Biometrika 1982 MILLER 521 31

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Biometr i lca (1982) , 6 9 , 3 , pp. 5 2 1 - 3 1 5 2 1Printed in Greed Br i ta in

Regression with censored dataB Y R U P E R T M I L L E R AND J E R R Y H A L P E R NDivision of Bio s ta t is t ics , Stan ford Univers i ty Med ica l Center, California , U.S.A.

SUMMARYThere are four regression techniques currently available for use with censored datawhich do not assume particular parametric families of survival distributions. They aredue to (i) Cox (19 72), (ii) Miller (19 76), (iii) B uckley & Jam es (19 79), and (iv) Ko ul,Susarla & Van Ryzin (1981). These four methods are described, and their performancescompared on the u pdated Stanford heart tra nsp lant da ta. Conclusions on the usefulnessof the four procedures are drawn.

So me k e y words: Censored data; Linear model; Proportional hazards model; Regression.

1. INTRODUCTIONWithin the past decade a variety of techniques have been proposed for handlingregression problems in which the d ependen t variable is subject to censoring. Some ofthese techniques rely upon normal theory, but others make virtually no assumptionabout the underlying distribution. The purpose of this paper is to compare four methodsof the latter type.One of these techniques (Cox, 1972) bases its approach on the proportional hazardsmodel. If F(t; x) and/(


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522 R. M I L L E R A N D J. H A L P E R N2. ESTIMATORS

2 1 . GeneralitiesLet Tlt...,Tn be the survival times for n patients. These may not all be observable dueto censoring times C\,...,Cn for each of the patients. The observable quantities are

Y^T^C,, Si = l(Ti < C), a:, = (*,...,*),where 'A ' denotes the minimum, / is the indicator function, and x{ is the transposedvector of independent variables associated with patient i. The random censoring modeladopted in this paper assumes that, given the xh the Tt and Ct are independentlydistributed according to the continuous distributions F(t\ xt) and 0(t; xt), respectively.In the above framework the survival times Tt are censored on the right by thecensoring times Ct. They could just as well be all left censored with Y( = Tt v C( and


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Regression wi th censore d da ta 5 2 3The method adopted in this paper is to estimate X0(t ) by Breslow's (1974) estimator

- l

for yuy-X) < t < yuW, where j / u ( 1 ) < yu(2 ) < are t he ordered dis tin ct uncensored ob-servations and d^i) is the number of deaths at yuW, and then to substitute 20(t ) and $ into(3). This estimator S(t; x) , which was proposed by C. Link in a 19 79 Stanford Universitytechnical report an d T siatis (1 9 8 1) , differs slightly from the K aplan & Meier (19 58 )product-limit estimator if $ = 0.

2-3. Mil ler 's estimatorMiller (1976) suggested minimizing the sum of squares

n \e 2dP{e;a,b) (4)with respect to a and the vector b , where fl(e; a , b ) is the product-limit estimator based on8h e t = e t(a,b) = y t a x tb for i = 1 , ...,n. Specifically,

l-f(e;a,b)= f l ( l - ^ o K Awhere e(1 ) < e(2 ) < ... are the ordered distinct values of e h n ( i) is the number at risk ate(i) , d (t ) is the number dying at e(l) , and S (i ) = 1 if d (i) > 0, = 0 otherwise. Expression (4)is a generalization of the usual sum of squares E (y t a xtb)2 for uncensored data.It is difficult to locate the infimum of (4) because it is a discontinuous function of b.Therefore, Miller proposed using an iterative sequence to calculate the estimate of theregression coefficient vector fi :

(5 )where\ V = i ( 6 )

= d i a g { * ; , ( & ) } , y = (y lt . . . , y H ) T .The limit of the sequence fi k for k = 0 , 1 , . . . , is the estimate of/?. Because of discontinu-ities in the weights as functions of fi k, the sequence may become trapped in a loop. If thevalues in the loop are not far apart, an average value over the loop can be used for theestimate.The weight w t($ k) in (6) is the size of the jump assigned to e = e((0, $k) = y i xi^k bythe Kaplan & Meier estim ator applied to e, ...,e. Inclusion of an intercep t estimate a k inthe estimate of e , is unnecessary since the weights are invariant under location shifts inthe data. If the largest e f is censored, the mass 1 ^( + oo; (),/?*) is unassigned to any e f.The convention which seems to work best for these estimators is to normalize the weightsso that they sum to one in (6) and (8), but to assign the remaining mass 1 fi( + oo; 0, fi k)to the largest e f in (7).Only the uncensored y, received nonzero weights in (5). For this reason it makes senseto use as a starting value /?0 the ordinary unweighted least squares estimator applied tojust the uncensored data.

