Extensions and applications of the Cox-Aalen survival model

A semiparametric Cox-Aalen transformation model with censored data

We propose a broad class of so-called Cox-Aalen transformation models that incorporate both multiplicative and additive covariate effects on the baseline hazard function within a transformation. The proposed models provide a highly flexible and versatile class of semiparametric models that include the transformation models and the Cox-Aalen model as special cases. Specifically, it extends the transformation models by allowing potentially time-dependent covariates to work additively on the baseline hazard and extends the Cox-Aalen model through a predetermined transformation function. We propose an estimating equation approach and devise an expectation-solving (ES) algorithm that involves fast and robust calculations. The resulting estimator is shown to be consistent and asymptotically normal via modern empirical process techniques. The ES algorithm yields a computationally simple method for estimating the variance of both parametric and nonparametric estimators. Finally, we demonstrate the performance of our procedures through extensive simulation studies and applications in two randomized, placebo-controlled human immunodeficiency virus (HIV) prevention efficacy trials. The data example shows the utility of the proposed Cox-Aalen transformation models in enhancing statistical power for discovering covariate effects.

CASANOVA: Permutation inference in factorial survival designs

We propose inference procedures for general factorial designs with time-to-event endpoints. Similar to additive Aalen models, null hypotheses are formulated in terms of cumulative hazards. Deviations are measured in terms of quadratic forms in Nelson-Aalen-type integrals. Different from existing approaches, this allows to work without restrictive model assumptions as proportional hazards. In particular, crossing survival or hazard curves can be detected without a significant loss of power. For a distribution-free application of the method, a permutation strategy is suggested. The resulting procedures' asymptotic validity is proven and small sample performances are analyzed in extensive simulations. The analysis of a data set on asthma illustrates the applicability.

Estimation of conditional cumulative incidence functions under generalized semiparametric regression models with missing covariates, with application to analysis of biomarker correlates in vaccine trials

This article studies generalized semiparametric regression models for conditional cumulative incidence functions with competing risks data when covariates are missing by sampling design or happenstance. A doubly-robust augmented inverse probability weighted complete-case (AIPW) approach to estimation and inference is investigated. This approach modifies IPW complete-case estimating equations by exploiting the key features in the relationship between the missing covariates and the phase-one data to improve efficiency. An iterative numerical procedure is derived to solve the nonlinear estimating equations. The asymptotic properties of the proposed estimators are established. A simulation study examining the finite-sample performances of the proposed estimators shows that the AIPW estimators are more efficient than the IPW estimators. The developed method is applied to the RV144 HIV-1 vaccine efficacy trial to investigate vaccine-induced IgG binding antibodies to HIV-1 as correlates of acquisition of HIV-1 infection while taking account of whether the HIV-1 sequences are near or far from the HIV-1 sequences represented in the vaccine construct.

Confidence bands in survival analysis

BACKGROUND

Providing estimates of uncertainty for statistical quantities is important for statistical inference. When the statistical quantity of interest is a survival curve, which is a function over time, the appropriate type of uncertainty estimate is a confidence band constructed to account for the correlation between points on the curve, we will call this a simultaneous confidence band. This, however, is not the type of confidence band provided in standard software, which is constructed by joining the confidence intervals at given time points.

METHODS

We show that this type of band does not have desirable joint/simultaneous coverage properties in comparison to simultaneous bands.

RESULTS

There are different ways of constructing simultaneous confidence bands, and we find that bands based on the likelihood ratio appear to have the most desirable properties. Although there is no standard software available in the three major statistical packages to compute likelihood-based simultaneous bands, we summarise and give code to use available statistical software to construct other simultaneous forms of bands, which we illustrate using a study of colon cancer.

CONCLUSIONS

There is a need for more user-friendly statistical software to compute simultaneous confidence bands using the available methods.

Equivalence tests under the Cox-Aalen model and the partly Aalen model

For the equivalence trial with survivor outcomes from two treatment groups, the most popular testing procedure is the test proposed by under the proportional hazards (PH) model. In this article, when the treatment effect is time invariant, we demonstrate that the result under the PH model can be extended to the Cox-Aalen model. When the treatment effect is time-variant, we propose an equivalent test for the differences of two cumulative hazard functions under the partly Aalen model. Simulation studies show that the proposed tests perform well in finite samples. We illustrate the proposed tests using bladder cancer data and the primary biliary cirrhosis (PBC) data.

