A two-part model for censored medical cost data

The two-part model is often used to analyse medical cost data which contain a large proportion of zero cost and are highly skewed with some large costs. The total medical costs over a period of time are often censored due to incomplete follow-up, making the analysis difficult as the censoring can be informative. We propose to apply the inverse probability weighting method on a two-part model to analyse right-censored cumulative medical costs with informative censoring. We also introduce a set of simple functionals based on the intermediate cost history to be applied with the efficiency augmentation technique. In addition, we propose a practical model-checking technique based on the cumulative residuals. Simulation studies are conducted to evaluate the finite sample performance of the proposed method. We use a data set on the cardiovascular disease (CVD)-related Medicare costs to illustrate our proposed method.     

A shared random effects model for censored medical costs and mortality

In this paper, we propose a model for medical costs recorded at regular time intervals, e.g. every month, as repeated measures in the presence of a terminating event, such as death. Prior models have related monthly medical costs to time since entry, with extra costs at the final observations at the time of death. Our joint model for monthly medical costs and survival time incorporates two important new features. First, medical cost and survival may be correlated because more 'frail' patients tend to accumulate medical costs faster and die earlier. A joint random effects model is proposed to account for the correlation between medical costs and survival by a shared random effect. Second, monthly medical costs usually increase during the time period prior to death because of the intensive care for dying patients. We present a method for estimating the pattern of cost prior to death, which is applicable if the pattern can be characterized as an additive effect that is limited to a fixed time interval, say b units of time before death. This 'turn back time' method for censored observations censors cost data b units of time before the actual censoring time, while keeping the actual censoring time for the survival data. Time-dependent covariates can be included. Maximum likelihood estimation and inference are carried out through a Monte Carlo EM algorithm with a Metropolis-Hastings sampler in the E-step. An analysis of monthly outpatient EPO medical cost data for dialysis patients is presented to illustrate the proposed methods.     

Joint modeling of medical cost and survival in complex sample surveys

Medical cost data are typically highly skewed to the right with a large proportion of zero costs. It is also common for these data to be censored because of incomplete follow‐up and death. In the case of censoring due to death, it is important to consider the potential dependence between cost and survival. This association can occur because patients who incur a greater amount of medical cost tend to be frailer and hence are more likely to die. To handle this informative censoring issue, joint modeling of cost and survival with shared random effects has been proposed. In this paper, we extend this joint modeling approach to handle a final feature of many medical cost data sets, i.e., Specifically, the fact that data were obtained via a complex survey design. Specifically, we extend the joint model by incorporating the sample weights when estimating the parameters and using the Taylor series linearization approach when calculating the standard errors. We use a simulation study to compare the joint modeling approach with and without these adjustments. The simulation study shows that parameter estimates can be seriously biased when information about the complex survey design is ignored. It also shows that standard errors based on the Taylor series linearization approach provide satisfactory confidence interval coverage. The proposed joint model is applied to monthly hospital costs obtained from the 2004 National Long Term Care Survey. Copyright © 2012 John Wiley & Sons, Ltd.     

Regression-adjusted matching and double-robust methods for estimating average treatment effects in health economic evaluation

Regression, propensity score (PS) and double-robust (DR) methods can reduce selection bias when estimating average treatment effects (ATEs). Economic evaluations of health care interventions exemplify complex data structures, in that the covariate–endpoint relationships tend to be highly non-linear, with highly skewed cost and health outcome endpoints. When either the regression or PS model is correct, DR methods can provide unbiased, efficient estimates of ATEs, but generally the specification of both models is unknown. Regression-adjusted matching can also protect against bias from model misspecification, but has not been compared to DR methods. This paper compares regression-adjusted matching to selected DR methods (weighted regression and augmented inverse probability of treatment weighting) as well as to regression and PS methods for addressing selection bias in cost-effectiveness analyses (CEA). We contrast the methods in a CEA of a pharmaceutical intervention, where there are extreme estimated PSs, hence unstable inverse probability of treatment (IPT) weights. The case study motivates a simulation which considers settings with functional form misspecification in the PS and endpoint regression models (e.g. cost model with log instead of identity link), stable and unstable PS weights. We find that in the realistic setting of unstable IPT weights and misspecifications to the PS and regression models, regression-adjusted matching reports less bias than DR methods. We conclude that regression-adjusted matching is a relatively robust method for estimating ATEs in applications with complex data structures exemplified by CEA.     

