
49.49
53.73
55.73
57.73
59.73
61.55
62.55
65.86
69.17
70.17
71.17
72.17
73.17
74.17
75.17
76.99
77.99
78.99
81.11
84.42
86.41
87.41
89.23
91.35
94.66
96.78
98.90

select
1
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0

2.4.1 PPS Sampling of PAR’s
The method of PPS sampling is used to select police jurisdictions as well as to select PAR’s. We will
augment the definitions which we used for selecting PJ’s by means of PPS sampling with a few more
definitions.

Definition 9.

1. SAMP LINGF RAMEij is the set of all PAR’s of the sampled PJ’s which are
examined on CONT DAT Ej and which describe a collision in which at least one automobile
or one light truck has to be towed away due to damage and which had occurred on a highway
in P SUi .

2. Let RANDij be the random number which is generated by a computer program for a U(0, 1)
probability distribution for P SUi. The same RANDij applies to all elements of P SUi on
CONT DAT Ej .
3. P JW GHTiln is the weight of P Jil of P SUi which is assigned to P ARijklm.
4. Let P ARW GHTijkln = ST RT W GHTk P JW GHTiln for that PAR of P SUi and P Jil which
was the nth PAR which was inspected on CONT DAT Ej .
Λ
ARW GHTijkln) = ' P ARW GHTijkln
5. cumsum(P
n≤Λ
n=1

6. CASELOADij is the sampling size for the j th week of P SUi .
'
P ARW GHTijkln
7. ORIGSAMPij =
over all elements∈SAM P LIN GF RAM Eij

8. INT ERV ALij =

ORIGSAM Pij
CASELOADij

9. BEGINij = INT ERV ALij RANDij
Once the sampling frame of the PSU has been ordered according to the SEQUENCENUMBER
within each stratum, elements of the sample are drawn systematically beginning with BEGINij
and at succeeding intervals, BEGINij + mINT ERV Alij , that is, the CDS sample for P SUi and
CONT DAT Ej is

Definition 10.
ARW GHTijkls) ≥ BEGINij +
Sij = {P ARijklν |ν = min {λ| cumsum(P
s≤λ
m INT ERV ALij } ∀m ≤ CASELOADij }

2.5 Certainties
After the PAR’s have been ordered alphabetically by strata and then within a stratum afterwards by
the SEQUENCENUMBER, the process of sampling PAR’s according to PPS sampling begins. Some
PAR’s will be classified as certainties when the probability of selection is 1. The process of identifying
certainties is the following decision tree:
11

1. If any PAR listed in the ordered sampling frame has a PAR weight greater than or equal to the
sampling interval, then it is selected with certainty.
2. Mark the case(s) as selected.
3. Recalculate the ORIGSAMP after removing the certainty case(s). Reduce the CASELOAD by
the number of certainty cases selected.
4. Repeat Steps 1, 2, and 3 while using the revised ORIGSAMP and CASELOAD at each iteration
until no more certainties have been discovered.
2.5.1 Sub-divided Police Jurisdictions
Some PJ’s are sub-divided due to the extremely large number of cases to list. Within such a selected
sub-division, half of the PAR’s might be selected. For example, only odd numbered PAR’s are se­
lected. In other PJ’s, every fifth PAR is selected. In these particular PJ’s, the selection of sub-divisions
constitutes another stage of sampling for a PAR. In spite of sub-dividing some PJ’s, a total of 171
PJ’s are eventually selected. The sub-sampling weights are listed in Table 7. These sub-weights are
referenced in the program CDSWGT.SAS by the variable ADJUST .
Table 7: Criteria for Sub-dividing Police Jurisdictions
ADJUSTijkl Condition
2
if PSU=3
2
else if PSU=72 and P J ∈ {1, 2, 3, 4, 5, 6}
5
else if PSU=72 and P J = 7
2
else if PSU=79 and P J ∈ {1, 7}
2
else if PSU=81 and P J ∈ {1, 2}
1
otherwise