Tolerance of young allis shad Alosa alosa (Clupeidae) to oxy-thermic stress

The ecology of the young stages of allis shad Alosa alosa is poorly documented, although they can be exposed to many pressures during their freshwater phase and their downstream migration. When passing through systems such as the Gironde-Garonne-Dordogne watershed (GGD, SW France), they can be subjected to high temperatures and low levels of oxygen (hypoxia). The aim of this work is to assess the tolerance of young Alosa alosa at four ages (c. 10, 30, 60 and 85 days old) by challenging them to different temperatures (18, 22, 26 and 28°C) together with decreasing oxygen saturation levels (from 100% to 30%). Survival of the 10-day-old individuals was not influenced by oxy-thermic conditions, but high stress levels were detected and perhaps this age class was too fragile regarding the constraint of the experimental design. Survival at 30 and at 60 days old was negatively influenced by the highest temperatures tested alone (from 26°C and from 28°C, respectively) but no effect was detected at 85 days old up to 28°C. A combined effect of temperature and oxygen level was highlighted, with heat accelerating survival decrease when associated with oxygen level depletion: essentially, survival was critical (<50%) at 30 days old at temperature ≥22°C together with 30% O₂ ; at 60 days old, at temperature = 28°C with 30% O₂ ; at 85 days old, at temperature ≥26°C with ≤40% O₂ . Tolerance to oxy-thermic pressures appeared to be greater among the migratory ages (60 and 85 days old) than among the 30-day-old group. Based on environmental data recorded in the GGD system and on our experimental results, an exploratory analysis allowed a discussion of the possible impact of past oxy-thermic conditions on the local population dynamics between 2005 and 2018. The oxy-thermic conditions that may affect Alosa alosa at ages when they migrate downstream (60 and 85 days old) were not frequently recorded in this period, except in cases of extreme episodes of heat together with hypoxia that occurred in some years, in summertime in the turbidity maximum zone of the Gironde estuary (particularly in the year 2006). Interestingly, oxy-thermic conditions that are likely to threaten the 30-day-old individuals occurred more frequently in the lower freshwater parts of the GGD system between the years 2005 and 2018. In the context of climate change, a general increase in temperature is predicted, as well as more frequent and severe hypoxic events, therefore we suggest that local Alosa alosa population recruitment could encounter critical oxy-thermic conditions more frequently in the future if no adaptive management of water resources occurs.

Factorial analyses of treatment effects under independent right-censoring

This paper introduces new effect parameters for factorial survival designs with possibly right-censored time-to-event data. In the special case of a two-sample design, it coincides with the concordance or Wilcoxon parameter in survival analysis. More generally, the new parameters describe treatment or interaction effects and we develop estimates and tests to infer their presence. We rigorously study their asymptotic properties and additionally suggest wild bootstrapping for a consistent and distribution-free application of the inference procedures. The small sample performance is discussed based on simulation results. The practical usefulness of the developed methodology is exemplified on a data example about patients with colon cancer by conducting one- and two-factorial analyses.

The wild bootstrap for multivariate Nelson-Aalen estimators

We rigorously extend the widely used wild bootstrap resampling technique to the multivariate Nelson-Aalen estimator under Aalen's multiplicative intensity model. Aalen's model covers general Markovian multistate models including competing risks subject to independent left-truncation and right-censoring. This leads to various statistical applications such as asymptotically valid confidence bands or tests for equivalence and proportional hazards. This is exemplified in a data analysis examining the impact of ventilation on the duration of intensive care unit stay. The finite sample properties of the new procedures are investigated in a simulation study.

