Meei Pyng Ng 1 and Ray Watson 1

Transcription

1 Aust N Z J Stat 444), 2002, DEALING WITH TIES IN FAILURE TIME DATA Meei Pyng Ng 1 and Ray Watson 1 University of Melbourne Summary In dealing with ties in failure time data the mechanism by which the data are observed should be considered If the data are discrete, the process is relatively simple and is determined by what is actually observed With continuous data, ties are not supposed to occur, but they do because the data are grouped into intervals even if only rounding intervals) In this case there is actually a non-identifiability problem which can only be resolved by modelling the process Various reasonable modelling assumptions are investigated in this paper They lead to better ways of dealing with ties between observed failure times and censoring times of different individuals The current practice is to assume that the censoring times occur after all the failures with which they are tied Key words: competing risk; discrete failure time; Kaplan Meier estimator; non-identifiability; tied observations 1 Introduction Discrete failure time data occur naturally and frequently The data may be genuinely discrete, such as counting data The failure time may be the number of trials till the occurrence of an event of interest, such as pregnancy in an in-vitro fertilization program IVF) or component failure in an industrial life-testing experiment Continuous data are sometimes modelled as discrete data; see eg Willett & Singer 1993) They can be regarded as continuous data grouped into intervals Indeed, all recorded continuous data are rounded or truncated, and so might be regarded as discrete data The importance of the discreteness is determined by the relative frequency of ties in the data Discrete failure time data have been modelled in the literature; see, for example, Cox 1972), Kalbfleisch & Prentice 1980) or Adams & Watson 1989) However, there has been little discussion on how tied failure times and censoring times should be handled When time is treated as continuous, tied observations supposedly cannot occur and so their occurrence cannot be handled theoretically In practice, tied observations do occur owing to the crudeness of measurements The usual convention, when there are failures and censorings occurring at the same time t, is to shift the censoring times infinitesimally to the right so that they occur after all the failures at time t This is commonly used when computing, for example, a Kaplan Meier estimate for the survival function Kaplan & Meier, 1958) When time is treated as discrete, then ties can occur with non-zero probability, and should be part of the modelling process However, it is usual to apply the convention used for continuous data to discrete data; see, for example, Kalbfleish & Prentice 1980 p 99) or Lawless Received July 2001; accepted September 2001 Author to whom correspondence should be addressed 1 Dept of Mathematics and Statistics, University of Melbourne, Parkville, Vic 3010, Australia mpn@unimelbeduau c Australian Statistical Publishing Association Inc 2002 Published by Blackwell Publishing Ltd

