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We consider estimation of the semiparametric additive hazards model with an unspecified baseline hazard function where the effect of a continuous covariate has a specific shape but otherwise unspecified. Such estimation is particularly useful for a unimodal hazard function, where the hazard is monotone increasing and monotone decreasing with an unknown mode. A popular approach of the proportional hazards model is limited in such setting due to the complicated structure of the partial likelihood. Our model defines a quadratic loss function, and its simple structure allows a global Hessian matrix that does not involve parameters. Thus, once the global Hessian matrix is computed, a standard quadratic programming method can be applicable by profiling all possible locations of the mode. However, the quadratic programming method may be inefficient to handle a large global Hessian matrix in the profiling algorithm due to a large dimensionality, where the dimension of the global Hessian matrix and number of hypothetical modes are the same order as the sample size. We propose the quadratic pool adjacent violators algorithm to reduce computational costs. The proposed algorithm is extended to the model with a time‐dependent covariate with monotone or U‐shape hazard function. In simulation studies, our proposed method improves computational speed compared to the quadratic programming method, with bias and mean square error reductions. We analyze data from a recent cardiovascular study. [ABSTRACT FROM AUTHOR]

In many phase 1 oncology trials of immunotherapies, no dose‐limiting toxicities are observed and the maximum tolerated dose cannot be identified. In these settings, dose‐finding can be guided by a biomarker of response rather than the occurrences of dose‐limiting toxicity. The recommended phase 2 dose can be defined as the dose with mean response equal to a prespecified value of a continuous response biomarker. To target the mean of a continuous biomarker, we build on the idea of the continual reassessment method and the quasi‐Bernoulli likelihood. We extend the design to a problem of finding the recommended phase 2 dose combination in a trial with multiple immunotherapies. [ABSTRACT FROM AUTHOR]

Studies on the health effects of environmental mixtures face the challenge of limit of detection (LOD) in multiple correlated exposure measurements. Conventional approaches to deal with covariates subject to LOD, including complete-case analysis, substitution methods, and parametric modeling of covariate distribution, are feasible but may result in efficiency loss or bias. With a single covariate subject to LOD, a flexible semiparametric accelerated failure time (AFT) model to accommodate censored measurements has been proposed. We generalize this approach by considering a multivariate AFT model for the multiple correlated covariates subject to LOD and a generalized linear model for the outcome. A two-stage procedure based on semiparametric pseudo-likelihood is proposed for estimating the effects of these covariates on health outcome. Consistency and asymptotic normality of the estimators are derived for an arbitrary fixed dimension of covariates. Simulations studies demonstrate good large sample performance of the proposed methods vs conventional methods in realistic scenarios. We illustrate the practical utility of the proposed method with the LIFECODES birth cohort data, where we compare our approach to existing approaches in an analysis of multiple urinary trace metals in association with oxidative stress in pregnant women. [ABSTRACT FROM AUTHOR]

In the analysis for competing risks data, regression modeling of the cause‐specific hazard functions has been usually conducted using the same time scale for all event types. However, when the true time scale is different for each event type, it would be appropriate to specify regression models for the cause‐specific hazards on different time scales for different event types. Often, the proportional hazards model has been used for regression modeling of the cause‐specific hazard functions. However, the proportionality assumption may not be appropriate in practice. In this article, we consider the additive risk model as an alternative to the proportional hazards model. We propose predictions of the cumulative incidence functions under the cause‐specific additive risk models employing different time scales for different event types. We establish the consistency and asymptotic normality of the predicted cumulative incidence functions under the cause‐specific additive risk models specified on different time scales using empirical processes and derive consistent variance estimators of the predicted cumulative incidence functions. Through simulation studies, we show that the proposed prediction methods perform well. We illustrate the methods using stage III breast cancer data obtained from the Surveillance, Epidemiology, and End Results (SEER) program of the National Cancer Institute. [ABSTRACT FROM AUTHOR]

