How to account for early overly small risk sets in the analysis of pregnancy outcome data?-Comparison of different methods for stabilizing the Aalen-Johansen estimator.

PURPOSE: In analyzing pregnancy data concerning drug exposure in the first trimester, the risk of spontaneous abortions is of primary interest. For estimating the cumulative incidence function, the Aalen-Johansen estimator is typically used, and competing risks such as induced abortion and livebirth are considered. However, the delayed study entry can lead to overly small risk sets for the first events. This results in large jumps in the estimated cumulative incidence function of spontaneous abortions or induced abortions using the Aalen-Johansen estimator, and consequently in an overestimation of the probability. METHODS: Several approaches account for early overly small risk sets. The first approach is conditioning on the event time being greater than the event time causing the large jump. Second, the events can be ignored by censoring them. Third, the events can be postponed until a large enough number is at risk. These three approaches are compared. RESULTS: All approaches are applied using data of 54 lacosamide-exposed pregnancies. The Aalen-Johansen estimate of the probability of spontaneous abortion is 22.64%, which is relatively large for only three spontaneous abortions in the dataset. The conditional approach and the ignore approach have an estimated probability of 7.17%. In contrast, the estimate of the postpone approach is 16.45%. In this small sample, bootstrapped confidence intervals seem more accurate. CONCLUSIONS: In the analyses of pregnancy data with rare events, the postpone approach is favorable as no events are excluded. However, the approach that ignores early events has the narrowest confidence interval.

Comparison of nonparametric estimators of the expected number of recurrent events.

We compare the performance of nonparametric estimators for the mean number of recurrent events and provide a systematic overview for different recurrent event settings. The mean number of recurrent events is an easily interpreted marginal feature often used for treatment comparisons in clinical trials. Incomplete observations, dependencies between successive events, terminating events acting as competing risk, or gaps between at risk periods complicate the estimation. We use survival multistate models to represent different complex recurrent event situations, profiting from recent advances in nonparametric estimation for non-Markov multistate models, and explain several estimators by using multistate intensity processes, including the common Nelson-Aalen-type estimators with and without competing mortality. In addition to building on estimation of state occupation probabilities in non-Markov models, we consider a simple extension of the Nelson-Aalen estimator by allowing for dependence on the number of prior recurrent events. We pay particular attention to the assumptions required for the censoring mechanism, one issue being that some settings require the censoring process to be entirely unrelated while others allow for state-dependent or event-driven censoring. We conducted extensive simulation studies to compare the estimators in various complex situations with recurrent events. Our practical example deals with recurrent chronic obstructive pulmonary disease exacerbations in a clinical study, which will also be used to illustrate two-sample-inference using resampling.

On estimation and cross-validation of dynamic treatment regimes with competing risks.

The optimal moment to start renal replacement therapy in a patient with acute kidney injury (AKI) remains a challenging problem in intensive care nephrology. Multiple randomized controlled trials have tried to answer this question, but these contrast only a limited number of treatment initiation strategies. In view of this, we use routinely collected observational data from the Ghent University Hospital intensive care units (ICUs) to investigate different prespecified timing strategies for renal replacement therapy initiation based on time-updated levels of serum potassium, pH, and fluid balance in critically ill patients with AKI with the aim to minimize 30-day ICU mortality. For this purpose, we apply statistical techniques for evaluating the impact of specific dynamic treatment regimes in the presence of ICU discharge as a competing event. We discuss two approaches, a nonparametric one - using an inverse probability weighted Aalen-Johansen estimator - and a semiparametric one - using dynamic-regime marginal structural models. Furthermore, we suggest an easy to implement cross-validation technique to assess the out-of-sample performance of the optimal dynamic treatment regime. Our work illustrates the potential of data-driven medical decision support based on routinely collected observational data.

G-estimation of structural nested mean models for competing risks data using pseudo-observations.

This article provides methods of causal inference for competing risks data. The methods are formulated as structural nested mean models of causal effects directly related to the cumulative incidence function or subdistribution hazard, which reflect the survival experience of a subject in the presence of competing risks. The effect measures include causal risk differences, causal risk ratios, causal subdistribution hazard ratios, and causal effects of time-varying exposures. Inference is implemented by g-estimation using pseudo-observations, a technique to handle censoring. The finite-sample performance of the proposed estimators in simulated datasets and application to time-varying exposures in a cohort study of type 2 diabetes are also presented.