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524 R. MILLER AND J. HALPERNFor the limiting value fi the associated estimate of the intercept is

=If the variation in /?due to an estimate of ft rather than the true /? being used in the

computation of the weights is ignored, then an estimate of the covariance matrix forfts")}-1. (8)

For fi to be consistent it is necessary that the censoring distributions G(t ; x) satisfyG{t;x) =G(t-a-xp). (9)

This assumption requires that, as x changes, the censoring distributions shift along thesame line as the survival distributions.2-4. Buck l ey &James ' s e s t ima t o r

The Buckley & James (1979) estimator exploits the following linear relationship:. (10)

Buckley & James substitute an estimate for the conditional expectation 2?(T, |T, >yt)based on the Kaplan & Meier estimator into the variable yt = yt)and then solve the usual least squares normal equations iteratively.Specifically, if


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Regression luith censored data 525In estimating the variability Buckley & James recommend using

where nu is the number of uncensored observations. Their estimator for the covariancematrix of /? isa2u{(X-lu)TA(X-Xu)}-1, (12)

where X" isthe matrix with elements n~1Y,i5ixij and A=diag(


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5 2 6 R . M ILLER A N D J . H A L P E R N2 - 6 . Othe r e s t ima to r s

A com puter routine which minimizes the sum of squares for all the uncensoredobservations and only the censored ones falling above the estimated regression line hasbeen introduced by J. Friedman and W. Stuetzle in a 1981 Stanford University technicalreport.Fo r norm al linear models Schmee & Hahn (19 79) introduc ed an estima tor which isidentical in spirit to B uckley & Jam es's estimato r. Th e difference is th at Schmee &Hahn base the computation of the conditional expectation E(T\T > y) in (10) on theassumption of a normal d istribution. N elson & Hahn (1972, 1973) treated the case wherethere are replicate observations for each value of the independent variable. Linearcombinations of order statistics are used to estimate the normal param eters. A memiya(1973) applied maximum likelihood theory to the normal linear model with censoring ata fixed upper, or lower, level, type I censoring. A mem iya extended earlier work by Tobin(1958) and Glasser (1965). Heckman (1976) gave an a d hoc two-stage estimator in thenormal linear model.Regression estimators with censored data have also been proposed for the exponentialdistribution where X(t; x) = k(x) for t > 0. Zippin & A rmitage (1 96 6) treated the linearmodel k(x) = (a + xfi)~ l in the special casep = 1, and Mantel & Myers (1971) studied themultidimensional case. Glasser (1967) considered the proportional hazards modelX{x) = Aoe xP in the special case p = 1. All three papers are based on the maximumlikelihood approach.

3 . RESULTSThe Stanford heart transp lantatio n program was begun in October 19 67. By F ebruary1980, 184 patien ts had received he art trans plants. A few of these had multiple

transplants. Their survival times, uncensored or censored in February 1980, aredisplayed in Table 1 along with the ir ages at the tim e of the first transp lant. Thepatients' T5 mismatch scores which measure the degree of tissue incompatabilitybetween the initial donor and recipient hearts with respect to HLA antigens are alsoincluded in Table 1.The data on patients who were admitted to the program but did not receivetrans plan ts and the pretran splan t waiting times for patients who did receive tran splantswere not collected for this study. Because of this there is no analysis of whether or nottransplantation prolongs survival.In analysing the T5 mismatch scores on the data up to March 1974 Miller (1976)and Crowley & Hu (1977) made a distinction between deaths primarily due to rejectionof the donor heart by the recipient's immune system and nonrejection related deaths.This distinction is no longer maintained because histological information is not alwaysavailable and the judgement of rejection death is largely subjective.For 27 of the 184 transplant patients the T5 mismatch scores are missing because thetissue typing was never completed. For the regression analyses reported here just the 157patients with complete records were used. Of these 157 patients 55 were still alive, i.e.were censored as of February 1980, and 102 were deceased, i.e. were uncensored.Table 2 gives the estimated coefficients and their estimated standard deviations in themultiple regression of the base 10 logarithm of the time to death against age and T5mismatch score. For the Buckley-James and Miller estimators the initial estimates, /?0,the one-step estimates, /fj, and the final estimates, /?, are all given. Xo truncation was

