Machine learning methods for leveraging baseline covariate information to improve the efficiency of clinical trials

Clinical trials are widely considered the gold standard for treatment evaluation, and they can be highly expensive in terms of time and money. The efficiency of clinical trials can be improved by incorporating information from baseline covariates that are related to clinical outcomes. This can be done by modifying an unadjusted treatment effect estimator with an augmentation term that involves a function of covariates. The optimal augmentation is well characterized in theory but must be estimated in practice. In this article, we investigate the use of machine learning methods to estimate the optimal augmentation. We consider and compare an indirect approach based on an estimated regression function and a direct approach that aims directly to minimize the asymptotic variance of the treatment effect estimator. Theoretical considerations and simulation results indicate that the direct approach is generally preferable over the indirect approach. The direct approach can be implemented using any existing prediction algorithm that can minimize a weighted sum of squared prediction errors. Many such prediction algorithms are available, and the super learning principle can be used to combine multiple algorithms into a super learner under the direct approach. The resulting direct super learner has a desirable oracle property, is easy to implement, and performs well in realistic settings. The proposed methodology is illustrated with real data from a stroke trial.     

A robust weighted Kaplan–Meier approach for data with dependent censoring using linear combinations of prognostic covariates

The weighted Kaplan–Meier (WKM) estimator is often used to incorporate prognostic covariates into survival analysis to improve efficiency and correct for potential bias. In this paper, we generalize the WKM estimator to handle a situation with multiple prognostic covariates and potential‐dependent censoring through the use of prognostic covariates. We propose to combine multiple prognostic covariates into two risk scores derived from two working proportional hazards models. One model is for the event times. The other model is for the censoring times. These two risk scores are then categorized to define the risk groups needed for the WKM estimator. A method of defining categories based on principal components is proposed. We show that the WKM estimator is robust to misspecification of either one of the two working models. In simulation studies, we show that the robust WKM approach can reduce bias due to dependent censoring and improve efficiency. We apply the robust WKM approach to a prostate cancer data set. Copyright 2010 John Wiley & Sons, Ltd.     

Statistical consequences of a successful lung allocation system – recovering information and reducing bias in models for urgency

The lung allocation system has reduced the number of waitlist deaths by ranking transplant candidates on the basis of a lung allocation score that requires estimation of the current 1‐year restricted mean waitlist survival (urgency). Fewer waitlist deaths and the systematic removal of candidates from the waitlist for transplantation present statistical challenges that must be addressed when using recent waitlist data. Multiple overlapping 1‐year follow‐up windows are used in a restricted mean model that estimates patient urgency on the basis of updated risk factors at the start of the window. In simulation studies, our proposed multiple imputation procedure was able to produce unbiased parameter estimates with similar efficiency to those obtained if censoring had never occurred. The analysis of 10,740 lung transplant candidates revealed that for most risk factors incorporating additional follow‐up windows produced more efficient estimates. Copyright © 2017 John Wiley & Sons, Ltd.     

Doubly robust inference procedure for relative survival ratio in population-based cancer registry data

Cancer registry system has been playing important roles in research and policy making in cancer control. In general, information on cause of death is not available in cancer registry data. To make inference on survival of cancer patients in the absence of cause of death information, the relative survival ratio is widely used in the population-based cancer research utilizing external life tables for the general population. Another difficulty arising in analyzing cancer registry data is informative censoring. In this article, we propose a doubly robust inference procedure for the relative survival ratio under a certain type of informative censoring, called the covariate-dependent censoring. The proposed estimator is doubly robust in the sense that it is consistent if at least one of the regression models for the time-to-death and for the censoring time is correctly specified. Furthermore, we introduced a doubly robust test assessing underlying conditional independence assumption between the time-to-death and the censoring time. This test is model based, but is doubly robust in the sense that at least one of the models for the time to event and for the censoring time is correctly specified, it maintains its nominal significance level. This notable feature entails us to make inference on cancer registry data relying on assumptions, which are much weaker than the existing methods and are verifiable empirically. We examine the theoretical and empirical properties of our proposed methods by asymptotic theory and simulation studies. We illustrate the proposed method with cancer registry data in Osaka, Japan.     

