Skip to main content

Advertisement

SpringerLink
Log in

Landmark estimation of survival and treatment effects in observational studies

  • Published:
Lifetime Data Analysis Aims and scope Submit manuscript

Abstract

Clinical studies aimed at identifying effective treatments to reduce the risk of disease or death often require long term follow-up of participants in order to observe a sufficient number of events to precisely estimate the treatment effect. In such studies, observing the outcome of interest during follow-up may be difficult and high rates of censoring may be observed which often leads to reduced power when applying straightforward statistical methods developed for time-to-event data. Alternative methods have been proposed to take advantage of auxiliary information that may potentially improve efficiency when estimating marginal survival and improve power when testing for a treatment effect. Recently, Parast et al. (J Am Stat Assoc 109(505):384–394, 2014) proposed a landmark estimation procedure for the estimation of survival and treatment effects in a randomized clinical trial setting and demonstrated that significant gains in efficiency and power could be obtained by incorporating intermediate event information as well as baseline covariates. However, the procedure requires the assumption that the potential outcomes for each individual under treatment and control are independent of treatment group assignment which is unlikely to hold in an observational study setting. In this paper we develop the landmark estimation procedure for use in an observational setting. In particular, we incorporate inverse probability of treatment weights (IPTW) in the landmark estimation procedure to account for selection bias on observed baseline (pretreatment) covariates. We demonstrate that consistent estimates of survival and treatment effects can be obtained by using IPTW and that there is improved efficiency by using auxiliary intermediate event and baseline information. We compare our proposed estimates to those obtained using the Kaplan–Meier estimator, the original landmark estimation procedure, and the IPTW Kaplan–Meier estimator. We illustrate our resulting reduction in bias and gains in efficiency through a simulation study and apply our procedure to an AIDS dataset to examine the effect of previous antiretroviral therapy on survival.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  • Amato DA (1988) A generalized Kaplan-Meier estimator for heterogenous populations. Commun Stat 17(1):263–286

    Article  MATH  Google Scholar 

  • Austin PC (2007) The performance of different propensity score methods for estimating marginal odds ratios. Stat Med 26(16):3078–3094

    Article  MathSciNet  Google Scholar 

  • Austin PC (2009) Balance diagnostics for comparing the distribution of baseline covariates between treatment groups in propensity-score matched samples. Stat Med 28:3083–3107

    Article  MathSciNet  Google Scholar 

  • Austin PC, Stuart EA (2015) Moving towards best practice when using inverse probability of treatment weighting (IPTW) using the propensity score to estimate causal treatment effects in observational studies. Stat Med 34(28):3661–3679

    Article  MathSciNet  Google Scholar 

  • Bai X, Tsiatis AA, O’Brien SM (2013) Doubly-robust estimators of treatment-specific survival distributions in observational studies with stratified sampling. Biometrics 69(4):830–839

    Article  MathSciNet  MATH  Google Scholar 

  • Beran R (1981) Nonparametric regression with randomly censored survival data. Technical report, University of California Berkeley

  • Bhatta L, Klouman E, Deuba K, Shrestha R, Karki DK, Ekstrom AM, Ahmed LA (2013) Survival on antiretroviral treatment among adult HIV-infected patients in Nepal: a retrospective cohort study in far-western region, 2006–2011. BMC Infect Dis 13(1):604

    Article  Google Scholar 

  • Breiman L, Friedman J, Stone CJ, Olshen RA (1984) Classification and regression trees. CRC Press, Boca Raton

    MATH  Google Scholar 

  • Cai T, Tian L, Wei LJ (2005) Semiparametric Box-Cox power transformation models for censored survival observations. Biometrika 92(3):619–632

    Article  MathSciNet  MATH  Google Scholar 

  • Cai T, Tian L, Uno H, Solomon S, Wei L (2010) Calibrating parametric subject-specific risk estimation. Biometrika 97(2):389–404

    Article  MathSciNet  MATH  Google Scholar 

  • Chen PY, Tsiatis AA (2001) Causal inference on the difference of the restricted mean lifetime between two groups. Biometrics 57(4):1030–1038

    Article  MathSciNet  MATH  Google Scholar 

  • Cook R, Lawless J (2001) Some comments on efficiency gains from auxiliary information for right-censored data. J Stat Plan Inference 96(1):191–202

    Article  MATH  Google Scholar 

  • Cox DR (1972) Regression models and life tables (with discussion). J R Stat Soc 34:187–220

