Constant Baseline Hazard Research Articles

Joint modelling of longitudinal ordinal and multi-state data.

Joint modeling of longitudinal and survival data is increasingly used in biomedical studies. However, existing joint models are not applicable to model the longitudinal ordinal responses with non-ignorable missing values caused by the occurrence of events in a multi-state process. In this article, we introduce a joint model for longitudinal ordinal measurements and multi-state data. Our proposed joint model consists of two sub-models: a proportional odds sub-model for longitudinal ordinal measurements and a multi-state sub-model with transition-specific proportional hazards for times of transitions between different health states, both linked by shared random effects. The model parameters were estimated employing the maximum likelihood method for a piecewise constant baseline hazard function. The proposed joint model is evaluated in a simulation study and, as an illustration, it is fitted to real data from people with human immunodeficiency virus.

Case weighted power priors for hybrid control analyses with time-to-event data.

We develop a method for hybrid analyses that uses external controls to augment internal control arms in randomized controlled trials (RCTs) where the degree of borrowing is determined based on similarity between RCT and external control patients to account for systematic differences (e.g., unmeasured confounders). The method represents a novel extension of the power prior where discounting weights are computed separately for each external control based on compatibility with the randomized control data. The discounting weights are determined using the predictive distribution for the external controls derived via the posterior distribution for time-to-event parameters estimated from the RCT. This method is applied using a proportional hazards regression model with piecewise constant baseline hazard. A simulation study and a real-data example are presented based on a completed trial in non-small cell lung cancer. It is shown that the case weighted power prior provides robust inference under various forms of incompatibility between the external controls and RCT population.

A flexible model based on piecewise linear approximation for the analysis of left truncated right censored data with covariates, and applications to Worcester Heart Attack Study data and Channing House data.

Left truncated right censored (LTRC) data arise quite commonly from survival studies. In this article, a model based on piecewise linear approximation is proposed for the analysis of LTRC data with covariates. Specifically, the model involves a piecewise linear approximation for the cumulative baseline hazard function of the proportional hazards model. The principal advantage of the proposed model is that it does not depend on restrictive parametric assumptions while being flexible and data-driven. Likelihood inference for the model is developed. Through detailed simulation studies, the robustness property of the model is studied by fitting it to LTRC data generated from different processes covering a wide range of lifetime distributions. A sensitivity analysis is also carried out by fitting the model to LTRC data generated from a process with a piecewise constant baseline hazard. It is observed that the performance of the model is quite satisfactory in all those cases. Analyses of two real LTRC datasets by using the model are provided as illustrative examples. Applications of the model in some practical prediction issues are discussed. In summary, the proposed model provides a comprehensive and flexible approach to model a general structure for LTRC lifetime data.

A Time-to-Event Exposure-Response Model for Amyloid-Related Imaging Abnormalities Following Administration of Aducanumab to Subjects With Early Alzheimer Disease.

Amyloid-related imaging abnormalities with edema (ARIA-E) have been reported in patients with early Alzheimer disease treated with aducanumab. ARIA-E incidence has been observed to be dependent on both dose and apolipoprotein E4 carrier status. A time-to-event (TTE) approach applying data from 2 phase 3 studies (studies 301 and 302) was used to describe the effect of aducanumab serum exposure on the instantaneous risk of 2 end points: the first incidence of ARIA-E and time to ARIA-E resolution. A total of 3251 subjects with 826 events supported the TTE model to characterize the first ARIA-E event. The TTE resolution model was supported by data from 768 of 826 subjects who had ARIA-E resolved. Relationships between drug concentrations and ARIA-E events were modeled with a hazard function dependent on time, aducanumab serum concentrations, attenuation of aducanumab exposure effects with time (ie, potential for tolerance to aducanumab exposure), study, and apolipoprotein E4 carrier status. The TTE model showed that ARIA-E incidence rates were higher during the first 200 days, followed by a reduction in rates. The change in event rate reflects the attenuation of drug effect, thereby providing support for the current proposed titration regimen. Time to ARIA-E resolution was characterized by a constant baseline hazard with a probability to resolution affected by baseline ARIA-E severity and aducanumab concentration. ARIA-E resolution was found to be driven primarily by baseline hazard and time and suggested that aducanumab concentration effect is a minor contributor to the time to resolution of ARIA-E.