Analysis of clustered matched-pair data

Evaluation of the performance of a new diagnostic procedure with respect to a standard procedure arises frequently in practice. The response of interest, often in a dichotomous form, is measured twice, once with each procedure. The two procedures are administered to either two matched individuals, or when practical, to the same individual. A large sample test for matched-pair data is the McNemar test. The main assumption of this test is independent paired responses; however, when more than one outcome from an individual is measured by each procedure, the data are clustered. Examples of such cases can be seen in dental and ophthalmology studies. Variance adjustment methods for the analysis of clustered matched-pair data have been proposed; however, because of unequal cluster sizes, variability of correlation structures within a cluster (within paired responses in a cluster as well as between paired responses in a cluster), and unequal success probabilities among the clusters, the performances of some available methods are not consistent. This research proposes a simple adjustment to the McNemar test for the analysis of clustered matched-pair data. Method of moments is used to calculate a consistent variance estimator. Using Monte Carlo simulation, the size and power of the proposed test are compared to those of two currently available methods. To illustrate practical application, clustered matched-pair data from two clinical studies are analysed.     

Inference for the median residual life function in sequential multiple assignment randomized trials

In survival analysis, median residual lifetime is often used as a summary measure to assess treatment effectiveness; it is not clear, however, how such a quantity could be estimated for a given dynamic treatment regimen using data from sequential randomized clinical trials. We propose a method to estimate a dynamic treatment regimen‐specific median residual life (MERL) function from sequential multiple assignment randomized trials. We present the MERL estimator, which is based on inverse probability weighting, as well as, two variance estimates for the MERL estimator. One variance estimate follows from Lunceford, Davidian and Tsiatis' 2002 survival function‐based variance estimate and the other uses the sandwich estimator. The MERL estimator is evaluated, and its two variance estimates are compared through simulation studies, showing that the estimator and both variance estimates produce approximately unbiased results in large samples. To demonstrate our methods, the estimator has been applied to data from a sequentially randomized leukemia clinical trial. Copyright © 2013 John Wiley & Sons, Ltd.     

Truncated tests for combining evidence of summary statistics

To date, thousands of genetic variants to be associated with numerous human traits and diseases have been identified by genome-wide association studies (GWASs). The GWASs focus on testing the association between single trait and genetic variants. However, the analysis of multiple traits and single nucleotide polymorphisms (SNPs) might reflect physiological process of complex diseases and the corresponding study is called pleiotropy association analysis. Modern day GWASs report only summary statistics instead of individual-level phenotype and genotype data to avoid logistical and privacy issues. Existing methods for combining multiple phenotypes GWAS summary statistics mainly focus on low-dimensional phenotypes while lose power in high-dimensional cases. To overcome this defect, we propose two kinds of truncated tests to combine multiple phenotypes summary statistics. Extensive simulations show that the proposed methods are robust and powerful when the dimension of the phenotypes is high and only part of the phenotypes are associated with the SNPs. We apply the proposed methods to blood cytokines data collected from Finnish population. Results show that the proposed tests can identify additional genetic markers that are missed by single trait analysis.     

Value of External Data in the Extrapolation of Survival Data: A Study Using the NJR Data Set

Background

Extrapolation of time-to-event data can be a critical component of cost-effectiveness analysis.

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.     

PedBLIMP: Extending Linear Predictors to Impute Genotypes in Pedigrees

Recently, Wen and Stephens (Wen and Stephens [2010] Ann Appl Stat 4(3):1158–1182) proposed a linear predictor, called BLIMP, that uses conditional multivariate normal moments to impute genotypes with accuracy similar to current state‐of‐the‐art methods. One novelty is that it regularized the estimated covariance matrix based on a model from population genetics. We extended multivariate moments to impute genotypes in pedigrees. Our proposed method, PedBLIMP, utilizes both the linkage‐disequilibrium (LD) information estimated from external panel data and the pedigree structure or identity‐by‐descent (IBD) information. The proposed method was evaluated on a pedigree design where some individuals were genotyped with dense markers and the rest with sparse markers. We found that incorporating the pedigree/IBD information can improve imputation accuracy compared to BLIMP. Because rare variants usually have low LD with other single‐nucleotide polymorphisms (SNPs), incorporating pedigree/IBD information largely improved imputation accuracy for rare variants. We also compared PedBLIMP with IMPUTE2 and GIGI. Results show that when sparse markers are in a certain density range, our method can outperform both IMPUTE2 and GIGI.     