Analysis of Generalized Semiparametric Regression Models for Cumulative Incidence Functions with Missing Covariates

The cumulative incidence function quantifies the probability of failure over time due to a specific cause for competing risks data. The generalized semiparametric regression models for the cumulative incidence functions with missing covariates are investigated. The effects of some covariates are modeled as non-parametric functions of time while others are modeled as parametric functions of time. Different link functions can be selected to add flexibility in modeling the cumulative incidence functions. The estimation procedures based on the direct binomial regression and the inverse probability weighting of complete cases are developed. This approach modifies the full data weighted least squares equations by weighting the contributions of observed members through the inverses of estimated sampling probabilities which depend on the censoring status and the event types among other subject characteristics. The asymptotic properties of the proposed estimators are established. The finite-sample performances of the proposed estimators and their relative efficiencies under different two-phase sampling designs are examined in simulations. The methods are applied to analyze data from the RV144 vaccine efficacy trial to investigate the associations of immune response biomarkers with the cumulative incidence of HIV-1 infection.

Smoothed Rank Regression for the Accelerated Failure Time Competing Risks Model with Missing Cause of Failure

This paper examines the accelerated failure time competing risks model with missing cause of failure using the monotone class rank-based estimating equations approach. We handle the non-smoothness of the rank-based estimating equations using a kernel smoothed estimation method, and estimate the unknown selection probability and the conditional expectation by non-parametric techniques. Under this setup, we propose three methods for estimating the unknown regression parameters based on 1) inverse probability weighting, 2) estimating equations imputation and 3) augmented inverse probability weighting. We also obtain the associated asymptotic theories of the proposed estimators and investigate the estimators' small sample behaviour in a simulation study. A direct plug-in method is suggested for estimating the asymptotic variances of the proposed estimators. A real data application based on a HIV vaccine efficacy trial study is considered.

Instrumental variable with competing risk model

In this paper, we discuss causal inference on the efficacy of a treatment or medication on a time-to-event outcome with competing risks. Although the treatment group can be randomized, there can be confoundings between the compliance and the outcome. Unmeasured confoundings may exist even after adjustment for measured covariates. Instrumental variable methods are commonly used to yield consistent estimations of causal parameters in the presence of unmeasured confoundings. On the basis of a semiparametric additive hazard model for the subdistribution hazard, we propose an instrumental variable estimator to yield consistent estimation of efficacy in the presence of unmeasured confoundings for competing risk settings. We derived the asymptotic properties for the proposed estimator. The estimator is shown to be well performed under finite sample size according to simulation results. We applied our method to a real transplant data example and showed that the unmeasured confoundings lead to significant bias in the estimation of the effect (about 50% attenuated). Copyright © 2017 John Wiley & Sons, Ltd.

Time-dependent impact of age and comorbidities on long-term overall survival after liver resection

BACKGROUND & AIMS

Advanced age and comorbidities are known to be associated with increased perioperative risks after liver resection. However, the precise impact of these variables on long-term overall survival (OS) remains unclear. Thus, the aim of this study was to evaluate the confounder-adjusted, time-dependent effect of age and comorbidities on OS following hepatectomy for primary and secondary malignancies.

METHODS

From a prospective database of 1.143 liver resections, 763 patients treated for primary and secondary malignancies were included. For time-varying OS calculations, a Cox-Aalen model was fitted. The confounder-adjusted hazard was compared with mortality tables of the German population.

RESULTS

Overall, age (P = 0.003) and comorbidities (P = 0.001) were associated with shortened OS. However, time-dependent analysis indicated that age and comorbidities had no impact on OS within 39 and 55 months after resection respectively. From this time on, a significant decline in OS was shown. Subgroup analysis indicated an earlier increase of the effect of age in patients with hepatocellular carcinoma (17 months) than in those with colorectal metastases (70 months). The confounder-adjusted hazard of 70-year-old patients was increased post-operatively but dropped 66 months after surgery, and the risk of death was comparable to the general population 78 months after resection. At this time, one-third of patients aged 70 years and older were still alive.

CONCLUSIONS

With regard to long-term outcome, liver resection for both primary and secondary malignancies should not be categorically denied due to age and comorbidities. This information should be considered for the patient selection process and informed consent.