2 468 MEEI PYNG NG AND RAY WATSON 1982 p 372) When applied to continuous data grouped into intervals, this convention in effect is equivalent to allowing censoring to occur only ust before the end of the interval If the interval is wide, there can be serious overestimation of the survival function The failure-before-censoring convention for continuous data makes the hazard estimate as small as possible among possible orderings of the apparently tied observations Another aspect of discrete failure time that has escaped attention is the operation of the censoring mechanism when the potential failure time and the potential censoring time are tied for an individual In a random censorship model, each individual has a latent failure time T and a latent censoring time W What is actually observed when T = W? The answer to this question affects how ties between failure times and censoring times of different individuals should be handled In Section 2, we discuss the various censoring mechanisms that can occur in situations when the failure time and potential censoring time are tied for an individual In one of these situations there is a non-identifiability problem, which is discussed in Section 3 Section 4 deals with estimation and various reasonable assumptions that can be used to overcome this problem The failure-before-censoring convention creates an inconsistency when applied to competing-risk problems In a competing-risk problem, failures are classified into types, according to the cause of failure A cause-specific hazard rate, λ c t), for failure type cc= 1,,l) is commonly estimated by a Kaplan Meier estimate, treating all failures of type not equal to c as censored observations A consequence of this is that the relation Ft) = l c=1 F c t), where F c t) = exp t 0 λ c u) du) and Ft) is the overall survival function, does not hold for the corresponding Kaplan Meier estimates when there are ties between different failure types This is because different orders of occurrence of the tied observations are used, depending on which of the F c t) is being estimated The competing-risk problem is considered further in Section 5, where we describe an approach that overcomes the inconsistency problem described above In this paper, we investigate how ties in failure time data should be handled It is shown that there are more appropriate ways of handling ties than the standard failure-before-censoring method 2 Censoring mechanism We consider a random censorship model for discrete failure time With each individual in the population is associated a failure time T and a potential censoring time W, and T and W are independent When T < W, T is observed; when T > W, W is observed What happens when T = W depends on the context of the problem We consider three cases Case 1 When T = W, both T and W are observed This situation can occur if T and W correspond to two failure types T and W may be time to two types of failure that correspond to two different components in a piece of equipment under test The equipment is put through cycles of use until failure occurs The data can be represented as Y i,δ i ), i = 1,,n, where Y i = mint i,w i ) and δ i = 0ifY i = W i <T i, δ i = 1ifY i = T i <W i and δ i = 2ifY i = W i = T i In this case therefore and only in this case) we actually observe ties: observations for which δ i = 2 These data are expressed per individual, but when it comes to estimation of the hazard c Australian Statistical Publishing Association Inc 2002

3 DEALING WITH TIES IN FAILURE TIME DATA 469 and survivor functions the data are aggregated at each time point Thus in this case we have observations on: w = freqy = t,δ = 0) = freqt > t,w = t ), d = freqy = t,δ = 1) = freqt = t,w >t ), m = freqy = t,δ = 2) = freqt = t,w = t ) Case 2 When T = W, T is observed and W is not observed This means that, having observed T = t, all we know about the censoring time is that W t This situation is actually a special case of Case 3, but it is important and occurs often It can occur, for example, in data from an IVF program, where T is the number of completed cycles of treatment before pregnancy, and W is the number of completed cycles of treatment before withdrawal from the program If T = W = t, then pregnancy would be observed after t cycles and it would not be known that the patient would have withdrawn from the program had the last treatment not succeeded The data can be represented, as in Case 1, by Y i,δ i ), i = 1,,n But here, δ i takes only values 0 and 1: δ i = 0ifY i = W i <T i and δ i = 1ifY i = T i W i In this case we have observations on: w = freqy = t,δ = 0) = freqt > t,w = t ), d = freqy = t,δ = 1) = freqt = t,w t ) Case 3 When T = W, one and only one of them is observed, and which of them is observed is a random event for any particular individual This situation models the case when the discrete time arises from continuous time grouped into intervals Let T and W denote the underlying continuous failure time and censoring time respectively Suppose the time axis is partitioned into k intervals [a 1,a ), = 1,,k, where a 0 = 0 and a k = When T and W both lie in the interval t = [a 1,a ), then T = W = t However, T is observed if T W ; otherwise, W is observed That is, an individual with both failure time and censoring time in the interval t = [a 1,a ) could be censored at time t because the individual is censored before it fails if W <T ) and it is never known that it would have failed by time a had it not been censored On the other hand, it could be observed to fail at t,ift <W, the probability of which is denoted by π = PrT <W a 1 T <a,a 1 W <a ) The data can be represented as in Case 2, with δ i = 1 if failure is observed and δ i = 0 if the datum is censored The situation is represented in Figure 1: d = freqt = t,w >t ), u = freqt = t,w = t ; T observed), d = d + u ; w = freqt > t,w = t ), v = freqt = t,w = t ; W observed), w = w + v ; m = freqt = t,w = t ) = u + v ; n = freqt t,w t ), the number in the risk set at time t ; so that k n = d + u + v + w + n +1 = d + w + n +1 and n = d l + w l ) l= c Australian Statistical Publishing Association Inc 2002

