Bootstrap-type confidence intervals for quantiles of the survival distribution

In this paper we outline and illustrate an easy to program method for analytically calculating both parametric and non-parametric bootstrap-type confidence intervals for quantiles of the survival distribution based on right censored data. This new approach allows for the incorporation of covariates within the framework of parametric models. The procedure is based upon the notion of fractional order statistics and is carried forth using a simple beta transformation of the estimated survival function (parametric or non-parametric). It is the only direct method currently available in the sense that all other methods are based on inverting test statistics or employing confidence intervals for other survival quantities. We illustrate that the new method has favourable coverage probabilities for median confidence intervals as compared to six other competing methods.     

Distribution-free inference on contrasts of arbitrary summary measures of survival

We present an approach for inference on contrasts of clinically meaningful functionals of a survivor distribution (e.g., restricted mean, quantiles) that can avoid strong parametric or semiparametric assumptions on the underlying structure of the data. In this multistage approach, we first use an adaptive predictive model to estimate conditional survival distributions based on covariates. We then estimate nonparametrically one or more functionals of survival from the covariate-specific survival curves and evaluated contrasts of those functionals. We find that the use of an adaptive nonparametric tree-based predictive model leads to minimal loss in precision when semiparametric assumptions hold and provides marked improvement in accuracy when those assumptions are invalid. Therefore, this work as a whole supports the use of survival summaries appropriate to a given medical application, whether that be, for example, the median or 75th percentile in some settings or perhaps a restricted mean in others. The approach is also compared with the Mayo R score for primary biliary cirrhosis prognosis.     

Estimating causal contrasts involving intermediate variables in the presence of selection bias

An important goal across the biomedical and social sciences is the quantification of the role of intermediate factors in explaining how an exposure exerts an effect on an outcome. Selection bias has the potential to severely undermine the validity of inferences on direct and indirect causal effects in observational as well as in randomized studies. The phenomenon of selection may arise through several mechanisms, and we here focus on instances of missing data. We study the sign and magnitude of selection bias in the estimates of direct and indirect effects when data on any of the factors involved in the analysis is either missing at random or not missing at random. Under some simplifying assumptions, the bias formulae can lead to nonparametric sensitivity analyses. These sensitivity analyses can be applied to causal effects on the risk difference and risk‐ratio scales irrespectively of the estimation approach employed. To incorporate parametric assumptions, we also develop a sensitivity analysis for selection bias in mediation analysis in the spirit of the expectation–maximization algorithm. The approaches are applied to data from a health disparities study investigating the role of stage at diagnosis on racial disparities in colorectal cancer survival. Copyright © 2016 John Wiley & Sons, Ltd.     

A hybrid method in combining treatment effects from matched and unmatched studies

The most common data structures in the biomedical studies have been matched or unmatched designs. Data structures resulting from a hybrid of the two may create challenges for statistical inferences. The question may arise whether to use parametric or nonparametric methods on the hybrid data structure. The Early Treatment for Retinopathy of Prematurity study was a multicenter clinical trial sponsored by the National Eye Institute. The design produced data requiring a statistical method of a hybrid nature. An infant in this multicenter randomized clinical trial had high‐risk prethreshold retinopathy of prematurity that was eligible for treatment in one or both eyes at entry into the trial. During follow‐up, recognition visual acuity was accessed for both eyes. Data from both eyes (matched) and from only one eye (unmatched) were eligible to be used in the trial. The new hybrid nonparametric method is a meta‐analysis based on combining the Hodges–Lehmann estimates of treatment effects from the Wilcoxon signed rank and rank sum tests. To compare the new method, we used the classic meta‐analysis with the t‐test method to combine estimates of treatment effects from the paired and two sample t‐tests. We used simulations to calculate the empirical size and power of the test statistics, as well as the bias, mean square and confidence interval width of the corresponding estimators. The proposed method provides an effective tool to evaluate data from clinical trials and similar comparative studies. Copyright © 2013 John Wiley & Sons, Ltd.     