Measurement error in continuous endpoints in randomised trials: Problems and solutions

In randomised trials, continuous endpoints are often measured with some degree of error. This study explores the impact of ignoring measurement error and proposes methods to improve statistical inference in the presence of measurement error. Three main types of measurement error in continuous endpoints are considered: classical, systematic, and differential. For each measurement error type, a corrected effect estimator is proposed. The corrected estimators and several methods for confidence interval estimation are tested in a simulation study. These methods combine information about error-prone and error-free measurements of the endpoint in individuals not included in the trial (external calibration sample). We show that, if measurement error in continuous endpoints is ignored, the treatment effect estimator is unbiased when measurement error is classical, while Type-II error is increased at a given sample size. Conversely, the estimator can be substantially biased when measurement error is systematic or differential. In those cases, bias can largely be prevented and inferences improved upon using information from an external calibration sample, of which the required sample size increases as the strength of the association between the error-prone and error-free endpoint decreases. Measurement error correction using already a small (external) calibration sample is shown to improve inferences and should be considered in trials with error-prone endpoints. Implementation of the proposed correction methods is accommodated by a new software package for R.     

Meta‐analysis of single‐arm survival studies: a distribution‐free approach for estimating summary survival curves with random effects

In epidemiologic studies and clinical trials with time‐dependent outcome (for instance death or disease progression), survival curves are used to describe the risk of the event over time. In meta‐analyses of studies reporting a survival curve, the most informative finding is a summary survival curve. In this paper, we propose a method to obtain a distribution‐free summary survival curve by expanding the product‐limit estimator of survival for aggregated survival data. The extension of DerSimonian and Laird's methodology for multiple outcomes is applied to account for the between‐study heterogeneity. Statistics I² and H² are used to quantify the impact of the heterogeneity in the published survival curves. A statistical test for between‐strata comparison is proposed, with the aim to explore study‐level factors potentially associated with survival. The performance of the proposed approach is evaluated in a simulation study. Our approach is also applied to synthesize the survival of untreated patients with hepatocellular carcinoma from aggregate data of 27 studies and synthesize the graft survival of kidney transplant recipients from individual data from six hospitals. Copyright © 2014 John Wiley & Sons, Ltd.     

Estimating the mean hazard ratio parameters for clustered survival data with random clusters

We consider a latent variable hazard model for clustered survival data where clusters are a random sample from an underlying population. We allow interactions between the random cluster effect and covariates. We use a maximum pseudo-likelihood estimator to estimate the mean hazard ratio parameters. We propose a bootstrap sampling scheme to obtain an estimate of the variance of the proposed estimator. Application of this method in large multi-centre clinical trials allows one to assess the mean treatment effect, where we consider participating centres as a random sample from an underlying population. We evaluate properties of the proposed estimators via extensive simulation studies. A real data example from the Studies of Left Ventricular Dysfunction (SOLVD) Prevention Trial illustrates the method. © 1997 by John Wiley & Sons, Ltd.     

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.     

A new strategy for meta-analysis of continuous covariates in observational studies

When several studies are available, a meta-analytic assessment of the effect of a risk or prognostic factor on an outcome is often required. We propose a new strategy, requiring individual participant data, to provide a summary estimate of the functional relationship between a continuous covariate and the outcome in a regression model, adjusting for confounding factors. Our procedure comprises three steps. First, we determine a confounder model. Ideally, the latter should include the same variables across studies, but this may be impossible. Next, we estimate the functional form for the continuous variable of interest in each study, adjusted for the confounder model. Finally, we combine the individual functions by weighted averaging to obtain a summary estimate of the function. Fractional polynomial methodology and pointwise weighted averaging of functions are the key components. In contrast to a pooled analysis, our approach can reflect more variability between functions from different studies and more flexibility with respect to confounders. We illustrate the procedure by using data from breast cancer patients in the Surveillance, Epidemiology, and End Results Program database, where we consider data from nine individual registries as separate studies. We estimate the functional forms for the number of positive lymph nodes and age. The former is an example where a strong prognostic effect has long been recognized, whereas the prognostic effect of the latter is weak or even controversial. We further discuss some general issues that are found in meta-analyses of observational studies.     