4 470 MEEI PYNG NG AND RAY WATSON W d =d' +u m = u + v d' t u w' w = w' + v v T t Figure 1 Observed data in relation to failure time and censoring time Usually we observe only n, w and d = 1,,k), as is assumed for Case 3 Case 1 is unusual as it models the truly discrete case where ties between individuals failure times and censoring times can be observed, so that m is available It is more common for only one of failure or censoring to be observed In Case 2, it is assumed that v = 0 which brings about a simplification of the data and the subsequent analysis The difficulty with most discrete or grouped data is the generally unknown) subdivision of the m tied observations Of course, if there are few ties then this is not a serious problem However, if ties are common, we need at least to consider the possible influence of various assumptions about the subdivision between failure times and censoring times, and if they are potentially influential they should be allowed for by appropriate modelling 3 Non-identifiability Let t 1 <t 2 < <t k be the mass points of T and W They may represent intervals to which continuous-time variables have been grouped Let P = PrT t ), p = PrT = t ) and λ = PrT = t T t ), for = 1,,k We note that P = 1 l=1 1 λ l ), p = 1 λ l=1 1 λ l ), and P 1 = k l=1 p l = 1 = λ k In a non-parametric setting, λ 1,,λ k 1 are the parameters of the distribution of T Similarly, let Q = PrW t ), q = PrW = t ) and κ = PrW = t W t ) In a survival study, the primary interest is in the distribution of the failure time T Since only censored observations Y, δ) are available, the distribution of T is said to be nonidentifiable if it is not uniquely determined by the oint distribution of Y, δ) It is well-known that the distribution of T is not identifiable unless T and W are independent However, it is not generally known that in discrete-time survival analysis the distribution of T is nonidentifiable when the censoring mechanism has a random element, as in Case 3 above We consider the case when T = W and either T or W is observed at random; T and W are assumed to be independent Let π denote the probability that T is observed, when T = W = t Then the oint distribution of Y, δ) is p 1 = PrY = t,δ = 1) = λ 1 1 π )κ ) P Q, p 0 = PrY = t,δ = 0) = κ 1 π λ ) P Q, 1) c Australian Statistical Publishing Association Inc 2002

5 DEALING WITH TIES IN FAILURE TIME DATA 471 where k =1 p 0 + p 1 ) = 1 Letting p0 = p 0 /P Q ) and p 1 = p 1 /P Q ), the above can be written as: p 1 = λ 1 1 π )κ ), p 0 = κ 1 π λ ) These can be solved for λ and κ to give 1 λ = 1 p 0 + π p 0 + p 1 ) p 0 + π p 0 + p 1 )) 2 4π p1, 2π κ = p 0 + p 1 λ 1 λ Noting that P and Q can be expressed in terms of {λ l,κ l ; l = 1,, 1}, the above are solutions of λ and κ in terms of {π l,p l0,p l1 ; l = 1,, 1} We note in particular that, for fixed values of {p l0,p l1 ; l = 1,,k}, we can vary the values of π and obtain different solutions of λ That is, the distribution of T is not determined uniquely by the oint distribution of Y, δ) For identifiability, we need to make assumptions on π Some reasonable assumptions are considered in Section 4 Another indication of the non-identifiability of the situation can be gleaned from considering the likelihood function L, which is given in the appendix It is seen that L is a function of the parameters {λ,κ,π : = 1,,k} It can be shown that the score equations log L λ = log L κ = log L π = 0 = 1, 2,,k) are dependent and hence cannot produce a unique solution for all the λ,κ and π Indeed, they contain only 2k independent equations and so only 2k parameters can be estimated 4 Estimation Interest usually centres on estimating λ and P Maximum likelihood principles can be applied We treat the three cases described in Section 2 separately, using the notation given in Section 2 Case 1 In this case, when T = W = t, both T and W are observed to occur and so m is observed Standard likelihood methods give: ˆλ = d + m n and seˆλ ) = ˆλ 1 ˆλ ) n These are the usual results for binomial estimates Case 2 Here, when T = W = t, only T is observed, and W is not: this is the failurebefore-censoring scenario The case can be handled in much the same way as Case 1, since d = d + m = freqt = t,w t ) is observed We obtain: ˆλ = d and seˆλ n ) = ˆλ 1 ˆλ ) n c Australian Statistical Publishing Association Inc 2002

