DATA EVALUATION AND METHODS RESEARCH 

PROPERWoF~E


PUBLICATIONSBRANCH 

EDITORIAL LIBRARY 

On the Mathematics 

of Competing Risks 

Series 2 

Number 77 

‘


This report reviews the sources and origins of competing risk theory 

and presents the probabilityy theory of competing risks and statistical estimation techniques and tests of hypotheses. 

DHEW Publication No. (PHS) 79-1351 

U.S. DEPARTMENT OF HEALTH, EDUCATION, AND WELFARE


Public Health Service


Office of the Assistant Secretary for Health


National Center for Health Statistics


Hyattsville, Md. January 1979

Library of Congress Cataloging in Publication Data 

Bimbaum, Z. William. 

On the mathematics of competing risks. 

(Vital and health statistics: Series 2, Data evaluation and methods research; no. 77) 

(DHEW publication; (PHS) 79-1351) 

Bibliography: p. 56 

1. Competing risks. 2. Estimation theory. 3. Statistical hypothesis testing. I. Title. 

11.	 Series: United States. National Center for Health Statistics. Vital and health statistics:


Series 2, Data evaluation and methods research; no. 77. 111. Series: United States. Dept.


of Health, Education, and Welfare. DHEW publication; (PHS) 79-1351.


RA409.U45 no. 77 [QA273] 312’.07’23s [362.1’01’82]


ISBN 0-8406 -0138-7 78-15122


For sale by the Superintendent of Documents, U.S. Government Printing O ffIce 

Washington, D. C. 20402 
NATIONAL CENTER FOR HEALTH STATISTICS 

DOROTHY P. RICE, Director 

ROBERT A. ISRAEL, Deputy Director


JACOB J. FELDMAN, Ph.D., Associate Director for Analysis


GAIL F. FISHER, Ph. D., Associate Director for the Cooperative Health Statistics System


ELIJAH L. WHITE, Associate Director for Data Systems


JAMES T. BAIRD, JR., Ph.D., Associate Director for International Statistics


ROBERT C. HUBER, Associate Director for Management


MONROE G. SIRKEN, Ph.D., Associate Director for Mat/zematica~ Statistics


PETER L. HURLEY, Associate Director for Operations


JAMES M. ROBEY, Ph.D., Associate Director for Program Development 

PAUL E. LEAVERTON, Ph. D., Associate Director-for Research 

.ALICE HAYWOOD, Information Officer 

OFFICE OF THE ASSOCIATE DIRECTOR 

FOR MATHEMATICAL STATISTICS 

MONROE G. SIRKEN, Ph.D., Associizte Director 

Vital and Health Statistics-Series 2-No. 77 

DH EW Publication No. (PHS) 79-1351 

Libraty of Congress Catalog Card Number 78-15122 
FOREWORD


In view of the many potential applications of the statistical theory of competing risks in the analysi; of health-and vital statistics, the National Center for 

Health Statistics was responsive to Dr. Birnbaum’s proposal to prepare an introductory report on the mathematics of competing risks which would clarify the 

concepts and unify the theory common to the several disciplines in which competing risk models have been applied. As a result, we have this excellent report. 

It has become Center policy to submit all reports in this publication series to 

peer review. In this regard, we should like to recognize the contributions of Professor Anthony J. Quinzi who served as technical reviewer of this report. Dr. 

Birnbaum joins me in thanking Dr. Quinzi for his careful review and for his many 

helpful suggestions and comments. 

Monroe G. Sirken 

Associate Director for 

Mathematical Statistics 

... 

Ill 
CONTENTS 

Foreword ............................................................................................................................................... 

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


1.1.	 Sources and Origins of Competing Risk Theory .....................................................................


1.1.1. Work of Daniel BemouW .............................................................................................


1.1.2. Actuarial Problems ......................................................................................................


1.1.3. Clinical Experiments ...................................................................................................


1.1.4. Competing Risks in Technology ..................................................................................


1.1.5. Other Areas .................................................................................................................


1.2.	 Comments on the Present State of the Theory .......................................................................


1.2.1. Duplication and Communication Gaps ........................................................................


1.2.2. Probabilistic Versus Statistical Approach .....................................................................


1.2.3. Choice of Topics for This Report ................................................................................


2. Probability Theory of Competing Risks .........................................................................................


2.1. Life Distributions ..................................................................................................................


2.2. Net (Potential) and Crude (Observable) Lives .........................................................................


2.3. General Theory: Net tives ......................................................................................................


2.4. General Theory: Cmde Lives ..................................................................................................


2.5. An Identity ............................................................................................................................


2.6.	 The Problem of Identifiability ................................................................................................


2.6.1. Formulation of Problem ..............................................................................................


2.6.2. The Case of Independent Net I.hes .............................................................................


2.6.3. The Case of Possibly Dependent Net Lives ..................................................................


2.6.4. An Interpretation of the IdentifiabWlty Problem: Some Inequalities ...........................


2.7. Summary and Comments on Identifiability ............................................................................


2.8. Multiple Decrement Tables Versus Questions About Net Lives ..............................................


3. Estimation Techniques ...................................................................................................................


3.1.	 Case of Independent Net Lh?es: The Kaplan-Meier Nonparametic Technique .......................


3.1.1. Statement of Problem .................................................................................................


3.1,2. Definitions and Notations ...........................................................................................


3.1.3. A General Estimation Procedure .................................................................................


3.1.4. The Kaplan-Meier Estimates ........................................................................................


3.1.5. Some Properties of the Kaplan-Meier Estimates ...........................................................


3.2. Some Actuarial Estimates and Their Relationships to Kaplan-Meier Estimates .......................


3.2.1. A Generdlzed Kaplan-Meier Procedure and Some Actuarial Estimates ........................


3.2,2, Conditions for p~b Being Consistent ...........................................................................


3.3.	 Life Table Estimates: Proportional Hazard Rates ...................................................................


3.3.1, Definitions and Assumptions .......................................................................................


3.3.2. Observable Random Variables: Maximum Likelihood Estimates of Cmde and Net


Probabilities ..............................................................................................................


3.3.3. Conditions for Consistency of Chkang’s Estimator of the Net Survival Probabilities ....


3.3.4. Examples of Competing Risks WMs Proportional Hazard Rates ...................................


4.	 Estimation for Parametric Models ..................................................................................................


4,1. The Likelihood Function: General Case .................................................................................


4,2. The Case of Independent Net Lives ........................................................................................


4.3. Comments on Parametric F~les of Ms.dtivariate Life Dktibutions ....................................


4.4. Concomitant Variables ...........................................................................................................


4.5. Estimation of the Ratio of Hazaxd Functions .........................................................................


... m 

1 

1 

1 

7 

7 

8 

9 

9 

9 

9 

10 

10 

10 

12 

16 

19 

19 

20 

20 

21 

23 

26 

29 

30 

30 

30 

30 

30 

31 

31 

32 

33 

33 

35 

39 

39 

;: 

44 

45 

45 

46 

47 

47 

48 

v 
5.	 Tests of Hypotheses .................................................................................................. ..................... 

5.1. Formulation of a Problem ...................................................................................................... 

5.2. Gehan’s Statistic ..................................................................................................................... 

5.3. Testing for Independence When Data Are Censored ............................................................... 

5.4. Large-Sample Tests .................................................................. ............................................... 

References ...... .............................................................................................................................. .......... 

1. 

2. 

3. 

4. 

5. 

6. 

A. 

LIST OF FIGURES 

Introductory page of Daniel Bernoufli’s monograph on hk theory of competing risks ................... 

Daniel Bemouilli’s originaf life tables ............................................................................................. 

Probability densities used in example D ......................................................................................... 

Regions of integration considered in equations (30) and (31) ........................................................ 

Graph of Kaplan-Meier estimate obtained in table A ........................................... ........................... 

Diagram of events defined by formulas (54) ....................................................... ............................ 

TABLE 

Example of a Kaplars-Meier estimate .............................................................................................. 

50 

50 

51 

54 

56 

57 

2 

5 

25 

27 

33 

36 

52 
ON THE , MATHEMATICS OF COMPETING RISKS

Z. William Birnbaum, University of Washington, Seattle 

1. INTRODUCTION 

1.1. SOURCES AND ORIGINS OF COMPETING RISK THEORY 

1.1.1. Work of Daniel Bernoulli 

Toward the middle of the 18th century, the question of mandatory vaccination against smallpox 

was widely discussed and attempts were made to evaluate its possible effects. In his “M6moir” published in 1776 (figure 1), Daniel Bernoullil gave this question the following specific formulation: 

Available life tables reflect the mortality of the population for which they were calculated, taking 

into account all causes of death including smallpox. How would these life tables change if, because of 

mu.ndatory vaccination, deaths from smallpox were entirely eliminated? 

As empirical sources of information, Bernoulli considered existing life tables, such as those computed in 1693 by the astronomer E. Halley2 from the records of the city of Breslau in Germany. 

These tables were based on data which reported age at death for each individual; in addition, it may 

have been possible to determine whether death was a result of smallpox or of other causes. Therefore, 

each individual may be considered as exposed to two “risks”: death from smallpox and death from 

other causes. These risks competed for his life in the sense that the risk which materialized first determined his life length. Thus for an individual who died of smallpox there was no way of telling how 

lcmg he would have lived had smallpox been eliminated–and it was just this missing information that 

Bernoulli needed to answer his question. 

The argument devised by Bernoulli is ingenious and rather simple. It is also open to serious criticisms as to assumptions made explicitly or by implication. These criticisms already pornt at most of 

the difficulties encountered later in dealing with competing risks. We shall outline Bernoulli’s argumcnt,l following closely the somewhat naive reasoning of his M6moir and only modifying his notation. 

Consider a population of 10 individuals of age 0, and let 1X denote the number of those among the 

/0 who m-e still alive at age x >0. Assume 1X known for x >0, for example from a life table. Clearly, 

[X is a decreasing function of x. 

Let s. be the number of those of the 10 who survive to age x >0 and who have been infected with 

and survived smallpox. They are immune from smallpox for the rest of their lives. Similarly, let t% 

be the number of those surviving to age x >0 who did not have smallpox, hence may still contract this 

disease. Clearly, 

lx=sx+ t%. 
ME A401RES 

DE 

MATHkMATIQUE 

ET 

DE PHYSIQUE, 

De i’Ann6e M. D C C LX. 

J5’SS2! I DUNE NO UVELLE A NALY3’E

De .Zamortaliri ct7uJ70par la petize Virok, a ties 

avantagts de [’hofzhtion pour Za prt$enir. 

Par M. DANIEL BERNOULLI. 

IN TRODucTIOiV APo Lo GET I~uE*. 

cE u x qhi ont fenti tout Yavantage de i’Inocuh:ion, ont 

imagind diffdrentes fagons de reprifenter cet zvmtoge, 

qui, quoique revenant au nhhe, ne iaiflht pm de fiire unt2 

* Cette Introduction n’a <tt faite clue ion:-tcmps apr& le M5noire ~ 

+tant du 16 Avril 1765. 

Mth. A760J A 

Figure 1. Introductory page of Daniel Bernouilli’s monograph on his theory of competing risks. 

2 
Assume that in a unit of time (e.g., a year) smallpox attacks 1 in n individuals alive, and causes 

the death of 1 in m of those attacked. Bernoulli assumed m and n to be known (he estimated both to 

be about 8), and independent of x. 

The number of those dying of all causes during a time interval (x, x + dx ) is 

1%– l%.~% = –all% . 

Among them are 

tx dx 

—	 = number of those who get smallpox and die of the disease (1) 

nm 

and 

tx dx 

–all% – — = number of those who die of other causes. (2) 

nm 

The number tx decreases during the time interval (x, x + dx ) by the number of those among the t% 

who get smallpox and by the number of those among the tx who die of causes other than smallpox, so 

that 

This can be rewritten 

‘d[lOg(m+ll
 ‘dx) 

which yields by integration 

m lx 

tx = 1 + exp [(x + c)/n] ‘ 

and since to = 10,one has 

exp (c/n) = m — 1, 

SOthat 

m lx 

t% = (3) 

1 -+-(m –1) exp (x/n) “ 

3 
Consider now the life table obtained from the lx by eliminating smallpox as a cause of death, and 

denote ZX = number of those surviving to age x >0 when smallpox is eliminated, with Z. = 10. In the 

presenc~ of smallpox, the numbers of those dying of each of the two causes during a time interval (x, 

x + dx) were given by equations (1) and (2). Without smallpox only equation (2) applies, but at x 

there are z% alive instead of lx, hence the number of those dying in a timb interval (x, x + d.x) without 

smallpox is 

-’%.=-: (“+3 

Substituting tx from equation (3), one obtains 

dz% dl% ‘x —=— -1-

Zx lx n[l + (m – 1) exp (x/n)] ‘ 

a differential equation with the solution 

z% m 

—= (4)

1% m — 1 + exp (=/n) “ 

This ratio is clearly greater than 1 for x >0, and it describes the improvement of the life table ZX over 

the original life table lx. Asymptotically, one has for x ~ co 

z% m 

<“ ~—1” 

Using this mathematical model with the values m = n = 8 for which he claimed some empirical 

justification, Daniel BernouIli calculated his Tables I and II reproduced as they appear in his original 

paper (figure 2). Column headings were added for present use. 

