On hazard ratio estimators by proportional hazards models in matched-pair cohort studies

Background

In matched-pair cohort studies with censored events, the hazard ratio (HR) may be of main interest. However, it is lesser known in epidemiologic literature that the partial maximum likelihood estimator of a common HR conditional on matched pairs is written in a simple form, namely, the ratio of the numbers of two pair-types. Moreover, because HR is a noncollapsible measure and its constancy across matched pairs is a restrictive assumption, marginal HR as “average” HR may be targeted more than conditional HR in analysis.

Methods

Based on its simple expression, we provided an alternative interpretation of the common HR estimator as the odds of the matched-pair analog of C-statistic for censored time-to-event data. Through simulations assuming proportional hazards within matched pairs, the influence of various censoring patterns on the marginal and common HR estimators of unstratified and stratified proportional hazards models, respectively, was evaluated. The methods were applied to a real propensity-score matched dataset from the Rotterdam tumor bank of primary breast cancer.

Results

We showed that stratified models unbiasedly estimated a common HR under the proportional hazards within matched pairs. However, the marginal HR estimator with robust variance estimator lacks interpretation as an “average” marginal HR even if censoring is unconditionally independent to event, unless no censoring occurs or no exposure effect is present. Furthermore, the exposure-dependent censoring biased the marginal HR estimator away from both conditional HR and an “average” marginal HR irrespective of whether exposure effect is present. From the matched Rotterdam dataset, we estimated HR for relapse-free survival of absence versus presence of chemotherapy; estimates (95% confidence interval) were 1.47 (1.18–1.83) for common HR and 1.33 (1.13–1.57) for marginal HR.

Conclusion

The simple expression of the common HR estimator would be a useful summary of exposure effect, which is less sensitive to censoring patterns than the marginal HR estimator. The common and the marginal HR estimators, both relying on distinct assumptions and interpretations, are complementary alternatives for each other.

     

Variance reduction in randomised trials by inverse probability weighting using the propensity score

In individually randomised controlled trials, adjustment for baseline characteristics is often undertaken to increase precision of the treatment effect estimate. This is usually performed using covariate adjustment in outcome regression models. An alternative method of adjustment is to use inverse probability‐of‐treatment weighting (IPTW), on the basis of estimated propensity scores. We calculate the large‐sample marginal variance of IPTW estimators of the mean difference for continuous outcomes, and risk difference, risk ratio or odds ratio for binary outcomes. We show that IPTW adjustment always increases the precision of the treatment effect estimate. For continuous outcomes, we demonstrate that the IPTW estimator has the same large‐sample marginal variance as the standard analysis of covariance estimator. However, ignoring the estimation of the propensity score in the calculation of the variance leads to the erroneous conclusion that the IPTW treatment effect estimator has the same variance as an unadjusted estimator; thus, it is important to use a variance estimator that correctly takes into account the estimation of the propensity score. The IPTW approach has particular advantages when estimating risk differences or risk ratios. In this case, non‐convergence of covariate‐adjusted outcome regression models frequently occurs. Such problems can be circumvented by using the IPTW adjustment approach. © 2013 The authors. Statistics in Medicine published by John Wiley & Sons, Ltd.     

Variance estimation when using inverse probability of treatment weighting (IPTW) with survival analysis

Propensity score methods are used to reduce the effects of observed confounding when using observational data to estimate the effects of treatments or exposures. A popular method of using the propensity score is inverse probability of treatment weighting (IPTW). When using this method, a weight is calculated for each subject that is equal to the inverse of the probability of receiving the treatment that was actually received. These weights are then incorporated into the analyses to minimize the effects of observed confounding. Previous research has found that these methods result in unbiased estimation when estimating the effect of treatment on survival outcomes. However, conventional methods of variance estimation were shown to result in biased estimates of standard error. In this study, we conducted an extensive set of Monte Carlo simulations to examine different methods of variance estimation when using a weighted Cox proportional hazards model to estimate the effect of treatment. We considered three variance estimation methods: (i) a naïve model‐based variance estimator; (ii) a robust sandwich‐type variance estimator; and (iii) a bootstrap variance estimator. We considered estimation of both the average treatment effect and the average treatment effect in the treated. We found that the use of a bootstrap estimator resulted in approximately correct estimates of standard errors and confidence intervals with the correct coverage rates. The other estimators resulted in biased estimates of standard errors and confidence intervals with incorrect coverage rates. Our simulations were informed by a case study examining the effect of statin prescribing on mortality. © 2016 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd.     

