Skip to main content
NCBI home page
Biometrika logoLink to Biometrika
. 2012 Sep 30;99(4):995–1000. doi: 10.1093/biomet/ass049

Proportional mean residual life model for right-censored length-biased data

Kwun Chuen Gary Chan 1, Ying Qing Chen 2, Chong-Zhi Di 3
PMCID: PMC3635658  PMID: 23843676

Abstract

To study disease association with risk factors in epidemiologic studies, cross-sectional sampling is often more focused and less costly for recruiting study subjects who have already experienced initiating events. For time-to-event outcome, however, such a sampling strategy may be length biased. Coupled with censoring, analysis of length-biased data can be quite challenging, due to induced informative censoring in which the survival time and censoring time are correlated through a common backward recurrence time. We propose to use the proportional mean residual life model of Oakes & Dasu (Biometrika 77, 409–10, 1990) for analysis of censored length-biased survival data. Several nonstandard data structures, including censoring of onset time and cross-sectional data without follow-up, can also be handled by the proposed methodology.

Keywords: Biased sampling, Bivariate survival data, Proportional hazards model, Renewal process

1. Introduction

The longitudinal cohort study design has frequently been used in epidemiologic studies to investigate disease association with risk factors. To assemble a group of study participants in such a design, a conceptually straightforward sampling strategy is incident cohort sampling. For example, to study the incubation period between HIV-1 infection and development of an AIDS symptom, a subject would be recruited into the incident cohort as soon as his or her HIV positive serostatus is confirmed, and then followed till an AIDS-defining illness occurs. Although incident cohort sampling is a valid design to study the incubation period, it may need enormous resources and effort for subject accrual and follow-up, particularly when the HIV-infection rate is typically low even among high-risk populations because active antiretroviral therapy is highly effective for disease prevention.

Cross-sectional prevalent cohort sampling may offer a quick alternative way to assemble the cohort when the prevalence of initiating events is reasonably high. That is, only those who have experienced the initiating events would be sampled. However, when the disease outcome is the event time between initiating and failure events, cross-sectional sampling is likely to oversample those who have longer event times. One particular type of such cross-sectional sampling is so-called intercept or length-biased sampling (Canfield, 1941; Cox, 1969), when the sampling probability is proportional to the length of event times. For instance, when disease incidence follows a stationary Poisson process over time, cross-sectional sampling yields length-biased event times (Wang, 1991; Asgharian et al., 2002). Coupled with potential censoring on the event times, induced informative censorship may further complicate the analysis of such length-biased event times because the survival time and censoring time are correlated through a common backward recurrence time.

Biased sampling has been studied at least as far back as Wicksell (1925) for the corpuscle problem. Prevalent cohort bias in cross-sectional sampling was first discussed by Neyman (1955). Cox (1969) studied biased sampling in quality control, and Zelen & Feinleib (1969) identified biased sampling in screening for chronic diseases. Vardi (1982) studied nonparametric estimation for length-biased data in the absence of censoring. For censored length-biased outcomes, nonparametric estimation of survival has been discussed by Vardi (1989), Asgharian et al. (2002) and Huang & Qin (2011). Regression methods based on the Cox models were developed for length-biased data (Wang, 1996; Qin & Shen, 2010); methods based on the accelerated failure time model were also recently proposed (Bergeron et al., 2008; Shen et al., 2009; Chen, 2010; Mandel & Ritov, 2010).

In this article, we focus instead on a new regression model for length-biased data, which relies on modelling mean residual life functions. This takes advantage of a natural relationship between the mean residual life function and a length-biased outcome. It allows an efficient use of data without resorting to artificial pseudolikelihood sampling or inverse probability of censoring weighting, as used in existing methodologies (Wang, 1996; Shen et al., 2009).

2. A new regression method

In a cross-sectional study, subjects are eligible to be sampled if, at the sampling time, they have experienced initiating events but not failure events. The event time, T say, for a typical subject has a natural decomposition T = U + V, where U is the time between the initiating event and when the subject is sampled, and V is the time from recruitment to the failure event. When T is considered to be a recurrence time of a renewal process, U and V are also called backward and forward recurrence times (Cox, 1962).