In Table I the first and second columns (x and 1X) are taken from Halley’s life table; the third column contains values of t% calculated from equation (3) with m = n = 8, and the fourth column lists 

the values SX = 1%– tx. The number of those contracting smallpox at age x appears in column 5 and is 

computed as follows: On the intuitive assumption that the number t% decreases uniformly during a 

year, consider (1/2)(tX + t%+~) as the number of those exposed to the risk of contracting smallpox at 

age x, and enter under column 5 the estimate ( l/2)(tX + t%+~)/8. Dividing this number by m = 8, one 

obtains the entry in column 6, an estimate of the number of those dying of smallpox in a year. Column 7 accumulates the numbers of smallpox deaths from column 6, and column 8 accumulates the 

deaths from other causes. 

In Table II, the first two columns contain Halley’s life table reflecting mortality from smallpox 

and from all other causes; the third column contains values of ZX computed according to equation (4). 

The fourth column lists the “gains” z% — lx. 

A number of criticisms were raised against certain of Bernoulli’s assumptions even before publication of his paper—some he ascribed to an “eminent mathematician, ” most likeIy d’Alembert, and he 
MiMOIRES DE L’ACADfiMIE ROYALE D E s S c I E N c E S.


TABLE 1.


(1) (2) (3) (4) (5) (6) (7) (8) 

x lx tx Sx 

AGES Survivms Iwlym Aymw eu Pryt M 133;=TS SO M*, E MORTS 

p.w felon px,cu ,1 pm Virolc Ft. ,Crule dmmorts pJ:.y::~~ 

mllh. M, I-my. per. wi-ole. pet. w%ol. r ,.::2 d~:d:: ~fi”~””” Yw’q 

o 1 ;00 I ~oo 0 

1 1000 896 I 0.4 r 37 17?1 17)1 z83 

2 8j\ 68$ I 70 99 1?.,4 =9,5 133 

3 798 $7! 22 7 78 9.7 3’9>2 47 

4 760 48 ~. 27.$ 66 8,3 47*5 30 

5 732 416 316 56 7,0 -$+,j 21 

6 710 359 3)-1 48 6,0 60,5 16 

7 692 3.11 381 42 ~,z 6j,7 12,8 

8 680 Z.7L 408 36 +,$ 70,2 7? 5 

9 670 237 433 32 4,0 7+,2 6 

10 66 I 208 4$3 2.8 3,s 77.7 .$.5 

11 653 182 4.71 ~4,4 3,0 80,7 5 

12 646 160 486 21,4 2,7 83,4 +33 

13 640 1+0 >00 18,7 2, 3 85,7 3>7 

J+ 634 123 jll 16,6 2, I 87,8 399 

15 628 108 520 14.+ 1,8 8$),6 4,2 

~6 622 94 528 1 =,6 x ,6 91,2 4*4 

17 616 83 533 11,0 I ,4 92,6 4,6 

18 610 72 538 9>7 I ,2 93,8 4,8 

19 604 63 $41 8,+ 1,0 9+,8 5 , 20 598 56 54= 7>4 0>9 9577 ‘j, r 

21 )-9% 48,$ $43 6,5 0,8 96,5 5 ?~ 

22 586 4275 543 5,6 0,7 . 97*2 513 

23 579 37 $42 ~,0 0,6 97,8 6,4 

24 57= 3~.4 s+~ 424 0>5 98,3, 6, ~ 

Figure 2. Daniel Bernouilli’s original life tables. (Column headings were added for present use.) 

discussed them at some length. The more obvious ones (e.g., the assumption that the probability of 

contracting smallpox as well as the probability of dying of smallpox are independent of age), could be 

replaced by more flexible and realistic assumptions without much difficulty. Other assumptions, such 

as the use of (1/2) (tX + t%+1) in estimating the number of those attacked by smallpox for COIUmn 5 

in Table I, have been reappearing throughout the centuries, and we will have to deal with them 

later on. 

5

MLMOIRES DE L’ACADIfMIE ROYALE D E s S c I E N c E s. 

TAB LE II. 

(1) (2) (3) (4) (1) (2) (3) (4) x lx *x x lx =x 

ALGES l!.tstnaturcl J2TAT Dil%. AGES Erst nmsrel ET A-r Difftr, 

par & pr & 

aun~$. vmiolique. non- vnridiq.on gains ,,”:c~c varirrtique. non- wrioliq. ou gain 

m 

o	 1300 1300 0 13 640 741,1 7+,1 

I I 000 1017,1 17,1 r+ 634 709,7 75,7 

2 855 881,8 26,8 15 628 705,0 77,0 

3 798 833,3 35,3 16 6’2‘2 yoo, l 78,1 

4 760 802>0 42,0 17 616 695,0 ~y,o 

5 752 779,8 47>8 18 610 689,6 79,6 

6 710 762,8 52,8 19 604 68*,O 80,0 

7 692 7+9*1 57s2 20 5y8 67%,2 80,2 

8 680 740,9 60,9 21 592 672,3 80,2. 

9 670 1 73+,4 I f86 I 666.2 I 80.2 

10 I 661 728>4 I -

CctteTabJe faitvoir dun coup d’o?[[ ~ COt7tbiCh fur I 300 enfan.s, fuppofis 

116s en mi?metemps, il cm rdleroit de vivans tl’mnde en annr5e jufqu’i 

I‘iige de vingt-cinq ans , en les fuppofant tons ii.ijets i Ja petite vr5role; & 

(:ombien ii cn rrikroit s’ik ftoient [OUS exempts de cette maiadie, avec 1a 

(:omparaifon& la dlfftknce des deux 6tats. 

m 

Figure 2. Daniel Bernouilli’s original life tables. (Column headings were added for present use.) -Con. 

Two features of Bernoulli’s approach are worth noting. One is the deterministic interpretation of 

all processes: The quantities lx, Sx, tx, z% are functions of age x. A probabilistic approach could 

hardly have been expected at that time. The other feature is the nonparametric treatment: Bernoulli 

made no assumptions about the functional form of 1X and, consequently, of the other functions of 

age. It is only with the arrival of the life-table models introduced by Gompertz in 18253 and 

Makeham in 18604 that specific functional forms of 1Xwere considered. 

6 

L 
1.1.2. Actuarial Problems 

More than one hundred years after Bernoulli’s M6moir,1 W. M. Makeharn in 1874s developed a 

theory of “multiple decremental forces,” and thereby gave a systematic foundation to the actuarial 

treatment of competing risk problems. At Makeharn’s time, the theory of probability had already 

berm developed and actuaries were formulating their statements in probabilistic terms. Their earlier 

work was what, in discussing Bernoulli’s approach, we referred to as nonparametric: They started out 

with life tables obtained from experience without ascribing to them a specific functional form and 

estimated all the probabilities and actuarial values required. In 1860, Makeharn;4 expanding on earlier 

work by B. Gompertz in 1825,3 introduced the impartant parametric model known as the Gompertzhidwham law of mortality which appeared to be particularly suitable as an approximation to many 

empirical life tables. 

Actuarial theory has traditionally used both approaches: nonparametric as well as applying the 

Gompcrtz-Makeham law. In classical works such as E. F. Spurgeon’s in 1932 (pp. 223-3206) both treatments are discussed in great detail, and it is pointed out that, if a life table follows Makeham’s law, the 

mathematical arguments become more elegant and manageable, 

The competing risks problems considered by actuaries were usually of a different kind and simpler 

than the question asked by Bernoulli. Such problems arose mostly when insurance policies had to be 

written on several lives, e.g., a policy for husband and wife that assured the payment of a benefit (such 

as a lump sum, or life annuity) after the death of one of them, to the surviving spouse. Here the 

premium to be charged is determined by the probability distribution of the time to the earlier death, 

hence could be computed if the joint probability distribution of both lives were available. Since life 

tables for pairs of lives, which would yield such joint probability y distributions, are difficult to obtain, 

actuarial discussions of competing risks proceed on the assumption that the lives involved are independent in the sense of probability theory. This assumption has been recognized as unrealistic under 

many circumstances. For example, two people living together are likely to be exposed to the same 

environment, including a number of hazards, which affects their lives in a similar manner. Nevertheless, little, if anything, has been done in actuarial theory to account for this kind of stochastic 

dependence. 

When life tables are used to estimate various probabiliti~s, without the benefit of parametric 

models such as the Gompertz-Makeharn law, actuaries have been facing the problem which Bernoulli 

had to deal with in calculating column 5 of Table I: If life-table-type data are available only at endpoints of fixed time intervals, how does one use them when they appear in the denominator of a 

formula for a probability? The following procedure, quite analogous to Bernoulli’s, was recommended 

by Spurgeon in 1932 (p. 3826). Denoting by E. the number of individuals alive at age x, and by 19% 

the number of the E% who die between ages x and x + 1, he states: 

“If a number of the Ex persons aged x, say w%, withdrew from observation during the year, it 

would not be known whether they survived to age x + 1 or died between the date of withdrawal and 

the attainment of that age. . . . Assuming that the withdrawals were equally distributed throughout 

the year. . .. the rate of mortality will be obtained by taking 

6% 9? 

9X = Ex – (1/2)Wx “ (5) 

1.1.3. Clinical Experiments 

A large number of clinical experimental studies have been carried out in a manner which, possibly 

somewhat oversimplified, can be described by the following scheme. 
Included in an experiment are N subjects (patients, animals). At the beginning of the experiment 

each of them is exposed to a treatment that is conjectured to be somehow related to an event referred 

to as “failure.” For each individual subject, the experiment can terminate in one of two ways: Either 

failure is observed or the individual is lost from observation before failure occurs. Observable is the 

time W from the beginning of the experiment to its termination. This time W is clearly the smaller of 

two quantities: 

T = time to failure 

L = time to loss from observation. 

Typical experiments of this kind deal with the effect of potentially carcinogenic agents. Experimental animals are treated with such an agent and some develop cancer; others are lost from observation either by death from other causes or by developing conditions that would invalidate the experiment. An aim of the experiment is to gather information on the time from the beginning of the 

treatment to the onset of cancer. This information could then be used, e.g., for comparisons with 

similar data for animals who have not been exposed to the agent under study; or it could be used t~o 

assess the effect of age or sex or some other factors on the time to the occurrence of cancer. The 

crucial difficulty in using data obtained from such experiments is that one does not know the time to 

the occurrence of cancer in those experimental subjects whose observation is terminated by the competing risk, i.e., Ioss from observation. Many studies overlooked this difficulty and arrived at ccmclusions that could be show t to have no validity (for an example, see the 1939 article by Bernstein, Birnbaum, and AchsT ). 

Another class of experimental studies in which loss from observation and, possibly, other competing risks make it difficult to evaluate the results, consists of experiments in which a treatment is 

used to delay the occurrence of some event. Research on various prophylactic treatments falls intl~ 

this class, as do studies on the effectiveness of birth-control procedures. 

Lack of familiarity with even the classical actuarial theory has often affected the validity of clinical 

research. It was only in the mid-1950’s that papers by statisticians such as Cornfields began to 

remedy this situation by clearly pointing out the problem and by presenting actuarial techniques to 

those engaged in medical research. 

A model that is substantially more complex than that of competing risks, and yet quite realistic, 

was introduced in 1951 by Fix and Neyman.g They considered the study of a population of healthy 

individuals who may leave that population either by becoming sick or by dying; but by becoming 

healthy, the sick may return to the population. Treating such health-sickness-death sequences as a 

stochastic process, they stimulated a succession of publications that go beyond the competing-risks 

concept. A survey of work in this direction may be found in Chiang’s bookl” published in 1960 

and in his reportll pubIished in 1974. 

The stochastic process approach has been particularly useful in constructing models for human 

reproduction. First systematically presented in 1963 by Shepps and Perrinl 2 and followed by a 

sequence of papers developing the mathematical theory and, as an alternative, proposing computer 

simulation (see Perrin’s 1967 studyl 3 ), these models have led to a wide range of approaches. In 1973, 

Shepps and Menken14 surveyed these developments, 

1.1.4. Competing Risks in Technology 

If a technological system, such as an electronic network or a mechanical device, consists of n components c1 , c2, . . . , cn in sefies, and each component has a random life length, then the life of the 

entire system will end with the failure of the shortest lived component. If one agrees to say that 

8 
system failure is a result of the kth risk when c~ is the shortest lived component, then the n different 

~isks are competing in the sense that one usually can observe only the life length of the system and 

the component whose failure coincides with the failure of the system; in most practical situations, 

however, one cannot tell how long the remaining n – 1 components would have lasted. And yet such 

knowledge is essential if one wishes to make some statements on the extent to which the life of the 

system can be prolonged or. maintained by one of various patterns of replacing or servicing components, For example, components can be replaced at fixed time intervals; or there is a finite supply 

of spare components that are used to replace a working component immediately when it fails, so that 

system failure can be postponed until all spares have been used and then a component fails; or a particular component can be “beefed up” to increase its life length. 

Such problems have become particularly important since the advent of systems with large numbers 

of comp orients, such as computers, contemporary aircraft, or the total equipment involved in the 

successful launching of a space rocket. Such advanced systems gave rise to a new discipline, the 

mathematical theory of reliability which has been developed mainly within the last two decades, most 

of it in the United States and the Soviet Union. Fortunately, those actively involved in this development represented a wide range of backgrounds and combined technological insight, knowledge of 

actuarial concepts and ability to use mathematical abstraction, so that much of the learning-bymistake phase mentioned in section 1.1.3 could be avoided. 

1.1.5. Other Areas 

The sources of competing-risk problems mentioned previously do not constitute an exhaustive list. 

They provide representative examples, but many others can be added. In the area of popuktim 

dynamics, birth and immigration and death and emigration exemplify various modes of increment or 

of decrement of a population, and lead to the posing of problems in demography or in the study of 

biological populations. 