6 472 MEEI PYNG NG AND RAY WATSON Case 3 In this case, when T = W = t, only one of T and W is observed and it may be either one We assume that T is observed with probability π Whether T or W is observed is really a separate sub-experiment: if u and v were observed then we would have ˆλ = d + u + v )/n, as above As u and v are not available, this approach fails; and we need to re-write the likelihood in terms of d and w Also, as explained above, we have insufficient data to estimate the π, and so we need to make assumptions about the π One straightforward assumption is to give π a specified value: π = π The likelihood equations then give If π For π n π λ2 n w + π d + w )) λ + d = 0 = 0, the equation is linear and we have > 0, the quadratic equation gives ˆλ 0) d = n w n ˆλ = n w + π d + w ) w + π d + w )) 2 4 π n d 2 π n Then, P can be estimated by ˆλ 1) = d /n, as in Case 2; and P ˆ 1) estimate of the survivor function For π Pˆ = 1 l=1 1 ˆλ l ) For π = 1 l=1 n l d l )/n l, we obtain: = 1 2 2n ˆλ h) + d w 2n + d w ) 2 8n d =, 2n = 1, the above simplifies to which is the Kaplan Meier which is the maximum likelihood estimate given by Elandt-Johnson & Johnson 1980 p 164) under the assumption that the underlying continuous failure time is uniformly distributed in the interval and that all censoring occurs at the mid-point of the interval For an approximate standard error for this estimate, refer to Elandt-Johnson & Johnson 1980 p164) Another approach is to assume an explicit expression for π as a function of λ and κ As Case 3 is intended to model grouped continuous data, a plausible approach is to make assumptions about the underlying continuous failure time T and censoring time W and derive the consequent expression for π One such assumption is that T and W are uniformly distributed in the interval t = [a 1,a ) This gives π = 2 1, which has been dealt with above Another assumption is that both T and W have hazard functions that are constant in the interval [a 1,a ) These can be considered as piecewise approximations to the unknown hazard functions of T and W More generally, one can assume that the hazards of T and W are proportional over the interval [a 1,a ) Then π can be expressed as a function of λ and κ : π = λ )1 κ ) ) log1 κ )) κ λ κ log 1 λ )1 κ ) ) 2) c Australian Statistical Publishing Association Inc 2002

7 DEALING WITH TIES IN FAILURE TIME DATA 473 Table 1 Survivor function estimates for suicide ideation data age in years): S 0 denotes the survivor function estimate for π = 0, S J the oint risk estimate, S h the estimate for π = 1 2, and S 1 the Kaplan Meier estimate π = 1) Age, t years) n t d t w t S 0 t) S J t) S h t) S 1 t) If this relation is assumed then the problem is identifiable It is shown in the appendix that the resulting maximum likelihood estimate of λ is given by with standard error given by ˆλ J ) n d w ) d /d +w ) = 1, 3) n seˆλ J ) ) 2 = 1 ˆλ ) 2 d 2 n d + w )n d w ) + d w n d w )) d + w ) 3 log2 n The corresponding survivor function estimate given by se ˆ P J ) ) 2 = ˆ P J ) 1 ) 2 l=1 P ˆ J ) 4) = 1 l=1 1 ˆλ J ) l ) has standard error d 2 n d + w )n d w ) + d w n d w )) d + w ) 3 log2 n 5) The estimate in 3) is the oint-risk estimate of λ cited by Kaplan & Meier 1958) They indicate that the oint-risk estimate is the maximum likelihood estimate when the censoring is random with rate proportional to the failure rate This is precisely the assumption made here Further, Ng 1993) showed the oint-risk estimate to be the geometric mean of the Kaplan Meier estimates corresponding to all the possible orderings of the tied failure and censoring times: thus it represents an average over the possible orderings The derivations of the oint-risk estimate and its standard error are given in the appendix When w = 0, the estimate in 3) reduces to the usual Kaplan Meier estimate, and the variance given in 5) also reduces to the usual Greenwood formula used as variance of the Kaplan Meier estimate of the survival function Kaplan & Meier, 1958) Indeed, all the estimates λ i), i = 0, 1,h,J reduce to the same expression when w = 0 In practice, w = 0 means that no censoring time may occur at the same time as any other failure time: censorings are assumed to occur between the failure times Example 1 The data on first suicide ideation used by Willett & Singer 1993) are significantly grouped, with many ties between failure times The data were collected retrospectively from c Australian Statistical Publishing Association Inc 2002