Causal inference with a mediated proportional hazards regression model

The natural direct and indirect effects in causal mediation analysis with survival data having one mediator is addressed by VanderWeele. He derived an approach for (1) an accelerated failure time regression model in general cases and (2) a proportional hazards regression model when the time-to-event outcome is rare. If the outcome is not rare, then VanderWeele did not derive a simple closed-form expression for the log-natural direct and log-natural indirect effects for the proportional hazards regression model because the baseline cumulative hazard function does not approach zero. We develop two approaches to extend VanderWeele’s approach, in which the assumption of a rare outcome is not required. We obtain the natural direct and indirect effects for specific time points through numerical integration after we calculate the cumulative baseline hazard by (1) applying the Breslow method in the Cox proportional hazards regression model to estimate the unspecified cumulative baseline hazard; (2) assuming a piecewise constant baseline hazard model, yielding a parametric model, to estimate the baseline hazard and cumulative baseline hazard. We conduct simulation studies to compare our two approaches with other methods and illustrate our two approaches by applying them to data from the ASsessment, Serial Evaluation, and Subsequent Sequelae in Acute Kidney Injury (ASSESS-AKI) Consortium.

Impact of a non-constant baseline hazard on detection of time-dependent treatment effects: a simulation study

BackgroundNon-proportional hazards are common with time-to-event data but the majority of randomised clinical trials (RCTs) are designed and analysed using approaches which assume the treatment effect follows proportional hazards (PH). Recent advances in oncology treatments have identified two forms of non-PH of particular importance - a time lag until treatment becomes effective, and an early effect of treatment that ceases after a period of time. In sample size calculations for treatment effects on time-to-event outcomes where information is based on the number of events rather than the number of participants, there is crucial importance in correct specification of the baseline hazard rate amongst other considerations. Under PH, the shape of the baseline hazard has no effect on the resultant power and magnitude of treatment effects using standard analytical approaches. However, in a non-PH context the appropriateness of analytical approaches can depend on the shape of the underlying hazard.MethodsA simulation study was undertaken to assess the impact of clinically plausible non-constant baseline hazard rates on the power, magnitude and coverage of commonly utilized regression-based measures of treatment effect and tests of survival curve difference for these two forms of non-PH used in RCTs with time-to-event outcomes.ResultsIn the presence of even mild departures from PH, the power, average treatment effect size and coverage were adversely affected. Depending on the nature of the non-proportionality, non-constant event rates could further exacerbate or somewhat ameliorate the losses in power, treatment effect magnitude and coverage observed. No single summary measure of treatment effect was able to adequately describe the full extent of a potentially time-limited treatment benefit whilst maintaining power at nominal levels.ConclusionsOur results show the increased importance of considering plausible potentially non-constant event rates when non-proportionality of treatment effects could be anticipated. In planning clinical trials with the potential for non-PH, even modest departures from an assumed constant baseline hazard could appreciably impact the power to detect treatment effects depending on the nature of the non-PH. Comprehensive analysis plans may be required to accommodate the description of time-dependent treatment effects.

Regression modelling of interval censored data based on the adaptive ridge procedure

A new method for the analysis of time to ankylosis complication on a dataset of replanted teeth is proposed. In this context of left-censored, interval-censored and right-censored data, a Cox model with piecewise constant baseline hazard is introduced. Estimation is carried out with the expectation maximisation (EM) algorithm by treating the true event times as unobserved variables. This estimation procedure is shown to produce a block diagonal Hessian matrix of the baseline parameters. Taking advantage of this interesting feature in the EM algorithm, a L 0 penalised likelihood method is implemented in order to automatically determine the number and locations of the cuts of the baseline hazard. This procedure allows to detect specific areas of time where patients are at greater risks for ankylosis. The method can be directly extended to the inclusion of exact observations and to a cure fraction. Theoretical results are obtained which allow to derive statistical inference of the model parameters from asymptotic likelihood theory. Through simulation studies, the penalisation technique is shown to provide a good fit of the baseline hazard and precise estimations of the resulting regression parameters.

