Risk‐adjusted monitoring of time to event in the presence of long‐term survivors

The use of control charts for monitoring schemes in medical context should consider adjustments to incorporate the specific risk for each individual. Some authors propose the use of a risk‐adjusted survival time cumulative sum (RAST CUSUM) control chart to monitor a time‐to‐event outcome, possibly right censored, using conventional survival models, which do not contemplate the possibility of cure of a patient. We propose to extend this approach considering a risk‐adjusted CUSUM chart, based on a cure rate model. We consider a regression model in which the covariates affect the cure fraction. The CUSUM scores are obtained for Weibull and log‐logistic promotion time model to monitor a scale parameter for nonimmune individuals. A simulation study was conducted to evaluate and compare the performance of the proposed chart (RACUF CUSUM) with RAST CUSUM, based on optimal control limits and average run length in different situations. As a result, we note that the RAST CUSUM chart is inappropriate when applied to data with a cure rate, while the proposed RACUF CUSUM chart seems to have similar performance if applied to data without a cure rate. The proposed chart is illustrated with simulated data and with a real data set of patients with heart failure treated at the Heart Institute (InCor), at the University of São Paulo, Brazil.     

Two-Stage Least Squares Analysis of Hierarchical Linear Models

This paper has two major objectives. The first is to present a two-stage least squares procedure for estimation of the parameters in a linear model whose parameters are in themselves linear functions of some hyperparameters. The second, and perhaps more important point, is that the new estimator can be shown to be generally more precise than either the Bayesian or the generalized single-stage least squares estimator reported by LINDLEY and SMITH (1072).     

Detecting change in an Australian flowering record: Comparisons of linear regression and cumulative sum analysis change point analysis

Linear regression and cumulative sum analysis (CUSUM) change point analyses were used to determine whether there had been a significant change in the first flowering date between 1983 and 2006 for 65 species. Both methods agreed that the first flowering date of 47 species did not change and that eight species had a significant change (P < 0.05) in their flowering. Three species shifted to later flowering and five species to earlier. Over the observation period, each method found that the average shift to later flowering was greater (37.4 days or approx. 1.56 days per year for CUSUM change point analysis and 51.4 days or 2.14 days per year for linear regression) than that to earlier flowering (28.4 days or approx. 1.20 days per year for change point analysis and 46.5 days or 1.97 days per year for linear regression). For the remaining 10 species the results of linear regression and change point methods differed. Each method found five species (three earlier flowering and two later) to have a significantly changed first flowering date over their observation period, where the other method did not. Some of these differences can be attributed to the fact that the CUSUM method can detect multiple change points whereas linear regression can not. Significant change points in first flowering date were identified for 13 species between the years 1987 to 1998. The most frequent year identified as a change point year was 1995. The two methods, although not interchangeable, had strong agreement (84.6%) in detecting shifts. This gives greater confidence that a change in flowering has occurred for eight species and equally importantly, that no change in first flowering date has occurred for 47 species.     

Estimating the distribution of a renewal process from times at which events from an independent process are detected

The analysis of length-biased data has been mostly limited to the interarrival interval of a renewal process covering a specific time point. Motivated by a surveillance problem, we consider a more general situation where this time point is random and related to a specific event, for example, status change or onset of a disease. We also consider the problem when additional information is available on whether the event intervals (interarrival intervals covering the random event) end within or after a random time period (which we call a window period) following the random event. Under the assumptions that the occurrence rate of the random event is low and the renewal process is independent of the random event, we provide formulae for the estimation of the distribution of interarrival times based on the observed event intervals. Procedures for testing the required assumptions are also furnished. We apply our results to human immunodeficiency virus (HIV) test data from public test sites in Seattle, Washington, where the random event is HIV infection and the window period is from the onset of HIV infection to the time at which a less sensitive HIV test becomes positive. Results show that the estimator of the intertest interval length distribution from event intervals ending within the window period is less biased than the estimator from all event intervals; the latter estimator is affected by right truncation. Finally, we discuss possible applications to estimating HIV incidence and analyzing length-biased samples with right or left truncated data.     