Targeted learning in real‐world comparative effectiveness research with time‐varying interventions

In comparative effectiveness research (CER), often the aim is to contrast survival outcomes between exposure groups defined by time‐varying interventions. With observational data, standard regression analyses (e.g., Cox modeling) cannot account for time‐dependent confounders on causal pathways between exposures and outcome nor for time‐dependent selection bias that may arise from informative right censoring. Inverse probability weighting (IPW) estimation to fit marginal structural models (MSMs) has commonly been applied to properly adjust for these expected sources of bias in real‐world observational studies. We describe the application and performance of an alternate estimation approach in such a study. The approach is based on the recently proposed targeted learning methodology and consists in targeted minimum loss‐based estimation (TMLE) with super learning (SL) within a nonparametric MSM. The evaluation is based on the analysis of electronic health record data with both IPW estimation and TMLE to contrast cumulative risks under four more or less aggressive strategies for treatment intensification in adults with type 2 diabetes already on 2+ oral agents or basal insulin. Results from randomized experiments provide a surrogate gold standard to validate confounding and selection bias adjustment. Bootstrapping is used to validate analytic estimation of standard errors. This application does the following: (1) establishes the feasibility of TMLE in real‐world CER based on large healthcare databases; (2) provides evidence of proper confounding and selection bias adjustment with TMLE and SL; and (3) motivates their application for improving estimation efficiency. Claims are reinforced with a simulation study that also illustrates the double‐robustness property of TMLE. Copyright © 2014 John Wiley & Sons, Ltd.     

Non-parametric methods for recurrent event data with informative and non-informative censorings

Recurrent event data are commonly encountered in health-related longitudinal studies. In this paper time-to-events models for recurrent event data are studied with non-informative and informative censorings. In statistical literature, the risk set methods have been confirmed to serve as an appropriate and efficient approach for analysing recurrent event data when censoring is non-informative. This approach produces biased results, however, when censoring is informative for the time-to-events outcome data. We compare the risk set methods with alternative non-parametric approaches which are robust subject to informative censoring. In particular, non-parametric procedures for the estimation of the cumulative occurrence rate function (CORF) and the occurrence rate function (ORF) are discussed in detail. Simulation and an analysis of data from the AIDS Link to Intravenous Experiences Cohort Study is presented.     

Informative censoring in relative survival

With changing the age distribution at the time of cancer diagnosis, the administrative censoring due to study end may be informative. This problem has been mentioned frequently in the relative survival field, and an estimator aiming to correct this problem has been developed. In this paper, we review the existing methods for estimation in relative survival, demonstrate their deficiencies, and propose weighting to correct both the recently introduced net survival estimator and the Ederer I estimator. Using simulations and real cancer registry data, we evaluate the magnitude of the informative censoring problem. We clarify the assumptions behind the reviewed methods and provide guidance to their usage in practice. Copyright © 2013 John Wiley & Sons, Ltd.     

Modeling survival data with informative cluster size

Analysis of clustered data focusing on inference of the marginal distribution may be problematic when the risk of the outcome is related to the cluster size, termed as informative cluster size. In the absence of censoring, Hoffman et al. proposed a within-cluster resampling method, which is asymptotically equivalent to a weighted generalized estimating equations score equation. We investigate the estimation of the marginal distribution for multivariate survival data with informative cluster size using cluster-weighted Weibull and Cox proportional hazards models. The cluster-weighted Cox model can be implemented using standard software. Simulation results demonstrate that the proposed methods produce unbiased parameter estimation in the presence of informative cluster size. To illustrate the proposed approach, we analyze survival data from a lymphatic filariasis study in Recife, Brazil.     