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Regression wi th censore d da ta 5 2 7used in computing the estimates of Koul e t a l . and no standard deviations could becalculated for them.D ue to its nonsignificance the T 5 m ismatch score was deleted from the an alysis. Theestimated regression coefficients and standard deviations with age alone for the differentmethods are practically identical to those for age in Table 2.

Figure l(a) graphs the survival times against the ages of the patients at transplant.The dashed line is the Buckley-James estimated regression line. The solid line is theestimate of the median survival time from Cox's method. That is, at each x the value of tfor which (t; x) equals \ is plotted. The median estimator was selected because it ismuch easier to compute at each po int than the mean estim ator. A lso, since the time scaleis irrelevant to Cox's method the median estimator seems more appropriate.From Table 2 and other analyses not presented here we concluded that the Cox andBuckley-James estimators are the superior estimators. Full discussion of this will begiven in the n ext section. However, the fits of these two m odels to the da ta are not ideal.This is particularly evident in Fig. l(a) for Cox's model at low values of x where themedian estimator exceeds all the da ta. A lso, the Cox and Buck ley- Jam es estim ators donot agree on the statistical significance of the age effect. The estimated slope from Cox'smethod is more than 2 5 stan dar d d eviations away from zero, whereas tha t for Buckley &Jam es's m ethod is less than 2 standa rd deviations. N ote that the two estimators haveopposite signs because of the parameterizations.In'an attempt to achieve better fit a quadratic age model /?i(age) + /? 2 ( ag e) 2 w a s tried.In this analysis the 5 patients with survival times less than 10 days were deleted. Thelogarithmic transformation was not perfect in symmetrizing the data, and it was feltth at these few very low values might have distorted the B uck ley- Jam es regressionestimates and inflated the error variance. The results for the Cox and BuckleyJamesmethods applied with the quadratic model are presented in Table 3. The data and thefitted regression functions are displayed in Fig. l(b).The agreement between the two estimation methods is very much better for thequ ad rati c m odel. The degrees of significance of the linear and qu ad rati c regress ioncoefficients are very similar for the two estim ato rs. A lso, the m ean and the medianregression functions fit the data very much better. The estimates for the extremeyounger and older ages are more reasonable.Can any distinction be drawn between the proportional hazards model and the linearmodel? D oes one fit the da ta bette r th an the other? To examine this q uestion we plottedthe ordinary residuals for the linear model and the generalized residuals for theproportional hazards model. N either graph indicated any serious model deficiency. T helinear residuals have a nonpronounced U-shape, which indicates some slight modelirregularity, but both models fit rather well.

4. D ISCUSSIONFrom their performances on the updated Stanford heart tran spla nt d ata we feel th atthe Cox and Buckley & Jam es e stimators are the two most reliable regression estimatorsfor use with censored data. The other two estimators, those of Koul e t a l . and of Miller,have methodological weaknesses.The estimator of Koul et al. is based on the assumption that the censoring distribution

G(t) does not vary with x. The peculiarity of the dependent variable values used in thistechnique makes it sensitive to departures from this assumption. Each censored

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5 2 8 R . M I L L E R A N D J . H A L P E R NT a b l e 1 . Stanford hear t t ransp lan t da ta Februar y 1 9 8 0 ; f irs t co lum n, pa t i en t nu m ber; secondco lumn, T, surv iva l t ime (days); th i rd co lumn, dead = 1 , a l ive = 0 ; four th co l um n , age a tt ime o f f i rs t t ransp lan t ; f i f th co lumn, T 5 misma t ch s co r eNo.12345678910