Locally efficient estimation of the quality-adjusted lifetime distribution with right-censored data and covariates

Zhao and Tsiatis (1997) consider the problem of estimation of the distribution of the quality-adjusted lifetime when the chronological survival time is subject to right censoring. The quality-adjusted lifetime is typically defined as a weighted sum of the times spent in certain states up until death or some other failure time. They propose an estimator and establish the relevant asymptotics under the assumption of independent censoring. In this paper we extend the data structure with a covariate process observed until the end of follow-up and identify the optimal estimation problem. Because of the curse of dimensionality, no globally efficient nonparametric estimators, which have a good practical performance at moderate sample sizes, exist. Given a correctly specified model for the hazard of censoring conditional on the observed quality-of-life and covariate processes, we propose a closed-form one-step estimator of the distribution of the quality-adjusted lifetime whose asymptotic variance attains the efficiency bound if we can correctly specify a lower-dimensional working model for the conditional distribution of quality-adjusted lifetime given the observed quality-of-life and covariate processes. The estimator remains consistent and asymptotically normal even if this latter submodel is misspecified. The practical performance of the estimators is illustrated with a simulation study. We also extend our proposed one-step estimator to the case where treatment assignment is confounded by observed risk factors so that this estimator can be used to test a treatment effect in an observational study.     

Monitoring surgical performance using risk-adjusted cumulative sum charts

The cumulative sum (CUSUM) procedure is a graphical method that is widely used for quality monitoring in industrial settings. More recently it has been used to monitor surgical outcomes whereby it 'signals' if sufficient evidence has accumulated that there has been a change in the surgical failure rate. A limitation of the standard CUSUM procedure in this context is that since it is simply based on the observed surgical outcomes, it may signal as a result of changes in the referral pattern, such as an increased proportion of high-risk patients, rather than due to a change in the actual surgical performance. We describe a new CUSUM procedure that adjusts for each patient's pre-operative risk of surgical failure through the use of a likelihood-based scoring method. The procedure is therefore ideally suited for settings where there is a variable mix of patients over time.     

On the Bayeaian Estimation with Changing Failure Rate

In certain-life experiments it may happen that the failure rate may change after some initial period of time. Thie meane that the failure rate during certain interval of time and that of thereafter will differ significantly. Thie suggests a failure model in which one failure rate is assumed to be operative during period of time, say[0, T]. and another failure rate ie operative thereafter, that is after time T. In thie paper, Bayesian point estimates of these fnilure rates are obtained when underlying probability law law is exponential and T is known.     

A partially parametric estimator of survival in the presence of randomly censored data

Many biological or medical experiments have as their goal to estimate the survival function of a specified population of subjects when the time to the specified event may be censored due to loss to follow-up, the occurrence of another event that precludes the occurrence of the event of interest, or the study being terminated before the event of interest occurs. This paper suggests an improvement of the Kaplan-Meier product-limit estimator when the censoring mechanism is random. The proposed estimator treats the uncensored observations nonparametrically and uses a parametric model only for the censored observations. One version of this proposed estimator always has a smaller bias and mean squared error than the product-limit estimator. An example estimating the survival function of patients enrolled in the Ohio State University Bone Marrow Transplant Program is presented.     