On the choice of time scales in competing risks predictions

In the standard analysis of competing risks data, proportional hazards models are fit to the cause-specific hazard functions for all causes on the same time scale. These regression analyses are the foundation for predictions of cause-specific cumulative incidence functions based on combining the estimated cause-specific hazard functions. However, in predictions arising from disease registries, where only subjects with disease enter the database, disease-related mortality may be more naturally modeled on the time since diagnosis time scale while death from other causes may be more naturally modeled on the age time scale. The single time scale methodology may be biased if an incorrect time scale is employed for one of the causes and an alternative methodology is not available. We propose inferences for the cumulative incidence function in which regression models for the cause-specific hazard functions may be specified on different time scales. Using the disease registry data, the analysis of other cause mortality on the age scale requires left truncating the event time at the age of disease diagnosis, complicating the analysis. In addition, standard Martingale theory is not applicable when combining regression models on different time scales. We establish that the covariate conditional predictions are consistent and asymptotically normal using empirical process techniques and propose consistent variance estimators for constructing confidence intervals. Simulation studies show that the proposed two time scales method performs well, outperforming the single time-scale predictions when the time scale is misspecified. The methods are illustrated with stage III colon cancer data obtained from the Surveillance, Epidemiology, and End Results program of National Cancer Institute.

Time-dependent effects of clinical predictors in unrelated hematopoietic stem cell transplantation

Hematopoietic stem cell transplantation is a multifactorial process. Some of the predictors exhibit time-dependent effects. We present a systematic analysis and description of selected clinical predictors influencing outcome in a time-dependent manner based on an analysis of registry data from the German Registry for Stem Cell Transplantation. A total of 14,951 patients with acute myeloid leukemia, acute lymphocytic leukemia, myelodysplastic syndrome and non-Hodgkin lymphoma transplanted with peripheral blood stem cells or bone marrow grafts were included. Multivariate Cox regression models were tested for time-dependent effects within each diagnosis group. Predictors not satisfying the proportional hazards assumption were modeled in a time-dependent manner, extending the Cox regression models. Similar patterns occurred in all diagnosis groups. Patients with a poor Karnofsky performance score (<80) had a high risk for early mortality until day 139 following transplantation (HR 2.42, CI: 2.19-2.68; P<0.001) compared to patients with a good Karnofsky performance score (80-100). Afterwards the risk reduced to HR 1.43, CI: 1.25-1.63; P<0.001. A lower mortality risk was found for patients after conditioning treatment with reduced intensity until day 120 post transplant (HR: 0.81 CI: 0.75-0.88; P<0.001). After this, a slightly higher risk could be shown for these patients. Similarly, patients who had received a PBSC graft exhibited a significantly lower mortality risk until day 388 post transplantation (HR 0.79, CI: 0.73-0.85; P<0.001), reversing to a significantly higher risk afterwards (HR 1.23, CI: 1.08-1.40; P=0.002). Integrating time dependency in regression models allows a more accurate description and quantification of clinical predictors to be made, which may help in risk assessment and patient counseling.

A Proportional Hazards Regression Model for the Sub-distribution with Covariates Adjusted Censoring Weight for Competing Risks Data

With competing risks data, one often needs to assess the treatment and covariate effects on the cumulative incidence function. Fine and Gray proposed a proportional hazards regression model for the subdistribution of a competing risk with the assumption that the censoring distribution and the covariates are independent. Covariate-dependent censoring sometimes occurs in medical studies. In this paper, we study the proportional hazards regression model for the subdistribution of a competing risk with proper adjustments for covariate-dependent censoring. We consider a covariate-adjusted weight function by fitting the Cox model for the censoring distribution and using the predictive probability for each individual. Our simulation study shows that the covariate-adjusted weight estimator is basically unbiased when the censoring time depends on the covariates, and the covariate-adjusted weight approach works well for the variance estimator as well. We illustrate our methods with bone marrow transplant data from the Center for International Blood and Marrow Transplant Research (CIBMTR). Here cancer relapse and death in complete remission are two competing risks.

