Estimating a time‐dependent concordance index for survival prediction models with covariate dependent censoring

Given a predictive marker and a time‐to‐event response variable, the proportion of concordant pairs in a data set is called concordance index. A specifically useful marker is the risk predicted by a survival regression model. This article extends the existing methodology for applications where the length of the follow‐up period depends on the predictor variables. A class of inverse probability of censoring weighted estimators is discussed in which the estimates rely on a working model for the conditional censoring distribution. The estimators are consistent for a truncated concordance index if the working model is correctly specified and if the probability of being uncensored at the truncation time is positive. In this framework, all kinds of prediction models can be assessed, and time trends in the discrimination ability of a model can be captured by varying the truncation time point. For illustration, we re‐analyze a study on risk prediction for prostate cancer patients. The effects of misspecification of the censoring model are studied in simulated data. Copyright © 2012 John Wiley & Sons, Ltd.     

A two‐stage approach for dynamic prediction of time‐to‐event distributions

Dynamic prediction uses longitudinal biomarkers for real‐time prediction of an individual patient's prognosis. This is critical for patients with an incurable disease such as cancer. Biomarker trajectories are usually not linear, nor even monotone, and vary greatly across individuals. Therefore, it is difficult to fit them with parametric models. With this consideration, we propose an approach for dynamic prediction that does not need to model the biomarker trajectories. Instead, as a trade‐off, we assume that the biomarker effects on the risk of disease recurrence are smooth functions over time. This approach turns out to be computationally easier. Simulation studies show that the proposed approach achieves stable estimation of biomarker effects over time, has good predictive performance, and is robust against model misspecification. It is a good compromise between two major approaches, namely, (i) joint modeling of longitudinal and survival data and (ii) landmark analysis. The proposed method is applied to patients with chronic myeloid leukemia. At any time following their treatment with tyrosine kinase inhibitors, longitudinally measured BCR‐ABL gene expression levels are used to predict the risk of disease progression. Copyright © 2016 John Wiley & Sons, Ltd.     

Estimation of covariate effects in generalized linear mixed models with a misspecified distribution of random intercepts and slopes

Generalized linear mixed models with random intercepts and slopes provide useful analyses of clustered and longitudinal data and typically require the specification of the distribution of the random effects. Previous work for models with only random intercepts has shown that misspecifying the shape of this distribution may bias estimates of the intercept, but typically leads to little bias in estimates of covariate effects. Very few papers have examined the effects of misspecifying the joint distribution of random intercepts and slopes. However, simulation results in a recent paper suggest that misspecifying the shape of the random slope distribution can yield severely biased estimates of all model parameters. Using analytic results, simulation studies and fits to example data, this paper examines the bias in parameter estimates due to misspecification of the shape of the joint distribution of random intercepts and slopes. Consistent with results for models with only random intercepts, and contrary to the claims of severe bias in a recent paper, we show that misspecification of the joint distribution typically yields little bias in estimates of covariate effects and is restricted to covariates associated with the misspecified random effects distributions. We also show that misspecification of the distribution of random effects has little effect on confidence interval performance. Coverage rates based on the model‐based standard errors from fitted likelihoods were generally quite close to nominal. Copyright © 2012 John Wiley & Sons, Ltd.     

An improved test of latent‐variable model misspecification in structural measurement error models for group testing data

We consider structural measurement error models for group testing data. Likelihood inference based on structural measurement error models requires one to specify a model for the latent true predictors. Inappropriate specification of this model can lead to erroneous inference. We propose a new method tailored to detect latent‐variable model misspecification in structural measurement error models for group testing data. Compared with the existing diagnostic methods developed for the same purpose, our method shows vast improvement in the power to detect latent‐variable model misspecification in group testing design. We illustrate the implementation and performance of the proposed method via simulation and application to a real data example. Copyright © 2009 John Wiley & Sons, Ltd.     