A multiple locus analysis of the collaborative study on the genetics of alcoholism data set

Parametric and nonparametric statistical methods have been applied to the alcohol dependence data set collected in the Collaborative Study on the Genetics of Alcoholism (COGA). Our nonparametric linkage analyses (NPL) were based on the S(all) statistic of GENEHUNTER [Kruglyak et al., 1996] and the improved NPL statistic of GENEHUNTER-PLUS [Kong and Cox, 1997]. Based on likely regions for alcohol susceptibility genes identified from our nonparametric analyses, we reanalyzed the data using several two-locus models. We used the TMLINK program [Lathrop and Ott, 1990] in the LINKAGE package for these parametric analyses.     

Parametric modelling of cost data in medical studies

The cost of medical resources used is often recorded for each patient in clinical studies in order to inform decision-making. Although cost data are generally skewed to the right, interest is in making inferences about the population mean cost. Common methods for non-normal data, such as data transformation, assuming asymptotic normality of the sample mean or non-parametric bootstrapping, are not ideal. This paper describes possible parametric models for analysing cost data.Four example data sets are considered, which have different sample sizes and degrees of skewness. Normal, gamma, log-normal, and log-logistic distributions are fitted, together with three-parameter versions of the latter three distributions. Maximum likelihood estimates of the population mean are found; confidence intervals are derived by a parametric BC(a) bootstrap and checked by MCMC methods. Differences between model fits and inferences are explored.Skewed parametric distributions fit cost data better than the normal distribution, and should in principle be preferred for estimating the population mean cost. However for some data sets, we find that models that fit badly can give similar inferences to those that fit well. Conversely, particularly when sample sizes are not large, different parametric models that fit the data equally well can lead to substantially different inferences. We conclude that inferences are sensitive to choice of statistical model, which itself can remain uncertain unless there is enough data to model the tail of the distribution accurately. Investigating the sensitivity of conclusions to choice of model should thus be an essential component of analysing cost data in practice.     

Bayesian nonparametric inference on quantile residual life function: Application to breast cancer data

There is often an interest in estimating a residual life function as a summary measure of survival data. For ease in presentation of the potential therapeutic effect of a new drug, investigators may summarize survival data in terms of the remaining life years of patients. Under heavy right censoring, however, some reasonably high quantiles (e.g., median) of a residual lifetime distribution cannot be always estimated via a popular nonparametric approach on the basis of the Kaplan-Meier estimator. To overcome the difficulties in dealing with heavily censored survival data, this paper develops a Bayesian nonparametric approach that takes advantage of a fully model-based but highly flexible probabilistic framework. We use a Dirichlet process mixture of Weibull distributions to avoid strong parametric assumptions on the unknown failure time distribution, making it possible to estimate any quantile residual life function under heavy censoring. Posterior computation through Markov chain Monte Carlo is straightforward and efficient because of conjugacy properties and partial collapse. We illustrate the proposed methods by using both simulated data and heavily censored survival data from a recent breast cancer clinical trial conducted by the National Surgical Adjuvant Breast and Bowel Project.     

Nonparametric statistical inference method for partial areas under receiver operating characteristic curves, with application to genomic studies

Recently ROC50 index-the area under the lower portion of the receiver operating characteristic (ROC) curve up to the first 50 false positives-has been increasingly widely used in genomic research. Unfortunately, statistical inferences on the ROC50 index are not commonly drawn due to a lack of handy statistical inference methods and/or software tools. In this paper, we reviewed developments in statistical methods for the partial areas under ROC curves and using nonparametric methods we derived a simple and direct variance calculation formula for the partial areas, different from existing methods in the literature. We have also verified our method through simulation studies and compared our method with existing bi-normal approaches. We have shown that the partial area has an asymptotic normal distribution using trimmed U-statistics theory. On the basis of this asymptotic normality, we have given formulas for the confidence interval and the test statistic and we reported on their application to a genomic study of sample size approximately 10,000.     