8 474 MEEI PYNG NG AND RAY WATSON Table 2 Survivor function estimates for Stanford heart transplant data grouped into months: S 0 denotes the survivor function estimate for π = 0, S J the oint risk estimate, S h the estimate for π = 1 2 and S 1 the Kaplan Meier estimate π = 1) S JD denotes the oint risk estimate based on the daily data and S KD the Kaplan Meier estimate based on the daily data t months) S 0 t) S J t) S h t) S 1 t) S JD t) S KD t) undergraduates, who reported the age at which they first thought about suicide An observation was censored if the student had not thought of suicide by the date of data collection Of the 417 students, 287 had considered suicide by the date of data collection, and 130 had not; the latter are censored observations There was no censoring until age 16 years The data and the survivor function estimates for t 16 are given in Table 1 The full dataset can be found in Willett & Singer 1993) who report the Kaplan Meier estimate of the survivor function Table 1 shows that there is a difference between the survivor function estimates For these data, the oint risk estimate or the estimate for π = 2 1 would seem more appropriate, as they lie mid-way between the two extremes Example 2 The survival times in days for 249 patients in the Stanford heart transplant trial are a commonly used dataset Here we use pre-transplant survival time, and patients given transplants are considered censored The data are available in Kleinbaum 1996 p 297) In this form there are few ties and no problem with using the standard Kaplan Meier survivor function estimate However, if these data were recorded in months 30-day intervals), there would be a significant proportion of ties and a corresponding discrepancy in the survivor function estimates, as indicated in Table 2 If the degree of grouping approaches this level then significant errors can be made by using the Kaplan Meier survivor function estimate Here S J and S h provide closer estimates; in this case S h is closest 5 Competing-risk problem A competing-risk problem gives rise to data of the form {T i,δ i,c i ); i = 1,,n} where T i is the observed failure time, δ i the censoring indicator and C i is the failure type, which does not enter into the likelihood if δ i = 0, assuming a homogeneous population Kalbfleisch & Prentice, 1980 Chapter 7) Let λ c t) = lim t 0 Prt T<t+ t, C = c T t)/ t denote the cause-specific hazard rate, c = 1,,lThen λt) = lim t 0 Prt T<t+ t T t)/ t is the overall hazard rate The hazard rates are related by λt) = l c=1 λ c t), assuming each study unit can c Australian Statistical Publishing Association Inc 2002

