NATL INST OF STAND & TECH

NIST

A11107 22bM37 PUBLICATIONS

NBSIR 73-154

A Summary of the Relationships

Between Accident indices and

Rates Following a Redefinition

of "Failure"

Dr. Alexander R. Craw

Technical Analysis Division

National Bureau of Standards

Washington, D. C. 20234

March 1973

Final Report

QC

100

.U56

73-154

1973

Prepared for National Highway Traffic Safety Administration

Department of Transportation

Washington, D. C. 20590

NBSIR 73-154

A SUMMARY OF THE RELATIONSHIPS

BETWEEN ACCIDENT INDICES AND

RATES FOLLOWING A REDEFINITION

OF "FAILURE"

Dr. Alexander R. Craw

Technical Analysis Division

National Bureau of Standards

Washington, D. C. 20234

March 1973

Final Report

Prepared for National Highway Traffic Safety Administration

Department of Transportation

Washington, D. C. 20590 '

U. S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts. Director

Note: This report to the National Highway Traffic Safety Administration

(NHTSA) has been prepared to serve as a mathematical appendix

to a report to be published by the NHTSA.

SUMMAEY

Usinp; Cerrelli's a8sociQ,tion of "failure" with active involvement

in the case of two vehicles accidents, one can define failure indices,

accident involvement indices, failure rates, accident involvement

rates for 1-vehicle, 2-vehicle and for the union class of 1- and

2-vehicle accidents. This note presents in a condensed form all of

the definitions of these measures, and a number of useful relationship

and interrelationships that exist between these different measures

These formulae should prove useful in the calculation of the resulting

indices and rates and of converting from one set of measures to anothe

TABLE OF CONTENTS

Page

1.0 INTRODUCTION 1

2.0 NOTATIONAL CONVENTIONS 3

3.0 DEFINITIONS ^

i+.O ACCIDENT INVOLVEMENT INDICES 5

h.l Definitions 5 h.2 Relationships 6

5.0 FAILURE INDICES ^

5.1 Definitions ^ 5.2 Relationships ^

6.0 FAILURE RATES 6

6.1 Definitions ^ 6.2 Relationships ^

7.0 ACCIDENT INVOLVEMENT RATES

7.1 Definitions

7.2 Relationships

8.0 INTERRELATIONSHIPS BETWEEN RATES 8

9.0 INTERRELATIONSHIPS BETWEEN INDICES 8

10.0 INTERRELATIONSHIPS BETWEEN RATES AND INDICES 9

11.0 INTERRELATIONSHIPS BETWEEN FRACTIONAL

DEVIATIONS FROM NORM OF INDICES AND RATES 9

12.0 DERIVATION OF INTERRELATIONSHIPS BETWEEN

INDICES 12

1 . 0 Introduction

In his recent paper: "Interpretation of the Accident Involvement

Index," ^ Ezio Cerrelli points out the pitfalls in interpretation

(primarily by administrators) commonly associated with the currently

used Accident Involvement Index (All). A most common misinterpretation

is that of considering it as an absolute measure of misperformance

of a category of driver-vehicle combination, when in fact it is a

relative measure. A second difficulty is that of interpreting each

involvement as a "failure" in some sense and using involvements to

measure the degree of "failure" for each category of driver-vehicle

combination.

Cerrelli then proceeds with a rationale for associating one

"failure" with each accident and then defines a Failure Index (Fl)

to be associated with each category of driver-vehicle. In the case

of a one-vehicle accident, "failure" is identified with "involvement,"

but in the case of a two-vehicle accident, "failure" is associated

only with an "active" involvement while passive involvement is considered

as "non- failure . " The basic relation is that "failure" implies

involvement, but involvement does not necessarily imply "failure."

This may be diagrammed for

1-vehicle accidents: 2-vehicle accidents:

Failure 1" Involvement Fail-uire . Involvement

^To be published as a National Highway Traffic Safety Administration

(NHTSA) Report.

1
As currently used the Accident Involvement Index is defined for

the composite set (union) of the sets of 1- and 2-vehicle accidents.