3 CDS Weights
The CDS weights are computed using the List Case file. This file contains the basic information about
the cumulative collection of sampled PAR’s for the year. A computer program produces from the List
Case file the weights which are to be used for producing the national estimates of the information
which is obtained from the researchers’ investigations. The usual procedure for producing estimates
from the CDS data is to calculate the frequency of each category of an investigation by means of
RATWGT. The definition of RATWGT will be discussed later. Each observation contributes a value
of one to the unweighted frequency counts, but when we use RATWGT, each observation is expanded
to the national level so that it contributes the RATWGT weighted value for that observation.
As before, additional definitions will facilitate the description of the method of calculating CDS
weights.

12

Definition 11.

1. Let P ARW GHTijkln be the sampling weight of P ARijkln which was drawn
from the sampling frame with respect to P SUi and P ARST RTijk for the j th CONTDATE.

2. SELECT EDijkln =

1 if P ARijkln was selected
0 otherwise

3. CERT AINT Yijkln =
4. SELECTij =

1 if P ARijkln was selected with probability 1
0 otherwise
'
SELECT EDijkln

over all elements∈SAM P LIN GF RAM E ij

'

5. CERTij =

CERT AINT Yijkln

over all elements∈SAM P LIN GF RAM E ij

6. CERT SUMij =

'

P ARW GHTijklnCERT AINT Yijkln

over all elements∈SAM P LIN GF RAM Eij

The variable SELECTij is the same as the caseload for P SUi on CONT DAT Ej . CERTij repre­
sents the number of cases of certainty which are contained in the sample for P SUi on CONT DAT Ej .
CERT SUMij is the total PARWGHT of all cases selected with probability 1 in P SUi on CONT DAT Ej .
We will define the step function, H(x), and three indicator variables as follows.

Definition 12. H(x) =

1
0

if x > 0
if x ≤ 0

1. CERT Aij = H(SELECTij CERTij )
2. NONCij = H(SELECTij )(1 − H(CERTij ))
3. OOP Sij = 1 − H(SELECTij )
When CERT Aij = 1, at least one PAR of P SUi which is examined on CONT DAT Ej is se­
lected with certainty while NONCij = 1 implies that no PAR’s of P SUi which are examined on
CONT DAT Ej are selected with certainty. Supposedly, CERT Aij and NONCij should account for
all selected PAR’s. Otherwise, it will be be flagged by OOP Sij .
These many definitions are needed to define the following important quantity:
SIijkln =

P JW GHTilnCERT AINT Yijkln CERT Aij +
(ORIGSAMPij − CERT SUMij )
(1 − CERT AINT Yijkln )CERT Aij +
(SELECTij − CERTij )
ORIGSAMPij
NONCij SELECT EDijkln
SELECTij

Equation (3) reflects the three situations in which a P ARijkln can be found.
1. A PAR itself is selected with probability 1, i.e. with certainty.
13

(3)

2. The set of PAR’s which have the same CONTDATE belonging to the same PSU has at least one

of its members which was selected with certainty.

3. A PAR is not associated with any PAR which was selected with certainty.
If P ARijkln was selected with probability 1, then its SIijkln is the same as its P JW GHTil . If the
set of PAR’s has members which were selected with certainty, they are removed from the other selected
PAR’s, so that the removal effectively reduces the sampling size by the number of certainties, that is,
SELECTij − CERTij . Furthermore, the total sum of the PARWGHT’s is reduced by the collective
sum of the PARWGHT’s of the certainties, that is, ORIGSAMPij − CERT SUMij . Finally, if a PAR
ORIGSAM P
sum of P ARW GHT ′ s
.
is not associated with a certainty in any way, then SI = SELECTij ij = N umber
of Selected P AR′ s
The expansion factor to the national level is called the National Inflation Factor (NIF). Its definition
is

Definition 13.
NIFijkln = P SUW GHTi SIijkln

(1 − CERT AINT Yij )
CERT AINT Yijkln +
ST RT W GHTk

SELECT EDijkln (4)