Testing and Correcting for Non-Random Selection Bias due to Censoring: An Application to Medical Costs

Censoring is a common problem with medical cost data. Methods from traditional survival analysis are not directly applicable to estimate medical costs since patients accumulate costs with different rate functions over time, leading to negatively biased estimates. Heckman's two-step estimator results in large variances when identical explanatory variables that influence selection are included in the structural equation, i.e. when there are no exclusion restrictions. This paper provides a systematic treatment of the correction for nonrandom sample selection bias of medical cost data where the selection rule is described by a censored regression model. The proposed method first uses the duration of time a patient is tracked for the selection, rather than a binary variable, namely whether or not the duration is censored. Second, using Tobit residuals instead of the inverse Mills ratio in the structural equation allows us to decrease large variances introduced by the Heckman model when there are no exclusion restrictions. We show that the resulting estimators are consistent and asymptotically normal. Simulation studies confirmed our results. Moreover, we derive a simple test to determine possible sample selection bias due to censoring. Data from a study on the medical cost of breast, prostate, colon, and lung cancer is used as an application of the method.     

Estimation of average treatment effects among multiple treatment groups by using an ensemble approach

In observational studies, generalized propensity score (GPS)–based statistical methods, such as inverse probability weighting (IPW) and doubly robust (DR) method, have been proposed to estimate the average treatment effect (ATE) among multiple treatment groups. In this article, we investigate the GPS-based statistical methods to estimate treatment effects from two aspects. The first aspect of our investigation is to obtain an optimal GPS estimation method among four competing GPS estimation methods by using a rank aggregation approach. We further examine whether the optimal GPS-based IPW and DR methods would improve the performance for estimating ATE. It is well known that the DR method is consistent if either the GPS or the outcome models are correctly specified. The second aspect of our investigation is to examine whether the DR method could be improved if we ensemble outcome models. To that end, bootstrap method and rank aggregation method are used to obtain the ensemble optimal outcome model from several competing outcome models, and the resulting outcome model is incorporated into the DR method, resulting in an ensemble DR (enDR) method. Extensive simulation results indicate that the enDR method provides the best performance in estimating the ATE regardless of the method used for estimating GPS. We illustrate our methods using the MarketScan healthcare insurance claims database to examine the treatment effects among three different bones and substitutes used for spinal fusion surgeries. We draw conclusions based on the estimates from the enDR method coupled with the optimal GPS estimation method.     

Regression analysis of incomplete medical cost data

The accumulation of medical cost over time for each subject is an increasing stochastic process defined up to the instant of death. The stochastic structure of this process is complex. In most applications, the process can only be observed at a limited number of time points. Furthermore, the process is subject to right censoring so that it is unobservable after the censoring time. These special features of the medical cost data, especially the presence of death and censoring, pose major challenges in the construction of plausible statistical models and the development of the corresponding inference procedures. In this paper, we propose several classes of regression models which formulate the effects of possibly time-dependent covariates on the marginal mean of cost accumulation in the presence of death or on the conditional means of cost accumulation given specific survival patterns. We then develop estimating equations for these models by combining the approach of generalized estimating equations for longitudinal data with the inverse probability of censoring weighting technique. The resultant estimators are shown to be consistent and asymptotically normal with simple variance estimators. Simulation studies indicate that the proposed inference procedures behave well in practical situations. An application to data taken from a large cancer study reveals that the Medicare enrollees who are diagnosed with less aggressive ovarian cancer tend to accumulate medical cost at lower rates than those with more aggressive disease, but tend to have higher lifetime costs because they live longer.     

How serious is bias in effect estimation in randomised trials with survival data given risk heterogeneity and informative censoring?

It is often assumed that randomisation will prevent bias in estimation of treatment effects from clinical trials, but this is not true of the semiparametric Proportional Hazards model for survival data when there is underlying risk heterogeneity. Here, a new formula is proposed for estimation of this bias, improving on a previous formula through ease of use and clarity regarding the role of the mid‐study cumulative hazard rate, shown to be an important factor for the bias magnitude. Informative censoring (IC) is recognised as a source of bias. Here, work on selection effects among survivors due to risk heterogeneity is extended to include IC. A new formula shows that bias in causal effect estimation under IC has two sources: one consequent on heterogeneity and one from the additional impact of IC. The formula provides new insights not previously shown: there may less bias under IC than when there is no IC and even, in principle, zero bias. When tested against simulated data, the new formulae are shown to be very accurate for prediction of bias in Proportional Hazards and accelerated failure time analyses which ignore heterogeneity. These data are also used to investigate the performance of accelerated failure time models which explicitly model risk heterogeneity (‘frailty models’) and G estimation. These methods remove bias when there is simple censoring but not with informative censoring when they may lead to overestimation of treatment effects. The new formulae may be used to help researchers judge the possible extent of bias in past studies. Copyright © 2017 John Wiley & Sons, Ltd.     