Comparing Cox and parametric models in clinical studies

Parametric models are only occasionally used in the analysis of clinical studies of survival although they may offer advantages over Cox's model. In this paper, we report experiences that we have made fitting parametric models to data sets from different clinical trials mainly performed at the Vienna University Medical School. We emphasize the role of residuals for discriminating among candidate models and judging their goodness of fit. The effect of misspecification of the baseline distribution on parameter estimates and testing has been explored. The results from parametric analyses have always been contrasted with those from Cox's model.     

Regression models using parametric pseudo-observations

Pseudo-observations based on the nonparametric Kaplan-Meier estimator of the survival function have been proposed as an alternative to the widely used Cox model for analyzing censored time-to-event data. Using a spline-based estimator of the survival has some potential benefits over the nonparametric approach in terms of less variability. We propose to define pseudo-observations based on a flexible parametric estimator and use these for analysis in regression models to estimate parameters related to the cumulative risk. We report the results of a simulation study that compares the empirical standard errors of estimates based on parametric and nonparametric pseudo-observations in various settings. Our simulations show that in some situations there is a substantial gain in terms of reduced variability using the proposed parametric pseudo-observations compared with the nonparametric pseudo-observations. The gain can be measured as a reduction of the empirical standard error by up to about one third; corresponding to an additional 125% larger sample size. We illustrate the use of the proposed method in a brief data example.     

Modeling continuous diagnostic test data using approximate Dirichlet process distributions

There is now a large literature on the analysis of diagnostic test data. In the absence of a gold standard test, latent class analysis is most often used to estimate the prevalence of the condition of interest and the properties of the diagnostic tests. When test results are measured on a continuous scale, both parametric and nonparametric models have been proposed. Parametric methods such as the commonly used bi-normal model may not fit the data well; nonparametric methods developed to date have been relatively complex to apply in practice, and their properties have not been carefully evaluated in the diagnostic testing context. In this paper, we propose a simple yet flexible Bayesian nonparametric model which approximates a Dirichlet process for continuous data. We compare results from the nonparametric model with those from the bi-normal model via simulations, investigating both how much is lost in using a nonparametric model when the bi-normal model is correct and how much can be gained in using a nonparametric model when normality does not hold. We also carefully investigate the trade-offs that occur between flexibility and identifiability of the model as different Dirichlet process prior distributions are used. Motivated by an application to tuberculosis clustering, we extend our nonparametric model to accommodate two additional dichotomous tests and proceed to analyze these data using both the continuous test alone as well as all three tests together.     

Analysis of small sample size studies using nonparametric bootstrap test with pooled resampling method

Experimental studies in biomedical research frequently pose analytical problems related to small sample size. In such studies, there are conflicting findings regarding the choice of parametric and nonparametric analysis, especially with non‐normal data. In such instances, some methodologists questioned the validity of parametric tests and suggested nonparametric tests. In contrast, other methodologists found nonparametric tests to be too conservative and less powerful and thus preferred using parametric tests. Some researchers have recommended using a bootstrap test; however, this method also has small sample size limitation. We used a pooled method in nonparametric bootstrap test that may overcome the problem related with small samples in hypothesis testing. The present study compared nonparametric bootstrap test with pooled resampling method corresponding to parametric, nonparametric, and permutation tests through extensive simulations under various conditions and using real data examples. The nonparametric pooled bootstrap t‐test provided equal or greater power for comparing two means as compared with unpaired t‐test, Welch t‐test, Wilcoxon rank sum test, and permutation test while maintaining type I error probability for any conditions except for Cauchy and extreme variable lognormal distributions. In such cases, we suggest using an exact Wilcoxon rank sum test. Nonparametric bootstrap paired t‐test also provided better performance than other alternatives. Nonparametric bootstrap test provided benefit over exact Kruskal–Wallis test. We suggest using nonparametric bootstrap test with pooled resampling method for comparing paired or unpaired means and for validating the one way analysis of variance test results for non‐normal data in small sample size studies. Copyright © 2017 John Wiley & Sons, Ltd.     