Estimating gene penetrance from family data

Family data are useful for estimating disease risk in carriers of specific genotypes of a given gene (penetrance). Penetrance is frequently estimated assuming that relatives' phenotypes are independent, given their genotypes for the gene of interest. This assumption is unrealistic when multiple shared risk factors contribute to disease risk. In this setting, the phenotypes of relatives are correlated even after adjustment for the genotypes of any one gene (residual correlation). Many methods have been proposed to address this problem, but their performance has not been evaluated systematically. In simulations we generated genotypes for a rare (frequency 0.35%) allele of moderate penetrance, and a common (frequency 15%) allele of low penetrance, and then generated correlated disease survival times using the Clayton‐Oakes copula model. We ascertained families using both population and clinic designs. We then compared the estimates of several methods to the optimal ones obtained from the model used to generate the data. We found that penetrance estimates for common low‐risk genotypes were more robust to model misspecification than those for rare, moderate‐risk genotypes. For the latter, penetrance estimates obtained ignoring residual disease correlation had large biases. Also biased were estimates based only on families that segregate the risk allele. In contrast, a method for accommodating phenotype correlation by assuming the presence of genetic heterogeneity performed nearly optimally, even when the survival data were coded as binary outcomes. We conclude that penetrance estimates that accommodate residual phenotype correlation (even only approximately) outperform those that ignore it, and that coding censored survival outcomes as binary does not substantially increase the mean‐square errror of the estimates, provided the censoring is not extensive. Genet. Epidemiol. 34: 373–381, 2010. © 2010 Wiley‐Liss, Inc.     

Estimating parsimonious models of longitudinal causal effects using regressions on propensity scores

Parsimony is important for the interpretation of causal effect estimates of longitudinal treatments on subsequent outcomes. One method for parsimonious estimates fits marginal structural models by using inverse propensity scores as weights. This method leads to generally large variability that is uncommon in more likelihood‐based approaches. A more recent method fits these models by using simulations from a fitted g‐computation, but requires the modeling of high‐dimensional longitudinal relations that are highly susceptible to misspecification. We propose a new method that, first, uses longitudinal propensity scores as regressors to reduce the dimension of the problem and then uses the approximate likelihood for the first estimates to fit parsimonious models. We demonstrate the methods by estimating the effect of anticoagulant therapy on survival for cancer and non‐cancer patients who have inferior vena cava filters. Copyright © 2013 John Wiley & Sons, Ltd.     

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

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

An investigation of error sources and their impact in estimating the time to the most recent ancestor of spatially and temporally distributed HIV sequences

This is an investigation of significant error sources and their impact in estimating the time to the most recent common ancestor (MRCA) of spatially and temporally distributed human immunodeficiency virus (HIV) sequences. We simulate an HIV epidemic under a range of assumptions with known time to the MRCA (tMRCA). We then apply a range of baseline (known) evolutionary models to generate sequence data. We next estimate or assume one of several misspecified models and use the chosen model to estimate the time to the MRCA. Random effects and the extent of model misspecification determine the magnitude of error sources that could include: neglected heterogeneity in substitution rates across lineages and DNA sites; uncertainty in HIV isolation times; uncertain magnitude and type of population subdivision; uncertain impacts of host/viral transmission dynamics, and unavoidable model estimation errors. Our results suggest that confidence intervals will rarely have the nominal coverage probability for tMRCA. Neglected effects lead to errors that are unaccounted for in most analyses, resulting in optimistically narrow confidence intervals (CI). Using real HIV sequences having approximately known isolation times and locations, we present possible confidence intervals for several sets of assumptions. In general, we cannot be certain how much to broaden a stated confidence interval for tMRCA. However, we describe the impact of candidate error sources on CI width. We also determine which error sources have the most impact on CI width and demonstrate that the standard bootstrap method will underestimate the CI width.     

The Extrapolation Performance of Survival Models for Data With a Cure Fraction: A Simulation Study