To study the association between T and a p-vector covariate Z, we consider the proportional mean residual life model (Oakes & Dasu, 1990),

m(t;Z)=m0(t)exp(βTZ), (1)

where m(t; Z) = E(T − t | T > t, Z) is the mean residual life function of T at t, m0(t) is an unspecified baseline mean residual life function and β is a p × 1 regression parameter. The primary goal is then to estimate β using cross-sectional length-biased data. For data arising from incident cohort sampling, methods have been developed, for example, by Maguluri & Zhang (1994) and Chen & Cheng (2005).

The mean residual life function itself is important in reliability, demographics and actuarial sciences, with many other applications. Due to its natural link to the hazard functions of backward and forward recurrence times, we can model mean residual life functions in regression analysis of length-biased data.

Let f (t, z) be the joint density function of T and Z. The sampling distribution of (T, Z) under length-biased sampling is t f (t, z)/μ, where μ is the mean of T. The sampling density of (V, Z) was also shown (Cox, 1969) to be

fV,Z(υ,z)=μ−1∫υ∞f(t,z)dt    (υ>0).

The conditional density of V given Z is therefore

fV|Z(υ|z)=fV,Z(υ,z)∫fV,Z(υ,z)dυ=F¯T(υ|z)μ(z) (2)

where F̄T (t | z) = pr(T > t | Z = z) is the conditional survival function, and μ(z) is the conditional mean of T given Z = z. Let λ (υ; z) be the hazard function for the forward recurrence time V with covariate z. It follows from (2) that λ (υ; z) = 1/m(υ; z), and it further follows from (1) that the sampling density of V satisfies a proportional hazards model

λ(υ;z)=λ0(υ)exp(γTz) (3)

where γ = −β. See, for example, Maguluri & Zhang (1994) for a related discussion. Under right censoring of survival time, we observe (U, X, Δ, Z), where X = min(V, C), Δ = I (V ⩽ C) and C is the time from recruitment to censoring. Note that C potentially censors the forward recurrence time and may be called the residual censoring time. Suppose (U, V) and C are conditionally independent given covariates Z. Since V follows a proportional hazards model, the assumptions on the censoring mechanism guarantee the validity of partial likelihood estimation using data (X, Δ, Z) to estimate β. Such an estimate, however, is inefficient, because information from U is not used. It also follows from Cox (1969) that the marginal sampling distribution of (U, Z) is the same as that of (V, Z), which also follows a proportional hazards model (3). Note that (U, V) are correlated in general, but (U, V) satisfy marginal proportional hazards models. We could treat {(Ui, 1), (Xi, Δi) : i = 1, . . . , n} as paired survival data, and methods for multivariate survival data can be applied to estimate β. In particular, we consider an estimating function for bivariate survival data that is equivalent to a partial likelihood score function under an independent working correlation between the backward and forward times,

SPL(γ)=∑i=1n{Zi−S(1)(γ,Ui)S(0)(γ,Ui)}+Δi{Zi−S(1)(γ,Xi)S(0)(γ,Xi)} (4)

where S(k)(γ,t)=n−1∑i=1nZ⊗k{I(Ui⩾t)+I(Xi⩾t)}exp(γTZi)(k=0,1,2) and for any vector a, a⊗0 = 1, a⊗1 = a and a⊗2 = aaT. Let γ̂ be a solution to the equation SPL(γ) = 0 and β̂ = −γ̂. The idea is similar to Lee et al. (1992) and Spiekerman & Lin (1998) for clustered survival data, and the validity of the method is based on the fact that S(k)(γ, t) converges to the same limit s(k)(γ, t) = E[Z⊗k {I(U ⩾ t) + I (X ⩾ t)} exp(γTZ)] regardless of whether or not (U, V) are independent. Since the problem is transformed to a bivariate survival model, asymptotic properties of β̂ follow from Spiekerman & Lin (1998) and its asymptotic variance can be estimated by a sandwich-type estimate, V̂ = Â−1B̂ Â−1, where

         A^=1n∑i=1n(S(2)(γ^;Ui)S(0)(γ^;Ui)−{S(1)(γ^;Ui)S(0)(γ^;Ui)}⊗2                                   +Δi[S(2)(γ^;Ui)S(0)(γ^;Xi)−{S(1)(γ^;Xi)S(0)(γ^;Xi)}⊗2]),         B^=1n∑i=1n[{Zi−S1(γ^;Ui)S0(γ^;Ui)}+Δi{Zi−S1(γ^;Xi)S0(γ^;Xi)}                                   −∑j=1n{hi(Uj,Ui)+Δjhi(Xj,Ui)+hi(Uj,Xi)+Δjhi(Xj,Xi)}]⊗2,hi(s,t)=eγ^ZiI(s⩽t){Zi−S(1)(γ^;s)/S(0)(γ^;s)}nS(0)(γ^;s).