Modeling cumulative incidence function for competing risks data

A frequent occurrence in medical research is that a patient is subject to different causes of failure, where each cause is known as a competing risk. The cumulative incidence curve is a proper summary curve, showing the cumulative failure rates over time due to a particular cause. A common question in medical research is to assess the covariate effects on a cumulative incidence function. The standard approach is to construct regression models for all cause-specific hazard rate functions and then model a covariate-adjusted cumulative incidence curve as a function of all cause-specific hazards for a given set of covariates. New methods have been proposed in recent years, emphasizing direct assessment of covariate effects on cumulative incidence function. Fine and Gray proposed modeling the effects of covariates on a subdistribution hazard function. A different approach is to directly model a covariate-adjusted cumulative incidence function, including a pseudovalue approach by Andersen and Klein and a direct binomial regression by Scheike, Zhang and Gerds. In this paper, we review the standard and new regression methods for modeling a cumulative incidence function, and give the sources of computer packages/programs that implement these regression models. A real bone marrow transplant data set is analyzed to illustrate various regression methods.

A general binomial regression model to estimate standardized risk differences from binary response data

Estimates of absolute risks and risk differences are necessary for evaluating the clinical and population impact of biomedical research findings. We have developed a linear-expit regression model (LEXPIT) to incorporate linear and nonlinear risk effects to estimate absolute risk from studies of a binary outcome. The LEXPIT is a generalization of both the binomial linear and logistic regression models. The coefficients of the LEXPIT linear terms estimate adjusted risk differences, whereas the exponentiated nonlinear terms estimate residual odds ratios. The LEXPIT could be particularly useful for epidemiological studies of risk association, where adjustment for multiple confounding variables is common. We present a constrained maximum likelihood estimation algorithm that ensures the feasibility of risk estimates of the LEXPIT model and describe procedures for defining the feasible region of the parameter space, judging convergence, and evaluating boundary cases. Simulations demonstrate that the methodology is computationally robust and yields feasible, consistent estimators. We applied the LEXPIT model to estimate the absolute 5-year risk of cervical precancer or cancer associated with different Pap and human papillomavirus test results in 167,171 women undergoing screening at Kaiser Permanente Northern California. The LEXPIT model found an increased risk due to abnormal Pap test in human papillomavirus-negative that was not detected with logistic regression. Our R package blm provides free and easy-to-use software for fitting the LEXPIT model.

Proportional hazards model for competing risks data with missing cause of failure

We consider the semiparametric proportional hazards model for the cause-specific hazard function in analysis of competing risks data with missing cause of failure. The inverse probability weighted equation and augmented inverse probability weighted equation are proposed for estimating the regression parameters in the model, and their theoretical properties are established for inference. Simulation studies demonstrate that the augmented inverse probability weighted estimator is doubly robust and the proposed method is appropriate for practical use. The simulations also compare the proposed estimators with the multiple imputation estimator of Lu and Tsiatis (2001). The application of the proposed method is illustrated using data from a bone marrow transplant study.

Simulating and analysing infectious disease data in a heterogeneous population with migration

Mathematical modelling of infectious diseases has gained growing attention in epidemiology during the last decades. The major benefits of simulating compartmental models are the prediction of the consequences of potential interventions, a deeper understanding of epidemic dynamics and clinical decision support. The main limitation is however that several parameters are based on uncertain expert guesses (default values) and are not estimated from the study data. In this paper we build a bridge between an extension of the well-known deterministic S-I-R (Susceptible-Infectious-Removed) model which can be described with differential equations and the stochastic counterpart which can be used for statistical inference if outbreak data on an individual level are available. The possibly time-dependent transmission rate as well as the (basic) reproduction number are the main epidemiological parameters of interest. Furthermore, one important type of heterogeneity is considered: individuals may vary due to their susceptibility, i.e., risk factors for infection may be investigated. A SAS computer program is provided to simulate outbreak data for this type of setting. The statistical analysis and typical challenges with epidemic data are discussed. Given data on an individual level, the Cox-Aalen survival model that is based on a multiplicative-additive hazard structure turned out to be a suitable tool for that purpose. The results give valuable information for epidemiologists, statisticians and public health researchers.