Cox Piecewise Constant Hazard Model with Bayesian Method

Cox PH model is one of the survival models that is widely used for analyzing time-to-event data. Cox PH model consists of two main components, the baseline hazard consisting of time-dependent component; and the exponential function accomodating explanatory variables. The baseline hazard is not estimated in the Cox PH model, thus not accommodating the need for hazard rate estimation. Therefore, in this paper we discuss the estimation of baseline hazard through piecewise constant hazard using Bayesian method. Gamma distribution is assumed for the piecewise constant baseline hazard, and normal distribution is assumed for the regression coefficient. Sampling from the posterior is conducted using Markov chain Monte Carlo through Gibbs sampling. Echocardiogram data containing 106 observations and 6 explanatory variables were used in analysis. The result showed that the baseline hazard functions were estimated and each of parameters in the model is converged as shown by the trace plot and posterior density plot.

A phase I-II design based on periodic and continuous monitoring of disease status and the times to toxicity and death.

A Bayesian phase I-II dose-finding design is presented for a clinical trial with four coprimary outcomes that reflect the actual clinical observation process. During a prespecified fixed follow-up period, the times to disease progression, toxicity, and death are monitored continuously, and an ordinal disease status variable, including progressive disease (PD) as one level, is evaluated repeatedly by scheduled imaging. We assume a proportional hazards model with piecewise constant baseline hazard for each continuous variable and a longitudinal multinomial probit model for the ordinal disease status process and include multivariate patient frailties to induce association among the outcomes. A finite partition of the nonfatal outcome combinations during the follow-up period is constructed, and the utility of each set in the partition is elicited. Posterior mean utility is used to optimize each patient's dose, subject to a safety rule excluding doses with an unacceptably high rate of PD, severe toxicity, or death. A simulation study shows that, compared with the proposed design, a simpler design based on commonly used efficacy and toxicity outcomes obtained by combining the four variables described above performs poorly and has substantially smaller probabilities of correctly choosing truly optimal doses and excluding truly unsafe doses.

Casual effect of methotrexate+etanercept/infliximab on survival ofpatients withrheumatoid arthritis.

Background and objectives: Following the discovery of new drugs, physicians and pharmaceutical companies have become interested in examining patients’ mortality and morbidity rates. In this respect, the effects of methotrexate (MTX)+etanercept/infliximab (ETA/INF) therapy on the survival of rheumatoid arthritis patients (RA) were evaluated in this study using marginal structural piecewise constant baseline hazard model.Patients and methods: According to the standard protocol, MTX is considered as the first-line treatment for RA patients. If there is no adequate response to MTX, biologic drugs will be added. To compare the survival rates of RA patients in MTX- and MTX+ETA/INF-treated groups, the piecewise constant baseline hazard model was fitted. Then, due to the existence of the time-dependent confounder (VAS) which was affected by previous treatment, the weight for each person-time was calculated via the inverse probability treatment weighting method. These weights were then used by marginal structural piecewise constant baseline hazard model. Finally, these models were compared.Results: The median (IQR) of the follow-up period in patients receiving MTX+ETN/INF and MTX was 11 (15.25) and 11 (31), respectively, and the 8-year survival rate was reported by 70% versus 68%, respectively. First, the piece-wise constant baseline hazard model was fitted. Fitting the given model showed that MTX+ETA/INF had a significant effect on patients’ survival (HR=0.789, 95% CI [0.634, 0.983]). Second, marginal structural piecewise constant baseline hazard model was fitted. But, the results of this model revealed that MTX+ETA/INF did not have a significant impact on patients’ survival (HR=0.968, 95% CI [0.860, 1.090]).Conclusion: Adjusting the pain score over time as a time-dependent confounder via marginal structural piecewise constant baseline hazard model, it has been demonstrated that MTX+ETA/INF does not have a significant effect on patients’ survival rates. Therefore, a significant difference can be found between survival rates of these groups using longitudinal studies.

Joint modeling of multiple repeated measures and survival data using multidimensional latent trait linear mixed model.