    MathSciNet  MATH  Google Scholar 

  • Du Y, Akritas M (2002) Uniform strong representation of the conditional Kaplan-Meier process. Math Methods Stat 11(2):152–182

    MathSciNet  MATH  Google Scholar 

  • Faucett CL, Schenker N, Taylor JM (2002) Survival analysis using auxiliary variables via multiple imputation, with application to AIDS clinical trial data. Biometrics 58(1):37–47

    Article  MathSciNet  MATH  Google Scholar 

  • Fine J, Jiang H, Chappell R (2001) On semi-competing risks data. Biometrika 88(4):907–919

    Article  MathSciNet  MATH  Google Scholar 

  • Finkelstein DM, Schoenfeld DA (1994) Analysing survival in the presence of an auxiliary variable. Stat Med 13(17):1747–1754

    Article  Google Scholar 

  • Fleming TR, Prentice RL, Pepe MS, Glidden D (1994) Surrogate and auxiliary endpoints in clinical trials, with potential applications in cancer and AIDS research. Stat Med 13(9):955–968

    Article  Google Scholar 

  • Garcia TP, Ma Y, Yin G (2011) Efficiency improvement in a class of survival models through model-free covariate incorporation. Lifetime Data Anal 17(4):552–565

    Article  MathSciNet  MATH  Google Scholar 

  • Gray R (1994) A kernel method for incorporating information on disease progression in the analysis of survival. Biometrika 81(3):527–539

    Article  MathSciNet  MATH  Google Scholar 

  • Griffin BA, Eibner C, Bird CE, Jewell A, Margolis K, Shih R, Slaughter ME, Whitsel EA, Allison M, Escarce JJ (2013) The relationship between urban sprawl and coronary heart disease in women. Health Place 20:51–61

    Article  Google Scholar 

  • Hammer S, Katzenstein D, Hughes M, Gundacker H, Schooley R, Haubrich R, Henry W, Lederman M, Phair J, Niu M et al (1996) A trial comparing nucleoside monotherapy with combination therapy in HIV-infected adults with CD4 cell counts from 200 to 500 per cubic millimeter. New Engl J Med 335(15):1081–1090

    Article  Google Scholar 

  • Hammer SM, Squires KE, Hughes MD, Grimes JM, Demeter LM, Currier JS, Eron JJ Jr, Feinberg JE, Balfour HH Jr, Deyton LR et al (1997) A controlled trial of two nucleoside analogues plus indinavir in persons with human immunodeficiency virus infection and CD4 cell counts of 200 per cubic millimeter or less. New Engl J Med 337(11):725–733

    Article  Google Scholar 

  • Hankey BF, Myers MH (1971) Evaluating differences in survival between two groups of patients. J Chron Dis 24(9):523–531

    Article  Google Scholar 

  • Harder VS, Stuart EA, Anthony JC (2010) Propensity score techniques and the assessment of measured covariate balance to test causal associations in psychological research. Psychol Methods 15(3):234

    Article  Google Scholar 

  • Hernán MÁ, Brumback B, Robins JM (2000) Marginal structural models to estimate the causal effect of zidovudine on the survival of HIV-positive men. Epidemiology 11(5):561–570

    Article  Google Scholar 

  • Higashi T, Shekelle PG, Adams JL, Kamberg CJ, Roth CP, Solomon DH, Reuben DB, Chiang L, MacLean CH, Chang JT et al (2005) Quality of care is associated with survival in vulnerable older patients. Ann Intern Med 143(4):274–281

    Article  Google Scholar 

  • Hill JL (2011) Bayesian nonparametric modeling for causal inference. J Comput Gr Stat 20(1):217–240

    Article  MathSciNet  Google Scholar 

  • Hirano K, Imbens GW (2004) The propensity score with continuous treatments. Applied bayesian modeling and causal inference from incomplete-data perspectives: an essential journey with donald rubin’s statistical family. Wiley, New York, pp 73–84

    Google Scholar 

  • Imai K, Ratkovic M (2014) Covariate balancing propensity score. J R Stat Soc 76(1):243–263

    Article  MathSciNet  Google Scholar 

  • Imai K, Van Dyk DA (2004) Causal inference with general treatment regimes. J Am Stat Assoc 99(467):854–866

    Article  MATH  Google Scholar 

  • Imbens GW (2000) The role of the propensity score in estimating dose-response functions. Biometrika 87(3):706–710