In the above discussion, we assume the observation of standard right-censored length-biased survival data, where censoring events affect only the forward recurrence times, and the backward recurrence times are completely observed. However, since the initial events happen before recruitment, they may not always be recorded in practice. For instance, if the date of disease diagnosis can only be obtained from a disease registry up to two years before recruitment, then every subject with U greater than two years is subject to left administrative censoring. Let CB be the time from censoring of the initial event to the time of recruitment, Y = min(U, CB) and ΔB = I (U ⩽ CB), and suppose we observe (Y, ΔB, X, Δ, Z). Further we assume that (U, V) and (CB, C) are conditionally independent given covariates Z. Under this scenario, we could modify (4) to estimate β using the paired survival data {(Yi, ΔBi), (Xi, Δi) : i = 1, . . . , n}, where γ̂ = −β̂ is estimated by solving S̃PL(γ) = 0, where

S˜PL(γ)=∑i=1nΔBi{Zi−S˜(1)(γ,Yi)S˜(0)(γ,Yi)}+Δi{Zi−S˜(1)(γ,Xi)S˜(0)(γ,Xi)},

with S˜(k)(γ,t)=n−1∑i=1nZ⊗k{I(Yi⩾t)+I(Xi⩾t)}exp(γTZi)(k=0,1,2).

The target parameter β can also be estimated when either backward or forward recurrence times are being collected. The former type of data can be collected in cross-sectional surveys without further follow-up (Allison, 1985). The latter type of data is collected when time of disease incidence cannot be ascertained retrospectively. The target parameter β from the mean residual lifetime model can be estimated by partial likelihood methods using only backward or forward recurrence times.

3. Numerical studies

We examined finite sample properties of the proposed estimator by simulation. For each simulation scenario, 5000 independent datasets were generated. For each dataset, we independently generated a Bernoulli variable Z1 with p = 0.5 and a U(0, 1) variable Z2. Conditioning on Z1 and Z2, T followed a mean residual life model mT (t) = (At + B) exp(β1Z1 + β2Z2). To obtain a length-biased sample, we generated random disease incidence time U0 from a U(0, 30) distribution; an observation was in the cross-sectional sample if −U0 + T ⩾ 0. Data were generated until the cross-sectional sample had 100 observations. We generated residual censoring time C from a U(0, 5) distribution. We considered three estimators of β = (β1, β2)T. The estimator β̂U maximized the partial likelihood using only backward recurrence times. The β̂V maximized the partial likelihood using censored forward recurrence times. The estimator β̂UV was the solution of SPL(−β) = 0. Table 1 shows that all the estimators have small bias and that using both backward and forward recurrence times greatly improves efficiency.

Table 1.

Comparisons among the proposed estimators under the residual time model m(t; z) = (At + B) exp(0.5z1 + 0.5z2)

(a) A = 0.1, B = 0.5
Estimator Parameter Bias × 103 SSE × 103 SEE × 103 Coverage (%)
β̂U β̂1 2 151 152 94.8
β̂2 −5 260 254 94.7
β̂V β̂1 2 163 167 95.8
β̂2 <1 281 284 95.3
β̂UV β̂1 −2 117 117 94.8
β̂2 −6 202 198 94.3
(b) A = 0.05, B = 1
Estimator Parameter Bias × 103 SSE × 103 SEE × 103 Coverage (%)
β̂U β̂1 1 160 152 94.0
β̂2 1 267 254 93.5
β̂V β̂1 −5 184 183 94.7
β̂2 −9 314 313 94.7
β̂UV β̂1 −4 124 118 94.2
β̂2 −7 212 199 92.4
Open in a new tab

SSE, the sampling standard deviation; SEE, the sample mean of standard error estimates; Coverage, the empirical coverage of approximate 95% confidence intervals.