In simplest terms, the NIF which is used in the operational CDS is basically the following:
'
P SUW GHTi
(ST RT W GHTk P JW GHTiln)
NIFijkln =
ST RT W GHTk
CASELOADij

(5)

3.1 Benchmarking
The CDS weights which represent the product of basically three stages of sampling are rectified to
known quantities in a method known as benchmarking. Recall that two PSU’s are drawn from each of
twelve PSU strata. The estimates which are published correspond to their respective PSU strata, called
PSUSTRAT as shown in Table 2, and they correspond to groups of PAR strata called COLSTRAT as
shown in Table 3.
The next step in calculating the CDS weights after the NIFijkln ’s have been calculated is the
implementation of a benchmarking method for rectifying the CDS weights to some known numbers
of crashes in non-sampled and sampled PJ’s with respect to PAR stratum and PSU. The process will
produce the final weight denoted by RATWGT in the analytical files.
The final CDS weight is

Definition 14.
RAT W GHTijkln =

SUMCASEAmζ
NIFijkln
SUMDENAmζ

(6)

where the quantities, SUMCASEAmζ and SUMDENAmζ depend on certain known quantities
and the number of unselected PAR’s from the current year’s sampling frame. RATWGT is calculated
with respect to PSUSRAT and to groups of PAR strata called COLSTRAT. In the computer program
for calculating the final CDS weights, PSUSTRAT is denoted by the SAS variable, PSUGRP, as shown
in Table 1. The definitions of PSUGRP and COLSTRAT appear in Tables 2 and 3.
14

Definition 15. Let RNik be the number of PAR’s of the union of non-sampled PJ’s which were left
over from the current year’s sampling frame and which were assigned to P SUi and to P ARST RTik .
Values of RN are given in Appendix II for 2007.
Let m designate the mth PSUGRP as defined in Table 2, and let ζ designate the ζ th COLSTRAT as
defined in Table 3, then
SUMDENAmζ =

NIFi.k.

(7)

(RNik + ADJUSTi.k. ) P SUW GHTi

(8)

k∈COLST RATζ i∈P SU GRPm

SUMCASEAmζ =
k∈COLST RATζ i∈P SU GRPm

where ADJUSTi.k. =

nj '
nl
'

ADJUSTijkl and ADJUSTijkl ’s are defined by Table 7. ADJUST

j=1 l=1

reflects the practice of sub-dividing a sampling frame of some PJ’s by taking every other PAR from
the sampling frame or every fifth PAR.

4 Trimming Weights
Though it is rarely employed, a process of trimming is performed. In order to mitigate excessively
large RAT W GTijkln ’s, they are rounded not to exceed a ceiling which is determined by examining
descriptive statistics of the RATWGT’s. The excess amount which exceeds the ceiling is distributed
uniformly over the remaining RATWGT’s.

Bibliography
[1] National Accident Sampling System, A Status Report, Volume I, Objectives of the National Ac­
cident Sampling System, National Highway Traffic Safety Administration, Washington, D.C.
20590, 1978.
[2] National Accident Sampling System, Pilot Study, Final Report, DOT HS-804 909, National High­
way Traffic Safety Administration, Washington, D.C. 20590, 1978.
[3] G. Binzer, H. J. Edmonds, , R. H. Hanson, D. R. Morganstein, and J. Waksberg, NASS Estima­
tion, Volumes 1: Technical Report, DOT HS 806 420, Westat, Inc., 1650 Research Boulevard,
Rockville, Maryland 20850, 1982.
[4] Nancy Bondy and Barbara Rhea, Research Note, Reweighting of the Primary Sampling Units in
the National Automotive Sampling System, U.S. Department of Transportation, National High­
way Traffic Safety Administration, 1200 New Jersey Avenue, S.E. Washington, D.C. 20590,
1997.
[5] National Safety Council, Manual on the Classification of Motor Vehicle Traffic Accidents Sixth
Edition (ANSI D 16.1 1996), National Safety Council, Itasca, Illinois, 1996.
15

[6] H. John Edmonds, Robert H. Hanson, David R. Morganstein, and Joseph Waksberg, National
Accident Sampling System Sample Design, Phases 2 and 3, Volumes I: Final Technical Report,
DOT HS-805 274, Westat, Inc., 1650 Research Boulevard, Rockville, Maryland 20850, 1979.
[7]

, National Accident Sampling System Sample Design, Phases 2 and 3, Volumes II: Ex­
hibits, DOT HS-805 275, Westat, Inc., 1650 Research Boulevard, Rockville, Maryland 20850,
1979.