Incorporating prognostic factors into causal estimators: A comparison of methods for randomised controlled trials with a time‐to‐event outcome

In randomised controlled trials, the effect of treatment on those who comply with allocation to active treatment can be estimated by comparing their outcome to those in the comparison group who would have complied with active treatment had they been allocated to it. We compare three estimators of the causal effect of treatment on compliers when this is a parameter in a proportional hazards model and quantify the bias due to omitting baseline prognostic factors. Causal estimates are found directly by maximising a novel partial likelihood; based on a structural proportional hazards model; and based on a ‘corrected dataset’ derived after fitting a rank‐preserving structural failure time model. Where necessary, we extend these methods to incorporate baseline covariates. Comparisons use simulated data and a real data example. Analysing the simulated data, we found that all three methods are accurate when an important covariate was included in the proportional hazards model (maximum bias 5.4%). However, failure to adjust for this prognostic factor meant that causal treatment effects were underestimated (maximum bias 11.4%), because estimators were based on a misspecified marginal proportional hazards model. Analysing the real data example, we found that adjusting causal estimators is important to correct for residual imbalances in prognostic factors present between trial arms after randomisation. Our results show that methods of estimating causal treatment effects for time‐to‐event outcomes should be extended to incorporate covariates, thus providing an informative compliment to the corresponding intention‐to‐treat analysis. Copyright © 2012 John Wiley & Sons, Ltd.     

The analysis of multivariate interval-censored survival data

This paper considers a marginal approach for the analysis of the effect of covariates on multivariate interval-censored survival data.Interval censoring of multivariate events can occur when the events are not directly observable but are detected by periodically performing clinical examinations or laboratory tests. The method assumes the marginal distribution for each event is based on a discrete analogue of the proportional hazards model for interval-censored data. A robust estimator for the covariance matrix is developed that accounts for the correlation between events. A simulation study comparing the performance of this method and a midpoint imputation approach indicates the parameter estimates from the proposed method are less biased. Furthermore, even when the events are only modestly correlated, ignoring the correlation can result in erroneous variance estimators. The method is illustrated using data from an ongoing clinical trial involving subjects with systemic lupus erythematosus.     

Simulating survival data with predefined censoring rates for proportional hazards models

The proportional hazard model is one of the most important statistical models used in medical research involving time‐to‐event data. Simulation studies are routinely used to evaluate the performance and properties of the model and other alternative statistical models for time‐to‐event outcomes under a variety of situations. Complex simulations that examine multiple situations with different censoring rates demand approaches that can accommodate this variety. In this paper, we propose a general framework for simulating right‐censored survival data for proportional hazards models by simultaneously incorporating a baseline hazard function from a known survival distribution, a known censoring time distribution, and a set of baseline covariates. Specifically, we present scenarios in which time to event is generated from exponential or Weibull distributions and censoring time has a uniform or Weibull distribution. The proposed framework incorporates any combination of covariate distributions. We describe the steps involved in nested numerical integration and using a root‐finding algorithm to choose the censoring parameter that achieves predefined censoring rates in simulated survival data. We conducted simulation studies to assess the performance of the proposed framework. We demonstrated the application of the new framework in a comprehensively designed simulation study. We investigated the effect of censoring rate on potential bias in estimating the conditional treatment effect using the proportional hazard model in the presence of unmeasured confounding variables. Copyright © 2016 John Wiley & Sons, Ltd.     

Analysis of accelerated failure time data with dependent censoring using auxiliary variables via nonparametric multiple imputation

We consider the situation of estimating the marginal survival distribution from censored data subject to dependent censoring using auxiliary variables. We had previously developed a nonparametric multiple imputation approach. The method used two working proportional hazards (PH) models, one for the event times and the other for the censoring times, to define a nearest neighbor imputing risk set. This risk set was then used to impute failure times for censored observations. Here, we adapt the method to the situation where the event and censoring times follow accelerated failure time models and propose to use the Buckley–James estimator as the two working models. Besides studying the performances of the proposed method, we also compare the proposed method with two popular methods for handling dependent censoring through the use of auxiliary variables, inverse probability of censoring weighted and parametric multiple imputation methods, to shed light on the use of them. In a simulation study with time‐independent auxiliary variables, we show that all approaches can reduce bias due to dependent censoring. The proposed method is robust to misspecification of either one of the two working models and their link function. This indicates that a working proportional hazards model is preferred because it is more cumbersome to fit an accelerated failure time model. In contrast, the inverse probability of censoring weighted method is not robust to misspecification of the link function of the censoring time model. The parametric imputation methods rely on the specification of the event time model. The approaches are applied to a prostate cancer dataset. Copyright © 2015 John Wiley & Sons, Ltd.     