A hybrid landmark Aalen-Johansen estimator for transition probabilities in partially non-Markov multi-state models.

Multi-state models are increasingly being used to model complex epidemiological and clinical outcomes over time. It is common to assume that the models are Markov, but the assumption can often be unrealistic. The Markov assumption is seldomly checked and violations can lead to biased estimation of many parameters of interest. This is a well known problem for the standard Aalen-Johansen estimator of transition probabilities and several alternative estimators, not relying on the Markov assumption, have been suggested. A particularly simple approach known as landmarking have resulted in the Landmark-Aalen-Johansen estimator. Since landmarking is a stratification method a disadvantage of landmarking is data reduction, leading to a loss of power. This is problematic for "less traveled" transitions, and undesirable when such transitions indeed exhibit Markov behaviour. Introducing the concept of partially non-Markov multi-state models, we suggest a hybrid landmark Aalen-Johansen estimator for transition probabilities. We also show how non-Markov transitions can be identified using a testing procedure. The proposed estimator is a compromise between regular Aalen-Johansen and landmark estimation, using transition specific landmarking, and can drastically improve statistical power. We show that the proposed estimator is consistent, but that the traditional variance estimator can underestimate the variance of both the hybrid and landmark estimator. Bootstrapping is therefore recommended. The methods are compared in a simulation study and in a real data application using registry data to model individual transitions for a birth cohort of 184 951 Norwegian men between states of sick leave, disability, education, work and unemployment.

NONPARAMETRIC INFERENCE FOR MARKOV PROCESSES WITH MISSING ABSORBING STATE.

This paper deals with the issue of nonparametric estimation of the transition probability matrix of a non-homogeneous Markov process with finite state space and partially observed absorbing state. We impose a missing at random assumption and propose a computationally efficient nonparametric maximum pseudolikelihood estimator (NPMPLE). The estimator depends on a parametric model that is used to estimate the probability of each absorbing state for the missing observations based, potentially, on auxiliary data. For the latter model we propose a formal goodness-of-fit test based on a residual process. Using modern empirical process theory we show that the estimator is uniformly consistent and converges weakly to a tight mean-zero Gaussian random field. We also provide methodology for simultaneous confidence band construction. Simulation studies show that the NPMPLE works well with small sample sizes and that it is robust against some degree of misspecification of the parametric model for the missing absorbing states. The method is illustrated using HIV data from sub-Saharan Africa to estimate the transition probabilities of death and disengagement from HIV care.

Multistate modeling and simulation of patient trajectories after allogeneic hematopoietic stem cell transplantation to inform drug development.

We present a case study for developing clinical trial scenarios in a complex progressive disease with multiple events of interest. The idea is to first capture the course of the disease in a multistate Markov model, and then to simulate clinical trials from this model, including a variety of hypothesized drug effects. This case study focuses on the prevention of graft-versus-host disease (GvHD) after allogeneic hematopoietic stem cell transplantation (HSCT). The patient trajectory after HSCT is characterized by a complex interplay of various events of interest, and there is no established best method of measuring and/or analyzing treatment benefits. We characterized patient trajectories by means of multistate models that we fitted to a subset of the Center for International Blood and Marrow Transplant Research (CIBMTR) database. Events of interest included acute GvHD of grade III or IV, severe chronic GvHD, relapse of the underlying disease, and death. The transition probability matrix was estimated using the Aalen-Johansen estimator, and patient characteristics were identified that were associated with different transition rates. In a second step, clinical trial scenarios were simulated from the model assuming various drug effects on the background transition rates, and the operating characteristics of different endpoints and analysis strategies were compared in these scenarios. This helped devise a drug development strategy in GvHD prevention after allogeneic HSCT. More generally, multistate models provide a rich framework for exploring complex progressive diseases, and the availability of a corresponding simulation machinery provides great flexibility for clinical trial planning.

Landmark estimation of transition probabilities in non-Markov multi-state models with covariates.