11121314151617181920212223242526272829303132333435363738394041424344454647484960

T1534662312664

1 3 5 0232791 0 2 4

10397301 9 6 1136183660

3 6 9 51 9 9 6

04754512 8 7 83 4 1 044994511 4 7 825489714851323

3 0 2 1662 9 8 42 7 2 3550

66652272 8 0 5256312 7 3 412632 4 7 4

D e a d /alive11111111111111111101111110111111111010111110110111

Age54404251485454564943564258335254446440494162495 0 {49453648473648

T51111-660-611-320-361-890872-051121132-761-380961-061-620471-580690-380910870872-09) 9 9 9075098000811-381-35108

4 6 99994 7 999952483 8 (4 9 (3 2 (324 8 (514519485326

1-61182>98>66>191-93>121121-68L-021-20L-68L-46

4 7 0-972 9 (5 6 !52

>6lM 61-70

No.51525354555657585960616263646566676809707172737475767778798081828384858687888990919293949596979899100

T1 3 8 454429482971 3 1 81 3 5 2505474316826L61142 3 1 3

1 6 3 4146482 1 2 72632 1 0 62932 0 2 52 0 0 62 0 0 01 9 9 51 9 4 5657311 8 6 6538

1 8 4 6681 7 7 81 7 2 29281 7 1 822407

1 6 3 81 6 1 2251 5 3 41 5 4 71 2 7 1441 2 4 71 2 3 2191

D e a d /alive

0100000110101001011100110

Age4652535342485446494751524340262345283549404330154547385538494944352740503927422848516244503246411842

T51 411-941-083-050-601-44O682-250-810-331-330-821-2099990-461-780-160-770-670-480-86O701-441-261-461-651-280-690-420-512-760-830-850-70O951121 771-641-591-000-431-250-531-710-181O51 71O430-701-74

No.101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144146146147148149150

T139312021378137327431134142381126412621261471193626481150451116

11071102195301040993950729121202841834265179332878175273886132

6636602219061961857656336549

Dead/alive01001101100010111100011000111001101000110011000010

Age4638414131335019455234473624535132481418393934433046494548484949211934204341124636423538475053414540

T50-9599991-651-38O580-361130-630-98O641-680-820-161151-74O992-250-650-540-251-350730840500-95999911099991-240-8699991-22O471-981021121-50O531-261-09047O751-041000-900822-259999O202-53

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Regression wi th censore d da ta 5 2 9Table

N o.1 511 521 5 31 5 41 5 51 5 61 571 5 81 5 91 6 0

1 (cont .)T

5 485 415 3 41 6 91 2 23 8 246 846 41 05

D e a d /alive0001110011

A ge3 0472 05 15 13 62 43 81 32 0

T 50-470-439 9 9 91-891-339 9 9 91-392-071-499 9 9 9

N o.1 6 11 6 21 6 31 6 41 6 51 6 61 6 71 6 81 6 91 70

T1 3 640 63 9 13 745 01 3 93 2 22 9 22 782 2

D e ad /alive1000110001

A ge5 53 92 7475 05 13 6434145

T 59 9 9 91 1 81 1 79 9 9 90 5 0O 9 61 731-400-989 9 9 9

N o.1 711 721 731 741 751 761 771 781 791 8 01 8 11 8 21 8 31 8 4

T2 3 11 451 8 81 761 3 81 491 1 91 079 88 96 05 621

D e a d /alive01001000000000

A ge5 25 05 22 9412 12 0461 92 71 32 73 92 7

T 59 9 9 90 9 69 9 9 91 729 9 9 99 9 9 99 9 9 99 9 9 99 9 9 99 9 9 99 9 9 99 9 9 99 9 9 99 9 9 9

F o r T5 m i s m a t c h s c o r e 9 9 9 9 d e n o t e s m i s s i n g v a l u e .

T a b l e 2. Regress ion es t imates and s tandard dev ia t ions , S D , for lo g 1 0 of t i m e to dea thversus a ge at t ransplant and T5mi sma t ch score with n = 1 57 Stanford hear t t ransplantpa t i en tsEstimator

Co xBuckley-JamesInitialOne-stepFinalMiller

InitialOne-stepFinalt

Koul-Susarla Van Ryzin| Loop with two values.