Originally Cerrelli defined separate failure indices for one-vehicle

and two-vehicle accidents. It was pointed out hy this author that

failure and involvement indices could be obtained for each set of

accident types (l-vehicle, 2-vehicle, composite), and that corresponding

rates (ratios) could be defined as well. The latter turn out to be

absolute rather than relative measures.

The purpose of this note is to present in a condensed form,

for possible rapid reference and use, the definitions, relationships

and interrelationships between traffic accident related indices and

rates that result from Cerrelli 's explicit identification of "failure"

in the case of two vehicle accidents. These formulas should prove

useful in the calculation of the indices and rates that result and

of converting from one set to another. The notation used is that

suggested to and employed by Cerrelli in his paper.

The reader should anticipate some general statements, such as:

a. the ratio of fractional deviations from the norms for failure

/indices! is the ratio of total involvements to total failures time

the fractional deviations from the norms for involvement (indices,.

\rates

This holds true for each type of accident: l-vehicle, 2-vehicle and

the union of 1- and 2- vehicle accidents.

2
"b. For tlie union of 1- and 2-vehicle accidents the

faccident involvement I index

1 failures j

^fractional deviations^ indices

[from norm for / rates

is equal to a weighted sum of the

; accident involvement! indices

\failure /

[fractional deviations'^ indices

\from norm for • rates

for 1- and 2-vehicle accidents.

2.0 Notational Conventions

Script letters (e.g.,|^) vill denote categories (classes)

or sets.

Script letters with subscripts denote subsets • v>

Upper case letters, with or without subscripts, denote integers: E^.

Lower case letters, with or without subscripts, denote proportions:

An unscripted single upper case letter denotes a summation

over the range of the subscript

E = Ej^^E2^ = l \-

k=l

1 refers to the set of 1-vehicle accidents;

2 refers to the set of 2-vehicle accidents

* (or u) refers to the union of the sets of 1- and 2-vehicle accidents

3
When double subscripts are used, the first subscript refers

to the number of vehicles in the class of accidents under consideration,

while the second subscript refers to a particular category.

All denotes an Accident Involvement Index;

AIR denotes an Accident Involvement Riate;

FI denotes a Fa,ilure Rate.

3.0 Definitions

= the nijimber of exposure xinits associated with category

e = the proportion of total exposure units associated with

K. category p-^ = E^/E

3.1 1-Vehicle Accidents

= the number of single vehicle accident involvements

the number of single vehicle accident failures associated

with

n

S + S_ + . . . + S =

I 12 S. n r , k

k=l

3.2 2-Vehicle Accidents

I, = the niimber of involvements in two vehicle accidents k

associated with ^k*

n

P = 1/2 = the total number of passive involvements.

= Pj^/P (assumption iii, Cerrelli -> = for 2-vehicle

accidents)
= the num'ber of active involvements in ^ ^

A = I - P = 1/2

K - Pi r In - P I, - e, 1/2

_k _ k k \ _ k k _ k k ... \ " A " 1/2 1/2 1/2

"-^^

\ Pk =

3.3 Union of 1- and 2-Vehicle Accidents

'^*k "'"k^ ~ ^k ''k ~ "total involvements associated with y^.

(or I*j = S + I

I*k = W^*

f

* ^*k ^ '^^ ~ ^k

~ total failures associated with

A^ (or A*) = S + A

^*k ^"^^ ^ " A*^/-^*

U . 0 Accident Involvement Indices

U.l Definitions

(AII),^ = ^= ^ = — ; (All) =^=1 3Jk e, e, S e, 1 e k k k

^k \ T/I (AIl)^^ = -Ap= ; (AII)^=i^=l 2k e, e, I e, 2 e

'k k'

^ ^k

5
k.2 Relationships

(AII)^ + (AII)^ = 2(AII)^

(AII),j^ = (Aii)^^ + J- (AID 2^

= "Weighted average of (AII)3_^ and (All)gj^

5.0 Failure Indices

5.1 Definitions

(FX)

(FI)

(FI)

Ik

2k

*k

A^/A

'k

e.