A regression-based method for estimating mean treatment cost in the presence of right-censoring

In many clinical trials and evaluations using medical care administrative databases it is of interest to estimate not only the survival time of a given treatment modality but also the total associated cost. The most widely used estimator for data subject to censoring is the Kaplan-Meier (KM) or product-limit (PL) estimator. The optimality properties of this estimator applied to time-to-event data (consistency, etc.) under the assumptions of random censorship have been established. However, whenever the relationship between cost and survival time includes an error term to account for random differences among patients' costs, the dependency between cumulative treatment cost at the time of censoring and at the survival time results in KM giving biased estimates. A similar phenomenon has previously been noted in the context of estimating quality-adjusted survival time. We propose an estimator for mean cost which exploits the underlying relationship between total treatment cost and survival time. The proposed method utilizes either parametric or nonparametric regression to estimate this relationship and is consistent when this relationship is consistently estimated. We then present simulation results which illustrate the gain in finite-sample efficiency when compared with another recently proposed estimator. The methods are then applied to the estimation of mean cost for two studies where right-censoring was present. The first is the heart failure clinical trial Studies of Left Ventricular Dysfunction (SOLVD). The second is a Health Maintenance Organization (HMO) database study of the cost of ulcer treatment.     

A Bayesian semiparametric joint hierarchical model for longitudinal and survival data

This article proposes a new semiparametric Bayesian hierarchical model for the joint modeling of longitudinal and survival data. We relax the distributional assumptions for the longitudinal model using Dirichlet process priors on the parameters defining the longitudinal model. The resulting posterior distribution of the longitudinal parameters is free of parametric constraints, resulting in more robust estimates. This type of approach is becoming increasingly essential in many applications, such as HIV and cancer vaccine trials, where patients' responses are highly diverse and may not be easily modeled with known distributions. An example will be presented from a clinical trial of a cancer vaccine where the survival outcome is time to recurrence of a tumor. Immunologic measures believed to be predictive of tumor recurrence were taken repeatedly during follow-up. We will present an analysis of this data using our new semiparametric Bayesian hierarchical joint modeling methodology to determine the association of these longitudinal immunologic measures with time to tumor recurrence.     

The performance of random coefficient regression in accounting for residual confounding

Greenland (2000, Biometrics 56, 915-921) describes the use of random coefficient regression to adjust for residual confounding in a particular setting. We examine this setting further, giving theoretical and empirical results concerning the frequentist and Bayesian performance of random coefficient regression. Particularly, we compare estimators based on this adjustment for residual confounding to estimators based on the assumption of no residual confounding. This devolves to comparing an estimator from a nonidentified but more realistic model to an estimator from a less realistic but identified model. The approach described by Gustafson (2005, Statistical Science 20, 111-140) is used to quantify the performance of a Bayesian estimator arising from a nonidentified model. From both theoretical calculations and simulations we find support for the idea that superior performance can be obtained by replacing unrealistic identifying constraints with priors that allow modest departures from those constraints. In terms of point-estimator bias this superiority arises when the extent of residual confounding is substantial, but the advantage is much broader in terms of interval estimation. The benefit from modeling residual confounding is maintained when the prior distributions employed only roughly correspond to reality, for the standard identifying constraints are equivalent to priors that typically correspond much worse.     

Nonparametric estimation of the concordance correlation coefficient under univariate censoring

Assessing agreement is often of interest in clinical studies to evaluate the similarity of measurements produced by different raters or methods on the same subjects. Lin's (1989, Biometrics 45, 255-268) concordance correlation coefficient (CCC) has become a popular measure of agreement for correlated continuous outcomes. However, commonly used estimation methods for the CCC do not accommodate censored observations and are, therefore, not applicable for survival outcomes. In this article, we estimate the CCC nonparametrically through the bivariate survival function. The proposed estimator of the CCC is proven to be strongly consistent and asymptotically normal, with a consistent bootstrap variance estimator. Furthermore, we propose a time-dependent agreement coefficient as an extension of Lin's (1989) CCC for measuring the agreement between survival times among subjects who survive beyond a specified time point. A nonparametric estimator is developed for the time-dependent agreement coefficient as well. It has the same asymptotic properties as the estimator of the CCC. Simulation studies are conducted to evaluate the performance of the proposed estimators. A real data example from a prostate cancer study is used to illustrate the method.     