    Article  MathSciNet  MATH  Google Scholar 

  • Kaplan EL, Meier P (1958) Nonparametric estimation from incomplete observations. J Am Stat Assoc 53(282):457–481

    Article  MathSciNet  MATH  Google Scholar 

  • Lagakos S (1988) The loss in efficiency from misspecifying covariates in proportional hazards regression models. Biometrika 75(1):156–160

    Article  MathSciNet  MATH  Google Scholar 

  • Lagakos S, Schoenfeld D (1984) Properties of proportional-hazards score tests under misspecified regression models. Biometrics 40:1037–1048

    Article  MathSciNet  MATH  Google Scholar 

  • Lee BK, Lessler J, Stuart EA (2010) Improving propensity score weighting using machine learning. Stat Med 29(3):337–346

    MathSciNet  Google Scholar 

  • Li Y, Taylor JM, Little RJ (2011) A shrinkage approach for estimating a treatment effect using intermediate biomarker data in clinical trials. Biometrics 67(4):1434–1441

    Article  MathSciNet  MATH  Google Scholar 

  • Liaw A, Wiener M (2002) Classification and regression by randomforest. R News 2(3):18–22

    Google Scholar 

  • Lin D (2000) On fitting cox’s proportional hazards models to survey data. Biometrika 87(1):37–47

    Article  MathSciNet  MATH  Google Scholar 

  • Lin D, Wei L (1989) The robust inference for the cox proportional hazards model. J Am Stat Assoc 84:1074–1078

    Article  MathSciNet  MATH  Google Scholar 

  • Lu X, Tsiatis A (2008) Improving the efficiency of the log-rank test using auxiliary covariates. Biometrika 95(3):679–694

    Article  MathSciNet  MATH  Google Scholar 

  • Marcus SM, Siddique J, Ten Have TR, Gibbons RD, Stuart E, Normand SLT (2008) Balancing treatment comparisons in longitudinal studies. Psychiatr Ann 38(12):805

    Article  Google Scholar 

  • McCaffrey DF, Ridgeway G, Morral AR (2004) Propensity score estimation with boosted regression for evaluating causal effects in observational studies. Psychol Methods 9(4):403

    Article  Google Scholar 

  • Mocroft A, Madge S, Johnson AM, Lazzarin A, Clumeck N, Goebel FD, Viard JP, Gatell J, Blaxhult A, Lundgren JD et al (1999) A comparison of exposure groups in the eurosida study: starting highly active antiretroviral therapy (HAART), response to HAART, and survival. JAIDS 22(4):369–378

    Google Scholar 

  • Murray S, Tsiatis A (1996) Nonparametric survival estimation using prognostic longitudinal covariates. Biometrics 52:137–151

    Article  MATH  Google Scholar 

  • Murray S, Tsiatis AA (2001) Using auxiliary time-dependent covariates to recover information in nonparametric testing with censored data. Lifetime Data Anal 7(2):125–141

    Article  MathSciNet  MATH  Google Scholar 

  • Nieto FJ, Coresh J (1996) Adjusting survival curves for confounders: a review and a new method. Am J Epidemiol 143(10):1059–1068

    Article  Google Scholar 

  • Normand SLT, Landrum MB, Guadagnoli E, Ayanian JZ, Ryan TJ, Cleary PD, McNeil BJ (2001) Validating recommendations for coronary angiography following acute myocardial infarction in the elderly: a matched analysis using propensity scores. J Clin Epidemiol 54(4):387–398

    Article  Google Scholar 

  • Pan Q, Schaubel DE (2008) Proportional hazards models based on biased samples and estimated selection probabilities. Can J Stat 36(1):111–127

    Article  MathSciNet  MATH  Google Scholar 

  • Parast L, Tian L, Cai T (2014) Landmark estimation of survival and treatment effect in a randomized clinical trial. J Am Stat Assoc 109(505):384–394

    Article  MathSciNet  Google Scholar 

  • Park Y, Wei LJ (2003) Estimating subject-specific survival functions under the accelerated failure time model. Biometrika 90:717–23

    Article  MathSciNet  MATH  Google Scholar 

  • Patel K, Williams PL, Seeger JD, McIntosh K, Van Dyke RB, Seage GR et al (2008) Long-term effectiveness of highly active antiretroviral therapy on the survival of children and adolescents with HIV infection: a 10-year follow-up study. Clin Infect Dis 46(4):507–515