1.2. COMMENTS ON THE PRESENT STATE OF THE THEORY 

1.2.1. Duplication and Communication Gaps 

In surveying the literature, one encounters instances in which researchers in some fields have been 

unaware of progress made in other areas, and either had to duplicate work already done or else did not 

use that work. Consequently, their conclusions were often weaker then they could have been, or 

sometimes were invalid. One of the main aims of this report is to focus on basic concepts common to 

ninny areas in which competing risks have been considered and to present the mathematical theory 

dealing with these concepts. 

1.2.2. Probabilistic Versus Statistical Approach 

In publications dealing with competing risks, much space has been devoted to the study of probabilistic models and their properties. Most of the traditional results are such that, when either a life 

table or some parameters of an underlying process are given, one can compute probabilities of various 

mwnts, or expectations of times to failure or of other random variables, their variances and covarianccs, and so forth. 

How to use empirical data to arrive at conclusions about the underlying probabilities was a question 

which for a long time was dealt with by rather primitive methods. Actuaries used smoothing (graduation) techniques to obtain life tables from raw mortality data, then estimated probabilities by expres9 
sions such as equation (5), and their practices were largely accepted and used by workers in such fields 

as vital statistics and clinical research. Only recently has the corresponding statistical theory begun to 

develop, and with this development came, among others, a critical investigation of the role of independence of the “lives” involved, a systematic discussion of the question of what information can be 

extracted from empirical data (problems of identifiability), of the advantages and disadvantages of 

traditional and of some newly proposed estimates, and of problems that can be stated in terms of tests 

of hypotheses. Part of this report will be devoted to such developments in statistical theory and to the 

resulting practical techniques. 

1.2.3. Choice of Topics for This Report 

This presentation does not aim at completeness in any sense of the word. Our chief aim is to 

formulate the mathematical concepts and the theory of competing risks common to various fields 

of practical application. Having done that, we state a number of problems and describe techniques for 

dealing with them. There is a large literature on such problems, and subjective choices had to be made 

in selecting those which appeared typical, illustrative, and of practical interest. 

In presenting details of mathematical derivations, too, some choices had to be made. As a rule, 

mathematical arguments are completely included when they are reasonably simple and representative 

of a method of approach. For derivations that require long and complicated arguments or the use of 

esoteric mathematical tools, the reader is most frequently referred to original sources. 

In many cases, the theory available at present is inadequate, mainly because it requires assumptions that are not satisfied in real situations, such as the assumption of independence of random quantities that in practice are likely to be dependent. In other cases, the theory deals with statistical 

estimates that are biased or not consistent. In still other cases, such as in dealing with problems of 

identifiability (see section 2.6 ), the theory leads to the negative result that the data, as they are usually 

available, cannot yield the information one would like to obtain. In such situations, where there seem 

to be no satisfactory answers, we at least try to formulate the problems and point out the difficulties. 

2. PROBABILITY THEORY OF COMPETING RISKS 

2.1 LIFE DISTRIBUTIONS 

In studying living organisms, one often considers their life length-the length of time from birth to 

death. Similarly, for technological devices, one may be interested in the length of time from the instant 

when the device was put into use to the instant when it failed. In either case one deals with a random 

variable capable of assuming only nonnegative values. Random variables of this kind are encountered 

in many other situations, such as incubation periods of infectious diseases, hospital stays, and the time 

an individual remains under observation or participates in an experiment. 

In the material that follows, we shall use the terms “life” or “life length” or “time to failure” { 

quite generally to mean a nonnegative random variable, i.e., a random variable X such that 

P(x< o) = o. 

When a value x of a life X is observed, we shall say that the individual failed (or died) at a time (or age) 

x, that is, that it was alive (functioning or “up”) at all times t: O < t <x, and that it is failed (or dead, 

or “down”) at all times t: x < t< CO. 

10 
The probability distribution function of a life X will be denoted by 

F~(x) =P(x< x), 

the probability that failure occurs not later than at time x. It will simplify many arguments if we 

assume that X has a probabilityy density, i.e., that there is a function fx(x)such that 

and 

The function 

f~(x) = o forx<O 

fx(x) >0 forx>O 

fx(x)dx J--m =1 

x 

Fx(x)= J fx(S) ds. 

o 

~x(x)= 1 -Fx(x) 

=P(x>x) 

will be called the “survival function, ” and the function 

d hjy(x)=- --& log Fx (x)= fx (x)/Fx(x) (6) 

is known as the “hazard function” for life X. The intuitive meaning of ?iX(x ) becomes clear when one 

writes the last equality as an equation between probabilityy elements 

fx (x) dx Ax(X) dx = ~x(x) 

and interprets the right-hand expression as the conditional probability that an individual, having survived to age x, fails in the time interval (x, x + dx). 

11

The relationships 

Fx(x)= 1 -Fx(x) 

(7) 

are easily verified and one observes that each of the functions fx(x),Fx (x ), ~x (x), Ax(x) determines 

all the others, hence any one of them describes the probability distribution of X. 

2.2 NET (POTENTIAL) AND CRUDE (OBSERVABLE) LIVES 

Example A .– We consider an electrical circuit described by the diagram 

62


c1


C3


in which the only components subject to failure are the contacts Cl, C2, C3. We assume that each of 

these contacts deteriorates with time until failure occurs, and we denote the corresponding component lives by T1, T2, T3, with probability densities 

fT1(.)Yf?’2 (.)J7’3(.). 

We furthermore assume that T1, T2, Tg are independent. Let W denote the time to failure of the circuit. Clearly the circuit fails when either Cl fails, or C2 and C3 both fail, whichever occurs first. We 

shall refer to the failure of Cl as “risk R 1,” and to the failure of both C2 and C3 as “risk R z ,“ and 

define 

xl = time to failure of Cl 

= T1 

X2 = time to failure of Cl and C2 

= max (T2, T3) 

Then the time to failure of the circuit is 

W=min(Xl ,X2) 

= min [Tl, max (T2, T8)] 

12

and its survival function can be computed as 

Fw(7.o)=PITI >zu,max (T~, Ts)>zu] 

J

=~ .

fT1(t) [u fT2 (t2 )fT3(t3)‘t3 ‘t2 ‘tl 

max(tz,ts)>w 1 

‘ImfTl(’’)lmf~2(t2)flf~3(t3)d’3dt’d” 

‘LmfTl(tl)ImfT3(’3)
rfT2(’2)d’’dt3dt’


Following an accepted terminology, we shall call Xl and X2 the “net” lives for risks R 1 and R2, 

respectively. In practice, only the “actual” life W is observed, together with the fact that the circuit 

fails due to risk R 1 or risk R2, so that the random variable observed is two-dimensional 

(W,J) 

where IV is the actual life and J = 1 or 2, according to which of the two risks “caused” the failure. 

The actually observed time to circuit failure due to risk R 1 is a random variable that has the conditional probability distribution of circuit life, given that failure is due to R 1. The random variable YI 

with this probability distribution is referred to as the “crude” life for R 1. It has the survival function 

~y,(yl) =P(W> yl IJ= 1) 

= P(W>yl, J=l) 

Pu= 1) 

One similarly defines the “crude” life Y2, due to R2. 

Example B .– A cohort of patients is under observation during a fixed period of time O < t < T. 

For each individual, observation may end due to either of the two “risks”: The individual develops a 

specified condition C, or the individual drops out of observation. We consider the random variables: 

.s1 = time until condition C occurs 

.Y2 = time to dropping out from observation. 

These two random variables will be referred to as the “net” lives for the first or second risk. 

We assume that Xl and X2 are independent, and that Xl has a probability density that increases 

with time according to the formula 

for O<xl S7 

fx, (xl) = :1’T2 

{ elsewhere 

13 
while X2 has the uniform probability density 

for O<xz<7 

fx, (~z) = 

{ 

:’7 elsewhere.= 

The actual time W during which an individual is under observation is the shorter of the two, X1 

and X2: 

W = min (XI, X2) = actual life. 

One computes its survival function 

F~(zu)=P(w>zu) 

=P(xl>W, Y2 >W) 

=~’~’(;)($dx,dx, 

= (T - W)(72 -UN) for O<w<r 

@ 

while one clearly has 

F~(w) = 1 forw<O 

Fw (w) = o for w > T. 

Since, for practical purposes, only the actual duration of observation W and the “cause” of termination (condition C or dropping out) cafi be observed, we consider the two-dimensional random variable 

(W,J)


where J = 1 when C occurs first, i.e., if Xl < X2, and J = 2 when the patient drops out before C 

occurs, i.e., if X2 <XI . 

‘These probability densities were chosen strictly for the simplicity of computations, without any pretense of corresponding to arealkticsituation. 

14 
The joint probabilityy distribution of (W, ]) can be computed as follows: for O < w < r we have€

P(W>W, J=l)=P(W<X1 <X2)


and similarly€

P(W>W, J=2)=P(7JJ<X2 <X1)


‘L’l:(+)(adx’dx’


= H++~H)


while for w <0 one has€

P(w>w, J=i)= ;


P(W>W, J=2)= ;


and for w > r onc has€

P(W>W, J=l)=P(W>W, J=2)


= o. 

The marginal probabilities of termination due to either of the two risks are, therefore: 

P(termination due to C) = P(j= 1)€

=P(W>O, J=l)


1€ =—€

3’ 

P(termination due to dropout) = P(J = 2) 

=P(W>O, J=2)


2
 =—
3“ 

15 
The conditional probability distribution of the time to termination, given that it occurs due to 

condition C, can be stated in form of the survival function 

P(w>wlj= 1)= ~@’’’’~-fl:)= 1, 

‘1-3(7+2(+’ ‘oro<w<” 

and a random variable Y1 that has this probability distribution, i.e., has the survival function 

17y Jyl)= l-3 

(?) ‘z(~)’ forO<y1<7 

is customarily referred to as the “crude” life due to the first risk. 

Similarly, the conditional probability distribution of the time to termination, given it occurs duc 

to dropping out, is described by 

P(W>ZU, ]=2) P(W>WIJ =2)= P(J=2) 

“-X9+W 

and the random variable Y2 that has this probability y distribution, i.e., has the survival function 

FyJy~) =P(Y2 >yz) 

is traditionally called the “crude” life due to the second risk. 

2.3. GENERAL THEORY: NET LIVES 

In general there may be any number k of “modes of failure” or “risks,” which will be denoted by 

R~, R2, ..., Rk. We assume that an individual organism, or system, or device, fails at the earliest occurrence of one of these modes of failure. To each of the risks corresponds a “net” or “potential” life; 

we shall interpret these net lives as a k-dimensional random vector 

(x,,~2,.,.,&)=x 

the coordinate Xj being the time to the occurrence of Rj. It should be noted that these net lives 

xl, . .. ,xk may be dependent random variables. 

16

� 

Since the system fails at the realization of the earliest risk, the time to failure for the system is 

W=min(X~, X2, ”-. ,X~), (8) 

called the “system-life” or “actual life.” 

We shall assume that the net lives have a joint probability density 

w =f(% >%!3.. . >%), 

so that the joint probabilityy distribution function may be written as 

F~(&)=Fg(xl >.. .>xk) 

‘~(xl<~l,”..,xk<xk) 

and the joint survival function as 

(9) 

Clearly, the marginal probability distribution function of the net life Xi is 

Fxi(X~)=P(Xi < Xi) 

‘Fx(w, . ..jm. Xijw, w). ,w) 

and the marginal survival function 

Fxi(Xi)=P(Xi > Xi) 

with xi in either case at the ith place in the lower expression. 

In view of equation (8), the survival function for the system life W is 

F~(w) =P(?’t’> w) 

=Fx(w, w,. ... w). 

17

From now on it will be assumed that, whenever the system fails, one can observe only its actual€

life and the mode of failure, i.e., the value of the two-dimensional random variable€

*€

(W, J) 

where W is the system life and J = j when 

W=mk(xl,xz,”””, xk) 

‘Xj, j=l,2, ”””, k. 

Since, under our assumptions,€

P(xj=Xj)=o for i + j, 

so that the probability of two or more net lives being equal is zero, the value of J is uniquely determined with probability one. 

The joint probability distribution of (W, J) can be expressed as 

F~,J(W,j) =P(W<w, J=j) 

= P(Xj <W, Xj<X~ fori#j) 

.

... [J”J f(x~, ”” “ , ‘k )n 

r#j 

“r ‘Xj forj=l, .””, k.. (lo) 

xj xj 1 

It follows that the probability distribution function for the actual life W is 

F~(w) =P(w<w)€

= & F~,J(W j).€

j=l


and the corresponding survival function€

k 

F~ (w) = 1- ~ Fw,j(w.o. 

j= 1 

One also has for the marginal probability that failure will occur at any time due to Rj the expression 

P(J = j) = FW,l(+CCJ,j) 

18

2.4. GENERAL THEORY: CRUDE LIVES 

Since we assume that only W and J, the actual life and the mode of failure, are observable, it is of 

interest to consider the conditional probability distribution of W given that J = j 

P(W<wl J=j)= P(W<zu, J=j) forj=l,2, .”. ,k. P(J=j) 

It has been customary to consider for each value of j a random variable Yj which has this probability 

distribution and hence corresponds to the observable time to failure due to risk Rj. This random 

variable is usually called the “crude” life for risk Rj; its probabilityy distribution function is 

‘Yj(Yj) = ‘(yj < Yj)


P(W<yj, J=j) = 

P(J = j) 

‘W,J (Yj> j) (12) = P(J = j) 

where the numerator can be computed according to equation (10) and the denominator according to 

equation (11 ). 

We shall consider the probability 

QYj(~)‘P(WS ~,J=j) 

= P(Xj > t, and Xj <Xi for Z’# j) 

=P(J=j and ~>t) 

= P(J = j) - ~w,J(~, j) 

This is the probabilityy of survival beyond age t and 

Qyj(o) = P(J = j and 

forj= l,””-. (13) 

then failure due to Rj. It should be noted that 

~> O)< P(J=j), 

hence QY,(0) need not be 1, so that in general QYj (t) is not a survival function according to the definition in section 2.1. 