(FI), = ^= 1

1 e

(FI) =^=1

d e

(FI), = —= 1

5.2 Relationships

S

6.0 Failxire Rates

6.1 Definitions

(FE),, =

i (FR)^

(FR)^

(FE),j^ (FR)^

6
6.2 Relationships

(FR)^ = (FR),^ + (FR)^^ ; (FR)^ = (FR)^ + (FR).

'Ik 2k

(FR) = y e, (FR) = weighted average over (FR)

' ^ J-k Ik

(FRjp = 1 e^{FR)^^ = -weighted average over (FR) 2k

(FRj^ = e^(FR)^^ = weighted average over (FR)

or

k^

(FR)^ = I e^(FR)j^, j = 1, 2, *

*k

7 . 0 Accident Involvement Rates

T.l Definitions

7.2 Relationships

(AIR)^ = S/E

(AIR)^ = I/E

(AIR)^ = ijE

(AIR)^ =

I ej^(AIR)^^ = weighted average of (AIR)

k

(AIR)^ = I e^(AIR)2^ = weighted average of (AIR)

k

(AIR)^ = I e^(AIR)^^ = weighted average of (AIR)

Ik

2k

«k

or (AIR), = I e^(AIR)^^, J = 1, 2,

j(AIR)^^= (AIR)^^ + (AIR)2j^ ; (AIR)^ + (AIR)^ = (AIR)^

7
8.0 Interrelationships betveen Rates

If we define passive rates : (P^^^i^;

(PR)

(FR)^

(AIR)^

(FR)^

(AIR)^

(FR)^

I

(AIR)

(PR)

2k

*k k/\

then (AIR)j^ = (FR)^^ + (PR)ji^, j = 1, 2, *

9.0 Interrelationships between Indices

(All) Ik

Ik

(AII)^^ = h[l + tFD^j^]

(All) *k

(FI)

(FI)

For J = 1, 2, *

(FR)

I tair:.

i

! ratio of total

equals the

failures to total

involvements

(FI),,^ = (Allj Ik

2k

*k =

2(AIl)2j^ - 1

ll (AII)^, . A

A k A

8
10. 0 Interrelationships between Rates and Indices

(FX)

(FI)

(FI)

(FR) Ik

Ik

2k

Cfr)^

(FR)

= (All)

2k

(FRl

(FR)

*k Tfr)7

(FI)

(FR)

jk " TfrJ

Ik

, j = 1, 2, *

(All)

(AIR)

Ik

Ik (AIR).

(AID

(All)

(AID

2k

*k

(AIR)

2k

(AIR),

(AIR)

(AIR)^

(AIR)

*k

jk (AIR)

= J = 1, 2,

11.0 Interrelationships between Fractional Deviations from Norm

of Indices and Rates

Since (Fl). = (All) = 1 for j = 1, 2, *

(FR)

J 0 (FI) - 1 (FI) - (FI).

(FR)j ^ (FRTj

I (FX) - (FI) (FR) - (FR) . . 2 * so|(Fl),,. - 1 = ^ = ^ 1 , J - 1, 2,

1W7

and in the same vay

(AII),T - (All). " (aXR)T, - (air)

|(AII)^3^ - 1 = Jk.

{All)

^ r -i 1=1 2 *l (AIR) . ' ' '

;

9
For J = 1,

(FI)^^ - (FI)^ (FR)^j^ - (FR)^ ^^^^^ik ~ ^^^^^1 *^TM^lk ^^^^^1

UTY[ " [fr]^ " (AIR)^j^ " (AIR)^

; i.e., all fractional deviations are the same,

I For j = 2,

(FD^j^ - (FI)^ (FR)^^^ - (FR)^ 2(AIl)2j^ - (AII)^ 2{Ain)^^ - (AIR}^

(FI)^ " " tmZ, " (All) 2 " (AIR)^

i.e., fractional deviations for failures are twice the fractional

I deviations for involvements

For j = *, (Fl)^j^ - ^ = a7 t(AIl)»^ - 1]

i.e., fractional deviations for failures is I*/A^ times the fractional

deviations for involvements. Note: I«/A^ > 1.

Note that I^/A^ is the ratio of total involvements to total

failures (active involvements).