Impairment caused by Amyotrophic lateral sclerosis (ALS) is multidimensional (e.g. bulbar, fine motor, gross motor) and progressive. Its multidimensional nature precludes a single outcome to measure disease progression. Clinical trials of ALS use multiple longitudinal outcomes to assess the treatment effects on overall improvement. A terminal event such as death or dropout can stop the follow-up process. Moreover, the time to the terminal event may be dependent on the multivariate longitudinal measurements. In this article, we develop a joint model consisting of a multidimensional latent trait linear mixed model (MLTLMM) for the multiple longitudinal outcomes, and a proportional hazards model with piecewise constant baseline hazard for the event time data. Shared random effects are used to link together two models. The model inference is conducted using a Bayesian framework via Markov chain Monte Carlo simulation implemented in Stan language. Our proposed model is evaluated by simulation studies and is applied to the Ceftriaxone study, a motivating clinical trial assessing the effect of ceftriaxone on ALS patients.

A joint model for recurrent events and a semi-competing risk in the presence of multi-level clustering.

Clinical trial designs often include multiple levels of clustering in which patients are nested within clinical sites and recurrent outcomes are nested within patients who may also experience a semi-competing risk. Traditional survival methods that analyze these processes separately may lead to erroneous inferences as they ignore possible dependencies. To account for the association between recurrent events and a semi-competing risk in the presence of two levels of clustering, we developed a semi-parametric joint model. The Gaussian quadrature with a piecewise constant baseline hazard was used to estimate the unspecified baseline hazards and the likelihood. Simulations showed that the proposed joint model has good statistical properties (i.e. <5% bias and 95% coverage) compared to the shared frailty and joint frailty models when informative censoring and multiple levels of clustering were present. The proposed method was applied to data from an AIDS clinical trial to investigate the impact of antiretroviral treatment on recurrent AIDS-defining events in the presence of a semi-competing risk of death and multi-level clustering and showed a significant dependency between AIDS-defining events and death at the patient level but not at the clinic level.

Job separation rates of immigrants and natives in the UK during the Great Recession

PurposeThe purpose of this paper is to assess the probability of job separations of immigrants and natives in the UK before and during the economic crisis of 2008.Design/methodology/approachA mixed proportional hazard duration model with a semi-parametric piecewise constant baseline hazard is used on a data sample of inflows into employment.FindingsIt is found that the crisis increased the probability of exits to unemployment for all groups, while immigrants from the new countries of the European Union seemed to have the lowest hazard towards unemployment even after controlling for their demographic and labour market characteristics. More specifically, even when we account for the fact that they tend to cluster in jobs that are most vulnerable to the business cycle, they are still less likely to exit dependent employment than natives. However, this migrant group is adversely affected by the crisis the most.Research limitations/implicationsPossible implications of out-migration of the lower performers are discussed.Originality/valueThis paper makes use of the panel element of the UK Quarterly Labour Force Survey, and uses duration analysis on the individual level to assess the labour market outcomes of natives and immigrants in the UK.

A two-phase model for chronic disease processes under intermittent inspection.

A model is developed for chronic diseases with an indolent phase that is followed by a phase with more active disease resulting in progression and damage. The time scales for the intensity functions for the active phase are more naturally based on the time since the start of the active phase, corresponding to a semi-Markov formulation. This two-phase model enables one to fit a separate regression model for the duration of the indolent phase and intensity-based models for the more active second phase. In cohort studies for which the disease status is only known at a series of clinical assessment times, transition times are interval-censored, which means the time origin for phase II is interval-censored. Weakly parametric models with piecewise constant baseline hazard and rate functions are specified, and an expectation-maximization algorithm is described for model fitting. Simulation studies examining the performance of the proposed model show good performance under maximum likelihood and two-stage estimation. An application to data from the motivating study of disease progression in psoriatic arthritis illustrates the procedure and identifies new human leukocyte antigens associated with the duration of the indolent phase. Copyright © 2017 John Wiley & Sons, Ltd.