ObjectivesCurative treatments can result in complex hazard functions. The use of standard survival models may result in poor extrapolations. Several models for data which may have a cure fraction are available, but comparisons of their extrapolation performance are lacking. A simulation study was performed to assess the performance of models with and without a cure fraction when fit to data with a cure fraction.MethodsData were simulated from a Weibull cure model, with 9 scenarios corresponding to different lengths of follow-up and sample sizes. Cure and noncure versions of standard parametric, Royston-Parmar, and dynamic survival models were considered along with noncure fractional polynomial and generalized additive models. The mean-squared error and bias in estimates of the hazard function were estimated.ResultsWith the shortest follow-up, none of the cure models provided good extrapolations. Performance improved with increasing follow-up, except for the misspecified standard parametric cure model (lognormal). The performance of the flexible cure models was similar to that of the correctly specified cure model. Accurate estimates of the cured fraction were not necessary for accurate hazard estimates. Models without a cure fraction provided markedly worse extrapolations.ConclusionsFor curative treatments, failure to model the cured fraction can lead to very poor extrapolations. Cure models provide improved extrapolations, but with immature data there may be insufficient evidence to choose between cure and noncure models, emphasizing the importance of clinical knowledge for model choice. Dynamic cure fraction models were robust to model misspecification, but standard parametric cure models were not.     

Sample size and robust marginal methods for cluster‐randomized trials with censored event times

In cluster‐randomized trials, intervention effects are often formulated by specifying marginal models, fitting them under a working independence assumption, and using robust variance estimates to address the association in the responses within clusters. We develop sample size criteria within this framework, with analyses based on semiparametric Cox regression models fitted with event times subject to right censoring. At the design stage, copula models are specified to enable derivation of the asymptotic variance of estimators from a marginal Cox regression model and to compute the number of clusters necessary to satisfy power requirements. Simulation studies demonstrate the validity of the sample size formula in finite samples for a range of cluster sizes, censoring rates, and degrees of within‐cluster association among event times. The power and relative efficiency implications of copula misspecification is studied, as well as the effect of within‐cluster dependence in the censoring times. Sample size criteria and other design issues are also addressed for the setting where the event status is only ascertained at periodic assessments and times are interval censored. Copyright © 2014 JohnWiley & Sons, Ltd.     

Interaction analysis under misspecification of main effects: Some common mistakes and simple solutions

The statistical practice of modeling interaction with two linear main effects and a product term is ubiquitous in the statistical and epidemiological literature. Most data modelers are aware that the misspecification of main effects can potentially cause severe type I error inflation in tests for interactions, leading to spurious detection of interactions. However, modeling practice has not changed. In this article, we focus on the specific situation where the main effects in the model are misspecified as linear terms and characterize its impact on common tests for statistical interaction. We then propose some simple alternatives that fix the issue of potential type I error inflation in testing interaction due to main effect misspecification. We show that when using the sandwich variance estimator for a linear regression model with a quantitative outcome and two independent factors, both the Wald and score tests asymptotically maintain the correct type I error rate. However, if the independence assumption does not hold or the outcome is binary, using the sandwich estimator does not fix the problem. We further demonstrate that flexibly modeling the main effect under a generalized additive model can largely reduce or often remove bias in the estimates and maintain the correct type I error rate for both quantitative and binary outcomes regardless of the independence assumption. We show, under the independence assumption and for a continuous outcome, overfitting and flexibly modeling the main effects does not lead to power loss asymptotically relative to a correctly specified main effect model. Our simulation study further demonstrates the empirical fact that using flexible models for the main effects does not result in a significant loss of power for testing interaction in general. Our results provide an improved understanding of the strengths and limitations for tests of interaction in the presence of main effect misspecification. Using data from a large biobank study “The Michigan Genomics Initiative”, we present two examples of interaction analysis in support of our results.     

Bayesian quantile regression‐based nonlinear mixed‐effects joint models for time‐to‐event and longitudinal data with multiple features

This article explores Bayesian joint models for a quantile of longitudinal response, mismeasured covariate and event time outcome with an attempt to (i) characterize the entire conditional distribution of the response variable based on quantile regression that may be more robust to outliers and misspecification of error distribution; (ii) tailor accuracy from measurement error, evaluate non‐ignorable missing observations, and adjust departures from normality in covariate; and (iii) overcome shortages of confidence in specifying a time‐to‐event model. When statistical inference is carried out for a longitudinal data set with non‐central location, non‐linearity, non‐normality, measurement error, and missing values as well as event time with being interval censored, it is important to account for the simultaneous treatment of these data features in order to obtain more reliable and robust inferential results. Toward this end, we develop Bayesian joint modeling approach to simultaneously estimating all parameters in the three models: quantile regression‐based nonlinear mixed‐effects model for response using asymmetric Laplace distribution, linear mixed‐effects model with skew‐t distribution for mismeasured covariate in the presence of informative missingness and accelerated failure time model with unspecified nonparametric distribution for event time. We apply the proposed modeling approach to analyzing an AIDS clinical data set and conduct simulation studies to assess the performance of the proposed joint models and method. Copyright © 2016 John Wiley & Sons, Ltd.     