In non-Markov multi-state models, the traditional Aalen-Johansen (AJ) estimator for state transition probabilities is generally not valid. An alternative, suggested by Putter and Spitioni, is to analyse a subsample of the full data, consisting of the individuals present in a specific state at a given landmark time-point. The AJ estimator of occupation probabilities is then applied to the landmark subsample. Exploiting the result by Datta and Satten, that the AJ estimator is consistent for state occupation probabilities even in non-Markov models given that censoring is independent of state occupancy and times of transition between states, the landmark Aalen-Johansen (LMAJ) estimator provides consistent estimates of transition probabilities. So far, this approach has only been studied for non-parametric estimation without covariates. In this paper, we show how semi-parametric regression models and inverse probability weights can be used in combination with the LMAJ estimator to perform covariate adjusted analyses. The methods are illustrated by a simulation study and an application to population-wide registry data on work, education and health-related absence in Norway. Results using the traditional AJ estimator and the LMAJ estimator are compared, and show large differences in estimated transition probabilities for highly non-Markov multi-state models.

Estimation of adjusted expected excess length-of-stay associated with ventilation-acquired pneumonia in intensive care: A multistate approach accounting for time-dependent mechanical ventilation.

The expected excess length-of-stay is an established concept to assess the health and economic impact of nosocomial, that is, hospital-acquired infections such as ventilation-acquired pneumonia in intensive care. Estimation must account for the timing of infection as in a multistate perspective, because common retrospective comparisons yield inflated estimates due to time-dependent bias. Since occurrence of ventilation-acquired pneumonia is closely linked to ventilation status, we suggest a multistate model incorporating time-dependent mechanical ventilation as additional states. The appeal is that the expected excess length-of-stay decomposes into extra days spent under ventilation and not under ventilation. This is not only highly relevant from a patient's perspective regarding quality of life, but also from an economic point of view, because ventilation is a major cost driver. The challenge is that estimation involves complex functionals of the matrix of transition probabilities, which in turn are based on the transition hazards. To address heterogeneity between patients, which is a common phenomenon in observational hospital epidemiology, we apply pseudovalue regression to adjust the ventilation-specific quantities for baseline confounding. The performance of our proposal is assessed by simulation and the methods are illustrated on data provided by 12 French intensive care units. Preliminary results indicate that the expected excess length-of-stay associated with ventilation-acquired pneumonia is mainly triggered by extra days spent under mechanical ventilation, and that the excess is most pronounced for intensive care patients with fewer comorbidities at baseline. We also find that such a decomposition is challenging for early times. Example code is provided.

Risk of second primary malignancies in women with breast cancer: Results from the European prospective investigation into cancer and nutrition (EPIC).

Women with a diagnosis of breast cancer are at increased risk of second primary cancers, and the identification of risk factors for the latter may have clinical implications. We have followed-up for 11 years 10,045 women with invasive breast cancer from a European cohort, and identified 492 second primary cancers, including 140 contralateral breast cancers. Expected and observed cases and Standardized Incidence Ratios (SIR) were estimated using Aalen-Johansen Markovian methods. Information on various risk factors was obtained from detailed questionnaires and anthropometric measurements. Cox proportional hazards regression models were used to estimate the role of risk factors. Women with breast cancer had a 30% excess risk for second malignancies (95% confidence interval-CI 18-42) after excluding contralateral breast cancers. Risk was particularly elevated for colorectal cancer (SIR, 1.71, 95% CI 1.43-2.00), lymphoma (SIR 1.80, 95% CI 1.31-2.40), melanoma (2.12; 1.63-2.70), endometrium (2.18; 1.75-2.70) and kidney cancers (2.40; 1.57-3.52). Risk of second malignancies was positively associated with age at first cancer, body mass index and smoking status, while it was inversely associated with education, post-menopausal status and a history of full-term pregnancy. We describe in a large cohort of women with breast cancer a 30% excess of second primaries. Among risk factors for breast cancer, a history of full-term pregnancy was inversely associated with the risk of second primary cancer.

Nonparametric estimation of transition probabilities in the non-Markov illness-death model: A comparative study.

Multi-state models are often used for modeling complex event history data. In these models the estimation of the transition probabilities is of particular interest, since they allow for long-term predictions of the process. These quantities have been traditionally estimated by the Aalen-Johansen estimator, which is consistent if the process is Markov. Several non-Markov estimators have been proposed in the recent literature, and their superiority with respect to the Aalen-Johansen estimator has been proved in situations in which the Markov condition is strongly violated. However, the existing estimators have the drawback of requiring that the support of the censoring distribution contains the support of the lifetime distribution, which is not often the case. In this article, we propose two new methods for estimating the transition probabilities in the progressive illness-death model. Some asymptotic results are derived. The proposed estimators are consistent regardless the Markov condition and the referred assumption about the censoring support. We explore the finite sample behavior of the estimators through simulations. The main conclusion of this piece of research is that the proposed estimators are much more efficient than the existing non-Markov estimators in most cases. An application to a clinical trial on colon cancer is included. Extensions to progressive processes beyond the three-state illness-death model are discussed.