In prevalent cohort studies where subjects are recruited at a cross‐section, the time to an event may be subject to length‐biased sampling, with the observed data being either the forward recurrence time, or the backward recurrence time, or their sum. In the regression setting, assuming a semiparametric accelerated failure time model for the underlying event time, where the intercept parameter is absorbed into the nuisance parameter, it has been shown that the model remains invariant under these observed data setups and can be fitted using standard methodology for accelerated failure time model estimation, ignoring the length bias. However, the efficiency of these estimators is unclear, owing to the fact that the observed covariate distribution, which is also length biased, may contain information about the regression parameter in the accelerated life model. We demonstrate that if the true covariate distribution is completely unspecified, then the naive estimator based on the conditional likelihood given the covariates is fully efficient for the slope. [ABSTRACT FROM AUTHOR]

The continual reassessment method (CRM) is a well‐known design for dose‐finding trials with the goal of estimating the maximum tolerated dose (MTD), the dose with a given probability of toxicity. The standard assumption is that the probability of toxicity monotonically increases with dose. We show that the CRM can still be consistent and correctly identify the MTD even when the dose‐toxicity curve is not monotone as long as there is monotonicity of the true toxicity probabilities right below and right above the true MTD. In the case of multiple therapies, where it is unclear how to order combinations of dose levels of multiple therapies, our findings provide insight into the performance of the partial order CRM (POCRM). To select the correct dose combination at the end of a trial, the POCRM does not have to select a monotone ordering of drug combinations. We illustrate the connection between our results for the CRM with a nonmonotone dose‐toxicity curve and the POCRM via simulations. [ABSTRACT FROM AUTHOR]

Instrumental variable (IV) analysis can be used to address bias due to unobserved confounding when estimating the causal effect of a treatment on an outcome of interest. However, if a proposed IV is correlated with unmeasured confounders and/or weakly correlated with the treatment, the standard IV estimator may be more biased than an ordinary least squares (OLS) estimator. Several methods have been proposed that compare the bias of the IV and OLS estimators relying on the belief that measured covariates can be used as proxies for the unmeasured confounder. Despite these developments, there is lack of discussion about approaches that can be used to formally test whether the IV estimator may be less biased than the OLS estimator. Thus, we have developed a testing framework to compare the bias and a criterion to select informative measured covariates for bias comparison and regression adjustment. We also have developed a bias‐correction method, which allows one to use an invalid IV to correct the bias of the OLS or IV estimator. Numerical studies demonstrate that the proposed methods perform well with realistic sample sizes. [ABSTRACT FROM AUTHOR]

In survival analysis, time-varying covariates are covariates whose value can change during follow-up. Outcomes in medical research are frequently subject to competing risks (events precluding the occurrence of the primary outcome). We review the types of time-varying covariates and highlight the effect of their inclusion in the subdistribution hazard model. External time-dependent covariates are external to the subject, can effect the failure process, but are not otherwise involved in the failure mechanism. Internal time-varying covariates are measured on the subject, can effect the failure process directly, and may also be impacted by the failure mechanism. In the absence of competing risks, a consequence of including internal time-dependent covariates in the Cox model is that one cannot estimate the survival function or the effect of covariates on the survival function. In the presence of competing risks, the inclusion of internal time-varying covariates in a subdistribution hazard model results in the loss of the ability to estimate the cumulative incidence function (CIF) or the effect of covariates on the CIF. Furthermore, the definition of the risk set for the subdistribution hazard function can make defining internal time-varying covariates difficult or impossible. We conducted a review of the use of time-varying covariates in subdistribution hazard models in articles published in the medical literature in 2015 and in the first 5 months of 2019. Seven percent of articles published included a time-varying covariate. Several inappropriately described a time-varying covariate as having an association with the risk of the outcome. [ABSTRACT FROM AUTHOR]