Semiparametric methods for multistate survival models in randomised trials

Transform methods have proved effective for networks describing a progression of events. In semi‐Markov networks, we calculated the transform of time to a terminating event from corresponding transforms of intermediate steps. Saddlepoint inversion then provided survival and hazard functions, which integrated, and fully utilised, the network data. However, the presence of censored data introduces significant difficulties for these methods. Many participants in controlled trials commonly remain event‐free at study completion, a consequence of the limited period of follow‐up specified in the trial design. Transforms are not estimable using nonparametric methods in states with survival truncated by end‐of‐study censoring. We propose the use of parametric models specifying residual survival to next event. As a simple approach to extrapolation with competing alternative states, we imposed a proportional incidence (constant relative hazard) assumption beyond the range of study data. No proportional hazards assumptions are necessary for inferences concerning time to endpoint; indeed, estimation of survival and hazard functions can proceed in a single study arm. We demonstrate feasibility and efficiency of transform inversion in a large randomised controlled trial of cholesterol‐lowering therapy, the Long‐Term Intervention with Pravastatin in Ischaemic Disease study. Transform inversion integrates information available in components of multistate models: estimates of transition probabilities and empirical survival distributions. As a by‐product, it provides some ability to forecast survival and hazard functions forward, beyond the time horizon of available follow‐up. Functionals of survival and hazard functions provide inference, which proves sharper than that of log‐rank and related methods for survival comparisons ignoring intermediate events. Copyright © 2013 John Wiley & Sons, Ltd.     

A practical introduction to Bayesian estimation of causal effects: Parametric and nonparametric approaches

Substantial advances in Bayesian methods for causal inference have been made in recent years. We provide an introduction to Bayesian inference for causal effects for practicing statisticians who have some familiarity with Bayesian models and would like an overview of what it can add to causal estimation in practical settings. In the paper, we demonstrate how priors can induce shrinkage and sparsity in parametric models and be used to perform probabilistic sensitivity analyses around causal assumptions. We provide an overview of nonparametric Bayesian estimation and survey their applications in the causal inference literature. Inference in the point‐treatment and time‐varying treatment settings are considered. For the latter, we explore both static and dynamic treatment regimes. Throughout, we illustrate implementation using off‐the‐shelf open source software. We hope to leave the reader with implementation‐level knowledge of Bayesian causal inference using both parametric and nonparametric models. All synthetic examples and code used in the paper are publicly available on a companion GitHub repository.     

Analysis of clustered and interval censored data from a community-based study in asthma

Many authors in recent years have proposed extensions of familiar survival analysis methodologies to apply in dependent data settings, for example, when data are clustered or subject to repeated measures. However, these extensions have been considered largely in the context of right censored data. In this paper, we discuss a parametric frailty model for the analysis of clustered and interval censored failure time data. Details are presented for the specific case where the underlying time to event data follow a Weibull distribution. Maximum likelihood estimates will be obtained using commercially available software and the empirical efficiency of these estimators will be explored via a simulation study. We also discuss a score test to make inferences about the magnitude and significance of over-dispersion in clustered data settings. These methods will be illustrated using data from the East Boston Asthma Study.     

On the use of nonparametric tests for comparing immunological Reverse Cumulative distribution curves (RCDCs)

Reverse Cumulative Distribution Curves (RCDCs) have proven to be a useful tool in summarizing immune response profiles in vaccine studies since their introduction by Reed, Meade, and Steinhoff (RMS) (1995). They are able to display virtually all of the treatment data and characterize summary statistics such as means or even their confidence intervals (CIs) that might be obscure. RMS mentioned their similarity to survival curves often used to summarize time-to-event data which are usually not normally distributed. The RCDCs, while intuitively pleasing and useful, contain important properties which allow for more powerful statistical applications. In this paper, we will suggest several widely used rank-based tests to compare the curves in the context of vaccine studies. These rank-based tests allow for comparisons between treatments, for stratified analyses, weighted analyses, and other modifications that make them the alternative of parametric analyses without the normality assumptions.Clinical trial identification: NCT01712984 and NCT01230957.     

