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1.  Quantile Regression for Recurrent Gap Time Data 
Biometrics  2013;69(2):375-385.
Evaluating covariate effects on gap times between successive recurrent events is of interest in many medical and public health studies. While most existing methods for recurrent gap time analysis focus on modeling the hazard function of gap times, a direct interpretation of the covariate effects on the gap times is not available through these methods. In this article, we consider quantile regression that can provide direct assessment of covariate effects on the quantiles of the gap time distribution. Following the spirit of the weighted risk-set method by Luo and Huang (2011, Statistics in Medicine 30, 301–311), we extend the martingale-based estimating equation method considered by Peng and Huang (2008, Journal of the American Statistical Association 103, 637–649) for univariate survival data to analyze recurrent gap time data. The proposed estimation procedure can be easily implemented in existing software for univariate censored quantile regression. Uniform consistency and weak convergence of the proposed estimators are established. Monte Carlo studies demonstrate the effectiveness of the proposed method. An application to data from the Danish Psychiatric Central Register is presented to illustrate the methods developed in this article.
PMCID: PMC4123128  PMID: 23489055
Clustered survival data; Data perturbation; Gap times; Quantile regression; Recurrent events; Within-cluster resampling
2.  Statistical Methods for Analyzing Right-censored Length-biased Data under Cox Model 
Biometrics  2009;66(2):382-392.
Length-biased time-to-event data are commonly encountered in applications ranging from epidemiologic cohort studies or cancer prevention trials to studies of labor economy. A longstanding statistical problem is how to assess the association of risk factors with survival in the target population given the observed length-biased data. In this paper, we demonstrate how to estimate these effects under the semiparametric Cox proportional hazards model. The structure of the Cox model is changed under length-biased sampling in general. Although the existing partial likelihood approach for left-truncated data can be used to estimate covariate effects, it may not be efficient for analyzing length-biased data. We propose two estimating equation approaches for estimating the covariate coefficients under the Cox model. We use the modern stochastic process and martingale theory to develop the asymptotic properties of the estimators. We evaluate the empirical performance and efficiency of the two methods through extensive simulation studies. We use data from a dementia study to illustrate the proposed methodology, and demonstrate the computational algorithms for point estimates, which can be directly linked to the existing functions in S-PLUS or R.
PMCID: PMC3035941  PMID: 19522872
Cox model; Dependent censoring; Estimating equation; Length-biased
3.  Empirical comparison of methods for analyzing multiple time-to-event outcomes in a non-inferiority trial: a breast cancer study 
Subjects with breast cancer enrolled in trials may experience multiple events such as local recurrence, distant recurrence or death. These events are not independent; the occurrence of one may increase the risk of another, or prevent another from occurring. The most commonly used Cox proportional hazards (Cox-PH) model ignores the relationships between events, resulting in a potential impact on the treatment effect and conclusions. The use of statistical methods to analyze multiple time-to-event events has mainly been focused on superiority trials. However, their application to non-inferiority trials is limited. We evaluate four statistical methods for multiple time-to-event endpoints in the context of a non-inferiority trial.
Three methods for analyzing multiple events data, namely, i) the competing risks (CR) model, ii) the marginal model, and iii) the frailty model were compared with the Cox-PH model using data from a previously-reported non-inferiority trial comparing hypofractionated radiotherapy with conventional radiotherapy for the prevention of local recurrence in patients with early stage breast cancer who had undergone breast conserving surgery. These methods were also compared using two simulated examples, scenario A where the hazards for distant recurrence and death were higher in the control group, and scenario B. where the hazards of distant recurrence and death were higher in the experimental group. Both scenarios were designed to have a non-inferiority margin of 1.50.
In the breast cancer trial, the methods produced primary outcome results similar to those using the Cox-PH model: namely, a local recurrence hazard ratio (HR) of 0.95 and a 95% confidence interval (CI) of 0.62 to 1.46. In Scenario A, non-inferiority was observed with the Cox-PH model (HR = 1.04; CI of 0.80 to 1.35), but not with the CR model (HR = 1.37; CI of 1.06 to 1.79), and the average marginal and frailty model showed a positive effect of the experimental treatment. The results in Scenario A contrasted with Scenario B with non-inferiority being observed with the CR model (HR = 1.10; CI of 0.87 to 1.39), but not with the Cox-PH model (HR = 1.46; CI of 1.15 to 1.85), and the marginal and frailty model showed a negative effect of the experimental treatment.
When subjects are at risk for multiple events in non-inferiority trials, researchers need to consider using the CR, marginal and frailty models in addition to the Cox-PH model in order to provide additional information in describing the disease process and to assess the robustness of the results. In the presence of competing risks, the Cox-PH model is appropriate for investigating the biologic effect of treatment, whereas the CR models yields the actual effect of treatment in the study.