9 DEALING WITH TIES IN FAILURE TIME DATA 475 fail because of only one failure type The overall survival function Ft) = exp t 0 λu)du) is related to the functions F c t) = exp t 0 λ c u)du) by Ft) = l c=1 F c t) We consider failure time as a continuous variable grouped into intervals; all observed continuous data are of this nature Then a competing-risk problem is similar to Case 3 as described in Section 2 with the added complication of failures being classified into types Suppose the observed data are d c = freqt = t,δ = 1,C = c), w = freqt = t,δ = 0), for = 1,,k and c = 1,,l Let n denote the number at risk at t, and d = c d c denote the total number of failures at time t The estimation of λt) and Ft) are as given in Section 4 for Case 3 The generalized maximum likelihood estimate of λ c t ) is obtained by treating all failures of types not equal to c as censored data Let d c = w + d c d d The estimates given for Case 3 can be extended to a competing-risk problem The extension of λ 0) and λ 1) is straightforward; λ h) becomes l ˆλ h) c t ) = l + 1)n ld c + d c ) 2 + 1)n ld c + d c 4l + 1)n d c 2n In this, we have replaced π = 2 1 by π censoring; λ J ) becomes = 1/l + 1) to account for the l failure types and The corresponding estimate of F c t) is n ˆλ J c ) d w ) dc/d +w ) t ) = 1 n ˆF c t) = t <t 1 ˆλ c t ) ) The relation Ft) = l c=1 F c t) holds for their estimates if there are no ties between the failure times of different types If ties occur, then the relation still holds for the oint-risk estimate ˆλ J c ) t ) but not for the other three The reason for this internal consistency of the oint-risk estimates can be seen by considering latent failure times of different failure types Let T c denote the time to failure of type c Assuming that T c c = 1,,l) are independent, the hazard function of T c is λ c t) Suppose G is a group of failure types to be aggregated into one type Let T G denote the time to failure of any type in G Then T G = min c G T c and its hazard rate is λ G t) = c G λ c t) This means that if all T c have proportional hazards in the interval [a,b), their hazards are also proportional to λ G t) for any group G In particular, by assuming all cause-specific hazard rates to be proportional and proportional to the hazard rate of censoring time W, we see that T = min T c has hazard proportional to W and T c has hazard proportional to T c = min{w,t d ; d c} Thus, the oint-risk formula is applicable to the estimation of λt ) and λ c t ) and the results are consistent within themselves On the other hand, if we assume c Australian Statistical Publishing Association Inc 2002

10 476 MEEI PYNG NG AND RAY WATSON T c and W to have uniform distribution within an interval [a,b), then T G does not have a uniform distribution within [a,b) Thus, λ h) does not extend to give internally consistent estimates of Ft) and F c t) 6 Discussion When failure time is genuinely discrete, the censoring mechanism can be identified and incorporated in the modelling process Estimation and inference are then straightforward In the more common case where the times are made discrete through rounding or grouping, assumptions are necessary to achieve identifiability of the survival function The various known estimates of the hazard rate Elandt-Johnson & Johnson, 1980 p 164; Kaplan & Meier, 1958; oint risk) can be derived under the same general structure by making different assumptions about π, the probability that failure is observed when there is a tie All the estimates considered are identical when there are no ties between failure times and censoring times Thus, they can be thought of as different ways of handling such ties The two extreme assumptions are π = 0 and π = 1 and they give bounds to the plausible estimates of the hazard rates and hence the survival function The Kaplan Meier corresponds to one extreme π = 1) In practice, if there are few such ties, all the estimates given in Section 4 are close and the Kaplan Meier estimate is a sensible simple choice, given that it is the current standard practice However, if there is a significant proportion of ties then this standard estimate can be quite wrong In that situation, ˆλ J ) or ˆλ h) are more moderate and are usually close to each other Either could be used We recommend that, as a matter of prudent practice, the estimates of the survival function corresponding to ˆλ 0) and ˆλ 1) should be routinely computed to check that they are not very different If the difference is small then the Kaplan Meier estimate ˆλ 1) ) can be used Otherwise ˆλ J ) should be used It is our view that ˆλ J ) has advantages over ˆλ h) It gives internally consistent estimates for the cause-specific hazards, when applied to competing-risk problems, as explained in Section 5 It is also the geometric mean of the Kaplan Meier estimates corresponding to all the possible orderings of the tied failure and censoring times see Ng, 1993) Further, it has a simpler algebraic expression than ˆλ h) and the expression for its standard error is more tractable For these reasons, ˆλ J ) is our preferred hazard rate estimate and might be better used as the default Nevertheless there is merit in computing the bounding estimates based on ˆλ 0) and ˆλ 1) to give an indication of possible error Appendix The maximum likelihood estimate of λ and its standard error as given in 3) and 4) are derived here Using the data description given for Case 3, the likelihood function is L = = k ) λ 1 ) d π )κ 1 λ l )1 κ l ) κ 1 π λ ) 1 λ l )1 κ l ) =1 l=1 k )) d λ 1 1 π )κ κ 1 π λ ) ) w 1 λ )1 κ ) ) n d w =1 l=1 ) w c Australian Statistical Publishing Association Inc 2002