When A = 0, the proportional mean residual life model is identical to a proportional hazards model with a target parameter of the same magnitude but a different sign. Table 2 compares the proposed estimator β̂UV with the partial likelihood estimator for left-truncated and right-censored data, β̂PL. The proposed estimator has better efficiency than the usual partial likelihood estimator under length-biased sampling because it takes advantage of the length-biased structure; the usual risk-set-based methods for left-truncated and right-censored data do not. In the presence of right censoring, the sampling variability of β̂UV is 39% and 44% of the sampling variability of β̂PL for β1 and β2 respectively. The results echo similar findings for nonparametric estimation in Wang (1989) and Asgharian et al. (2002). A reviewer pointed out that U and V are independent when T is exponentially distributed and that the asymptotic variance of β̂UV should be half of the asymptotic variance of β̂PL if there is no censoring. In small sample simulations without censoring, the sampling variances of β̂UV are 45% and 52% of the sampling variances of β̂PL for β1 and β2, respectively, close to the theoretical prediction.

Table 2.

Comparison of the proposed estimator β̂UVwith the partial likelihood estimator β̂PL

(a) U(0,5) censoring
Estimator Parameter Bias × 103 SSE × 103 SEE × 103 Coverage (%)
βUV β1 7 113 113 95.2
β2 −2 194 191 94.8
βPL β1 13 181 178 95.0
β2 1 291 300 95.6
(b) No censoring
Estimator Parameter Bias × 103 SSE × 103 SEE × 103 Coverage (%)
βUV β1 2 107 107 94.8
β2 −9 181 178 94.9
βPL β1 7 160 156 94.3
β2 −5 251 258 95.4
Open in a new tab

SSE, the sampling standard deviation; SEE, the sample mean of standard error estimates; Coverage, the empirical coverage of approximate 95% confidence intervals.

4. Discussion

For the usual right-censored data collected from an incident population, semiparametric models on the mean residual life function are identifiable only under a strong assumption that the right tail of the survival distribution is fully identified. This assumption is violated when administrative censoring is present after a limited study period. Cross-sectional sampling alleviates this common identifiability problem in residual time regression models. With the induced proportional hazards structure, the mean residual life ratio is identifiable for length-biased survival data when follow-up is limited. In fact, the parameter β in model (1) can even be estimated by partial likelihood methods when we collect only backward or forward recurrence times. The proposed methodology is applicable when one adopts a retrospective study based on a cross-sectional sample and to cases where onset of disease cannot be ascertained retrospectively. For prevalent cohort data, Wang et al. (1993) advocated the use of proportional hazards models for population survival time over proportional hazards models for forward recurrence time, because the former has a direct interpretation for the incident study population. Under length-biased sampling, the relative hazard for forward recurrence time indeed has a useful interpretation as the inverse of the relative mean residual life for the underlying population.

A reviewer mentioned that the joint distribution of (U, V) depends only on U + V; see Vardi (1989) who considered nonparametric likelihood estimation of the survival function for length-biased right-censored data. However, under the semiparametric proportional mean residual life model, the likelihood function does not lead to tractable analysis. In fact, it is unclear to the authors how the functional nuisance parameter m0(t) can be eliminated. Moreover, likelihood estimation for the semiparametric proportional mean residual lifetime model is an open problem even for standard survival data in the absence of length-biased sampling. However, using the proportional hazards structure of U and V, a functional nuisance parameter can be eliminated using risk-set arguments, and censoring in both directions can also be handled naturally.

Acknowledgments

The authors thank the editor, an associate editor, two reviewers and Prof. Mary Lou Thompson for their helpful comments and suggestions, which greatly improved this paper.