PMCID: PMC3610213  PMID: 23517401
Non-inferiority; Cox model; Correlation; Marginal model; Frailty model; Competing risks
4.  Statistical modelling for recurrent events: an application to sports injuries 
British Journal of Sports Medicine  2012;48(17):1287-1293.
Injuries are often recurrent, with subsequent injuries influenced by previous occurrences and hence correlation between events needs to be taken into account when analysing such data.
This paper compares five different survival models (Cox proportional hazards (CoxPH) model and the following generalisations to recurrent event data: Andersen-Gill (A-G), frailty, Wei-Lin-Weissfeld total time (WLW-TT) marginal, Prentice-Williams-Peterson gap time (PWP-GT) conditional models) for the analysis of recurrent injury data.
Empirical evaluation and comparison of different models were performed using model selection criteria and goodness-of-fit statistics. Simulation studies assessed the size and power of each model fit.
The modelling approach is demonstrated through direct application to Australian National Rugby League recurrent injury data collected over the 2008 playing season. Of the 35 players analysed, 14 (40%) players had more than 1 injury and 47 contact injuries were sustained over 29 matches. The CoxPH model provided the poorest fit to the recurrent sports injury data. The fit was improved with the A-G and frailty models, compared to WLW-TT and PWP-GT models.
Despite little difference in model fit between the A-G and frailty models, in the interest of fewer statistical assumptions it is recommended that, where relevant, future studies involving modelling of recurrent sports injury data use the frailty model in preference to the CoxPH model or its other generalisations. The paper provides a rationale for future statistical modelling approaches for recurrent sports injury.
PMCID: PMC4145455  PMID: 22872683
5.  Model Checking Techniques for Assessing Functional Form Specifications in Censored Linear Regression Models 
Statistica Sinica  2012;22(2):509-530.
In this paper we develop model checking techniques for assessing functional form specifications of covariates in censored linear regression models. These procedures are based on a censored data analog to taking cumulative sums of “robust” residuals over the space of the covariate under investigation. These cumulative sums are formed by integrating certain Kaplan-Meier estimators and may be viewed as “robust” censored data analogs to the processes considered by Lin, Wei & Ying (2002). The null distributions of these stochastic processes can be approximated by the distributions of certain zero-mean Gaussian processes whose realizations can be generated by computer simulation. Each observed process can then be graphically compared with a few realizations from the Gaussian process. We also develop formal test statistics for numerical comparison. Such comparisons enable one to assess objectively whether an apparent trend seen in a residual plot reects model misspecification or natural variation. We illustrate the methods with a well known dataset. In addition, we examine the finite sample performance of the proposed test statistics in simulation experiments. In our simulation experiments, the proposed test statistics have good power of detecting misspecification while at the same time controlling the size of the test.
PMCID: PMC3697158  PMID: 23825917
Censored linear regression; Goodness-of-fit; Partial linear model; Partial residual; Quantile regression; Resampling method; Rank estimation
6.  Network-based Survival Analysis Reveals Subnetwork Signatures for Predicting Outcomes of Ovarian Cancer Treatment 
PLoS Computational Biology  2013;9(3):e1002975.
Cox regression is commonly used to predict the outcome by the time to an event of interest and in addition, identify relevant features for survival analysis in cancer genomics. Due to the high-dimensionality of high-throughput genomic data, existing Cox models trained on any particular dataset usually generalize poorly to other independent datasets. In this paper, we propose a network-based Cox regression model called Net-Cox and applied Net-Cox for a large-scale survival analysis across multiple ovarian cancer datasets. Net-Cox integrates gene network information into the Cox's proportional hazard model to explore the co-expression or functional relation among high-dimensional gene expression features in the gene network. Net-Cox was applied to analyze three independent gene expression datasets including the TCGA ovarian cancer dataset and two other public ovarian cancer datasets. Net-Cox with the network information from gene co-expression or functional relations identified highly consistent signature genes across the three datasets, and because of the better generalization across the datasets, Net-Cox also consistently improved the accuracy of survival prediction over the Cox models regularized by or . This study focused on analyzing the death and recurrence outcomes in the treatment of ovarian carcinoma to identify signature genes that can more reliably predict the events. The signature genes comprise dense protein-protein interaction subnetworks, enriched by extracellular matrix receptors and modulators or by nuclear signaling components downstream of extracellular signal-regulated kinases. In the laboratory validation of the signature genes, a tumor array experiment by protein staining on an independent patient cohort from Mayo Clinic showed that the protein expression of the signature gene FBN1 is a biomarker significantly associated with the early recurrence after 12 months of the treatment in the ovarian cancer patients who are initially sensitive to chemotherapy. Net-Cox toolbox is available at
Author Summary
Network-based computational models are attracting increasing attention in studying cancer genomics because molecular networks provide valuable information on the functional organizations of molecules in cells. Survival analysis mostly with the Cox proportional hazard model is widely used to predict or correlate gene expressions with time to an event of interest (outcome) in cancer genomics. Surprisingly, network-based survival analysis has not received enough attention. In this paper, we studied resistance to chemotherapy in ovarian cancer with a network-based Cox model, called Net-Cox. The experiments confirm that networks representing gene co-expression or functional relations can be used to improve the accuracy and the robustness of survival prediction of outcome in ovarian cancer treatment. The study also revealed subnetwork signatures that are enriched by extracellular matrix receptors and modulators and the downstream nuclear signaling components of extracellular signal-regulators, respectively. In particular, FBN1, which was detected as a signature gene of high confidence by Net-Cox with network information, was validated as a biomarker for predicting early recurrence in platinum-sensitive ovarian cancer patients in laboratory.