Transition probability estimates for non-Markov multi-state models.

Non-parametric estimation of the transition probabilities in multi-state models is considered for non-Markov processes. Firstly, a generalization of the estimator of Pepe et al., (1991) (Statistics in Medicine) is given for a class of progressive multi-state models based on the difference between Kaplan-Meier estimators. Secondly, a general estimator for progressive or non-progressive models is proposed based upon constructed univariate survival or competing risks processes which retain the Markov property. The properties of the estimators and their associated standard errors are investigated through simulation. The estimators are demonstrated on datasets relating to survival and recurrence in patients with colon cancer and prothrombin levels in liver cirrhosis patients.

Survival analysis for AdVerse events with VarYing follow-up times (SAVVY): summary of findings and assessment of existing guidelines.

BACKGROUND: The SAVVY project aims to improve the analyses of adverse events (AEs) in clinical trials through the use of survival techniques appropriately dealing with varying follow-up times and competing events (CEs). This paper summarizes key features and conclusions from the various SAVVY papers. METHODS: Summarizing several papers reporting theoretical investigations using simulations and an empirical study including randomized clinical trials from several sponsor organizations, biases from ignoring varying follow-up times or CEs are investigated. The bias of commonly used estimators of the absolute (incidence proportion and one minus Kaplan-Meier) and relative (risk and hazard ratio) AE risk is quantified. Furthermore, we provide a cursory assessment of how pertinent guidelines for the analysis of safety data deal with the features of varying follow-up time and CEs. RESULTS: SAVVY finds that for both, avoiding bias and categorization of evidence with respect to treatment effect on AE risk into categories, the choice of the estimator is key and more important than features of the underlying data such as percentage of censoring, CEs, amount of follow-up, or value of the gold-standard. CONCLUSIONS: The choice of the estimator of the cumulative AE probability and the definition of CEs are crucial. Whenever varying follow-up times and/or CEs are present in the assessment of AEs, SAVVY recommends using the Aalen-Johansen estimator (AJE) with an appropriate definition of CEs to quantify AE risk. There is an urgent need to improve pertinent clinical trial reporting guidelines for reporting AEs so that incidence proportions or one minus Kaplan-Meier estimators are finally replaced by the AJE with appropriate definition of CEs.

A connection between survival multistate models and causal inference for external treatment interruptions.

Recently, treatment interruptions such as a clinical hold in randomized clinical trials have been investigated by using a multistate model approach. The phase III clinical trial START (Stimulating Targeted Antigenic Response To non-small-cell cancer) with primary endpoint overall survival was temporarily placed on hold for enrollment and treatment by the US Food and Drug Administration (FDA). Multistate models provide a flexible framework to account for treatment interruptions induced by a time-dependent external covariate. Extending previous work, we propose a censoring and a filtering approach both aimed at estimating the initial treatment effect on overall survival in the hypothetical situation of no clinical hold. A special focus is on creating a link to causal inference. We show that calculating the matrix of transition probabilities in the multistate model after application of censoring (or filtering) yields the desired causal interpretation. Assumptions in support of the identification of a causal effect by censoring (or filtering) are discussed. Thus, we provide the basis to apply causal censoring (or filtering) in more general settings such as the COVID-19 pandemic. A simulation study demonstrates that both causal censoring and filtering perform favorably compared to a naïve method ignoring the external impact.

Factors Affecting the Clinical Course of Follicular Lymphoma: A Multistate Survival Analysis Using Individual Patient Data from Eight Multicenter Randomized Clinical Trials.