References

  1. Allison PD. Survival analysis of backward recurrence times. J Am Statist Assoc. 1985;80:315–22. [Google Scholar]
  2. Asgharian M, M’Lan CE, Wolfson DB. Length-biased sampling with right censoring: An unconditional approach. J Am Statist Assoc. 2002;97:201–10. [Google Scholar]
  3. Bergeron PJ, Asgharian M, Wolfson DB. Covariate bias induced by length-biased sampling of failure times. J Am Statist Assoc. 2008;103:737–42. [Google Scholar]
  4. Canfield RH. Application of the line interception method in sampling range vegetation. J. Forestry. 1941;39:388–94. [Google Scholar]
  5. Chen YQ. Semiparametric regression in size-biased sampling. Biometrics. 2010;66:149–58. doi: 10.1111/j.1541-0420.2009.01260.x. [DOI] [PMC free article] [PubMed] [Google Scholar]
  6. Chen YQ, Cheng S. Semiparametric regression analysis of mean residual life with censored survival data. Biometrika. 2005;92:19–29. [Google Scholar]
  7. Cox DR. Renewal Theory. London: Methuen; 1962. [Google Scholar]
  8. Cox DR. Some sampling problems in technology. In: Johnson NL, Smith H, editors. New Developments in Survey Sampling. New York: Wiley; 1969. pp. 506–27. [Google Scholar]
  9. Huang C-Y, Qin J. Nonparametric estimation for length-biased and right-censored data. Biometrika. 2011;98:177–86. doi: 10.1093/biomet/asq069. [DOI] [PMC free article] [PubMed] [Google Scholar]
  10. Lee EW, Wei LJ, Amato DA. Cox-type regression analysis for large numbers of small groups of correlated failure time observations. In: Klein JP, Goel PK, editors. Survival Analysis: State of the Art. Dordrecht: Kluwer Academic Publishers; 1992. pp. 237–47. [Google Scholar]
  11. Maguluri G, Zhang C-H. Estimation in the mean residual life regression model. J. R. Statist. Soc B. 1994;56:477–89. [Google Scholar]
  12. Mandel M, Ritov Y. The accelerated failure time model under biased sampling. Biometrics. 2010;66:1306–8. doi: 10.1111/j.1541-0420.2009.01366_1.x. [DOI] [PubMed] [Google Scholar]
  13. Neyman J. Statistics: servant of all science. Science. 1955;122:401–6. doi: 10.1126/science.122.3166.401. [DOI] [PubMed] [Google Scholar]
  14. Oakes D, Dasu T. A note on residual life. Biometrika. 1990;77:409–10. [Google Scholar]
  15. Qin J, Shen Y. Statistical methods for analyzing right-censored length-biased data under Cox model. Biometrics. 2010;66:382–92. doi: 10.1111/j.1541-0420.2009.01287.x. [DOI] [PMC free article] [PubMed] [Google Scholar]
  16. Shen Y, Ning J, Qin J. Analyzing length-biased data with semiparametric transformation and accelerated failure time models. J Am Statist Assoc. 2009;104:1192–202. doi: 10.1198/jasa.2009.tm08614. [DOI] [PMC free article] [PubMed] [Google Scholar]
  17. Spiekerman CF, Lin DY. Marginal regression models for multivariate failure time data. J Am Statist Assoc. 1998;93:1164–75. [Google Scholar]
  18. Vardi Y. Nonparametric estimation in the presence of length bias. Ann Statist. 1982;10:616–20. [Google Scholar]
  19. Vardi Y. Multiplicative censoring, renewal processes, deconvolution and decreasing density: nonparametric estimation. Biometrika. 1989;76:751–61. [Google Scholar]
  20. Wang M-C. A semiparametric model for randomly truncated data. J Am Statist Assoc. 1989;84:742–8. [Google Scholar]
  21. Wang M-C. Nonparametric estimation from cross-sectional survival data. J Am Statist Assoc. 1991;86:130–43. [Google Scholar]
  22. Wang M-C. Hazards regression analysis for length-biased data. Biometrika. 1996;83:343–54. [Google Scholar]
  23. Wang M.-C, Brookmeyer R, Jewell NP. Statistical models for prevalent cohort data. Biometrics. 1993;49:1–11. [PubMed] [Google Scholar]
  24. Wicksell SD. The corpuscle problem: a mathematical study of a biometric problem. Biometrika. 1925;17:84–99. [Google Scholar]
  25. Zelen M, Feinleib M. On the theory of screening for chronic diseases. Biometrika. 1969;56:601–14. [Google Scholar]

Articles from Biometrika are provided here courtesy of Oxford University Press

RESOURCES