Analyzing medical costs with time‐dependent treatment: The nested g‐formula

As medical expenses continue to rise, methods to properly analyze cost outcomes are becoming of increasing relevance when seeking to compare average costs across treatments. Inverse probability weighted regression models have been developed to address the challenge of cost censoring in order to identify intent‐to‐treat effects (i.e., to compare mean costs between groups on the basis of their initial treatment assignment, irrespective of any subsequent changes to their treatment status). In this paper, we describe a nested g‐computation procedure that can be used to compare mean costs between two or more time‐varying treatment regimes. We highlight the relative advantages and limitations of this approach when compared with existing regression‐based models. We illustrate the utility of this approach as a means to inform public policy by applying it to a simulated data example motivated by costs associated with cancer treatments. Simulations confirm that inference regarding intent‐to‐treat effects versus the joint causal effects estimated by the nested g‐formula can lead to markedly different conclusions regarding differential costs. Therefore, it is essential to prespecify the desired target of inference when choosing between these two frameworks. The nested g‐formula should be considered as a useful, complementary tool to existing methods when analyzing cost outcomes.     

Regression with a right-censored predictor using inverse probability weighting methods

In a longitudinal study, measures of key variables might be incomplete or partially recorded due to drop-out, loss to follow-up, or early termination of the study occurring before the advent of the event of interest. In this paper, we focus primarily on the implementation of a regression model with a randomly censored predictor. We examine, particularly, the use of inverse probability weighting methods in a generalized linear model (GLM), when the predictor of interest is right-censored, to adjust for censoring. To improve the performance of the complete-case analysis and prevent selection bias, we consider three different weighting schemes: inverse censoring probability weights, Kaplan-Meier weights, and Cox proportional hazards weights. We use Monte Carlo simulation studies to evaluate and compare the empirical properties of different weighting estimation methods. Finally, we apply these methods to the Framingham Heart Study data as an illustrative example to estimate the relationship between age of onset of a clinically diagnosed cardiovascular event and low-density lipoprotein among cigarette smokers.     

Methods for the analysis of informatively censored longitudinal data.

This paper describes the problem of informative censoring in longitudinal studies where the primary outcome is rate of change in a continuous variable. Standard approaches based on the linear random effects model are valid only when the data are missing in a non-ignorable fashion. Informative censoring, which is a special type of non-ignorably missing data, occurs when the probability of early termination is related to an individual subject's true rate of change. When present, informative censoring causes bias in standard likelihood-based analyses, as well as in weighted averages of individual least-squares slopes. This paper reviews several methods proposed by others for analysis of informatively censored longitudinal data, and outlines a new approach based on a log-normal survival model. Maximum likelihood estimates may be obtained via the EM algorithm. Advantages of this approach are that it allows general unbalanced data caused by staggered entry and unequally-timed visits, it utilizes all available data, including data from patients with only a single measurement, and it provides a unified method for estimating all model parameters. Issues related to study design when informative censoring may occur are also discussed.     

Joint modeling of recurrent events and a terminal event adjusted for zero inflation and a matched design

In longitudinal studies, matched designs are often employed to control the potential confounding effects in the field of biomedical research and public health. Because of clinical interest, recurrent time‐to‐event data are captured during the follow‐up. Meanwhile, the terminal event of death is always encountered, which should be taken into account for valid inference because of informative censoring. In some scenarios, a certain large portion of subjects may not have any recurrent events during the study period due to nonsusceptibility to events or censoring; thus, the zero‐inflated nature of data should be considered in analysis. In this paper, a joint frailty model with recurrent events and death is proposed to adjust for zero inflation and matched designs. We incorporate 2 frailties to measure the dependency between subjects within a matched pair and that among recurrent events within each individual. By sharing the random effects, 2 event processes of recurrent events and death are dependent with each other. The maximum likelihood based approach is applied for parameter estimation, where the Monte Carlo expectation‐maximization algorithm is adopted, and the corresponding R program is developed and available for public usage. In addition, alternative estimation methods such as Gaussian quadrature (PROC NLMIXED) and a Bayesian approach (PROC MCMC) are also considered for comparison to show our method's superiority. Extensive simulations are conducted, and a real data application on acute ischemic studies is provided in the end.     