    Article  Google Scholar 

  • Robins JM, Hernán MÁ, Brumback B (2000) Marginal structural models and causal inference in epidemiology. Epidemiology 11(5):550–560

    Article  Google Scholar 

  • Rosenbaum PR, Rubin DB (1983a) Assessing sensitivity to an unobserved binary covariate in an observational study with binary outcome. J R Stat Soc 45:212–218

  • Rosenbaum PR, Rubin DB (1983b) The central role of the propensity score in observational studies for causal effects. Biometrika 70(1):41–55

    Article  MathSciNet  MATH  Google Scholar 

  • Rosenbaum PR, Rubin DB (1984) Reducing bias in observational studies using subclassification on the propensity score. J Am Stat Assoc 79(387):516–524

    Article  Google Scholar 

  • Rotnitzky A, Robins J (2005) Inverse probability weighted estimation in survival analysis. Encycl Biostat 4:2619–2625

    Google Scholar 

  • Stuart EA, Lee BK, Leacy FP (2013) Prognostic score-based balance measures can be a useful diagnostic for propensity score methods in comparative effectiveness research. J Clin Epidemiol 66(8):S84–S90

    Article  Google Scholar 

  • Therneau TM (2000) Modeling survival data: extending the Cox model. Springer, New York

    Book  MATH  Google Scholar 

  • Thomsen BL, Keiding N, Altman DG (1991) A note on the calculation of expected survival, illustrated by the survival of liver transplant patients. Stat Med 10(5):733–738

    Article  Google Scholar 

  • Tian L, Cai T, Goetghebeur E, Wei L (2007) Model evaluation based on the sampling distribution of estimated absolute prediction error. Biometrika 94(2):297–311

    Article  MathSciNet  MATH  Google Scholar 

  • Tian L, Cai T, Zhao L, Wei LJ (2012) On the covariate-adjusted estimation for an overall treatment difference with data from a randomized comparative clinical trial. Biostatistics 13(2):256–273

    Article  Google Scholar 

  • van der Laan MJ (2014) Targeted estimation of nuisance parameters to obtain valid statistical inference. Int J Biostat 10(1):29–57

    MathSciNet  Google Scholar 

  • Van Houwelingen J, Putter H (2012) Dynamic prediction in clinical survival analysis. CRC Press, New York

    MATH  Google Scholar 

  • Wood E, Hogg RS, Yip B, Harrigan PR, O’Shaughnessy MV, Montaner JS (2003) Is there a baseline CD4 cell count that precludes a survival response to modern antiretroviral therapy? AIDS 17(5):711–720

    Article  Google Scholar 

  • Xie J, Liu C (2005) Adjusted Kaplan-Meier estimator and log-rank test with inverse probability of treatment weighting for survival data. Stat Med 24(20):3089–3110

    Article  MathSciNet  Google Scholar 

  • Zhang M (2015) Robust methods to improve efficiency and reduce bias in estimating survival curves in randomized clinical trials. Lifetime Data Anal 2014:1–19

    MathSciNet  MATH  Google Scholar 

  • Zhang M, Schaubel DE (2012a) Contrasting treatment-specific survival using double-robust estimators. Stat Med 31(30):4255–4268

    Article  MathSciNet  Google Scholar 

  • Zhang M, Schaubel DE (2012b) Double-robust semiparametric estimator for differences in restricted mean lifetimes in observational studies. Biometrics 68(4):999–1009

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang M, Tsiatis AA, Davidian M (2008) Improving efficiency of inferences in randomized clinical trials using auxiliary covariates. Biometrics 64(3):707–715

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao L, Cai T, Tian L, Uno H, Solomon S, Wei L (2010) Stratifying subjects for treatment selection with censored event time data from a comparative study. Harvard University Biostatistics Working Paper Series, p 122

  • Zhu Y, Coffman DL, Ghosh D (2015) A boosting algorithm for estimating generalized propensity scores with continuous treatments. J Causal Inference 3(1):25–40

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. RAND Corporation, 1776 Main Street, Santa Monica, CA, 90403, USA

    Layla Parast & Beth Ann Griffin

Authors
  1. Layla Parast
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Beth Ann Griffin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Layla Parast.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 324 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Parast, L., Griffin, B.A. Landmark estimation of survival and treatment effects in observational studies. Lifetime Data Anal 23, 161–182 (2017). https://doi.org/10.1007/s10985-016-9358-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10985-016-9358-z

Keywords

Search

Navigation