2.5. AN IDENTITY 

Forj=l,2,. ””, konehas 

a–
 ~ Qyj(~) = ~ F4(x1, ..., % )X1=X2... ‘xk=t” (14) I 19

Proof: Without loss of generality, we prove this identity for j = 1. From equation (13) 

QYl(t) ‘P(X1 >tand Xl <Xi fori#l) 

/-= “m .CO 

=,, ,%1...jx1f(x.,s2,..., $k)dsds2ds,2ds, (15) 

hence, 

m 

f(t, s*,”””, s~)ds~ ““” dsz . ; QYl(~)=- Jt

“.” J“t


But from equation (9) one obtains for the right-hand side of equation (14) 

~ F~(xl, . . ..xk) x~=x*. ..=x~=t ax 1 

=_... 

. J“ Jf(xl, s2,. ””, sk)dsk” ””ds21x1=”””=xk =t 

%2 Xk 

=-H

.

... 

t 

. 

f(t, s2, ”””, sk)dsk .“”ds2 

t 

which concludes the proof. 

Identity (14) is due to Tsiatis (1975).15 Under the assumption of independent net lives, Berman 

(1963)16 gave an identity equivalent with equation (15) and explored its consequences, as described in 

section 2.6.2. 

2.6. THE PROBLEM OF IDENTIFIABILITY 

2.6.1. Formulation of Problem 

Since only the actual life W and the mode of failure are observable, the question arises how much 

information can such observations yield about the probabilityy distribution of net lives. One would 

like to learn as much as possible about that probabilityy distribution since, as we have seen, all other 

probability distributions of our theory follow from it. 

If there are sufficiently many observations of W and of J, then it should be possible to estimate 

very closely the probabilities P(J = j) of the modes of failure and FYj (t) for the crude lives. Appropriate estimation techniques will be discussed in section 3; for the time being we shall assume that 

practically unlimited numbers of observations were available so that P(J = j) and FYj(t) for 

j=l, z,... , k are known. 

20 
The problem of identifiability can now be stated: If P(J = j) and FYj (t) are given forj ~ 1,2, “ o“ , k, 

dom this determine the joint probability distribution of the net lives Xl, X2, . “ “ , X~ ? 

2.6.2. The Case of independent Net Lives 

We have observed that the net lives need not be independent. In fact, the assumption of their 

independence is in many practical situations obviously unrealistic. 

In this section, however, we make the assumption that X1 , X2, “ - - , Xk are independent, that is 

or equivalently 

From this, using the definition (6) of the hazard function, one has 

(16) 

and identity (14) yields 

d a– ~ Qyi(t)= ~F~(xlY”””~xk) x~=%z=”””=xk, =t 

i 

= -xjq(t)~~(t, “““,t). (17) 

cTsing equation (7) we have 

~xi(t)=exp[-~,hxi(s) d.] 

;md with the notation 

-&Ixj(x) = A(x) 

j=l 

21 
we may write 

Fx(t, — ”””, t) =fi F$(t) j=l


t = exp D- A(S) ds .

0 1

Since the survival function for the actual life W is 

F~(t)=P(xl>t, ””’, x~>t) 

=F--(t, ”””, t), 

we see that A(x) = XW(x), and we note this fact as follows: 

Lemma.– If the net lives are independent, then the hazard rate of the actual life W is 

Together with equation (17) this yields 

(18) 

(19) 

Now, when the P(~ = ~) and Fyj(yj) are known for j = 1, “ “ “ , k, then by equations (12) and(13) 

QYj(~) are know% and relationship (20) is a system of k equations with k unknown functions Axt(.t). 

To solve it, we sum equations (20) over i 

hence, 

& Qyi(~) =exp - t A(s) ds +.. 

i=1 [J 0
 1

In view of 
wc have c = O, and from equation (20) 

()$ ~Qyi(~) 

h~i(t) = - ~ fori=l, ”””, k, (21) 

~ @i(t) 

so that, knowing the QYi(t) for i = 1, “ “ “ , k, we can compute all hazard rates Axi(t),and by equation 

(7) all ~.yi(t)and Fxi(t). 

To summarize, if the net lives are independent, then the probabilities P(W < w, J = j), or the probabilities P(J =j) and Fyj (t) for j = 1, “ “ “ , k, determine all ~x. (t) and all Fxj (t), hence, ~SO the joint 

probability distribution of all net lives Xl, . “ “ , Xk. This result~is due to Berm~”16 

2.6.3. The Case of Possibly Dependent Net Lives 

The assumption of independence of the net lives was used to obtain equations (16) and (17), and 

to derive formulas (21) which show how, given the probabilities QYj (t) for the crude lives (or, equivalently, the probabilities P(J) and Fyj (~j)), one CZUIcompute the probability distributions Of the net 

lives. 

This suggests two questions: 

(a) Can formula (21) be used to obtain the distributions of net lives from those of crude lives if it 

is not known that the net lives are independent? 

(b) If it is not assumed that X1, ” - “ , X~ are independent, do the P(J) and Fyj (Yj) > ~ = 1> “ “ “ ~ k 

determine the joint distribution of Xl, -00, -%? 

The following example, given by Tsiatis,15 shows that the answer to question (a) is negative. 

Example C. –Let the joint survival function of (Xl, X2 ) be of the form 

F~(xl, xz)=exp(-kl -w2 -TC1X2) (22) 

where A >0, p >0, 0< T < AN. Clearly, Xl, X2 are independent if and only if T = O. The marginal survival functions for the net lives are 

FX1(X1)=exp (-til) 

(23) 

~xz(xz) = exp (- W2) 

and do not depend on T. 

By equation (14) 

: QY1 (~)=-(1 + ~t) exp (- M - W - ‘Yt2) 

~ QY2(t)= -(P+ -F) exp (- kt - pt - @ ). 

Integrating each of these identities one obtains for QY1 (t)and QY2 (t) expressions that depend on all 

three parameters A, p, T in such a manner that, substituted in the right-hand side of equation (21), 

23

they would lead to hazard rates XXI (t), AX2 (t) which depend on allthreeparameters and hence on 7. 

These hazard rates would, therefore, differ from”the correct ones which correspond to equations (23) 

and which are independent of ~. Thus, it is shown that equation (21) cannot be used in the case of 

dependent net lives. 

The next example, due to Rose,17 answers question (b), also in the negative. 

Example D.–Let k = 2 and 

1 for O<xl<land O<x2<l 

fxl,xz(~l Y~2) = { ~ elsewhere, 

i.e., let the net lives Xl, X2 be independent and each uniformly distributed on [0, 1] . 

Consider now the family of two-dimensional probability densities, obtained by modifying 

I!xl,x Z(x1,X2) in a square with sides Of length a, with O~ a < (1/2) (see figure 3)’ 

o when O<xl<~ and l-~<x2<l 

o	 when g <xl<a and l-a<x2 <l-~ 

2 

af X1,X2 (xl> 62)= 

2 when O<xl <~ and l-a<x2 <l-~ 

2

I

2 when~ 

2 

<xl<a and l-~<x2<l 

2 

L .fXI,X2(~I,x2) everywhere else. (24) 

Clearly .f!xl, X2 (X1, XZ) is contained in this family, since 

ofx~, x2(w$~2) =fxl, x2(~l>x2). 

One verifies that, for every ~fxl, X2 (x 1, X2), the net lives have marginal probability densities which are 

uniform on the unit interval 

fOr O< Xj<l afX~(xj) = ~ forj=l,2, { elsewhere 

hence are the same for all a, O < a < 1/2. For a >0 these net lives are not independent, for in the entire square O<xl < a/2, 1 -a/2< x2 <l one has 

a~X~,Xz(xl, x2)= 0#.fX~(xl) “ afX2(x2) = 1 

when O<a < 1/2. 

24 
B 

xl J; ‘ )n , 

A +N// ‘ +.,/ ‘ 

,f=o ,f=2 

/// 

~f= 

/

/

/ 

1 

/ 

/ 

/

/

,“

//

/ 

/ 

/ 

/ 

/ 

/ 

///

/ / / /~ 

1- a/2 

.f=2 .f=o 

1 -a 

/:/ f=l 

/ / /’ 

~f= 

/ 

1 

/

//’ 

/ /

/ / 

/ / / / / 

/ /’ /

/ / / / 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

I I ~ X2 

a/2 a 

Figure 3, Probability densities used in example D. 

The joint survival function of the net lives is 

3X1,X2(X1,X*)= (1 -X1)(1 -X2) for O<x~<x~+a <l, (25) 

and is equal to other expressions in other parts of the unit square. However, to use equation (14) for 

computing 

: QY1 (t) and $ QY~ (~), 

it is enough to know the joint survival function in an open set containing the diagonal x 1 = x z and 

equation (25 ) provides this information. Hence, by equations (14) and (25), we have 

: QY1(t) = -(1 -t) 

25 
and 

~2 

QY1(t)= ~ - t + QY1 (0) 

~2 

=1 -t+ P(J=l) for O< t<l. 

Since P(J = 1) = P(X1 <X2 ) = 1/2 independently of a, we have by equations (12) and (13) for the 

crude life Y1 

~Y~(Yl) =2Y1 -Y! for O<yl <l. (26) 

By an analogous argument one obtains 

Fy2(y2)=2y2 - y; for O<y2 <1. (27) 

Obviously 

o for Yj <0 

Fyj (Yj)= ‘{~ foryj>l, j=l,2. 

In example D, FYI (yl ) and FY2 (y2 ) as well as P(J = 1), P~ = 2) are the same for all values 

O < a < 1/2, and the same is, therefore, true of FW,J (w, j). The knowledge of these probabilities, 

therefore, does not identify the joint probability distribution of the net lives Xl , X2, and the answer 

to question (b) is negative. 

2.6.4. An Interpretation of the Identifiability Problem: Some Inequalities 

We limit ourselves to the case of two competing risks, k = 2, and consider the probabilities defined 

in equations (9) and (13) which, in our case, we denote simply by 

F(x~, x2)= P(x*>x~, x2>x2) (28) 

and 

QI(x1) ‘P(XI >x1 and XI <X2) 

(29) 

Q2(~2) ‘P(X2 >~2 and X2 <Xl). 

We assume as before that Q1 (x 1 ) and @ (X2) can be estimated, hence will be considered as known, 

and restate the problem of identifiability in the forni: To what extent do the functions Q1 (x 1) and 

Q(q) determine the joint surv@l functi~n ~(xl, X2) of the net lives, or at I:ast one of the marginal 

survival functions P(Xl >Xl ) = Fxl (xl ) = F(x I , O), P(X2 >X2) =~x2(x2) = F(O, X2)? 

26

As indicated on figure 4, ~(xl , X2) is the integral of the joint probability density $(x1, X2) over 

the region contained in the right angle between bold lines: 

i7(x1 ,x2) =JJ7(x1, x2)dx1 dx2, 

L 

while Q1 (X1 ) is the integral over the interior of the 45° angle bounded by broken lines 

Ql(~l)=	 ~~ ~(~l,~z)dxl dxz 

;/‘ 

find ~ (X2) is the integral over the interior of the 45° angle bounded by dotted lines 

Q2(x2)= /\ f(xw2)dxl dx2. 

L._ 

8 

Figure 4. Regions of integration considered in equations (30) and (31). 

27 
By inspection one has 

F(X1,X2)=Q1(X1)+Q(X2)-P(A) (30) 

where P(A) is the probability of (Xl, X2) falling into the triangle denoted by A in figure 4 or, more 

explicitly, 

~(xl, x2)= Q1(x1)+@(x2)-P(xl <X1 <X2 <X2) (31) 

when xl <X2 (as in figure 4) or 

~(xl, x2)= Q1(xl)+ Q(x2)-P(x2 <X2< X1 <x1) (32) 

when X2 < x 1. As an immediate consequence one obtains the inequalities 

QI [ITMX (x1,x2)] + @ [m= (x1,x2)] ~F(XI,X2) <Q2(x2) + Q1(x1). (33) 

Writing 

Q1 (0)= P(X1 <X2 ) = probability that failure occurs due to the first risk= pl 

@ (0)= P(X2 <Xl ) = probability that failure occurs due to the second risk= p2 

and recalling that survival probabilities for the net lives are 

~xl(xl) =qxl, 0), 

FX2(X2) = F(O, X2), 

one obtains from equation (33) the further inequalities 

Q1(W)+Q2(W) -X1( WFQ1(W)+P2 (34) 

QI (x2) ‘Q2(x2) ~~x,(x2)~P1 +Q2(x2). (35) 

The inequalities (32), (33), and (34) were obtained in 1975 by Arthur V. Peterson, Jr.,18 by a formal 

argument. 

The identities (30), (31), and (32) shed additional light on the question of identifiability: If only 

crude lives are observable, the probabilities P(A) cannot be estimated, and this is the reason why 

QI (XI) and 43 (x2 ) do not determine @l, X2) in the generalcase. It may be worth noting, however, 

that according to identity (30) 

~(x, x) = Q1 (x) + Q2(x) (36) 

i.e., the knowledge of both crude-life distributions determines the survival function F(x, x) = P(W > x) 

of W, the actual life, without the assumption of independence. 

28 
2.7.SUMMARY AND COMMENTS ON IDENTIFIABILITY 

We have seen~hat the joint probability distribution of the net lives Xl, . . . . Xk, as given by the 

survival function Fx (x 1, . . . , Xk ), determines the probabilities of the crude lives, i.e., all 

P(J = j) and FYj(y~) forj=l, . . ..k 

or, equivalently, all 

P(W<w, J=j) = Fw,j(w)j) forj=l, . . ..k. 