On the use of survival analysis techniques to estimate medical care costs.

Measurement of treatment costs is important in the evaluation of medical interventions. Accurate cost estimation is problematic, when cost records are incomplete. Methods from the survival analysis literature have been proposed for estimating costs using available data. In this article, we clarify assumptions necessary for validity of these techniques. We demonstrate how assumptions needed for valid survival analysis may be violated when these methods are applied to cost estimation. Our observations are confirmed through simulations and empirical data analysis. We conclude that survival analysis approaches are not generally appropriate for the analysis of medical costs and review several valid alternatives.     

Bootstrap confidence intervals for medical costs with censored observations

Medical costs data with administratively censored observations often arise in cost-effectiveness studies of treatments for life-threatening diseases. Mean of medical costs incurred from the start of a treatment until death or a certain time point after the implementation of treatment is frequently of interest. In many situations, due to the skewed nature of the cost distribution and non-uniform rate of cost accumulation over time, the currently available normal approximation confidence interval has poor coverage accuracy. In this paper, we propose a bootstrap confidence interval for the mean of medical costs with censored observations. In simulation studies, we show that the proposed bootstrap confidence interval had much better coverage accuracy than the normal approximation one when medical costs had a skewed distribution. When there is light censoring on medical costs (< or =25 per cent), we found that the bootstrap confidence interval based on the simple weighted estimator is preferred due to its simplicity and good coverage accuracy. For heavily censored cost data (censoring rate > or =30 per cent) with larger sample sizes (n > or =200), the bootstrap confidence intervals based on the partitioned estimator has superior performance in terms of both efficiency and coverage accuracy. We also illustrate the use of our methods in a real example.     

Inference for cumulative incidence functions with informatively coarsened discrete event-time data

We consider the problem of comparing cumulative incidence functions of non-mortality events in the presence of informative coarsening and the competing risk of death. We extend frequentist-based hypothesis tests previously developed for non-informative coarsening and propose a novel Bayesian method based on comparing a posterior parameter transformation with its expected distribution under the null hypothesis of equal cumulative incidence functions. Both methods use estimates derived by extending previously published estimation procedures to accommodate censoring by death. The data structure and analysis goal are exemplified by the AIDS Link to the Intravenous Experience (ALIVE) study, where researchers are interested in comparing incidence of human immunodeficiency virus seroconversion by risk behavior categories. Coarsening in the forms of interval and right censoring and censoring by death in ALIVE is thought to be informative; thus, we perform a sensitivity analysis by incorporating elicited expert information about the relationship between seroconversion and censoring into the model.     

Comparing treatments via the propensity score: stratification or modeling?

In observational studies of treatments or interventions, propensity score (PS) adjustment is often useful for controlling bias in estimation of treatment effects. Regression on PS is used most often and can be highly efficient, but it can lead to biased results when model assumptions are violated. The validity of stratification on PS depends on fewer model assumptions, but this approach is less efficient than regression adjustment when the regression assumptions hold. To investigate these issues, we compare stratification and regression adjustments in a Monte Carlo simulation study. We consider two stratification approaches: equal frequency strata and an approach that attempts to choose strata that minimize the mean squared error (MSE) of the treatment effect estimate. The regression approach that we consider is a generalized additive model (GAM) that estimates treatment effect controlling for a potentially nonlinear association between PS and outcome. We find that under a wide range of plausible data generating distributions the GAM approach outperforms stratification in treatment effect estimation with respect to bias, variance, and thereby MSE. We illustrate each approach in an analysis of insurance plan choice and its relation to satisfaction with asthma care.