An evaluation of resampling methods for assessment of survival risk prediction in high-dimensional settings

Resampling techniques are often used to provide an initial assessment of accuracy for prognostic prediction models developed using high-dimensional genomic data with binary outcomes. Risk prediction is most important, however, in medical applications and frequently the outcome measure is a right-censored time-to-event variable such as survival. Although several methods have been developed for survival risk prediction with high-dimensional genomic data, there has been little evaluation of the use of resampling techniques for the assessment of such models. Using real and simulated datasets, we compared several resampling techniques for their ability to estimate the accuracy of risk prediction models. Our study showed that accuracy estimates for popular resampling methods, such as sample splitting and leave-one-out cross validation (Loo CV), have a higher mean square error than for other methods. Moreover, the large variability of the split-sample and Loo CV may make the point estimates of accuracy obtained using these methods unreliable and hence should be interpreted carefully. A k-fold cross-validation with k = 5 or 10 was seen to provide a good balance between bias and variability for a wide range of data settings and should be more widely adopted in practice.     

Simulation from a known Cox MSM using standard parametric models for the g‐formula

It is routinely argued that, unlike standard regression‐based estimates, inverse probability weighted (IPW) estimates of the parameters of a correctly specified Cox marginal structural model (MSM) may remain unbiased in the presence of a time‐varying confounder affected by prior treatment. Previously proposed methods for simulating from a known Cox MSM lack knowledge of the law of the observed outcome conditional on the measured past. Although unbiased IPW estimation does not require this knowledge, standard regression‐based estimates rely on correct specification of this law. Thus, in typical high‐dimensional settings, such simulation methods cannot isolate bias due to complex time‐varying confounding as it may be conflated with bias due to misspecification of the outcome regression model. In this paper, we describe an approach to Cox MSM data generation that allows for a comparison of the bias of IPW estimates versus that of standard regression‐based estimates in the complete absence of model misspecification. This approach involves simulating data from a standard parametrization of the likelihood and solving for the underlying Cox MSM. We prove that solutions exist and computations are tractable under many data‐generating mechanisms. We show analytically and confirm in simulations that, in the absence of model misspecification, the bias of standard regression‐based estimates for the parameters of a Cox MSM is indeed a function of the coefficients in observed data models quantifying the presence of a time‐varying confounder affected by prior treatment. We discuss limitations of this approach including that implied by the ‘g‐null paradox’. Copyright © 2013 John Wiley & Sons, Ltd.     

A multiple imputation approach for clustered interval‐censored survival data

Multivariate interval‐censored failure time data arise commonly in many studies of epidemiology and biomedicine. Analysis of these type of data is more challenging than the right‐censored data. We propose a simple multiple imputation strategy to recover the order of occurrences based on the interval‐censored event times using a conditional predictive distribution function derived from a parametric gamma random effects model. By imputing the interval‐censored failure times, the estimation of the regression and dependence parameters in the context of a gamma frailty proportional hazards model using the well‐developed EM algorithm is made possible. A robust estimator for the covariance matrix is suggested to adjust for the possible misspecification of the parametric baseline hazard function. The finite sample properties of the proposed method are investigated via simulation. The performance of the proposed method is highly satisfactory, whereas the computation burden is minimal. The proposed method is also applied to the diabetic retinopathy study (DRS) data for illustration purpose and the estimates are compared with those based on other existing methods for bivariate grouped survival data. Copyright © 2010 John Wiley & Sons, Ltd.     

Analysis of childhood brain tumour data in New York City using frailty models.