PMCID: PMC3605061  PMID: 23555212
7.  A basis approach to goodness-of-fit testing in recurrent event models 
A class of tests for the hypothesis that the baseline hazard function in Cox’s proportional hazards model and for a general recurrent event model belongs to a parametric family C ≡ {λ0(·; ξ): ξ ∈ Ξ} is proposed. Finite properties of the tests are examined via simulations, while asymptotic properties of the tests under a contiguous sequence of local alternatives are studied theoretically. An application of the tests to the general recurrent event model, which is an extended minimal repair model admitting covariates, is demonstrated. In addition, two real data sets are used to illustrate the applicability of the proposed tests.
PMCID: PMC1563443  PMID: 16967104
Counting process; Goodness-of-fit test; Minimal repair model; Neyman’s test; Nonhomogeneous Poisson process; Repairable system; Score test
8.  Mark-specific proportional hazards model with multivariate continuous marks and its application to HIV vaccine efficacy trials 
For time-to-event data with finitely many competing risks, the proportional hazards model has been a popular tool for relating the cause-specific outcomes to covariates (Prentice and others, 1978. The analysis of failure time in the presence of competing risks. Biometrics 34, 541–554). Inspired by previous research in HIV vaccine efficacy trials, the cause of failure is replaced by a continuous mark observed only in subjects who fail. This article studies an extension of this approach to allow a multivariate continuum of competing risks, to better account for the fact that the candidate HIV vaccines tested in efficacy trials have contained multiple HIV sequences, with a purpose to elicit multiple types of immune response that recognize and block different types of HIV viruses. We develop inference for the proportional hazards model in which the regression parameters depend parametrically on the marks, to avoid the curse of dimensionality, and the baseline hazard depends nonparametrically on both time and marks. Goodness-of-fit tests are constructed based on generalized weighted martingale residuals. The finite-sample performance of the proposed methods is examined through extensive simulations. The methods are applied to a vaccine efficacy trial to examine whether and how certain antigens represented inside the vaccine are relevant for protection or anti-protection against the exposing HIVs.
PMCID: PMC3520499  PMID: 22764174
Competing risks; Failure time data; Goodness-of-fit test; HIV vaccine trial; Hypothesis testing; Mark-specific relative risk; Multivariate data; Partial likelihood estimation; Semiparametric model; STEP trial
9.  An Estimating Function Approach to the Analysis of Recurrent and Terminal Events 
Biometrics  2013;69(2):366-374.
In clinical and observational studies, the event of interest can often recur on the same subject. In a more complicated situation, there exists a terminal event (e.g. death) which stops the recurrent event process. In many such instances, the terminal event is strongly correlated with the recurrent event process. We consider the recurrent/terminal event setting and model the dependence through a shared gamma frailty that is included in both the recurrent event rate and terminal event hazard functions. Conditional on the frailty, a model is specified only for the marginal recurrent event process, hence avoiding the strong Poisson-type assumptions traditionally used. Analysis is based on estimating functions that allow for estimation of covariate effects on the recurrent event rate and terminal event hazard. The method also permits estimation of the degree of association between the two processes. Closed-form asymptotic variance estimators are proposed. The proposed method is evaluated through simulations to assess the applicability of the asymptotic results in finite samples and the sensitivity of the method to its underlying assumptions. The methods can be extended in straightforward ways to accommodate multiple types of recurrent and terminal events. Finally, the methods are illustrated in an analysis of hospitalization data for patients in an international multi-center study of outcomes among dialysis patients.
PMCID: PMC3692576  PMID: 23651362
Cox model; Frailty; Marginal rate function; Multivariate survival; Relative risk; Semiparametric methods
10.  Dynamic regression hazards models for relative survival 
Statistics in medicine  2008;27(18):3563-3584.