A proportional hazards regression model for the subdistribution with right-censored and left-truncated competing risks data

With competing risks failure time data, one often needs to assess the covariate effects on the cumulative incidence probabilities. Fine and Gray proposed a proportional hazards regression model to directly model the subdistribution of a competing risk. They developed the estimating procedure for right-censored competing risks data, based on the inverse probability of censoring weighting. Right-censored and left-truncated competing risks data sometimes occur in biomedical researches. In this paper, we study the proportional hazards regression model for the subdistribution of a competing risk with right-censored and left-truncated data. We adopt a new weighting technique to estimate the parameters in this model. We have derived the large sample properties of the proposed estimators. To illustrate the application of the new method, we analyze the failure time data for children with acute leukemia. In this example, the failure times for children who had bone marrow transplants were left truncated.

SAS macros for estimation of direct adjusted cumulative incidence curves under proportional subdistribution hazards models

The cumulative incidence function is commonly reported in studies with competing risks. The aim of this paper is to compute the treatment-specific cumulative incidence functions, adjusting for potentially imbalanced prognostic factors among treatment groups. The underlying regression model considered in this study is the proportional hazards model for a subdistribution function [1]. We propose estimating the direct adjusted cumulative incidences for each treatment using the pooled samples as the reference population. We develop two SAS macros for estimating the direct adjusted cumulative incidence function for each treatment based on two regression models. One model assumes the constant subdistribution hazard ratios between the treatments and the alternative model allows each treatment to have its own baseline subdistribution hazard function. The macros compute the standard errors for the direct adjusted cumulative incidence estimates, as well as the standard errors for the differences of adjusted cumulative incidence functions between any two treatments. Based on the macros' output, one can assess treatment effects at predetermined time points. A real bone marrow transplant data example illustrates the practical utility of the SAS macros.

Flexible survival regression modelling

Regression analysis of survival data, and more generally event history data, is typically based on Cox's regression model. We here review some recent methodology, focusing on the limitations of Cox's regression model. The key limitation is that the model is not well suited to represent time-varying effects. We start by considering classical and also more recent goodness-of-fit procedures for the Cox model that will reveal when the Cox model does not capture important aspects of the data, such as time-varying effects. We present recent regression models that are able to deal with and describe such time-varying effects. The introduced models are all applied to data on breast cancer from the Norwegian cancer registry, and these analyses clearly reveal the shortcomings of Cox's regression model and the need for other supplementary analyses with models such as those we present here.

Flexible competing risks regression modeling and goodness-of-fit

In this paper we consider different approaches for estimation and assessment of covariate effects for the cumulative incidence curve in the competing risks model. The classic approach is to model all cause-specific hazards and then estimate the cumulative incidence curve based on these cause-specific hazards. Another recent approach is to directly model the cumulative incidence by a proportional model (Fine and Gray, J Am Stat Assoc 94:496-509, 1999), and then obtain direct estimates of how covariates influences the cumulative incidence curve. We consider a simple and flexible class of regression models that is easy to fit and contains the Fine-Gray model as a special case. One advantage of this approach is that our regression modeling allows for non-proportional hazards. This leads to a new simple goodness-of-fit procedure for the proportional subdistribution hazards assumption that is very easy to use. The test is constructive in the sense that it shows exactly where non-proportionality is present. We illustrate our methods to a bone marrow transplant data from the Center for International Blood and Marrow Transplant Research (CIBMTR). Through this data example we demonstrate the use of the flexible regression models to analyze competing risks data when non-proportionality is present in the data.

Inference for outcome probabilities in multi-state models

In bone marrow transplantation studies, patients are followed over time and a number of events may be observed. These include both ultimate events like death and relapse and transient events like graft versus host disease and graft recovery. Such studies, therefore, lend themselves for using an analytic approach based on multi-state models. We will give a review of such methods with emphasis on regression models for both transition intensities and transition- and state occupation probabilities. Both semi-parametric models, like the Cox regression model, and parametric models based on piecewise constant intensities will be discussed.

A flexible semiparametric transformation model for survival data

I suggest an extension of the semiparametric transformation model that specifies a time-varying regression structure for the transformation, and thus allows time-varying structure in the data. Special cases include a stratified version of the usual semiparametric transformation model. The model can be thought of as specifying a first order Taylor expansion of a completely flexible baseline. Large sample properties are derived and estimators of the asymptotic variances of the regression coefficients are given. The method is illustrated by a worked example and a small simulation study. A goodness of fit procedure for testing if the regression effects lead to a satisfactory fit is also suggested.