Comparison of methods for estimating the attributable risk in the context of survival analysis

BackgroundThe attributable risk (AR) measures the proportion of disease cases that can be attributed to an exposure in the population. Several definitions and estimation methods have been proposed for survival data.MethodsUsing simulations, we compared four methods for estimating AR defined in terms of survival functions: two nonparametric methods based on Kaplan-Meier’s estimator, one semiparametric based on Cox’s model, and one parametric based on the piecewise constant hazards model, as well as one simpler method based on estimated exposure prevalence at baseline and Cox’s model hazard ratio. We considered a fixed binary exposure with varying exposure probabilities and strengths of association, and generated event times from a proportional hazards model with constant or monotonic (decreasing or increasing) Weibull baseline hazard, as well as from a nonproportional hazards model. We simulated 1,000 independent samples of size 1,000 or 10,000. The methods were compared in terms of mean bias, mean estimated standard error, empirical standard deviation and 95% confidence interval coverage probability at four equally spaced time points.ResultsUnder proportional hazards, all five methods yielded unbiased results regardless of sample size. Nonparametric methods displayed greater variability than other approaches. All methods showed satisfactory coverage except for nonparametric methods at the end of follow-up for a sample size of 1,000 especially. With nonproportional hazards, nonparametric methods yielded similar results to those under proportional hazards, whereas semiparametric and parametric approaches that both relied on the proportional hazards assumption performed poorly. These methods were applied to estimate the AR of breast cancer due to menopausal hormone therapy in 38,359 women of the E3N cohort.ConclusionIn practice, our study suggests to use the semiparametric or parametric approaches to estimate AR as a function of time in cohort studies if the proportional hazards assumption appears appropriate.

Log-normal frailty models fitted as Poisson generalized linear mixed models

Background and objectivesThe equivalence of a survival model with a piecewise constant baseline hazard function and a Poisson regression model has been known since decades. As shown in recent studies, this equivalence carries over to clustered survival data: A frailty model with a log-normal frailty term can be interpreted and estimated as a generalized linear mixed model with a binary response, a Poisson likelihood, and a specific offset. Proceeding this way, statistical theory and software for generalized linear mixed models are readily available for fitting frailty models. This gain in flexibility comes at the small price of (1) having to fix the number of pieces for the baseline hazard in advance and (2) having to “explode” the data set by the number of pieces. MethodsIn this paper we extend the simulations of former studies by using a more realistic baseline hazard (Gompertz) and by comparing the model under consideration with competing models. Furthermore, the SAS macro %PCFrailty is introduced to apply the Poisson generalized linear mixed approach to frailty models. ResultsThe simulations show good results for the shared frailty model. Our new %PCFrailty macro provides proper estimates, especially in case of 4 events per piece. ConclusionsThe suggested Poisson generalized linear mixed approach for log-normal frailty models based on the %PCFrailty macro provides several advantages in the analysis of clustered survival data with respect to more flexible modelling of fixed and random effects, exact (in the sense of non-approximate) maximum likelihood estimation, and standard errors and different types of confidence intervals for all variance parameters.

Joint modeling of longitudinal zero-inflated count and time-to-event data: A Bayesian perspective.

Longitudinal zero-inflated count data are encountered frequently in substance-use research when assessing the effects of covariates and risk factors on outcomes. Often, both the time to a terminal event such as death or dropout and repeated measure count responses are collected for each subject. In this setting, the longitudinal counts are censored by the terminal event, and the time to the terminal event may depend on the longitudinal outcomes. In the study described herein, we expand the class of joint models for longitudinal and survival data to accommodate zero-inflated counts and time-to-event data by using a Cox proportional hazards model with piecewise constant baseline hazard. We use a Bayesian framework via Markov chain Monte Carlo simulations implemented in the BUGS programming language. Via an extensive simulation study, we apply the joint model and obtain estimates that are more accurate than those of the corresponding independence model. We apply the proposed method to an alpha-tocopherol, beta-carotene lung cancer prevention study.

Phase I study of veliparib with gemcitabine and radiation therapy in patients with borderline resectable and locally advanced unresectable pancreatic cancer.