11 DEALING WITH TIES IN FAILURE TIME DATA 477 Using the expression for π given in 2), 1 1 λ )1 κ ) ) log1 κ ) κ 1 π λ ) = log 1 λ )1 κ ) ) = α 1 β ), A1) log1 λ ) where α = 1 1 λ )1 κ ) and β = log 1 λ )1 κ ) ) By 1), α 1 β ) is the probability that an individual is censored at t given that the individual is at risk at t Similarly, 1 1 λ )1 κ ) ) log1 λ ) λ 1 1 π )κ ) = log 1 λ )1 κ ) ) = α β A2) This is the probability that an individual is observed to fail at t, given that the individual is still at risk at t Asn is the number of individuals at risk at time t,wehave d d d d n = Bin,α β ), w n = Bin,α 1 β )) and d + w n = Bin,α ) Using A1) and A2), the likelihood function simplifies to k L = 1 α ) n d w α d +w β d 1 β )w =1 from which we obtain the maximum likelihood estimates ˆα = d + w d and ˆβ n = d + w The estimate in 3) is then obtained using the relation λ = 1 1 α ) β The asymptotic variances and covariance of ˆα and ˆβ can be found by evaluating the information matrix We note that 2 log L α 2 = n d w 1 α ) 2 d + w α 2, 2 log L β 2 = d β 2 w 1 β ) 2, and all other second-order partial derivatives are zero Further, conditional on n, the number at risk at time t, 2 log L ) n 2 E α 2 = α 1 α ), E log L ) n α β 2 = β 1 β ) Thus the information matrix is diagonal and its inverse gives var ˆα ) α 1 α ), var ˆβ n ) β 1 β ) and cov ˆα α n, ˆβ ) 0 Then using the relation ˆλ = 1 1 ˆα ) ˆβ, the asymptotic variance of ˆλ is found to be as given by 5) References Adams, GG & Watson, RK 1989) A discrete time parametric model for the analysis of failure time data Aust J Statist 31, Cox, DR 1972) Regression models and life tables with discussion) J Roy Stat Soc Ser B 34, c Australian Statistical Publishing Association Inc 2002

12 478 MEEI PYNG NG AND RAY WATSON Elandt-Johnson, RC & Johnson, NL 1980) Survival Models and Data Analysis New York: Wiley Kalbfleisch, JD & Prentice, RL 1980) The Statistical Analysis of Failure Time Data New York: Wiley Kaplan, EL & Meier, P 1958) Non-parametric estimation from incomplete observations J Amer Statist Assoc 53, Kleinbaum, DG 1996) Survival Analysis New York: Springer Lawless, JF 1982) Statistical Models and Methods for Lifetime Data New York: Wiley Ng, MP 1993) A relation between product-limit estimate and oint-risk estimate Comm Statist Theory Methods 22, Willett, JB & Singer, JD 1993) Investigating onset, cessation, relapse and recovery: why you should, and how you can, use discrete-time survival analysis to examine event occurrence J Consulting and Clinical Psychology 61, c Australian Statistical Publishing Association Inc 2002