Intercepta SD (a)

2-783143-232-032-582-572-540 7 2

0 3 5

051036

Ageh0-030

- 0 - 0 0 7- 0 0 1 3- 0 0 1 5O001- O O 0 2- O O 0 1OOOOO 0 2 4

0011

O008

0011O008

T5h0167

-O034-0011-O003

O061O0600072O040O251

0183

0134

O1910135

observation is decreased to zero, and each uncensored observation is inflated by thefactor {1 (?(y,)}~1. In some cases the observations are being moved away from theregression line in order to estimate it. With the Stanford heart transplant data theproportionately larger number of censored observations in the younger ages creates aproportionately larger number of zeros for small x. This gives a positive slope to theestimated regression line whereas everyone feels that increasing age is deleterious andthe slope should be negative. The implication for other data sets is th a t if the censoredobservations are not spread evenly over the ranges of the independent variables they cantip the estimated regression plane in a false direction or unduly enhance its slope.


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5 3 0 R . M I L L E R A X D J . H A L P E R N

IO 3

a " IO 2

1 020 40

A g e ( y e ars)

i i i I i i . . i i i i i i i

IO 2

1 0 '

1 ' !

60

Fig. 1. (a) (left). Scatterplot of log1 0 survival time (days) versus age at transplant (years) for 157Stanford heart transplant patients. Patien ts denoted by x are deceased; those by + were alive inFebruary 198 0. D ashed line, Buckley & James linear regression line; solid line, Cox proportionalhazards regression median.(b) (right). As Fig. l(a) but for quadratic regression with 152 patients who survived at least10 days.T a b l e 3 . Regress ion es t ima tes and s tandard dev ia t ions f o r lo g 1 0 o f tim e t o d ea thversus age and age square d a t t im e o f t r ansp lan t w i th n = 1 5 2 Stanford hear tt r ansp lan t pa t i en ts who surv ived a t l eas t 1 0 days

EstimatorCoxBuckley James

Interceptof S D ( a )

1-35 0-71

A geh-0-146

0-107

S D 0 . )0-0550O37

A ge 2h0 - 0 0 2 3

- 0 - 0 0 1 7

3D (fi2)0 1, computational expressions arecurrently available only when p = 1. A lso, further guidelines and experience are neededin the proper choice of the truncation point M n.Miller's estimator can also be thrown off by the censoring pattern. The asymptoticconsistency of the estimator requires condition (9) on the censoring distributions, andthis condition is not satisfied for the Stanford heart transplant data. While the Millerestimator does not seem to be biased so much by violations of (9) as the estimator ofKoul e t a l . by violations of G(t; x) = G(t), it is influenced by the censoring pattern.Both the Buckley & James and Miller iterative sequences of estimators can becometrapped in loops and fail to converge. This happened in Table 1 for the Miller estimator.Our stud ies show t ha t th e loops are less frequent and less severe for the B uckley & Jam esestimator than the Miller estimator. Why this should be the case was discussed byBuckley & Jam es (1979) .In conclusion, the Cox and Buckley & Jam es estimators are not dependent onparticular censoring patterns and have proved to be reliable estimators. Theoreticalvalida tion is lacking for B uckley & Jam es's v ariance estim ator, but use of it seemsjustified since it gives empirically sensible results and is supported by Monte Carlostudies. Both the Cox and Buckley & James estimators require about the same amountof programming and computing time. The choice between them should depend on theappropriateness of the proportional hazards model or the linear model for the data.

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Regression vr ith censored da ta 5 3 1This research was supported by a N ational Institu tes of Health Research G rant. Wethan k N an Laird for discovering two transcription errors in the d ata .