A data-adaptive strategy for inverse weighted estimation of causal effects

In most nonrandomized observational studies, differences between treatment groups may arise not only due to the treatment but also because of the effect of confounders. Therefore, causal inference regarding the treatment effect is not as straightforward as in a randomized trial. To adjust for confounding due to measured covariates, the average treatment effect is often estimated by using propensity scores. Typically, propensity scores are estimated by logistic regression. More recent suggestions have been to employ nonparametric classification algorithms from machine learning. In this article, we propose a weighted estimator combining parametric and nonparametric models. Some theoretical results regarding consistency of the procedure are given. Simulation studies are used to assess the performance of the newly proposed methods relative to existing methods, and a data analysis example from the Surveillance, Epidemiology and End Results database is presented.     

A fair comparison of tree-based and parametric methods in multiple imputation by chained equations

Multiple imputation by chained equations (MICE) has emerged as a leading strategy for imputing missing epidemiological data due to its ease of implementation and ability to maintain unbiased effect estimates and valid inference. Within the MICE algorithm, imputation can be performed using a variety of parametric or nonparametric methods. Literature has suggested that nonparametric tree-based imputation methods outperform parametric methods in terms of bias and coverage when there are interactions or other nonlinear effects among the variables. However, these studies fail to provide a fair comparison as they do not follow the well-established recommendation that any effects in the final analysis model (including interactions) should be included in the parametric imputation model. We show via simulation that properly incorporating interactions in the parametric imputation model leads to much better performance. In fact, correctly specified parametric imputation and tree-based random forest imputation perform similarly when estimating the interaction effect. Parametric imputation leads to slightly higher coverage for the interaction effect, but it has wider confidence intervals than random forest imputation and requires correct specification of the imputation model. Epidemiologists should take care in specifying MICE imputation models, and this paper assists in that task by providing a fair comparison of parametric and tree-based imputation in MICE.     

Practicable confidence intervals for current status data

Although confidence intervals (CIs) for binary isotonic regression and current status survival data have been well studied theoretically, their practical application has been limited, in part because of poor performance in small samples and in part because of computational difficulties. Ghosh, Banerjee, and Biswas (2008, Biometrics 64 , 1009‐1017) described three approaches to constructing CIs: (i) the Wald‐based method; (ii) the subsampling‐based method; and (iii) the likelihood‐ratio test (LRT)‐based method. In simulation studies, they found that the subsampling‐based method and LRT‐based method tend to have better coverage probabilities than a simple Wald‐based method that may perform poorly in realistic sample sizes. However, software implementing these approaches is currently unavailable. In this article, we show that by using transformations, simple Wald‐based CIs can be improved with small and moderate sample sizes to have competitive performance with LRT‐based method. Our simulations further show that a simple nonparametric bootstrap gives approximately correct CIs for the data generating mechanisms that we consider. We provide an R package that can be used to compute the Wald‐type and the bootstrap CIs and demonstrate its practical utility with two real data analyses. Copyright © 2012 John Wiley & Sons, Ltd.     

Semiparametric varying‐coefficient model for interval censored data with a cured proportion

Varying‐coefficient models have claimed an increasing portion of statistical research and are now applied to censored data analysis in medical studies. We incorporate such flexible semiparametric regression tools for interval censored data with a cured proportion. We adopted a two‐part model to describe the overall survival experience for such complicated data. To fit the unknown functional components in the model, we take the local polynomial approach with bandwidth chosen by cross‐validation. We establish consistency and asymptotic distribution of the estimation and propose to use bootstrap for inference. We constructed a BIC‐type model selection method to recommend an appropriate specification of parametric and nonparametric components in the model. We conducted extensive simulations to assess the performance of our methods. An application on a decompression sickness data illustrates our methods. Copyright © 2013 John Wiley & Sons, Ltd.