The converse is true if the net lives Xl , . . . . X~ are independent. In general however, i.e., when it is 

not assumed that the net lives are independent, the probability distributions of the crude lives do not 

determine the probability distribution of the net lives. 

This brings us to a serious practical problem. In competing risk situations, the only observable 

quantities are the actual life W and the mode of failure J (i.e., the crude lives), and the best one can 

hope for is to obtain good estimates of their probability distributions. How does one from then on 

draw conclusions about the net lives in cases where the assumption of independence is unrealistic? 

Clearly, the knowledge of the probability distribution of W and J will have to be supplemented by 

some further information, either stated in form of further assumptions, or drawn from additional 

{Observations. 

To see what some additional assumptions can contribute, let us return to example C, section 

X.6.3. In that example we assumed that Qy ~ (t) and QY2 (t) were known and we attempted, unsucccssfLdly, to determine ~x ~, x * (Xl, X2) by applying equation (21). We did not, however, make use of 

;my assumptions on the form of ~x ~, ~ ~ (xl , X2). Let us now assume again that Qy ~ (t), QY.2 (t) are 

linown and use the additional knowledge that ~x ,x* (xl, X2 ) is of the form given by equation (22). 

Now all one needs to determine ~x ~, x * (xl, X2‘) are the values of the three parameters h, p, and T. 

But this should be quite feasible: If QYI (t) is a known function of t,then the left side in the identity 

+ QYl(~)=-(~+T~) exP (-~t-Pt-7t2) (37) 

is also a known function. Entering any three different values t1, t2,t3 in equation (37), one obtains 

three equations that then can be solved for k, p, and y. In fact, knowing the probability distribution 

for only one crude life, together with the assumption that ~x ~, X2(x ~, X2) belongs to the parametric 

family of equation (22), is sufficient to determine completely the joint distribution of the net lives. 

This example points to one effective way for overcoming the difficulty due to the fact that, in 

general, the probability distribution of net lives is not determined by the probability distributions of 

crude lives: One assumes that ~x (E) belongs to a specified parametric family. Thisdoes, of course) 

raise another question: How doe; one choose a parametric model which is appropriate for the phenomena under investigation? In some cases a thorough knowledge of the physiological or technological processes underlying the functioning and failure of the systems may lead to the proper choice of 

such a parametric model. If there are no such guidelines, one may be tempted to choose a parametric 

model for its mathematical simplicity and manageability-a situation, and responsibility, not unfamiliar to those who deal with applied mathematics.b 

bProfessor A. J. Quinzi brought to our attention the 1977 and 1978 references Langberg, Proschan, and 

(~uimi,19-21 in which the authors developed a general theory of converting models with dependent random variables 

29 
2.8. MULTIPLE DECREMENT TABLES VERSUS Questions ABOUT NET LlVES 

When data on crude lives are available, it is important forsome applications toinvestigate mtinly 

the joint distributions of these crude lives. An important example of studies of this kind is the analysis 

of mortality data by different causes of death. Most of the work in this direction was done by actuaries who have developed techniques for constructing and using “multiple decrement tables. ” 

This report will, from now on, deal only with questions that go in the direction initiated by Daniel 

Bernoulli and are of the following general kind: When empirical data are available on crude lives in a 

competing risks situation, what inferences can be drawn from these data about the net lives and their 

probability distributions, and what assumptions are needed to make such inferences possible? 

3. ESTIMATION TECHNIQUES 

3.1. CASE OF INDEPENDENT NET LIVES: 

THE KAPLAN-MEIER NONPARAMETRIC TECHNIQUE 

3.1.1. Statement of Problem 

We have seen that the probability distributions of the crude lives determine the probabilit y laws of 

the net lives when one makes the assumption that the net lives are independent, or assumes a specified 

parametric model for the probability distribution of the net lives. We shall now discuss the case of 

independent net lives (hence will not assume any specific parametric form for their distribution) and 

present some of the known estimation procedures. 

3.1.2. Definitions and Notations 

In 1958, Kaplan and Meier22 dealt with the following situation: Consider two competing risks 

which, to aid one’s intuition, will be referred to as “death” and “loss from observation.” For each 

individual there is a two-dimensional random variable (T, L) with 

T = “time to death”= “life length” 

L = “time to loss from observation” = “limit of observation,” 

and T and L are assumed independent. At time t = O a cohort of n(0) individuals is placed in observation. Each individual remains under observation until the earlier of the two events (death or loss), 

occurs so that observation records show only values of the actual life W = min (T, L) and the event 

that occurred at time W. Using these recorded observations, one wishes to estimate either one of the 

two net survival functions, say that of T 

~(t) =P(T> t). (38) 

into models with independent variables. As one of the applications, they obtained a set of assumptions which are different from assuming either independence or a parametric model, and are sufficient for a positive answer to the problem of identifiability. 

30 
The recorded observations determine the nonnegative, decreasing, integer-valued function defined 

by 

n(t) = number of individuals alive and in observation at time t, (39) 

with the convention that a death occurring at t is already subtracted in computing n(t), while a loss at 

t is not yet subtracted. In other words, deaths at t are treated as if they occurred slightly before t,and 

losses from observation at t are counted as if they occurred slightly after t.Some consequences of this 

convention are these properties of n(t): 

n(t) is continuous at t if neither death nor loss occurs at t, 

n(t - O) - n(t) = number of deaths occurring at t, (40) 

n(t) - n(t + O) = number of losses occurring at t. 

3.1.3. A GeneraI Estimation Procedure 

A number of estimation techniques for ~(t) can be obtained as special cases of the following 

general scheme. 

Step (a). –The time axis is divided in some disjoint intervals 

(O, U~], (U~, U~], . . ..(Uj_~. Uj], . . . . 

We introduce the notations 

Pj = P(T> Uj) = ~(uj), (41) 

and 

(42) 

Step (b). –We decide to use some estimates ~j for pl~ j = 1, 2, . . . . 

Step (c).–For any t >0, we use as an estimate for F(t) the statistic 

ii(t)= n ~j. (43) 

Uj<t 

In this general scheme, it is left open how the time intervals (~j_l, ~j] are chosen and what the 

estimates ~j are. By choosing the division points ~j equidistant and assuming a specific parametric 

model for ~(t), one can use standard curvefitting techniques to estimate the parameters, and the 

resulting ~j and R(t) will be actuarial estimates corresponding to the parametric model. 

3.1.4. The Kaplan-Meier Estimates 

The following specific procedures will now be used in carrying out steps (a) and (b) of the preceding section. 

The recorded times of deaths and of losses are arranged in one increasing sequence of observed 

aCtUdtiVeS, Ul<U2<”””<Uj <“”” , and these uj will be used as the endpoints of the intervals 

31 
---

(uj_l, ~j]. For the sake of sirn@icit~ it will be assumed that only one of the events, death or loss, 

occurs at each~j. 

‘ We write q = TZ(Ui+ O) = number of individuals at risk (abe and in observation) immediately 

after uj, and use as an estimate of the conditional probability pj the statistic defined by 

1 if a loss occurred at~j,~= 1, 2, . . . (44) ,, 

fij= ‘j-l ‘1 

if a death occurred at uj, ~ = 1, 2, ~._. . 

{ nj_l 

The estimate (43) for ~(t) now becomes 

ii(t) =11 ~j. (45) 

Uj<t 

This estima~e will be referred LO as the “Kaplan-Meier estimate for ~(t).” It clearly is a step function such that H(0)A= 1 and that H(t) remains unchanged when t increases across a value Uj at which a 

10SSoccurred, and H(t) decreases by a factor (nj_l - l)/nj_l when t crosses a value uj at which a death 

occurred. 

A small-size example of this procedure, obtained by a slight modification of an example given in 

1958 by Kaplan and Meier (p. 46422) is presented in table A and figure 5. 

3.1.5. Some Properties of the Kaplan-Meier Estimates 

As the example of tabIe 1 shows, the Kaplan-Meier estimate fi(t) remains undetermined beyond 

the last actual life when the last event recorded is a loss. In our example we can ordy conclude that 

() < ~(t) < 21/80 for t >12.1. Had the last recorded event been a death, then we would have had 

fi(t) = O from then on. 

Wh:n the data are large; some grouping maybe desirable before carrying out the calculations leading to H(t). Several ways of doing this were suggested by Kaplan and Meier. 

It may be important for some practical applications to note that 

to consider individuals entering into observation after the beginning 

counting each such entrance as a negative loss, i.e., one records the 

number of individuals in observation by 1, and otherwise treats the 

Table A. Example of a Kaplan-Meier estimate 

= 

j ‘j Eventsl Ilj fij A(Uj) 

o	 0.0 8 1 1 

1 0.8 A 7 7/8 718 

2 1.0 k 6 1 718 

3 2.7 A 5 1 718 

4 3.1 A 4 4/5 7110 

5.4 A 3 314 21140 

: 7.0 k 2 1 21/40 

7 9.2 A 1 1/2 21/80 

8 12.1 ii o 1 21/80 

1A death is indicated by A, a loss ‘rem 

observation by 1. 

the K-M procedure permits one 

of their lives. This is done by 

time of entrance, increases the 

event as a loss. Such entering 

32 
/) 

A 

1


~


8


7


E


21


z


21
 m
 Tii5


i I 1 I I J *U


o
 2 4 6 8 10 12 

Figure 5. Graph of Kaplan-Meier estimate obtained in table A. 

individuals can later on disappear from observation and will then be counted as ordinary losses. It 

should be stressed, however, that the assumption of independence of life length and time to loss is 

being used whether the loss is positive (individual drops out from observation) or negative (individual 

enters into observation). 

Under the assumption of independence of T and L, Kaplan and Meier showed that ~l(t) is that 

member of the class of all survival functions which maximizes the likelihood function for the recorded 

yalues and that it is consistent. They also derived several approximate expressions for the variance of 

H(t). For these derivations, the reader is referred to the original paper. Much further theoretical 

research has been done on the asymptotic properties of the K-M estimates (Breslow and Crowley23 in 

1974, Crowley24 in 1975, and AalenZ5 in 1976). A generalization to the case of more than two competing risks was introduced in 1972 by Heel.26 

3.2. SOME ACTUARIAL ESTIMATES AND THEIR RELATIONSHIPS 

TO KAPLAN-MEIER ESTIMATES 

3.2.1. A Generalized K-M Procedure and Some Actuarial Estimates 

The procedure described in section 3.1.4 can be obtained as a special case of the following more 

general way of carrying out the steps listed in section 3.1.3. 

Step (a). –Given a recorded sequence of deaths and losses and the corresponding actual lives, we 

choose ul<uz<. ..<uj< ... arbitrarily, but so that each interval (uj- 1, uj ] contains only deaths, 

or only losses, not both. The uj’s need not be actual lives. 

;, 

Ln 33

Step (b).–We introduce as estimates of the pj the statistics defined by 

$~ = 1, (46) 

and, forj=l,2, . . ..by 

when (uj.. 1, uj] contains 8 deaths 

(47) 

when (~j- 1, Uj ] contains losses. 

Step (c).–We define 

i(t) =11 ~j. (48) 

Uj< t 

This procedure clearly reduces to the Kaplan-Meier estimate when one chooses 

?.41<242<... Uj <... 

to be all actual lives. 

For any i;terval (a, b] ,0 <a < b, one has according to definition (48) the statistic 

. 

;a,b=~ ;j=g!!j (49) 

(t Cuj<b H(a) 

as an estimate of the conditional probability 

~(b) ~a,b = -. 

F(a) 

To describe a class of approximations to ja,b which are often referred to as “actuarial” estimates, 

we consider an interval (a, b ] and the observed values: 

n = n (a + O) = number of individuals under risk immediately after a 

6 = number of deaths in (a, b] 

X = number of losses in (a, b] . 

We assume that these three numbers are known, but the order in which deaths and losses follow each 

other is not known. 

If all 8 deaths happened to precede all A losses then the estimate (49) could be obtained by choosing one division point and this estimate would be 

n-ti 

~a,b = — . (50) n 

34 
Similarly, if all k losses preceded all 6 deaths, this estimate would have the value 

(51) 

One readily verifies the inequality 

(52) 

If the order of occurrence of deaths and losses were completely known, then ~a,b could be computed according to (49). 

None of the three quantities #a,b, ~a,b, ~a,b can be justified as an estimate of ~a,b when only 

?Z, 6, ~ are given but nothing is known about the order in which the events occurred, as is the case in 

many practical situations. In these cases, it has been customary to use as an approximation to pa,bthe 

estimate 

* n-h/2-6 

(53) ‘a*b = n - ~/2 

sometimes referred to as the “adjusted-observed” acturial estimatec which satisfies the inequalities 

3.2.2. Conditions for p~,b Being Consistent 

The estimate (53) is one of the most frequently used so-called “actuarial” estimates, and it is of 

some interest to determine if, as the number of observations increases, it tends in probability to 

~a,b, i.e., if it is a consistent eStimate Of Pa,b. This matter was clarified in 1974 by Breslow and 

Crowley,zg and the following is a presentation of their findings. 

As before, we assume that the life T of an individual and the time L to loss from observation are 

independent random variables. Let their survival probabilities be 

~=(t) =P(T>t)= 1 ‘~=(t) 

~L(t) =P(L >t) = 1 -F~(t). 

For any given interval (a, b], we wish to estimate the conditional probability 

~=(b) .. Pa,b =	 ~ 

F~ (a) 

by using estimate (53). 

cThe reader may compare this with Bernoulli’s estimates for column (5) in his Table I (figure 2) which we mentioned in section 1.1.1, and Spurgeon’s recommendation of formula (5). 