This paper demonstrates the flexibility of a general approach for the analysis of discrete time competing risks data that can accommodate complex data structures, different time scales for different causes, and nonstandard sampling schemes. The data may involve a single data source where all individuals contribute to analyses of both cause-specific hazard functions, overlapping datasets where some individuals contribute to the analysis of the cause-specific hazard function of only one cause while other individuals contribute to analyses of both cause-specific hazard functions, or separate data sources where each individual contributes to the analysis of the cause-specific hazard function of only a single cause. The approach is modularized into estimation and prediction. For the estimation step, the parameters and the variance-covariance matrix can be estimated using widely available software. The prediction step utilizes a generic program with plug-in estimates from the estimation step. The approach is illustrated with three prognostic models for stage IV male oral cancer using different data structures. The first model uses only men with stage IV oral cancer from population-based registry data. The second model strategically extends the cohort to improve the efficiency of the estimates. The third model improves the accuracy for those with a lower risk of other causes of death, by bringing in an independent data source collected under a complex sampling design with additional other-cause covariates. These analyses represent novel extensions of existing methodology, broadly applicable for the development of prognostic models capturing both the cancer and noncancer aspects of a patient's health. [ABSTRACT FROM AUTHOR]

Recurrent event data frequently occur in longitudinal studies when subjects experience more than one event during the observation period. Often, the occurrence of subsequent events is associated with the experience of previous events. Such dependence is commonly ignored in the application of standard recurrent event methodology. In this paper, we utilize a Cox-type regression model with time-varying triggering effect depending on the number and timing of previous events to enhance both model fit and prediction. Parameter estimation and statistical inference is achieved via the partial likelihood. A statistical test procedure is provided to assess the existence of the triggering effects. We demonstrate our approach via comprehensive simulation studies and a real data analysis on chronic pseudomonas infections in young cystic fibrosis patients. Our model provides significantly better predictions than standard recurrent event models. [ABSTRACT FROM AUTHOR]

Propensity-score matching is a popular analytic method to remove the effects of confounding due to measured baseline covariates when using observational data to estimate the effects of treatment. Time-to-event outcomes are common in medical research. Competing risks are outcomes whose occurrence precludes the occurrence of the primary time-to-event outcome of interest. All non-fatal outcomes and all cause-specific mortality outcomes are potentially subject to competing risks. There is a paucity of guidance on the conduct of propensity-score matching in the presence of competing risks. We describe how both relative and absolute measures of treatment effect can be obtained when using propensity-score matching with competing risks data. Estimates of the relative effect of treatment can be obtained by using cause-specific hazard models in the matched sample. Estimates of absolute treatment effects can be obtained by comparing cumulative incidence functions (CIFs) between matched treated and matched control subjects. We conducted a series of Monte Carlo simulations to compare the empirical type I error rate of different statistical methods for testing the equality of CIFs estimated in the matched sample. We also examined the performance of different methods to estimate the marginal subdistribution hazard ratio. We recommend that a marginal subdistribution hazard model that accounts for the within-pair clustering of outcomes be used to test the equality of CIFs and to estimate subdistribution hazard ratios. We illustrate the described methods by using data on patients discharged from hospital with acute myocardial infarction to estimate the effect of discharge prescribing of statins on cardiovascular death. [ABSTRACT FROM AUTHOR]

Abstract: In statistical modelling, it is often of interest to evaluate non‐negative quantities that capture heterogeneity in the population such as variances, mixing proportions and dispersion parameters. In instances of covariate‐dependent heterogeneity, the implied homogeneity hypotheses are nonstandard and existing inferential techniques are not applicable. In this paper, we develop a quasi‐score test statistic to evaluate homogeneity against heterogeneity that varies with a covariate profile through a regression model. We establish the limiting null distribution of the proposed test as a functional of mixtures of chi‐square processes. The methodology does not require the full distribution of the data to be entirely specified. Instead, a general estimating function for a finite dimensional component of the model, that is, of interest is assumed but other characteristics of the population are left completely unspecified. We apply the methodology to evaluate the excess zero proportion in zero‐inflated models for count data. Our numerical simulations show that the proposed test can greatly improve efficiency over tests of homogeneity that neglect covariate information under the alternative hypothesis. An empirical application to dental caries indices demonstrates the importance and practical utility of the methodology in detecting excess zeros in the data. [ABSTRACT FROM AUTHOR]