Meta‐analysis of time‐to‐event outcomes from randomized trials using restricted mean survival time: application to individual participant data

Meta‐analysis of time‐to‐event outcomes using the hazard ratio as a treatment effect measure has an underlying assumption that hazards are proportional. The between‐arm difference in the restricted mean survival time is a measure that avoids this assumption and allows the treatment effect to vary with time. We describe and evaluate meta‐analysis based on the restricted mean survival time for dealing with non‐proportional hazards and present a diagnostic method for the overall proportional hazards assumption. The methods are illustrated with the application to two individual participant meta‐analyses in cancer. The examples were chosen because they differ in disease severity and the patterns of follow‐up, in order to understand the potential impacts on the hazards and the overall effect estimates. We further investigate the estimation methods for restricted mean survival time by a simulation study. Copyright © 2015 John Wiley & Sons, Ltd.     

Robust versus consistent variance estimators in marginal structural Cox models

In survival analyses, inverse‐probability‐of‐treatment (IPT) and inverse‐probability‐of‐censoring (IPC) weighted estimators of parameters in marginal structural Cox models are often used to estimate treatment effects in the presence of time‐dependent confounding and censoring. In most applications, a robust variance estimator of the IPT and IPC weighted estimator is calculated leading to conservative confidence intervals. This estimator assumes that the weights are known rather than estimated from the data. Although a consistent estimator of the asymptotic variance of the IPT and IPC weighted estimator is generally available, applications and thus information on the performance of the consistent estimator are lacking. Reasons might be a cumbersome implementation in statistical software, which is further complicated by missing details on the variance formula. In this paper, we therefore provide a detailed derivation of the variance of the asymptotic distribution of the IPT and IPC weighted estimator and explicitly state the necessary terms to calculate a consistent estimator of this variance. We compare the performance of the robust and consistent variance estimators in an application based on routine health care data and in a simulation study. The simulation reveals no substantial differences between the 2 estimators in medium and large data sets with no unmeasured confounding, but the consistent variance estimator performs poorly in small samples or under unmeasured confounding, if the number of confounders is large. We thus conclude that the robust estimator is more appropriate for all practical purposes.     

Dealing with competing risks: testing covariates and calculating sample size

It is universally agreed that Kaplan-Meier estimates overestimate the probability of the event of interest in the presence of competing risks. Kalbfleisch and Prentice recommend using the cumulative incidence as an estimate of the probability of an event of interest. However, there is no consensus on how to test the effect of a covariate in the presence of competing risks. Using simulations, this paper illustrates that the Cox proportional hazards model gives valid results when employed in testing the effect of a covariate on the hazard rate and when estimating the hazard ratio. A method to calculate the sample size for testing the effect of a covariate on outcome in the presence of competing risks is also provided.     

A simulation study of finite-sample properties of marginal structural Cox proportional hazards models

Motivated by a previously published study of HIV treatment, we simulated data subject to time‐varying confounding affected by prior treatment to examine some finite‐sample properties of marginal structural Cox proportional hazards models. We compared (a) unadjusted, (b) regression‐adjusted, (c) unstabilized, and (d) stabilized marginal structural (inverse probability‐of‐treatment [IPT] weighted) model estimators of effect in terms of bias, standard error, root mean squared error (MSE), and 95% confidence limit coverage over a range of research scenarios, including relatively small sample sizes and 10 study assessments. In the base‐case scenario resembling the motivating example, where the true hazard ratio was 0.5, both IPT‐weighted analyses were unbiased, whereas crude and adjusted analyses showed substantial bias towards and across the null. Stabilized IPT‐weighted analyses remained unbiased across a range of scenarios, including relatively small sample size; however, the standard error was generally smaller in crude and adjusted models. In many cases, unstabilized weighted analysis showed a substantial increase in standard error compared with other approaches. Root MSE was smallest in the IPT‐weighted analyses for the base‐case scenario. In situations where time‐varying confounding affected by prior treatment was absent, IPT‐weighted analyses were less precise and therefore had greater root MSE compared with adjusted analyses. The 95% confidence limit coverage was close to nominal for all stabilized IPT‐weighted but poor in crude, adjusted, and unstabilized IPT‐weighted analysis. Under realistic scenarios, marginal structural Cox proportional hazards models performed according to expectations based on large‐sample theory and provided accurate estimates of the hazard ratio. Copyright © 2012 John Wiley & Sons, Ltd.     