Time-dependent ROC curves for censored survival data and a diagnostic marker

ROC curves are a popular method for displaying sensitivity and specificity of a continuous diagnostic marker, X, for a binary disease variable, D. However, many disease outcomes are time dependent, D(t), and ROC curves that vary as a function of time may be more appropriate. A common example of a time-dependent variable is vital status, where D(t) = 1 if a patient has died prior to time t and zero otherwise. We propose summarizing the discrimination potential of a marker X, measured at baseline (t = 0), by calculating ROC curves for cumulative disease or death incidence by time t, which we denote as ROC(t). A typical complexity with survival data is that observations may be censored. Two ROC curve estimators are proposed that can accommodate censored data. A simple estimator is based on using the Kaplan-Meier estimator for each possible subset X > c. However, this estimator does not guarantee the necessary condition that sensitivity and specificity are monotone in X. An alternative estimator that does guarantee monotonicity is based on a nearest neighbor estimator for the bivariate distribution function of (X, T), where T represents survival time (Akritas, M. J., 1994, Annals of Statistics 22, 1299-1327). We present an example where ROC(t) is used to compare a standard and a modified flow cytometry measurement for predicting survival after detection of breast cancer and an example where the ROC(t) curve displays the impact of modifying eligibility criteria for sample size and power in HIV prevention trials.     

Estimating survival and association in a semicompeting risks model

Summary .   In many follow-up studies, patients are subject to concurrent events. In this article, we consider semicompeting risks data as defined by Fine, Jiang, and Chappell (2001, Biometrika 88 , 907–919) where one event is censored by the other but not vice versa. The proposed model involves marginal survival functions for the two events and a parametric family of copulas for their dependency. This article suggests a general method for estimating the dependence parameter when the dependency is modeled with an Archimedean copula. It uses the copula-graphic estimator of Zheng and Klein (1995, Biometrika 82 , 127–138) for estimating the survival function of the nonterminal event, subject to dependent censoring. Asymptotic properties of these estimators are derived. Simulations show that the new methods work well with finite samples. The copula-graphic estimator is shown to be more accurate than the estimator proposed by Fine et al. (2001) ; its performances are similar to those of the self-consistent estimator of Jiang, Fine, Kosorok, and Chappell (2005, Scandinavian Journal of Statistics 33, 1–20). The analysis of a data set, emphasizing the estimation of characteristics of the observable region, is presented as an illustration.     

One-step targeted maximum likelihood estimation for time-to-event outcomes

Researchers in observational survival analysis are interested in not only estimating survival curve nonparametrically but also having statistical inference for the parameter. We consider right-censored failure time data where we observe n independent and identically distributed observations of a vector random variable consisting of baseline covariates, a binary treatment at baseline, a survival time subject to right censoring, and the censoring indicator. We assume the baseline covariates are allowed to affect the treatment and censoring so that an estimator that ignores covariate information would be inconsistent. The goal is to use these data to estimate the counterfactual average survival curve of the population if all subjects are assigned the same treatment at baseline. Existing observational survival analysis methods do not result in monotone survival curve estimators, which is undesirable and may lose efficiency by not constraining the shape of the estimator using the prior knowledge of the estimand. In this paper, we present a one-step Targeted Maximum Likelihood Estimator (TMLE) for estimating the counterfactual average survival curve. We show that this new TMLE can be executed via recursion in small local updates. We demonstrate the finite sample performance of this one-step TMLE in simulations and an application to a monoclonal gammopathy data.     