INTRODUCTION/BACKGROUND: Leveraging the Follicular Lymphoma Analysis of Surrogacy Hypothesis database of individual patient data from first-line clinical trials, we studied the clinical course of follicular lymphoma (FL) and investigated clinical factors associated with FL outcomes. PATIENTS AND METHODS: We examined 2428 patients from 8 randomized trials using multistate survival models with 4 states: induction treatment, progression, death from FL, and death from other causes. We utilized Aalen-Johansen estimator and Cox models to assess the likelihood of FL outcomes and quantify predictors' effects. RESULTS: Two-year progression, FL-related death, and death from other causes estimates were 26.5%, 3.4% and 1.4%, respectively. FL-associated deaths were the primary cause of mortality within 10 years of follow-up. Male sex (hazard ratio: 1.25; 95% confidence interval: 1.05-1.47), > 4 involved nodal areas (1.51; 1.23-1.86), elevated LDH (1.20; 1.01-1.43), low hemoglobin (1.44; 1.15-1.81), and elevated ß-2 levels (1.23; 1.02-1.47) increased risk of progression. CD20-targeting agents reduced risks for progression (0.29; 0.22-0.39), death from FL (0.05; 0.01-0.20), and death from other causes without progression (0.13; 0.05-0.33) and following progression (0.52; 0.30-0.92). Estimated 2-year progression rates were 22.3% and 43.5% with or without CD20-targeting agents, respectively. Two-year FL-associated mortality rate was 8.3% among patients without CD20-targeting agents, 5.4% with B-symptoms, 4.9% with elevated LDH, and 9.1% with low hemoglobin. CONCLUSION: This study identified independent contributions of baseline clinical factors to distinct outcomes for patients with FL following first-line therapy on a clinical trial. Similar analytical approaches are needed to increase understanding of factors that influence FL outcomes in other settings.

Survival analysis for AdVerse events with VarYing follow-up times (SAVVY)-estimation of adverse event risks.

BACKGROUND: The SAVVY project aims to improve the analyses of adverse events (AEs), whether prespecified or emerging, in clinical trials through the use of survival techniques appropriately dealing with varying follow-up times and competing events (CEs). Although statistical methodologies have advanced, in AE analyses, often the incidence proportion, the incidence density, or a non-parametric Kaplan-Meier estimator are used, which ignore either censoring or CEs. In an empirical study including randomized clinical trials from several sponsor organizations, these potential sources of bias are investigated. The main purpose is to compare the estimators that are typically used to quantify AE risk within trial arms to the non-parametric Aalen-Johansen estimator as the gold-standard for estimating cumulative AE probabilities. A follow-up paper will consider consequences when comparing safety between treatment groups. METHODS: Estimators are compared with descriptive statistics, graphical displays, and a more formal assessment using a random effects meta-analysis. The influence of different factors on the size of deviations from the gold-standard is investigated in a meta-regression. Comparisons are conducted at the maximum follow-up time and at earlier evaluation times. CEs definition does not only include death before AE but also end of follow-up for AEs due to events related to the disease course or safety of the treatment. RESULTS: Ten sponsor organizations provided 17 clinical trials including 186 types of investigated AEs. The one minus Kaplan-Meier estimator was on average about 1.2-fold larger than the Aalen-Johansen estimator and the probability transform of the incidence density ignoring CEs was even 2-fold larger. The average bias using the incidence proportion was less than 5%. Assuming constant hazards using incidence densities was hardly an issue provided that CEs were accounted for. The meta-regression showed that the bias depended mainly on the amount of censoring and on the amount of CEs. CONCLUSIONS: The choice of the estimator of the cumulative AE probability and the definition of CEs are crucial. We recommend using the Aalen-Johansen estimator with an appropriate definition of CEs whenever the risk for AEs is to be quantified and to change the guidelines accordingly.

State transition modeling of complex monitored health data.

This article considers the analysis of complex monitored health data, where often one or several signals are reflecting the current health status that can be represented by a finite number of states, in addition to a set of covariates. In particular, we consider a novel application of a non-parametric state intensity regression method in order to study time-dependent effects of covariates on the state transition intensities. The method can handle baseline, time varying as well as dynamic covariates. Because of the non-parametric nature, the method can handle different data types and challenges under minimal assumptions. If the signal that is reflecting the current health status is of continuous nature, we propose the application of a weighted median and a hysteresis filter as data pre-processing steps in order to facilitate robust analysis. In intensity regression, covariates can be aggregated by a suitable functional form over a time history window. We propose to study the estimated cumulative regression parameters for different choices of the time history window in order to investigate short- and long-term effects of the given covariates. The proposed framework is discussed and applied to resuscitation data of newborns collected in Tanzania.