TPS487 Background: Targeted inhibition of PARP1/2 is one way to further exploit the well-known synergy between gemcitabine (G) and radiotherapy (RT) in locally advanced pancreatic cancer (LAPC). PARP1/2 inhibitors, such as veliparib (V), have shown excellent anti-tumor activity when used with other cytotoxic therapies. This synergy may be further exploited in pancreatic cancer by targeting tumors with pre-existing defects in double- strand DNA repair. Extrapolating from our own laboratory findings (Tuli et al, Transl Oncol, 2014) and based on our hypothesis that PARP1/2 inhibition with G and RT will result in enhanced tumor control, we are conducting an IRB- approved phase I study in patients with borderline resectable and LAPC. Methods: The primary objective of this study is to determine the maximum tolerated dose (MTD) of V, which is defined as the dose level resulting in a probability (θ = 0.4) that a dose limiting toxicity (DLT) will occur within six weeks. Treatment cycle is 3 weeks followed by weekly evaluation for an additional 3 weeks. G (1000 mg/m2) is administered on days 1, 8, 15. RT (36 Gy) is given in 15 fractions (2.4 Gy/day). V is administered BID in 20 mg increments beginning at a dose of 20 mg. Inclusion criteria: histolopathological diagnosis of borderline resectable or LAPC, age &gt; 18 years, KPS &gt; 70%, life expectancy &gt; 6 months, normal organ and marrow function, and negative pregnancy test. Dose escalation follows a Bayesian escalation without control (EWOC) design, where time to DLT is modeled using a proportional hazards model with constant baseline hazard rate. Secondary objectives: 1) measure clinical activity using RECIST 1.1 for PFS, OS; 2) evaluate pre-treatment tumor biopsy specimen and longitudinal blood samples for baseline levels of DNA repair proteins (ERCC1, XRCC1, PAR, etc.) as potential prognostic, predictive and correlative biomarkers; and 3) assess BRCA1/2, PALB2, PTEN germline and somatic mutations using validated gene sequencing, immunohistochemical and quantitative PCR methods. Since December 2013, 26 patients have consented with 22 enrolled. Clinical trial information: NCT01908478.

Comparison Between Two Controlled Multiple Imputation Methods for Sensitivity Analyses of Time-to-Event Data With Possibly Informative Censoring

Controlled imputation methods provide general and flexible sensitivity analyses to address nonignorable missing data. For time-to-event data with possibly informative censoring, we compare two popular methods for imputing the censored event time conditional on the time of follow-up discontinuation. One is the delta-adjusted method that specifies that the hazard of having an event for subjects who discontinued before the time point is multiplicatively increased relative to the hazard for subjects who continued beyond the time point. The other is the reference-based method that specifies that the hazard for experimental subjects who discontinued lies between the hazard for experimental subjects who continued and the hazard for the reference control (e.g., placebo) subjects. We consider both piecewise constant and nonparametric baseline hazard functions, Bayesian and frequentist imputations, and Rubin’s and bootstrap variances for the multiple imputation estimator. We show that both the reference-based and delta-adjusted sensitivity analyses control the one-sided Type I error rate (in the direction of a difference favoring the experimental treatment). In addition, when the bootstrap variance is used for inference, the reference-based sensitivity analysis has better power than the delta-adjusted sensitivity analysis for the same underlying treatment effect.

Population Pharmacokinetics and Antimalarial Pharmacodynamics of Piperaquine in Patients With Plasmodium vivax Malaria in Thailand.

Dihydroartemisinin-piperaquine is an effective drug in the treatment of Plasmodium falciparum and P. vivax malaria. The objective of this study was to evaluate the population pharmacokinetics and pharmacodynamics of piperaquine in patients with P. vivax malaria in Thailand after a standard regimen of dihydroartemisinin-piperaquine to determine whether residual piperaquine prevents or delays the emergence of P. vivax relapse. Sparse blood samples were collected from 116 patients. Piperaquine pharmacokinetics were described well by a three-compartment distribution model. Relapsing P. vivax malaria was accommodated by a constant baseline hazard (8.94 relapses/year) with the addition of a surge function in a fixed 3-week interval and a protective piperaquine effect. The results suggest that a large proportion of the first relapses were suppressed completely by residual piperaquine concentrations and that recurrences resulted mainly from emergence of the second or third relapse or from reinfection. This suggests a significant reduction in P. vivax morbidity when using dihydroartemisinin-piperaquine compared with other antimalarial drugs with shorter terminal postprophylactic effects.