For j = 1, this ratio is 1

j = 2, this ratio is 2

j = *, this ratio is I#/A^

10
so we can generalize

The ratio of fractional deviations from the norms for failure

is the ratio of total involvements to total failures times the

fractional deviations from the norm of involvements. This holds

for j = 1, 2, *.

[ For j = * ,

- 1 = i f^^^^lk - ^ i ^^^^)2k - 1]> A, = S . A

I / failures \

j i.e., the fractional deviation of\involvement J from the

j norm for the union of the classes of 1- and 2-vehicle accidents

i

j is equal to the weighted sum of the fractional deviations of

I [failures |

I [involvements j from the norm for 1- and 2-vehicle accidents.

I (AII)^^ -1=1 t(AIl)^^ - 1] + f

[{All)^^ - 1], = S + I

One can combine the ahove resTolts with those from sections

h.2 and 5-2 to state more generally:

For the union of 1- and 2-vehicle accidents the

jaccident involvement! index

i failure J

\ is equal to a weighted sum of the

jjfractional deviations! indices | •(from norm for j rates i

[accident involvement \ indices ! (failures / |

i [fracti onal deviations! indices i (from norm for J rates

for 1- and 2-vehicle accidents.

11
12.0 Derivation of Interrelationships between Indices

or

and

Starting frora

= sir f^I'lk * STT '«l)2k' ^ = 2A (Sec. k.2)

S+I 1 S+A '"^-^'Ik S+A '"•^^'2k

S+A

S+I I|_sIa S+A <''^'lk * il I '1 ^ ^^I'2k>] (Sec. 9)

S+A r S . ^ A ^ A 1 M sTa ^^^^Ik ^ S^A ^^^^2k ^ sTaJ (Sec. 5-2)

(FI) *k

_4 M f^i>*k " sir = i; (^i)*k " i' A, = s . A

I, = S + I

(Aii),^ = ^

(n),^ +

A* + A

[(FI)^,, - 1] + I^ —

A* A^ + A - I^

(^^^*k - \

= i7 - 1^ ^ —i;

Since A^+A-I^=S+2A-S-I=2A-I=0, the last term

falls out J and

A^

(AII)^^ - 1 = — [(FI)^^ -1] (Sec. 10)

12
FORM NBS-IHA (1-71) U.S. DEPT. OF COMM. 1 . PU B LIC AT ION OR RE PORT NO. 2. Gov't Accession

BIBLIOGRAPHIC DATA NBSIR-T3-15U

3. Recipient s Accession No.

4. TITLE AND SUBTITLE

A Siimmary of the Relationships Between Accident Indices

ana Kates rolloving a Kedefmltion oi railure

5. Publication Date

6. Performing Organization Code

7. AUTHOR(S)

Alexander R. Craw

8. Performing Organization

NBSIR 73-L54

' 9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/ Work Unit No. U3IUU96

11. Contract/Grant No.

12. Sponsoring Organization Name and Address

National Highway Traffic Safety Administration

Department of Transportation

Washington, D.C, 20590

13. Type of Report Si Period

Covered

Final

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.) Using Cerrelli's association of * failure* with active involvements in

the case of two vehicles accidents, one can define failure indices, accident

involvement indices, failure rates, accident involvement rates for 1-vehicle,

2-vehicle and for the union class of 1- and 2-vehicle accidents. This note

presents in a condensed form all of the definitions of these measures, and a number of useful relationships and interrelationships that exist between these

different measures. These formulae should prove useful in the calculation of

the resulting indices and rates and of converting from one set of measures to

another

.

17. KEY WORDS (Alphabetical order, separated by semicolons)

accident indices; accident rates; fsdlvire indices; failure rates; highway accidents

18. AVAILABILITY STATEMENT

HUNLIMITED.

1 1 FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE

TO NTIS.

19. SECURITY CLASS

(THIS REPORT)

UNCL ASSIFIED

21. NO. OF PAGES

20. SECURITY CLASS

(THIS PAGE)

UNCLASSIFIED

22. Price

USCOMM-DC 66244-P71


I

I