Analysis of clustered failure time data with cure fraction using copula

Clustered survival data in the presence of cure has received increasing attention. In this paper, we consider a semiparametric mixture cure model which incorporates a logistic regression model for the cure fraction and a semiparametric regression model for the failure time. We utilize Archimedean copula (AC) models to assess the strength of association for both susceptibility and failure times between susceptible individuals in the same cluster. Instead of using the full likelihood approach, we consider a composite likelihood function and a two-stage estimation procedure for both marginal and association parameters. A Jackknife procedure that takes out one cluster at a time is proposed for the variance estimation of the estimators. Akaike information criterion is applied to select the best model among ACs. Simulation studies are performed to validate our estimating procedures, and two real data sets are analyzed to demonstrate the practical use of our proposed method.     

Assessment of the trace organic molecular composition of industrial and municipal wastewater effluents by capillary gas chromatography/real-time high-resolution mass spectrometry: a preliminary report

In summary, the use of HRGC/HRMS and ECC can provide valuable information on the organic constituents of refinery wastewater samples, even when present in extremely complex mixtures. In many cases, the low-resolution (nominal mass) spectra would not have been interpretable without the availability of the corresponding high-resolution (accurate mass and elemental composition) information, since, even with the use of capillary columns, the components of these complex mixtures were not chromatographically resolved. It is apparent that additional components in these fractions could be identified if additional data processing followed by detailed analysis of the ECC data set is carried out incorporating the retention index information derivable from the series of n-alkanes present, although the incomplete chemical fractionation of the samples is a complicating factor in these particular wastewater extracts.     

Non-parametric estimation of survival function for recurrent events data

Recurrent events when we deal with survival studies demand a different methodology from what is used in standard survival analysis. The main problem that we found when we make inference in these kind of studies is that the observations may not be independent. Thus, biased and inefficient estimators can be obtained if we do not take into account this fact. In the independent case, the interocurrence survival function can be estimated by the generalization of the limit product estimator (Pe?a et al. (2001)). However, if data are correlated, other models should be used such as frailty models or an estimator proposed by Wang and Chang (1999), that take into account the fact that interocurrence times were or not correlated. The aim of this paper has been the illustration of these approaches by using two real data sets.     

Adaptive Dunnett tests for treatment selection

Clinical trials incorporating treatment selection at pre-specified interim analyses allow to integrate two clinical studies into a single, confirmatory study. In an adaptive interim analysis, treatment arms are selected based on interim data as well as external information. The specific selection rule does not need to be pre-specified in advance in order to control the multiple type I error rate. We propose an adaptive Dunnett test procedure based on the conditional error rate of the single-stage Dunnett test. The adaptive procedure uniformly improves the classical Dunnett test, which is shown to be strictly conservative if treatments are dropped at interim. The adaptive Dunnett test is compared in a simulation with the classical Dunnett test as well as with adaptive combination tests based on the closure principle. The method is illustrated with a real-data example.     

Nonparametric estimation of broad sense agreement between ordinal and censored continuous outcomes

The concept of broad sense agreement (BSA) has recently been proposed for studying the relationship between a continuous measurement and an ordinal measurement. They developed a nonparametric procedure for estimating the BSA index, which is only applicable to completely observed data. In this work, we consider the problem of evaluating BSA index when the continuous measurement is subject to censoring. We propose a nonparametric estimation method built upon a derivation of a new functional representation of the BSA index, which allows for accommodating censoring by plugging in the nonparametric survival function estimators. We establish the consistency and asymptotic normality for the proposed BSA estimator. We also investigate an alternative approach based on the strategy of multiple imputation, which is shown to have better empirical performance with small sample sizes than the plug-in method. Extensive simulation studies are conducted to evaluate our proposals. We illustrate our methods via an application to a Surgical Intensive Care Unit study.