[8] National
Center
for
Statistics
and
Analysis,
NASS
Brochure,
http://www.nhtsa.dot.gov/portal/site/nhtsa/menuitem.331a23559ab04dd24ec86e10dba046a0/,
National Highway Traffic Safety Administration, Washington, D.C. 20590, 2009.
[9] R. H. Hanson, H. John Edmonds, L. Mohadjer, M. D. Rhoads, A. Chu, and D. R. Morganstein,
National Accident Sampling System Estimation, Final Report, Westat, Inc., 1650 Research Boule­
vard, Rockville, Maryland 20850, 1985.
[10] Transportation Safety Institute, NASS Sampling, Part One, U.S. Department of Transportation,
Oklahoma City, Oklahoma, 2000.
[11] Charles J. Kahane, National Accident Sampling System, Selection of Primary Sampling Units,
DOT HS-802 063, National Highway Traffic Safety Administration, Washington, D.C. 20590,
1976.
[12] R. Kaplan and A. Wolfe, Design for NASS: Supplemental Information for Planning the National
Accident Sampling System, DOT-HS-801-989, Highway Safety Research Institute, The Univer­
sity of Michigan, Ann Arbor, Michigan 48105, 1976.
[13] Eugene Lunn, Mike Brick, Ernst Meyer, Vern Roberts, Jim Hedlund, Jim Fell, Glenn Parsons, and
Russell Smith, National Accident Sampling System, A Status Report, Volume III, Implementation
of NASS Subsystems, National Highway Traffic Safety Administration, Washington, D.C. 20590,
1978.
[14] Ernst Meyer, National Accident Sampling System, A Status Report, Volume II, Plan for a Pilot
Study, National Highway Traffic Safety Administration, Washington, D.C. 20590, 1978.
[15] J. O’Day, A. Wolfe, and R. Kaplan, Design for NASS: A National Accident Sampling System.
Volume I. Text., Highway Safety Research Institute, The University of Michigan, Ann Arbor,
Michigan 48105, 1976.
[16]

, Design for NASS: A National Accident Sampling System. Volume II - Appendices, DOT­
HS-4-00890, Highway Safety Research Institute, The University of Michigan, Ann Arbor, Michi­
gan 48105, 1976.

[17] Terry S. T. Shelton, National Accident Sampling System, General Estimates System, Technical
Note, 1988 to 1990, DOT-HS-807-796, National Highway Traffic Safety Administration, Wash­
ington, D.C. 20590, 1991.

16

[18] Russell A. Smith, James Fell, and Charles J. Kahane, FY 1977 Implementation of the National

Accident Sampling System, DOT HS-802 260, National Highway Traffic Safety Administration,

Washington, D.C. 20590, 1977.

[19] Russell A. Smith and Eugene Lunn, National Accident Sampling System, A Status Report, Vol­
ume IV, Implementation Schedule and Resource Requirements, National Highway Traffic Safety

Administration, Washington, D.C. 20590, 1978.