35 
-------------------- ------

Instead of intervals (a, b ] we may, without loss of generality, consider the intervals 

Ig=(o,g], g>o 

with arbitrary $. We now define the following events (see figure 6): 

6~={o<T<&, L>g} (individual dies in It, not due to be lost in 1:), 

8~={o<7’<L<g} (individual dies in 15, before it is due for loss which is in lE), 

/.J={o<L<g, L<T} (individual is lost in Ig and before its death), (55) 

T,={o<T, L>g} (individual is not due for loss in It ), 

~,={o<T, L<g} (individual is due for loss in lE). 

L 

62 

I *T 

Figure 6. Diagram of events defined by formulas in (55). 

36 
N(~t all of these events are observable; however, the event p is observable and so are the following 

unions 

s=a~ua~ (individual dies in IE before getting lost) 

-Y= ’Y1U72 (individual is alive and present, hence under risk, at O). 

Ctmsider now the random variables. 

iV1 = number of individuals for which Y1 occurs 

N2 = number of individuals for which 72 occurs 

D 1 = number of individuals for which 61 occurs (56) 

D2 = number of individuals for which 6 z occurs 

M = number of individuals for which p occurs. 

N = N1 + N2 = number of individuals for which -y occurs, i.e., number under risk at O 

D = D1 + D2 = number of individuals for which 8 occurs, i.e., whose deaths occur in It, before 

they are lost. 

The random variables M, N, and D are observable. The probabilities of the events in (55) are 

P(61) ‘~=($)[1 -FL(:)] 

E 

P(/J) = [1 - F=(s)] dFL (s) (57) 

./ o 

P(Y2 ) = FL($) 

und, assuming the number N known, each of the random variables in (56) has binomial probability 

distribution with parameters N and the corresponding probability in (57). 

The statistic (53) can now be written 

N- MJ2-D 

‘:,t = N -M/2 = 1 -q; ,E 

37 
where 

D N1 D1 + Nz - M/2 D2 

(58) ‘~~~= N-M/2 = N-M/2 “ NI N - M/2 “ N2 -M/2 “ 

As N + CO,each of the quantities N1 /N, D1 /N, D2 /N, M/N tends in probability to the corresponding 

probability in (57) hence the right side of equation (58) tends to 

1 -FL ($) ~ FT(~)[l -FL(t)] 

t’ 1 -FL (t) 

1- (1/2) \ (1 -FT) dFL 

o 

t E 

FL ($) - (1/2) f (1 - FT) dFL FT dFL 

o / 0 

+ x . (59) 

1- (1/2) \g (1 -FT) dFL FL(E) - (1/2) f (1 -Fz-) d~L 

o 0 

We assume FL (~) < 1, which must be true if there are any individuals left in observation after ~. 

Expression (59) then reduces to a weighted mean of the form 

i 

Jo 

FT dFL 

A* FT(~)+(l-A)o (60) 

FL(E) - (1/2) ~g (1 - FT) dFL 

o 

with O< A <1. Since p~,t is consistent if qj,gtends in probability to FT (g), i.e., when the expression 

(60) is equal to FT($), we obtain 

~ 

Jo

FT(&) = for all ~ >0 (61) 

t 

FL (.&)- (1/2)\ (1 - FT) dFL 

o 

FT dFL 

as a necessary and sufficient condition for p~,~being consistent for every It. Abbreviating 

(62) 

we rewrite equation (61) as 

(63) 

38 
From equation (62) one verifies that ~ 

a cliffercntial equation which has the solution 

G


forc>O. (64) (~_ G)2 ‘CFL 

Solving equation (64) for G in terms of FL and substituting in equation (63) one has 

“(g)=1-&) (65) 

WC see, therefore, that relationship (65) between the probability distribution functions F~ and FL 

k necessary and sufficient for the statistic (53) being a consistent estimate of ~a,b for all intervals 

((l, b]. 

In practical situations, it appears quite unrealistic to assume that T and L are not only independent 

but even have probability distributions related by the very stringent condition (65). One arrives, 

therefore, at the disappointing conclusion that the “adjusted observed” estimates of the probabilities 

Pa,6 are in general not consistent. It would be interesting to obtain bounds on the asymptotic bias of 

these estimates, but such bounds do not appear to be known. 

3.3. LIFE-TABLE ESTIMATES: PROPORTIONAL HAZARD RATES 

3.3.1. Definitions and Assumptions 

\Vcconsider risks Rl, R2, . . . ,Rr, . . . , Rh. The time axis is divided by points 

o<u~<zq<. ..<uj <... 

CAOst’tl O?lce j-or all (e.g., the last days of consecutive calendar years) into fixed time intervals 

lj = (Uj-1, Uj] forj==l,2, . . . . One starts out with a cohort of N individuals, and observes the 

numbers 

~rj = number of failures due to Rr in lj. (66) 

Wc make the assumption that the net lives Xl, X2, . . . , Xk , corresponding to the k risks, are 

independent random variables, and wish to estimate the “net probabilities” 

9rj‘P(X, ‘Uj IXr >Uj_~). 

(67) 

Let A,(t) denote the hazard rate for the net life X,, r = 1, 2, . . . , k and A(t) the hazard rate for the 

iiCtUd life IV = min (X1 , . . . , Xh ). Since the X. are independent, we have according to equation (19) 

k 

A(t) ==. Ar(t). (68) 

39 
The probability that in the presence of all competing risks an individual, alive at zfj. 1, will survive 

to zq can be written 

Pj ‘P(l’V> Uj I U’> Uj_~ ) ‘P(Jt’> Uj)/P(J’V> Uj_~ ) 

=exp[-[:l~(s)ds]=l-qj (69) 

The “crude” probability that an individual, alive at ~j_l , will fail of risk R, during Ij is 

and can be written 

(70) (&j . J“’“exp[-~_,~(s)ds]~(t)dI 

Uj-1


We now make, in addition to the assumption of independent net lives, the following assumption of 

proportional hazard rates (PHR): For each interval ~ there are constants Clj, Czj, “ “ ., Crj, .00, Ck j 

such that 

h,(t)


—	 . qj for t Glj. (71) A(t) 

Under this assumption, the crude probabilities of equation (70) can be expressed as 

Qj=’rj/:l’’p[-~_, ~(s)ds]~(t)dt 

=cj{l-exp[-(:,~(s)ds]) 

hence 

Q.j = C~jqj forr=l, . . ..k.j=l,2, ..., (72) ‘ 

where qj is defined by equation (69) as the probability that an individual, alive at uj- 1, will fail in Ij. 

From equations (71) and (72) one has 

‘r(t) Qrj —= —


qj = “j for t in lj. (73) X(t) 
Summing over r and using equation (68) one sees that 

+ Qrj = qj. (74) 

Equation (74) follows directly from the definition of the Qj and does not require the pHR assumption. Using the PHR assumption, the net probability qrj defined by equation (67) Cm be written 

q~j = 1 

-exp[-[:,~+)ds] 

=1-exp [ -c,J [;l ~(s)ds] 

and by equation (72) 

qrj = 1 ‘P~rj’qj, (75) 

which expresses the net probability qrj in terms of the probabilities ~j and qj = 1- ~j$ and the crude 

probability Qj. 

3.3.2. Observable Random Variables: Maximum Likelihood 

Estimates of Crude and Net Probabilities 

We assume that, as is most often the case in life-table-type studies, the records contain only the 

numbers 

/j = number of those alive immediately after ~j_ ~ 

forj=l, 2,. ..; r=l, k.. ,k. (76) 

drj = number of those failing in Ij due to R, 

Clearly, 

(77) 

Given that an individual survived ~j_l, the probabilities that it will fail due to R 1, . . . . Rr, . . . . Rk 

are, respectively, Qlj9. . . 3Qrj, . . . ~~ j, and the probability that it will survive beyond uj is Pj. Therefore, the conditional joint probability y distribution of dlj, . . . . drj, . . . . dkj, lj+1, given Zj, is the multinominal expression 

lj ! 

dlj ! 0- ‘dkj!L’j+l! 

Q;? WQkjpjdk j lj+I . (78) 

41

When for an interval Ij the observed frequencies $, dlj, . . . . dkj and lj+ 1 are available from observation, then it is well known that the maximum likelihood estimates for the probabilities appearing in 

expression (78 ) are the relative frequencies 

O.rj=~forr=l,2, . . ..k 

(79) 

and expressions for variances and covariances can be explicitly obtained (p. 25310). 

Thus far, the arguments have been straightforward and fairly simple, but they yielded only the 

estimates (79) of the crude probabilities. To estimate the net probabilities qrj, Chiang recommends 

that the values (79) can be substituted in equation (75), so that one obtains the estimates 

/ lj+ 1 drj/dj 

= by () (80) 

j 

where dj = ~ - $+1 = total number of failures in Ij. Since in most practical situations one is mainly 

interested in the net probabilities, the question arises whether the estimates (80) have the usual 

desirable properties and in particular whether they are consistent. 

3.3.3. Conditions for Consistency of Chiang’s Estimator 

of the Net Survival Probabilities 

As in section 3.2.2, we consider the case of k = 2 competing risks, call the corresponding lives T = 

time to failure and L = time to loss from observation, assume that T and L are independent random 

variables, and denote their survival probabilities by ~~ (t) and FL (t) and their probability densities by 

fT(t) and fL(t).Again, without loss of generality, we fix our attention on an interval 15 = (O, $] , &>0, 

and for given initial size N of the cohort use the estimator (80). 

Using the events defined by equations (55) and the random variables in (56) we rewrite (80) in the 

form 

~T~=’-(N-:-M)D’(D+M) 

(81) 

Again, as in section 3.2.2 we conclude that with N + ~ each of the quotients in this expression 

tends in probability to the corresponding probability in (57), hence ~~,E tends in probability to 

9* =1 - [1 -P(8~)-P@2) -P(/.L)] [P(t7~)+P(8~)] /[P(6~)+P(6Jl.P(#)] (82) 

42 
As can be seen from figure 6, the sets 6 = 61 U62 and P correspond to the symmetric, observable ~ 

events “failure before $ and before loss” and “loss before .$and before failure,” and their probabilities 

P(6), P(IJ) satisfy the equation 

P(a) +P(/.l) +P(L > g? T> .$) =P(a) +P(p) +F~(g)F~(&) = 1, 

so that equation (82) can be written 

q* = 1- [FL ($)~~ ($)] 1P(8)]/[1 ‘FL (@~(E)] (83) 

Therefore, for ~T, ~ to be consistent, it is necessary and sufficient that 

[~(~)]/[1 ‘FL(@T(~)] = ~=(~)

[~~ (~) “ ~=(~)] 

which reduces to 

[FL (g)]~(~) = [FT(g)]~(~L (84) 

If the PHR assumption (71) is satisfied, which in our case of two independent competing risks means 

h~(.s) = CA(S), XL (s) = (1 - c))+), 

then 

‘T(u) =exp[-crA(s)dsl’

‘L(u) =exp[-(’+(s)d’l 

and 

P(/..l)= -

1

t 

~T (U) d~’ (U) 

~=() 

=(1-c) { l-exp [J’wds]) -

E 

P(6) = - ~’ (U) d~T (U) 

f 

“i-exi-lwdsl} 

and Ime verifies that condition (84) is satisfied. We conclude that under the PHR assumption, estimate 

(80) proposed by Chiang k consistent. 

43 
3.3.4. Examples of Competing Risks With Proportional Hazard Rates 

We have seen that for data of the life-table type, under the assumptions of independence and 

proportional hazard rates, Chiang’s estimate (80) is consistent. Leaving aside the question of how 

realistic these assumptions may be in specific practical situations, one may still wish to see nontrivial 

examples of independent competing risks which satisfy the PHR assumption. l?o~owing H. A. David in 

1970,ZT we present a further discussion of the PHR property (71), which leads to the construction 

of such examples. 

Writing equation (71) as 

h(t) = %j~(~) fOr , Uj.1 <tGUj (85)

and integrating from uj- 1 to t(< Uj), one obtains according to equation (6) the relationships 

Fjr(t) F~ (t) 

10g ~w (uj_l ) .F~,(Uj-~ ) = % 10!4

where W = min (Xl, . . . . Xk ). This shows that 

[1Fq.(t) l/crj 

(86) 

‘Xr (Uj-l) 

is independent of r for fixed j and u;_l < t < u;. 

Consider now the more restrictive special c~se of the PHR assumption (85) 

Ir(t)= Crx(t) for O<t<~; r=l, . . ..k. (87) 

Then expression (86) can be written 

[Fxr(t)]1/”= [Fxl (t)]1/” for O<t<-; r=2, . . ..k 

or 

[F&(t)]“/c’=Fx,(t) for O<t<W; r=2, . . ..k. 

Ifthe cl,..., Ck are so chosen that 

~r=: forr=2, . . ..k 

are positive integers, then one has 

~xl (~)=lxx,(t)] “ for O<t<~; r=2, . . ..k. (88) 

hence Xl can be represented, for each r = 2, . . . , k, as the minimum of p, independent identically 

distributed random variables with survival function ~xr (t). 