Discrimination slope, defined as the slope of a linear regression of predicted probabilities of event derived from a prognostic model on the binary event status, has recently gained popularity as a measure of model performance. It is as a building block for the integrated discrimination improvement that equals the difference in discrimination slopes between the two models being compared. Several authors have pointed out that it does not make sense to apply the integrated discrimination improvement and discrimination slope when working with mis-calibrated models, whereas others have raised concerns about the ability of improving discrimination slope without adding new information. In this paper, we show that under certain assumptions the discrimination slope is asymptotically related to two other R-squared measures, one of which is a rescaled version of the Brier score, known to be proper. Furthermore, we illustrate how a simple recalibration makes the slope equal to the rescaled Brier R-squared metric. We also show that the discrimination slope can be interpreted as a measure of reduction in expected regret for the Gini-Brier regret function. Using theoretical and practical examples, we illustrate how all of these metrics are affected by different levels of model mis-calibration. In particular, we demonstrate that simple recalibration ascertaining calibration in-the-large and calibration slope equal to 1 are not sufficient to correct for some forms of mis-calibration. We conclude that R-squared metrics, including the discrimination slope, offer an attractive choice for quantifying model performance as long as one accounts for their sensitivity to model calibration. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In survival analysis, a competing risk is an event whose occurrence precludes the occurrence of the primary event of interest. Outcomes in medical research are frequently subject to competing risks. In survival analysis, there are 2 key questions that can be addressed using competing risk regression models: first, which covariates affect the rate at which events occur, and second, which covariates affect the probability of an event occurring over time. The cause-specific hazard model estimates the effect of covariates on the rate at which events occur in subjects who are currently event-free. Subdistribution hazard ratios obtained from the Fine-Gray model describe the relative effect of covariates on the subdistribution hazard function. Hence, the covariates in this model can also be interpreted as having an effect on the cumulative incidence function or on the probability of events occurring over time. We conducted a review of the use and interpretation of the Fine-Gray subdistribution hazard model in articles published in the medical literature in 2015. We found that many authors provided an unclear or incorrect interpretation of the regression coefficients associated with this model. An incorrect and inconsistent interpretation of regression coefficients may lead to confusion when comparing results across different studies. Furthermore, an incorrect interpretation of estimated regression coefficients can result in an incorrect understanding about the magnitude of the association between exposure and the incidence of the outcome. The objective of this article is to clarify how these regression coefficients should be reported and to propose suggestions for interpreting these coefficients. [ABSTRACT FROM AUTHOR]

In studies with survival or time-to-event outcomes, a competing risk is an event whose occurrence precludes the occurrence of the primary event of interest. Specialized statistical methods must be used to analyze survival data in the presence of competing risks. We conducted a review of randomized controlled trials with survival outcomes that were published in high-impact general medical journals. Of 40 studies that we identified, 31 (77.5%) were potentially susceptible to competing risks. However, in the majority of these studies, the potential presence of competing risks was not accounted for in the statistical analyses that were described. Of the 31 studies potentially susceptible to competing risks, 24 (77.4%) reported the results of a Kaplan-Meier survival analysis, while only five (16.1%) reported using cumulative incidence functions to estimate the incidence of the outcome over time in the presence of competing risks. The former approach will tend to result in an overestimate of the incidence of the outcome over time, while the latter approach will result in unbiased estimation of the incidence of the primary outcome over time. We provide recommendations on the analysis and reporting of randomized controlled trials with survival outcomes in the presence of competing risks. © 2017 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd. [ABSTRACT FROM AUTHOR]

Predicting the occurrence of an adverse event over time is an important issue in clinical medicine. Clinical prediction models and associated points-based risk-scoring systems are popular statistical methods for summarizing the relationship between a multivariable set of patient risk factors and the risk of the occurrence of an adverse event. Points-based risk-scoring systems are popular amongst physicians as they permit a rapid assessment of patient risk without the use of computers or other electronic devices. The use of such points-based risk-scoring systems facilitates evidence-based clinical decision making. There is a growing interest in cause-specific mortality and in non-fatal outcomes. However, when considering these types of outcomes, one must account for competing risks whose occurrence precludes the occurrence of the event of interest. We describe how points-based risk-scoring systems can be developed in the presence of competing events. We illustrate the application of these methods by developing risk-scoring systems for predicting cardiovascular mortality in patients hospitalized with acute myocardial infarction. Code in the R statistical programming language is provided for the implementation of the described methods. © 2016 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd. [ABSTRACT FROM AUTHOR]