Joint models for efficient estimation in proportional hazards regression models

In survival studies, information lost through censoring can be partially recaptured through repeated measures data which are predictive of survival. In addition, such data may be useful in removing bias in survival estimates, due to censoring which depends upon the repeated measures. Here we investigate joint models for survival T and repeated measurements Y, given a vector of covariates Z. Mixture models indexed as f (T/Z) f (Y/T,Z) are well suited for assessing covariate effects on survival time. Our objective is efficiency gains, using non-parametric models for Y in order to avoid introducing bias by misspecification of the distribution for Y. We model (T/Z) as a piecewise exponential distribution with proportional hazards covariate effect. The component (Y/T,Z) has a multinomial model. The joint likelihood for survival and longitudinal data is maximized, using the EM algorithm. The estimate of covariate effect is compared to the estimate based on the standard proportional hazards model and an alternative joint model based estimate. We demonstrate modest gains in efficiency when using the joint piecewise exponential joint model. In a simulation, the estimated efficiency gain over the standard proportional hazards model is 6.4 per cent. In clinical trial data, the estimated efficiency gain over the standard proportional hazards model is 10.2 per cent.     

Estimating restricted mean treatment effects with stacked survival models

The difference in restricted mean survival times between two groups is a clinically relevant summary measure. With observational data, there may be imbalances in confounding variables between the two groups. One approach to account for such imbalances is estimating a covariate‐adjusted restricted mean difference by modeling the covariate‐adjusted survival distribution and then marginalizing over the covariate distribution. Because the estimator for the restricted mean difference is defined by the estimator for the covariate‐adjusted survival distribution, it is natural to expect that a better estimator of the covariate‐adjusted survival distribution is associated with a better estimator of the restricted mean difference. We therefore propose estimating restricted mean differences with stacked survival models. Stacked survival models estimate a weighted average of several survival models by minimizing predicted error. By including a range of parametric, semi‐parametric, and non‐parametric models, stacked survival models can robustly estimate a covariate‐adjusted survival distribution and, therefore, the restricted mean treatment effect in a wide range of scenarios. We demonstrate through a simulation study that better performance of the covariate‐adjusted survival distribution often leads to better mean squared error of the restricted mean difference although there are notable exceptions. In addition, we demonstrate that the proposed estimator can perform nearly as well as Cox regression when the proportional hazards assumption is satisfied and significantly better when proportional hazards is violated. Finally, the proposed estimator is illustrated with data from the United Network for Organ Sharing to evaluate post‐lung transplant survival between large‐volume and small‐volume centers. Copyright © 2016 John Wiley & Sons, Ltd.     

Marginal estimation for multi-stage models: waiting time distributions and competing risks analyses.

We provide non-parametric estimates of the marginal cumulative distribution of stage occupation times (waiting times) and non-parametric estimates of marginal cumulative incidence function (proportion of persons who leave stage j for stage j' within time t of entering stage j) using right-censored data from a multi-stage model. We allow for stage and path dependent censoring where the censoring hazard for an individual may depend on his or her natural covariate history such as the collection of stages visited before the current stage and their occupation times. Additional external time dependent covariates that may induce dependent censoring can also be incorporated into our estimates, if available. Our approach requires modelling the censoring hazard so that an estimate of the integrated censoring hazard can be used in constructing the estimates of the waiting times distributions. For this purpose, we propose the use of an additive hazard model which results in very flexible (robust) estimates. Examples based on data from burn patients and simulated data with tracking are also provided to demonstrate the performance of our estimators.     

Power and sample size calculation for log‐rank test with a time lag in treatment effect

The log‐rank test is the most powerful non‐parametric test for detecting a proportional hazards alternative and thus is the most commonly used testing procedure for comparing time‐to‐event distributions between different treatments in clinical trials. When the log‐rank test is used for the primary data analysis, the sample size calculation should also be based on the test to ensure the desired power for the study. In some clinical trials, the treatment effect may not manifest itself right after patients receive the treatment. Therefore, the proportional hazards assumption may not hold. Furthermore, patients may discontinue the study treatment prematurely and thus may have diluted treatment effect after treatment discontinuation. If a patient's treatment termination time is independent of his/her time‐to‐event of interest, the termination time can be treated as a censoring time in the final data analysis. Alternatively, we may keep collecting time‐to‐event data until study termination from those patients who discontinued the treatment and conduct an intent‐to‐treat analysis by including them in the original treatment groups. We derive formulas necessary to calculate the asymptotic power of the log‐rank test under this non‐proportional hazards alternative for the two data analysis strategies. Simulation studies indicate that the formulas provide accurate power for a variety of trial settings. A clinical trial example is used to illustrate the application of the proposed methods. Copyright © 2009 John Wiley & Sons, Ltd.     