R E F E R E N C E SAME MI YA, T. (1973). Regression analysis when the dependent variable is truncated normal. Econometrica 41 ,9 9 7 - 1 0 1 6 .BRESLOW, N . (19 74). Covariance analysis of censored survival d ata. Biometrics 8 0 , 8 9 - 9 9 .BUCKLEY, J . & J A M E S , I. (1979). Linear regression with censored data. Biometrika 6 6 , 4 2 9 - 3 6 .Cox, D . R. (1 972 ). Regression m odels and life-tables (with discussion). J. R. Statist . Soc. B 84, 187-202.Cox, D . R. (19 75). Partial likelihood. B i o m e t r i k a 6 2 , 2 6 9 - 7 6 .CROWLEY, J . & Hu, M. (1977). Covariance analysis of heart transplant survival data. J. Am. Statist. Assoc.72 , 2 7 - 3 6 .D E MP ST E R, A . P ., L A I R D , N . M. & R U B I N , D . B. (19 77). Maximum likelihood from incomplete data v ia the EMalgorithm (with discussion). J. R. Statist . Soc. B 8 9 , 1 - 2 2 .E F R O N , B. (1977). The efficiency of Cox's likelihood function for censored data. J. Am . Statist . Assoc . 72 ,5 5 7 - 6 5 .GLASSER, M. (1965). Regression analysis with dependent variable censored. Biometrics 2 1 , 300-7.GLASSER, M. (1967). Exponential survival with covariance. J. Am . Sta t is t . Assoc . 6 2 , 5 6 1 - 8 .H E C K M A N , J. J. (1976). The common structure of statistical methods for truncation, sample selection, andlimited dependent variables and a simple estimator for such models. Ann. Earn. Soc. Meas. 5 , 4 75 - 9 2 .KA L BF LE I SCH, J . & P RE NT I CE , R. L. (1 972 ). Contribution to discussion of paper by D . R. Cox. J. R. Statist .Soc . B 3 4 , 2 1 5 - 6 .K A P L A N , E. L. & ME I E R, P. (1 95 8). N onparametric estimation from incomplete observations. J. Am. Statist .Assoc. 5 3 , 4 5 7 - 8 1 .K O U L , H ., SUSARL A, V. & V A N R Y Z I N , J. (1981). Regression analysis with randomly right censored data.Ann. Statist . 9 , 1 2 7 6 - 8 8 .M A N T E L , N . & MYE RS, M. (1971). Problems of convergence of maximum likelihood iterative procedures inmultiparameter situations. J. Am. Statist . Assoc. 6 6 , 4 8 4 - 9 1 .MILLER, R. G. (1976). Least squares regression with censored data. Biometrika 6 8 , 4 4 9 - 6 4 .X E L S O N , W. & H A H N , G. J. (1972). Linear estimation of a regression relationship from censored data. PartISimple methods and their applications. Technometrics 14, 247-69 .X E L S O N , W. & H A H N , G. J. (1973). Linear estimation of a regression relationship from censored data. PartIIBest linear unbiased estimation and theory. Technometrics 15 , 133-50.O A K E S , D . (19 77). The asymptotic information in censored survival data . Biometrika 6 4 , 4 41 - 8 .PETO, R. (1 972 ). Contribution to discussion of paper by D . R. Cox. J. R. Statist . Soc. B 34, 205-7.SCHMEE, J . & H A H N , G. J. (1979). A simple method forregressionanalysis with censored da ta. Technometrics2 1 , 4 1 7 - 3 2 .T O B I N , J. (1958). Estimation of relationships for limited dependent variables. Econometrica 2 6 , 2 4 - 3 6 .T SI AT I S, A . (19 81 ). A large sample study of Cox's regression model. Ann. Statist . 9 , 9 3 - 1 0 8 .T URNBUL L , B. W. (19 74). N onparam etric estimation of a survivorship function with doubly censored d ata.J. Am. Statist . Assoc. 6 9 , 1 6 9 - 7 3 .T URNBUL L , B. W. (1976). The empirical distribution function with arbitrarily grouped, censored andtruncated data. J. R . Statist . Soc . B 3 8 , 2 9 0 - 5 .Z I P P IN , C.& ARMI T AGE , P. (1966). Use of concomitant variables and incomplete survival information in theestimation of an exponential survival parameter. Biometrics 2 2 , 6 6 5 - 7 2 .

[Rece ived Ju ly 1 9 8 1 . Revised Apri l 1 9 8 2 ]


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