44

I’I ) ld~tain a nice example of net lives that satisfy relation (88), David considers among others the 

ftimily of Weibull distributions 

P(x> t;ck!, p) = F(t; 0!, f?) = exp [- w?] (89) 

Ivhich lIW the well known property that the minimum of independent identically distributed random 

vwiablcs of this family has again a Weibull distribution (his discussion goes further, to consider all 

three known classes of extreme-value distributions of the minimum). Choosing c 1, C2, . . . . Ch so that 

i’t)rr=2, ..o, k the ratios p, are positive integers, and assuming for Xl a Weibull distribution (89), 

OIICobtains an example of independent net lives, each with a WeibuU distribution, which satisfy condition (87), i.e., have proportional hazard rates on the entire time axis. For suggestions of further 

cxw-nplcs the reader may wish to consult David’s 1970 paper. Z7 

4. ESTIMATION FOR PARAMETRIC MODELS 

4.1. THE LIKELIHOOD FUNCTION: GENERAL CASE 

WC shall now assume that the joint probability distribution of the net lives belongs to a specified 

~~~tmnlctric family. Let the joint probability density of these net lives Xl, . . . . Xk be 

~~(~; ~)=~(xl,..., xh; o,%).,%). (90) 

I( ~rfixed j, let 

Tj = P(min Xi = Xj) forj=l, . . ..k. (91) 

i=l ,...,k 

bc the probability that failure due to risk Rj is observed. Writing in equation (11) fx(~;Q)instead of 

f(.Y*,. . . , XIJ), one obtains 

;In[l Ihc pn~bobility density of the “crude” life Yjj as defined in section 2.4, can be written 

. 

.fI”j(,)’j;fl)= $ J“ Jf(.x~,.. ., Xj-1 , I“j> .Xj+. 1> . . ., Xk; el, . . .t3m)~ dxi. (93) 0.­

>’j Yj i+j 

\VhcII H individuals are observed to failure, and nj of them fail due to Rj, j = 1, . . . . k, then the 

jt}int probability density of the observed (crude) life lengths lj, r = 1, . . . ,nj, conditional on the ~, 

is 

fiqr:rim... -f(x,,..., xj_~3yjr, xj+~, . . . ,Xk;e)m d~i. 

j= I ~= 1 H Yjr ?jr
 i#j

45

Since the nj have a multinominal probability distribution 

g(?q,.... ?z~)= 721 n (/:1~.1J)(@?) 

the likelihood function of the observed (crude) life lengths is 

f(x~ , ‘*O>xj-l~yjr>xj+lj...txk ;h. . .,d~)~ d~i. (94) 

i#j 

This expression, given in 1971 by Moeschberger and David, 28 does not require the assumption that the 

net lives are independent, but only the assumption of a specified parametric form of the joint probabilityy density (equation 9 O) of the net lives. Whenever such a parametric family is specified and the 

values of the crude lives yjr are available, the likelihood function 194) may be used to obtain maximum likelihood estimates of the parameters 01, . . . Om by the usual procedure of computing the 

partial derivatives 

ilL aln L 

~ ‘r M. ‘ () 

equating them to zero, and solving the resulting system of m equations for 01, . . . , 19m. However, 

even if the density (90) is of a reasonably simple form, one would expect in carrying out these steps 

to encounter difficulties which, at best, maybe overcome by using computers. 

4.2. THE CASE OF INDEPENDENT NET LIVES 

If one assumes that the net lives Xl, . . . . xk are independent, with probability densities fi(xj ;~) 

and survival functions ~j (~j; @ ), then equation (93) becomes 

fi(Yj; Q) fYj(Yi; ‘) = flj~(yj; 0, r+WY1; 42) (95) 

1=1 

and the likelihood function (94) takes the form 

1 

~1 h 

‘) 

~~[ 

1=1k L = ~“! (‘/j=lj=, r=, 

~(yjr; f?)]/[~j(yjr ;@)]m ~(Yfi;d)” (96) 

This expression is much less complicated than (94). Moreover, for simple parametric models, it 

lends itself to manageable treatment for obtaining maximum likelihood estimates of the parameters 

@=(el, . . ..em). 

46 
4.3.	 COMMENTS ON PARAMETRIC FAMILIES OF 

MULTIVARIATE LIFE DISTRIBUTIONS 

The maximum likelihood estimation procedure under the assumption of independent net lives, as 

just described, has been actually used. The reader may find technical details as well as completely 

worked out numerical examples in the 1971 paper by Herman and Pate112g or in Moeschberger and 

llavid28 for the cases when the net lives are assumed to have independent exponential or WeibuI1 

distributions, Moeschberger and David consider also the multivariate exponential distribution introduced in 1967 by Marshall and Olkin~O which deals with dependent net lives, and suggest a way of 

proceeding in this cased 

The NIarshall-Olkin multivariate exponential distribution is one of the few multivariate life distributions admitting dependence, A multivariate Weibull distribution has been proposed by several 

authors, and a rather general method for obtaining new multivariate life distributions was outlined by 

Lce and Thompson. 31 To this writer’s knowledge, little work has been done on the use of such distributions. It would seem desirable to construct more parametric families of multivariate life distributimls, justify their use either b y deriving them from some plausible assumptions, or by at least showing 

that they agree reasonably well with empirical data, and to develop in detail the maximum likelihood 

estimation techniques for their parameters. 

4.4. CONCOMITANT VARIABLES 

In some practical situations, the available empirical data contain more information than just the 

observed crude lives (i.e., actual lives and the risk recorded at failure), and it appears desirable to use 

this additional information. An important example is the situation when values of “concomitant 

variables” have been observed. Considerable advances have been made recently in developing techniques for using concomitant variables and, while a systematic presentation of this area of research 

~vould require a separate monograph, it may be useful to give here an example of the concepts involved and mention some of the pertinent papers. 

In 1965 Feigl and Zelen32 considered the study of data on leukemia patients which, for each 

patient, contain the observed time from diagnosis to death (life length) and the white blood cell count 

at di+qnosis (concomitant variable). They assumed that the probability distribution of life length Tat 

diasnosis depends on the white cell count x in such a“way that the probability density of T is 

exp(- M) fort20 

F(t) = o 

{ fort<O 

where 

E(T) = ~ =a+bx. 

The problem was to estimate the parameters a, b which determine the dependence of T on x; knowing 

these parameters would clearly help in estimating the life expectancy of a patient for whom a value of 

‘The Marshall-Olkin distribution is also discussed in Langberg, Proschan, and Quinzi’s 1978 paper.20 

47 
x was observed. Feigl and Zelen derived the maximum likelihood procedure for this problem, computed the asymptotic variance-covariance matrix, and applied the theory to numerical data. In 1966 

Zippin and Armitage33 extended the Feigl-Zelen model to the case in which not all patients had died 

at the time when the study was concluded, thus introducing a competing risk. 

The important paper of 1972 by COX34 considered a very general model dealing with both censoring (competing risks) and concomitant variables. This paper was followed by a number of studies, sL[ch 

as Kalbfleisch and Prentice35 in 1973, Lagakos36 in 1976, and Holford37 also in 1976. 

4.5. ESTIMATION OF THE RATIO OF HAZARD FUNCTIONS 

As an example of a problem which requires estimating a parameter from data with compctin~ 

risks, we refer to a 1975 paper by Crowley24 in which he discussed the following situation. 

Survival data were obtained independently for two different populations, each affected by conlpeting risks. There is reason to assume that the two net life distributions are of the same type and that 

the difference between them is due only to the fact that their hazard rates differ by a constant i’tictor. 

How can one estimate that factor? 

Let Xl denote the net life in the first population, and T1 the competing life, so that only the 

actual life 

WI = min (X1, T1 ) 

and an indicator variable 

1 when Xl < T1 

al= ~ 

{ when Xl > I“l 

can be observed, and let X2, T2, W2, 6 * be similarly defined for the second population. IVC shall say 

that IVl is an uncensored value when ~ 1 = 1, and a censored value when 61 = O, and use similar terms 

for W*. 

Available are the samples of (WI, 61) 

(wlj,81j), ~=1,2, . . ..nl (97) 

and of (W2, 62) 

(w*~,6*~), k=l,2,. ... n*. (98) 

We assume that (1) Xl, T1, X2, and T2 are independent and (2) the hazard rates of .Y1 and .Yq diflcr 

only by a constant factor, i.e., 

Assumption (99) is equivalent to the relationship between survival functions 

~x2 (~)= [Fxl (t)]0. (100) 

If these assumptions are satisfied, then O >1 implies that X2 is stochastically smaller than .Y1. 

48 
Crowley considered several possibilities of presenting the data (individual sample values are available, or data are grouped in time intervals, etc.) and discussed several estimators. We shall describe one 

of thcm, for the case when the exact individual values (97) and (98) are available. Let 

nll = ~ ~lj = number of uncensored values of WI 

j= 1 

nl 

7L12 G	 ~ (1 - ~lj) = number of censored values of W1 

j=l 

so that nl 1 1-n Iz = nl and, similarly, 

n2 

72*, = 5 82, ?’222= ~ (1 - tiz~), 

k=l k=l


with n~l + 7222= n2. We denote the uncensored values of W1 by Wl, 1, Wl, z, . . . , Wl,nil; the censored VdUCS of W1 by 1$’1,~11+1, ~l,nll+zj . . . j WI,. II+ .I ~ ; the uncensored values of W2 by 

w~, 1, W2,2, . . . . w@21 ; and the censored values of Wz by Wz,~~1+1, . . ., ‘2, n21+n22. 

The likelihood function (96) becomes now 

.nI ! nll nll+n12 

7211! 7212!IT fxI (wlf-)FTI(Wlr) IT f~~(wlr)~xl (WI.) 

).=1 r=nll+l 

x 

Tql !7222! ~=1 MW2SR-2(W2S) ‘2H2 fT2(w2s)~x2(w2s). (101) ‘2! 1%


s=n21+l 

To carry out the procedure leading to a solution of the likelihood equation for 0, one must assume 

a specific functional expression for ~xl (t). Following Crowley, we illustrate this procedure by choosing 

Fxl (t) = exp (- At) (102) 

hence 

~X2 (t) = exp (- I!IM). (103) 

Writing expression (101 ) for these specific survival functions and differentiating log L with respect 

to 1 and to 0, one arrives at the likelihood equations 

7221 n2 

—e -~-X ~=1 w2*”o, 

49

and the estimators for h and 0 obtained by solving these equations are 

i=n~~ /’ & WI,. 

/ r=l 

nl 

6 = (?z21/n4~ WI, s W2* . (104)

,=1 / ~=1 

Estimate (104) is consistent, since it can be written: 

and this tends in probability to 

P(x,< T,) E(wl) = J

-FT2(t) 

o


P(X1< 1-~)E(W2)
 - &l (~)
d~xl (t) o- t d[~x2 (t)~T2 (t)] 

fo J 

Jm ~~z (t) d[exp (- ON)] o t d[exp (- ~t)~r-l (~)1 

o . J“ = . 

- ~T1(t)d[ap (- ~~)] o t d[exp (- 6~t)~T2 (t)] 

Jo J 

= e. 

This argument makes use of equations (102) and (103), i.e., of the assumption of exponential net 

lives. In general, consistency and asymptotic normality of the estimate obtained from the likelihood 

function (10 1) will follow from known properties of maximum likelihood estimates. 

5. TESTS OF HYPOTHESES 

5.1. FORMULATION OF A PROBLEM 

In clinical studies dealing with the com~arison . of two treatments one often wishes to test the

hypothesis that these treatments are equally effective, i.e., that lives (times from beginning of treatment to the onset of some condition or to death) have the same probability distribution under both 

treatments, against some alternative indicating that one of the treatments is superior. When there are 

50 
no competing risks, a number of statistical tests are readily 

Wilcoxon-Mann-Whitney test. The case when patients under 

peting risk, such as loss from observation (censoring on the 

Gchan.38 Similar techniques had been proposed in 1962 in 

1965 Gehan40 extended the test procedure to doubly censored 

available for this purpose, such as the 

each treatment are exposed to a comright), was first considered in 1965 by 

an unpublished thesis by Gilbert.39 In 

data. A further study of properties of 

the Gehan statistic was made in 1967 by Efron41 who proposed additional test statistics. Breslow42 

studied (in 1970) the more general problem of comparing k >2 treatments and testing the hypothesis 

that they are equally effective, when each of the available k samples is subject to competing risks and 

the competing net lives have possibly different probability y distributions. 

5.2. GEHAN’S STATISTIC 

In presenting the contents of Gehan,38 we shall follow some of the arguments used by Efron.4 1 

L:t .Y and U be independent lives corresponding to competing risks, with survival functions ~X (t) 

and Fu (t); and let Y and V be another pair of independent lives corresponding to competing risks, 

with survival functions ~Y (t), ~v (t). To avoid complications due to the possibility of ties, we shall 

assume that all survival functions are continuous. Available are independent random samples 

Xl,xZ,....xm Ofx, hence with survival function ~x 

Ul, uz, . . ..umofu. hence with survival function ~u 

Yl, Y2, . . .. Ynof Y. hence with survival function ~y 

Vl, vz, . . ..vnofv. hence with survival function ~v. 

Since we deal with competing risks, we can observe only 

X; = min (Xi,U;) (105) 

md 

(106) 

fori= 1, 2, . . . . m, and similarly 

YJ*= min (Yj, Vj) (107) 

and 

1 if Ylf = Yj 

~j = forj=l,2, . . ..n. (108) 

{ O if YJ~= Vj 

51

Clearly, the random variables X;, Y;, are mutually independent, and their survival functions are 

F~; (s) = F* (S)FU (s) fori=l,2, . . ..m 

(109) 

Fyf (s)= Fy (S)FV (.$) forj=l,2, . . ..n. 

while the & and ej are mutually independent Bernoulli variables with probabilities 

P(8~ = 1) ‘P(Xj < Ui) 

. 

=. ~U (s) d~x (s) 

J o 

(110) 

.P(Cj = 1) ‘P(Yj’< Vj) 

. 

=- ~v(s) d~y (s). 

J o 

In general, X; and bi are not independent, and neither are y;, cj. . . 

‘The following notation used by Efron simplifies writing some of the arguments: For two independent random variables S, T, we shall write 