This paper uses frailty models to analyse overall survival and progression-free survival times for children with a brain tumour. We are interested in how surgery resection affects survival times. We are also interested in how strong a child's progression-free survival correlates with his/her overall survival and if the association differs with age. Traditionally the frailty is modelled parametrically and a maximum likelihood approach is used to estimate the parameters of interest. However, the result is sensitive to the misspecification of the frailty distribution and the currently developed algorithms for the maximum likelihood approach do not allow the association parameter to depend on covariates. Xue formulates a Poisson regression model and applies an estimating equation approach to obtain a consistent estimate of the covariate effect on survival. This paper extends that approach to obtain consistent and efficient estimates of the association parameter as well as the covariate effect and to allow the association parameter to depend on the covariates. The approach does not require the specification of the frailty distribution. The performance of the method is evaluated through simulation studies. We apply this method to a childhood brain tumour data set in New York City.     

Between–within models for survival analysis

A popular way to control for confounding in observational studies is to identify clusters of individuals (e.g., twin pairs), such that a large set of potential confounders are constant (shared) within each cluster. By studying the exposure–outcome association within clusters, we are in effect controlling for the whole set of shared confounders. An increasingly popular analysis tool is the between–within (BW) model, which decomposes the exposure–outcome association into a ‘within‐cluster effect’ and a ‘between‐cluster effect’. BW models are relatively common for nonsurvival outcomes and have been studied in the theoretical literature. Although it is straightforward to use BW models for survival outcomes, this has rarely been carried out in practice, and such models have not been studied in the theoretical literature. In this paper, we propose a gamma BW model for survival outcomes. We compare the properties of this model with the more standard stratified Cox regression model and use the proposed model to analyze data from a twin study of obesity and mortality. We find the following: (i) the gamma BW model often produces a more powerful test of the ‘within‐cluster effect’ than stratified Cox regression; and (ii) the gamma BW model is robust against model misspecification, although there are situations where it could give biased estimates. Copyright © 2013 John Wiley & Sons, Ltd.     

Bias and inference from misspecified mixed‐effect models in stepped wedge trial analysis

Many stepped wedge trials (SWTs) are analysed by using a mixed‐effect model with a random intercept and fixed effects for the intervention and time periods (referred to here as the standard model). However, it is not known whether this model is robust to misspecification. We simulated SWTs with three groups of clusters and two time periods; one group received the intervention during the first period and two groups in the second period. We simulated period and intervention effects that were either common‐to‐all or varied‐between clusters. Data were analysed with the standard model or with additional random effects for period effect or intervention effect. In a second simulation study, we explored the weight given to within‐cluster comparisons by simulating a larger intervention effect in the group of the trial that experienced both the control and intervention conditions and applying the three analysis models described previously. Across 500 simulations, we computed bias and confidence interval coverage of the estimated intervention effect. We found up to 50% bias in intervention effect estimates when period or intervention effects varied between clusters and were treated as fixed effects in the analysis. All misspecified models showed undercoverage of 95% confidence intervals, particularly the standard model. A large weight was given to within‐cluster comparisons in the standard model. In the SWTs simulated here, mixed‐effect models were highly sensitive to departures from the model assumptions, which can be explained by the high dependence on within‐cluster comparisons. Trialists should consider including a random effect for time period in their SWT analysis model. © 2017 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd.     

A comparison of change point models with application to longitudinal lung function measurements in children with cystic fibrosis

Cystic fibrosis (CF) is a hereditary lung disease characterized by loss of lung function over time. Lung function in CF is believed to decline at a higher rate during the adolescence period. It has been also hypothesized that there is a subgroup of individuals for whom lung disease remains relatively stable with only a slight decline over their lifetime. Using data from the University of Colorado CF Children's Registry, we investigate four change point models to model the decline of lung function in children and adolescents: (i) a two‐component mixture random change point model, (ii) a two‐component mixture‐fixed change point model, (iii) a random change point model, and (iv) a fixed change point model. The models are investigated through posterior predictive simulation at the individual and population levels, and a simulation study examining the effects of model misspecification. The data support the mixed random change point model as the preferred model, with roughly 30% of adolescents experiencing a steady decline of 0.5 %FEV₁ per year and 70% experiencing an increase in decline of 4.4 %FEV₁ per year beginning on average at 14.6 years of age. Copyright © 2016 John Wiley & Sons, Ltd.     

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.