Estimating net survival: the importance of allowing for informative censoring

Net survival, the one that would be observed if cancer were the only cause of death, is the most appropriate indicator to compare cancer mortality between areas or countries. Several parametric and non-parametric methods have been developed to estimate net survival, particularly when the cause of death is unknown. These methods are based either on the relative survival ratio or on the additive excess hazard model, the latter using the general population mortality hazard to estimate the excess mortality hazard (the hazard related to net survival). The present work used simulations to compare estimator abilities to estimate net survival in different settings such as the presence/absence of an age effect on the excess mortality hazard or on the potential time of follow-up, knowing that this covariate has an effect on the general population mortality hazard too. It showed that when age affected the excess mortality hazard, most estimators, including specific survival, were biased. Only two estimators were appropriate to estimate net survival. The first is based on a multivariable excess hazard model that includes age as covariate. The second is non-parametric and is based on the inverse probability weighting. These estimators take differently into account the informative censoring induced by the expected mortality process. The former offers great flexibility whereas the latter requires neither the assumption of a specific distribution nor a model-building strategy. Because of its simplicity and availability in commonly used software, the nonparametric estimator should be considered by cancer registries for population-based studies.     

Regression models for the restricted residual mean life for right‐censored and left‐truncated data

The hazard ratios resulting from a Cox's regression hazards model are hard to interpret and to be converted into prolonged survival time. As the main goal is often to study survival functions, there is increasing interest in summary measures based on the survival function that are easier to interpret than the hazard ratio; the residual mean time is an important example of those measures. However, because of the presence of right censoring, the tail of the survival distribution is often difficult to estimate correctly. Therefore, we consider the restricted residual mean time, which represents a partial area under the survival function, given any time horizon τ, and is interpreted as the residual life expectancy up to τ of a subject surviving up to time t. We present a class of regression models for this measure, based on weighted estimating equations and inverse probability of censoring weighted estimators to model potential right censoring. Furthermore, we show how to extend the models and the estimators to deal with delayed entries. We demonstrate that the restricted residual mean life estimator is equivalent to integrals of Kaplan–Meier estimates in the case of simple factor variables. Estimation performance is investigated by simulation studies. Using real data from Danish Monitoring Cardiovascular Risk Factor Surveys, we illustrate an application to additive regression models and discuss the general assumption of right censoring and left truncation being dependent on covariates. Copyright © 2017 John Wiley & Sons, Ltd.     

Estimation in multi‐arm two‐stage trials with treatment selection and time‐to‐event endpoint

We consider estimation of treatment effects in two‐stage adaptive multi‐arm trials with a common control. The best treatment is selected at interim, and the primary endpoint is modeled via a Cox proportional hazards model. The maximum partial‐likelihood estimator of the log hazard ratio of the selected treatment will overestimate the true treatment effect in this case. Several methods for reducing the selection bias have been proposed for normal endpoints, including an iterative method based on the estimated conditional selection biases and a shrinkage approach based on empirical Bayes theory. We adapt these methods to time‐to‐event data and compare the bias and mean squared error of all methods in an extensive simulation study and apply the proposed methods to reconstructed data from the FOCUS trial. We find that all methods tend to overcorrect the bias, and only the shrinkage methods can reduce the mean squared error. © 2017 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.     

Estimating cumulative incidence functions in competing risks data with dependent left-truncation

Both delayed study entry (left-truncation) and competing risks are common phenomena in observational time-to-event studies. For example, in studies conducted by Teratology Information Services (TIS) on adverse drug reactions during pregnancy, the natural time scale is gestational age, but women enter the study after time origin and upon contact with the service. Competing risks are present, because an elective termination may be precluded by a spontaneous abortion. If left-truncation is entirely random, the Aalen-Johansen estimator is the canonical estimator of the cumulative incidence functions of the competing events. If the assumption of random left-truncation is in doubt, we propose a new semiparametric estimator of the cumulative incidence function. The dependence between entry time and time-to-event is modeled using a cause-specific Cox proportional hazards model and the marginal (unconditional) estimates are derived via inverse probability weighting arguments. We apply the new estimator to data about coumarin usage during pregnancy. Here, the concern is that the cause-specific hazard of experiencing an induced abortion may depend on the time when seeking advice by a TIS, which also is the time of left-truncation or study entry. While the aims of counseling by a TIS are to reduce the rate of elective terminations based on irrational overestimation of drug risks and to lead to better and safer medical treatment of maternal disease, it is conceivable that women considering an induced abortion are more likely to seek counseling. The new estimator is also evaluated in extensive simulation studies and found preferable compared to the Aalen-Johansen estimator in non–misspecified scenarios and to at least provide for a sensitivity analysis otherwise.