. 

P(F. >F~)=- ~S (%) d~T (Z) 

J -w 

=.P(S> T). (111) 

For every observed four values X;, bi, Yj~, ~j, we define a scoring function Q(X:, ~i~ Y;, ~j) b 

assigning to it the value 1 when the four values establish unequivocally that Xi > Yj, the value O ~vh~m 

the four values establish unequivocally that Xi < Yj, and the value 1/2 in all other cases. More formally 

1 when ~j = 1 and X; > Y,? 

Q(J? , ~i, ~ , ~j) = O when ~i = 1 and x; < y]; (112) 

{ 1/2 otherwise. 

One verifies that this definition of the scoring function is equivalent with 

1 when Yj < min (Xi, Ui, Vj) 

Q(x~, ~i, y;, ~j) = 0 

when Xi <rein (Yj, ~j, ~i) (113) 

1/2 when Ui < min (-xi, yj, ~j) 

{ 1/2 when Vj < min (Xi, Yj, ~i ). 

52 

I 
� 

Ckhan’s test statistic is defined in terms of this scoring function by the expression 

(114) 

Using equation (11 3) and the notation introduced in equation (111), we have 

1 ——— 

~[Q(x; , %, y;, q) = 1] ‘~(~y ~~x~~~~) 

P[Q(.Y; , iji, Y;, q) = O] ‘P(FX < ~Y~V~U) 

P[Q(X; , 6;, Yj~, ej) = 1/2] =P(~U <~x~y~v) +P(~v <FXFYFU). 

IIcmcc the expectation of W can be written 

E(w) =F@y <F.jyF~F~) + (1/2) “ [F’(FU <F-jy FyFv) +P(FV <Fx FyF~)]. (115) 

Under the null hypothesis (H. ) 

Fx(t) = Fy (t) for all t> O, 

(111(2has 

P(Fy < FxF~Fv) = P(FX <Fy F~Tv) 

;Ind L’qUNti[ln (1 15) becomes 

E(W) = (1/2) [P(Fx <FyF~F~) +P(Fy <FxF~Fv) 

+ P(FU <FxFyF~) +P(FJ7 <Fx7yF~)] = 1/2. 

II sh( mld bC notecl that equation (1 16) is true for any two censoring distributions Fu, 

liy a similar argument one obtains for the variance of W under the null hypothesis 

sit~n 

\TarHO(~!~)= ~2mn L [3P(Fj <FuF~)+(n - l)P(~j <&~,) 

+ (m - l)P(F} < F;FV)] . 

(116) 

FY ..


(H. ) the expres­


(117) 

‘Illis ~,xpression for the variance of W was obtained in 1962 by Gilbert39 and is quoted in 1967 in 

I{[n)n’s ~UpLT.Al 

“Gdmn obtained equation ( 116) only under the additional assumption that the censoring distributions ~u(t) and 

~l. ([) arc the same. In 1967 Mantel 43 ob~emed ~at this ~sumption is not needed and he cormmmicmd *is ‘act 

wdicr to Ekm for inclusion in his 1967 report.4* 

53 
To test Ho, one would compute W from the available data and compare it with the expected value 

1/2. To judge the significance of the difference W - 1/2, one has to keep in mind two facts pointed 

out by Efron: 

The statistic W is not nonparametric, since, even under Ii.: ~X = ~Y, its variance (117) depends 

on the relationship between ~X, ~u, and ~v.€

W is asymptotically nonparametric in the following sense: when m + ~, n + W, so that€

[m/(m + 1)] + X where O < h <1, then under Ho the probability distribution function of 

+== [w-(w)] tendsto~{oj Ww[wv+ +y+ “;]} (118)

where€

\– –-, 

‘22 =P(FfiFo >F:), 

that is, the random variable <w (W - 1/2) is asymptotically normal with expectation O and variante 

(1/12) x 

Furthermore, Efron suggests consistent estimates for a? and o ~, but does not elaborate on the way to€

compute them.€

Under the additional assumption ~u = ~v, i.e., of the same probability distribution for the two 

censoring variables, Gehan38 proposed in 1965 the use of W for a conditional test of Ho of a conlbinatorial nature. Mante148 offers a simple combinatorial formulation of Gehan’s conditional test statistic, 

and discusses its properties under the assumption ~u = ~v and also under less restrictive assumptions. 

5.3. TESTING FOR INDEPENDENCE WHEN DATA ARE CENSORED 

An important problem of hypothesis testing, treated extensively in 1974 by Brown, Hollander, and 

Korwar,44 may be stated as follows. 

Consider the random variables X, Y with a bivariate life distribution, i.e., such that 

P(.x>o, y>o)=l. 

If a sample€

(x~, Y~), (x*, Y2),. ... (xn, Yn) (120) 

is available, then there are a number of well known tests of the hypothesis that X and 1’ are indepen-€

dent or, more generally, uncorrelated. A new problem arises when to one or both of the two lives l’, 1’€

corresponds a competing life (censoring variable), U in competition with X, and V in competition with€

54 
1’, so that one can only observe 

where .Y~, ~i, Y?, q are defined in equations (105)-(108). 

Brown, Hollander, and Korwar44 considered data from a heart transplant program, and explored 

hypotheses such as that sex (X) and survival time after transplant (Y) are independent, where Y is 

censored by 

V = time to closing date of the research program. 

To test the hypothesis of independence of X and Y when only Y is censored (a generalization to 

the case of X and Y both censored is straightforward), the authors modified Kendall’s rank correlation statistic which for a complete sample (120) is defined as 

S = & ~ aijbij (122) 

i=l j=l 

1 if Xi > Xj 

aij = o if Xi = Xj (123) 

{ -1 if Xi < Xj 

and 

1 if Yi > Yj 

bij = () if Yi = Yj (124) 

{ -1 if Yi < Yj. 

IVflen 1’ is censored, i.e., the sample (121) is of the form 

(xl, y~,~1),. . ..(xi. yf, ~i), (xn,y~,~n),~n), (125) 

then the aij are still defined by equations (123), but equations (124) are modified to read 

1 if Y~>Yl~and~j=l, orY~ =Y~andei=O 

b~j= -1 if Y; < YJ~and~i= l,or Y; = Yj* andej=O (126) 

{ o in all other cases 

ancl the test statistic of equation (122) is replaced by 

(127) 

55 
I

If X and Y are independent, then S‘ has a conditional probability y distribution of a purely ccmlbinatorial nature, hence is nonparametric and can be used to test the hypothesis of independence n f .S and 

Y, provided the following additional assumption is satisfied: 

Assumption A: When X and Y are independent, then X and (Y* ,e) are independent. 

If assumption A hoIds, critical values of S’ can be calculated by a combinatorial arLgument when n 

is small. For large n, again under assumption A and when the hypothesis is true, the statistic S‘ is ccmditionally asymptotically normal with conditional expectation E(S’) = O and a conditional variance 

that is given by Brown, Hollander, and Korwar44 and is a function of the a~j and bjj. A further sttitistic 

for testing independence of X, Y when one or both variables are censored, based on Kaplan-Mcicr 

extimates of the survival functions for the net lives X and Y, is proposed in their paper. The reader is 

referred to the original paper for a description of this test, as well as of still another “pscuducunditional” test. 

5.4. LARGE-SAMPLE TESTS 

In preceding sections we gave examples of tests of hypotheses about net lives which arc obscrvcci 

in the presence of competing risks. These examples dealt with tests designed for specific problems, and 

the sampling distributions of the test statistics could in some cases be computed exactly for snmll 

sample sizes, and in all cases can be obtained for large samples from the asymptotic normality of’ the 

test statistics. Clearly, additional large sample tests can be obtained in those competing risk situations 

for which estimates of parameters are available and are known to be asymptotically normal. 

000


56 
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57 
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26Hoe1, J). G.: A representation of mortality data by competing risks. Biometrics 28:475-488, 1972. 

27 David, H. A-: On Chiang’s proportionality assumption in the theory of competing risks. Biometrics 26:336-339, 

1970. 

28Moeschberger, M. L., and David, H. A.: Life tests under competing causes of failure and the theory of competing 

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‘g Herman, R. J., and Patell, R. K. N.: Maximum likelihood estimation for multi-risk model. Technomctrics 13:385-

396, 1971. 

3 OM=sh~l, A. w., and C)lkin, 1.: A multivariate exponential distribution.
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31 Lee, L., and Thompson, W. A., Jr.: Results on failure time and pattern for the series system, in Frank Proschan and 

R. J. Serfling, eds. Reliability an$ Biometry. Philadelphia. Society for Industrial and Applied Mathematics, 1974. 

pp. 291-302. 

32 Feigl, P., and Zelen, M.: Estimation of exponential survival probabilities with concomitant information. Biometrics 

21:826-838, 1965. 

33 Zippin, C., and Armitage, P.: Use of concomitant variables and incomplete survival information in the estimation 

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35 KAbfleisch, F. 0., and prentice, R. L.: Marginal likelihood based on Cox’s regression and life model. Biometrika 

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36 Lagakos, S. W.: A stochastic model for censored survival data in the presence of an auxiliary variable. Biometric~ 

32:551-559, 1976. 

37Holford, T. R.: Life tables with concor,,it~t information. Biometrics 32:587-597, 1976. 

38 Gehm, E. A.: A generalized Wilcoxon test for comparing arbitrarily single-censored samples. Biometrika 52:203-

223, 1965. 

39Gilbe~t,J. p.: Random Censorship. Unpublished doctoral thesis, University of Chicago, 1962. 

40 Gehm, E. A.: A gener~ized two-5
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41 Efron, B.: The two-sample problem with censored data. Proc. Fifth Berkeley Symp. Math. Stat. Probab. 4:831 ~ 

853, 1967. 

4Z Breslow, N.: A generalized Kruskal-Wallis test for comparing K samples subject to unequal patterns of censorship. 

Biometrika 57:579-594, 1970. 

AsMantel, N.: Ranking procedures for arbitrarily restricted observation. Biometrics 23:65-78, 1967. 

44 Bro-, B. w., Hollander, M., ad Kowar, R. M.: Nonparametric tests for censored data, with application to heart 

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trial and Applied Mathematics, 1974. pp. 327-354. 

58 *U.S. WVSRMENT PRINTING O??ICE : 1979 O-2 Bl-259/l 
VITAL AND HEALTH STATISTICS Series 

Series 1.	 Programs and Collection Procedures. –Reports which describe the general programs of the National 

Center for Health Statistics and its offices and divisions and data collection methods used and include 

definitions and other material necessary for understanding the data. 

Series 2.	 Data Evacuation and Methods Research. –Studies of new statistical methodology including experimental tests of new survey methods, studies of vital statistics collection methods, new analytical 

techniques, objective evaluations of reliability of collected data, and contributions to statistical theory. 

Series 3.	 Analytical Studies. –Reports presenting analytical or interpretive studies based on vital and health 

statistics, carrying the analysis further than the expository types of reports in the other series. 

Series 4.	 Documents and Committee Reports. –Finaf reports of major committees concerned with vital and 

heafth statistics and documents such as recommended model vital registration laws and revised birth 

and death certi~lcates. 

Series 10.	 Data Fro m the Health Interview Survey. –Statistics on illness, accidental injuries, disability, use of 

hospital, medical, dental, and other services, and other health-related topics, afl based on data collected 

in a continuing national household interview survey. 

Series 11.	 Data From the Health Examination Survey and the Health and Nutrition Examination Survey. –Data 

from direct examination, testing, and measurement of national samples of the civilian noninstitutionalized population provide the basis for two types of reports: (1) estimates of the medically defined 

prevalence of specific diseases in the United States and the distributions of the population with respect 

to physical, physiological, and psychological characteristics and (2) analysis of relationships among the 

various measurements without reference to an explicit finite universe of persons. 

Series 12.	 Data From the Institutionalized Population Surveys. –Discontinued effective 1975. Future reports from 

these surveys will be in Series 13. 

Series 13.	 Data on Health Resources Utilization. –Statistics on the utilization of health manpower and facilities 

providing long-term care, ambulatory care, hospital care, and family planning services. 

Series 14.	 Data on Health Resources: Manpower and Facilities. –Statistics on the numbers, geographic distribution, and characteristics of health resources including physicians, dentists, nurses, other health 

occupations, hospitafs, nursing homes, and outpatient facilities. 

Series 20.	 Data on Mortality. –Various statistics on mortafity other than as included in regular annual or monthly 

reports. Speciaf analyses by cause of death, age, and other demographic variables; geographic and time 

series analyses; and statistics on characteristics of deaths not available from the vital records based on 

sample surveys of those records. 

Series 21.	 Data on Natality, Marriage, and Divorce. –Various statistics on natafity, marriage, and divorce other 

than as included in regular annual or monthly reports. Special analyses by demographic variables; 

geographic and time series analyses; studies of fertility; and statistics on characteristics of births not 

available from the vital records based on sample surveys of those records. 

Series 22.	 Data From the National Mortality and Natality Survey s.–Discontinued effective 1975. Future reports 

from these sample surveys based on vital records will be included in Series 20 and 21, respectively. 

Series 23.	 Data From the National Survey of Family Growth. –Statistics on fertility, family formation and dissolution, family planning, and related matemaf and infant health topics derived from a bienniaf survey 

of a nationwide probability sample of ever-m~ied women 15-44 years of age. 

For a list of titles of reports published in these series, write to: Scientific and Technical Information Branch 

National Center for Health Statistics


Public Health Service


Hyattsville, Md. 20782