17

5 Appendix I: List of PSU’s

PSU
2
3
4
5
6
8
9
11
12
13
41
43
45
48
49
72
73
74
75
76
78
79
81
82

PSUGRP
9
1
9
5
1
5
7
10
6
10
3
11
7
11
3
2
6
2
8
12
12
4
8
4

Table 8:
PSUSTRAT
3
1
3
2
1
2
8
6
5
6
7
9
8
9
7
4
5
4
11
12
12
10
11
10

18

Number of PJ Strata
8
9
8
10
9
13
10
7
7
7
4
6
3
8
2
7
7
7
5
8
5
4
8
2

6 Appendix II: Non-sampled PJ Counts
PSU
2
3
4
5
6
8
9
11
12
13
43
73
75
76
78
79
81

A
1
0
0
2
12
6
3
0
0
0
0
1
0
0
3
1
1

B
6
1
5
5
8
21
7
0
9
0
1
4
1
0
3
2
1

C
5
121
13
45
32
29
78
1
8
0
1
4
3
1
2
37
9

D
30
205
19
93
81
70
141
0
16
1
4
44
2
3
7
43
21

Table 9:
E
F
46
162
527
809
120
187
404
944
627 1668
485 1059
155
379
0
7
64
162
1
6
22
82
95
274
10
24
6
5
6
19
1116 1208
65
237

G
182
871
341
1139
552
1488
384
13
189
5
90
311
37
15
20
2100
397

H
417
1319
650
2344
1054
2580
921
33
434
22
198
850
103
15
65
2461
1139

PSUGRP
9
1
9
5
1
5
7
10
6
10
11
6
8
12
12
4
8

There are 17 PSU’s in this table even though 24 PSU’s were sampled between 2002 and
2007. Other PSU’s did not contain any non-sampled PJ’s; therefore there are no entries for them in
this table. Those PSU’s which do not contain any non-sampled PJ’s are: 41, 45, 48, 49, 72, 74, and 82.

19

7 Appendix III: Definitions of PAR strata.
Stratum A - crashes in which at least one occupant of a towed CDS applicable late model year vehicle had
a police reported injury of ”K” (fatal injury).
Stratum B - crashes not qualifying for Stratum A in which at least one occupant of a towed CDS applicable
non-late model year vehicle had a police reported injury of ”K” (fatal injury).
Stratum J - crashes not qualifying for Strata A or B in which at least one occupant of a towed CDS applica­
ble late model year vehicle had a police reported injury of ”A” (incapacitating injury) AND was
transported to a treatment facility for treatment AND was admitted overnight to a hospital. If
the crash involved more than one CDS applicable vehicle, at least two CDS applicable vehicles
must be towed.
Stratum K - crashes not qualifying for Strata A, B or J in which at least one occupant of a towed CDS
applicable non-late model year vehicle had a police reported injury of ”A” (incapacitating in­
jury) AND was transported to a treatment facility for treatment AND was admitted overnight
to the hospital. If the crash involved more than one CDS applicable vehicle, at least two CDS
applicable vehicles must be towed.
Stratum C - crashes not qualifying for Strata A, B, J or K in which at least one occupant of a towed CDS
applicable late model year vehicle had a police reported injury of ”A” (incapacitating injury)
AND was transported to a treatment facility for treatment. If the crash involved more than one
CDS applicable vehicle, then at least two CDS applicable vehicles must be towed.
Stratum D - crashes not qualifying for Strata A, B, J, K, or C in which at least one occupant of a towed
CDS applicable non-late model year vehicle had a police reported injury of ”A” (incapacitating
injury) AND was transported to a treatment facility for treatment. If the crash involved more
than one CDS applicable vehicle, then at least two CDS applicable vehicles must be towed.
Stratum E - crashes not qualifying for Strata A, B, J, K, C or D in which at least one occupant of a towed
CDS applicable late model year vehicle was transported to a treatment facility for treatment.
Stratum F - crashes not qualifying for Strata A, B, J, K, C, D or E in which at least one occupant of a towed
CDS applicable non-late model year vehicle was transported to a treatment facility for treatment.
Stratum G - crashes not qualifying for Strata A, B, J, K, C, D, E or F which involve at least one CDS
applicable late model year vehicle that was towed from the scene.
Stratum H - crashes not qualifying for Strata A, B, J, K, C, D, E, F or G which involve at least one CDS
applicable non-late model year vehicle that was towed from the scene.

20

DOT HS 811 327
May 2010