Indexing variability: A case study with climate change impacts on ecosystems

We developed a methodology to index statistical changes in the variance of ecological measures over time. While ecological indicators are used to assess ecosystem health, vulnerability, risk and damage to ecosystems, their primary focus has been on changes to the mean of the ecological state or process. Little work has been done on incorporating variability into ecological indices. The methodology developed here is based on the Modified Levene's test of variance and a moving block where an initial time period (block) of ecosystem behavior is compared to the moving block. This allows for the detection of not only shifts in variance but also the magnitude of the shift on a continuous basis. Our results compared well with the benchmarked results from the Centered Cumulative Sum of Squares Algorithm (CUSUM) for detection of variance changes in fixed time series. The output from this methodology is a continuous stream of parameters (significant variance shift, magnitude of the shift, and the direction of the variance shift) suitable for indexing variance or integrating into an index measuring ecosystem change. Results suggest that the block interval widths should be at least 50 and that a smoothing factor of five be used to avoid false positives. We used modeled vegetation carbon output to analyze the utility of the methodology, illustrating that different model assumptions and CO₂ regimes affect the variability of the ecological response. The degree of risk resource managers may want to explore can be altered by choices in the block length, the length of the string used to smooth variance shifts, the assumption that the initial period has constant variance and the alpha levels used to determine statistical significance.     

Accuracy of rate estimation using relaxed-clock models with a critical focus on the early metazoan radiation

In recent years, a number of phylogenetic methods have been developed for estimating molecular rates and divergence dates under models that relax the molecular clock constraint by allowing rate change throughout the tree. These methods are being used with increasing frequency, but there have been few studies into their accuracy. We tested the accuracy of several relaxed-clock methods (penalized likelihood and Bayesian inference using various models of rate change) using nucleotide sequences simulated on a nine-taxon tree. When the sequences evolved with a constant rate, the methods were able to infer rates accurately, but estimates were more precise when a molecular clock was assumed. When the sequences evolved under a model of auto-correlated rate change, rates were accurately estimated using penalized likelihood and by Bayesian inference using lognormal and exponential models of rate change, while other models did not perform as well. When the sequences evolved under a model of uncorrelated rate change, only Bayesian inference using an exponential rate model performed well. Collectively, the results provide a strong recommendation for using the exponential model of rate change if a conservative approach to divergence time estimation is required. A case study is presented in which we use a simulation-based approach to examine the hypothesis of elevated rates in the Cambrian period, and it is found that these high rate estimates might be an artifact of the rate estimation method. If this bias is present, then the ages of metazoan divergences would be systematically underestimated. The results of this study have implications for studies of molecular rates and divergence dates.     

Enhanced Cumulative Sum Charts for Monitoring Process Dispersion

The cumulative sum (CUSUM) control chart is widely used in industry for the detection of small and moderate shifts in process location and dispersion. For efficient monitoring of process variability, we present several CUSUM control charts for monitoring changes in standard deviation of a normal process. The newly developed control charts based on well-structured sampling techniques - extreme ranked set sampling, extreme double ranked set sampling and double extreme ranked set sampling, have significantly enhanced CUSUM chart ability to detect a wide range of shifts in process variability. The relative performances of the proposed CUSUM scale charts are evaluated in terms of the average run length (ARL) and standard deviation of run length, for point shift in variability. Moreover, for overall performance, we implore the use of the average ratio ARL and average extra quadratic loss. A comparison of the proposed CUSUM control charts with the classical CUSUM R chart, the classical CUSUM S chart, the fast initial response (FIR) CUSUM R chart, the FIR CUSUM S chart, the ranked set sampling (RSS) based CUSUM R chart and the RSS based CUSUM S chart, among others, are presented. An illustrative example using real dataset is given to demonstrate the practicability of the application of the proposed schemes.     

Asymptotically efficient two-step estimators of the hazards ratio for follow-up studies and survival data

Asymptotically efficient estimators of a common hazard rate ratio (for follow-up studies) and the proportional hazards ratio (for survival studies) are obtained by a single iteration of the "Mantel-Haenszel" estimator appropriate for each setting. Estimators of their variance are also developed. The two-step estimator for survival data and its variance estimator are shown by simulation to be minimally biased and the estimator is shown to be efficient relative to the Cox partial likelihood estimator in small samples.