A natural way of modelling relative survival through regression analysis is to assume an additive form between the expected population hazard and the excess hazard due to the presence of an additional cause of mortality. Within this context, the existing approaches in the parametric, semiparametric and non-parametric setting are compared and discussed. We study the additive excess hazards models, where the excess hazard is on additive form. This makes it possible to assess the importance of time-varying effects for regression models in the relative survival framework. We show how recent developments can be used to make inferential statements about the non-parametric version of the model. This makes it possible to test the key hypothesis that an excess risk effect is time varying in contrast to being constant over time. In case some covariate effects are constant, we show how the semiparametric additive risk model can be considered in the excess risk setting, providing a better and more useful summary of the data. Estimators have explicit form and inference based on a resampling scheme is presented for both the non-parametric and semiparametric models. We also describe a new suggestion for goodness of fit of relative survival models, which consists on statistical and graphical tests based on cumulative martingale residuals. This is illustrated on the semiparametric model with proportional excess hazards. We analyze data from the TRACE study using different approaches and show the need for more flexible models in relative survival.
PMCID: PMC2737139  PMID: 18338318
11.  Impact of Cyclooxygenase Inhibitors in the Women's Health Initiative Hormone Trials: Secondary Analysis of a Randomized Trial 
PLoS Clinical Trials  2006;1(5):e26.
We evaluated the hypothesis that cyclooxygenase (COX) inhibitor use might have counteracted a beneficial effect of postmenopausal hormone therapy, and account for the absence of cardioprotection in the Women's Health Initiative hormone trials. Estrogen increases COX expression, and inhibitors of COX such as nonsteroidal anti-inflammatory agents appear to increase coronary risk, raising the possibility of a clinically important interaction in the trials.
The hormone trials were randomized, double-blind, and placebo-controlled. Use of nonsteroidal anti-inflammatory drugs was assessed at baseline and at years 1, 3, and 6.
The Women's Health Initiative hormone trials were conducted at 40 clinical sites in the United States.
The trials enrolled 27,347 postmenopausal women, aged 50–79 y.
We randomized 16,608 women with intact uterus to conjugated estrogens 0.625 mg with medroxyprogesterone acetate 2.5 mg daily or to placebo, and 10,739 women with prior hysterectomy to conjugated estrogens 0.625 mg daily or placebo.
Outcome Measures:
Myocardial infarction, coronary death, and coronary revascularization were ascertained during 5.6 y of follow-up in the estrogen plus progestin trial and 6.8 y of follow-up in the estrogen alone trial.
Hazard ratios with 95% confidence intervals were calculated from Cox proportional hazard models stratified by COX inhibitor use. The hazard ratio for myocardial infarction/coronary death with estrogen plus progestin was 1.13 (95% confidence interval 0.68–1.89) among non-users of COX inhibitors, and 1.35 (95% confidence interval 0.86–2.10) among continuous users. The hazard ratio with estrogen alone was 0.92 (95% confidence interval 0.57–1.48) among non-users of COX inhibitors, and 1.08 (95% confidence interval 0.69–1.70) among continuous users. In a second analytic approach, hazard ratios were calculated from Cox models that included hormone trial assignment as well as a time-dependent covariate for medication use, and an interaction term. No significant interaction was identified.
Use of COX inhibitors did not significantly affect the Women's Health Initiative hormone trial results.
Editorial Commentary
Background: As part of a set of studies known as the Women's Health Initiative trials, investigators aimed to find out whether providing postmenopausal hormone therapy (estrogen in the case of women who had had a hysterectomy, and estrogen plus progestin for women who had not had a hysterectomy) reduced cardiovascular risk as compared to placebo. Earlier observational studies had suggested this might be the case. The trials found that postmenopausal hormone therapy did not reduce cardiovascular risk in the groups studied. However, there was a concern that medication use outside the trial with nonsteroidal anti-inflammatory drugs (NSAIDs), and specifically the type of NSAID known as COX-2 inhibitors, could have affected the findings. This concern arose because it is known that COX-2 inhibition lowers levels of prostacyclin, a molecule thought to be beneficial to cardiovascular health, whereas estrogen increases prostacyclin levels. Evidence from randomized trials and observational studies has also shown that patients treated with some COX-2 inhibitors are at increased risk of heart attacks and strokes; the cardiovascular safety of other NSAIDs is also the focus of great attention. Therefore, the authors of this paper aimed to do a statistical exploration of the data from the Women's Health Initiative hormone trials, to find out whether NSAID use by participants in the trials could have affected the trials' main findings.
What this trial shows: In this reanalysis of the original data from the trials, the investigators found that the effects of hormone therapy on cardiovascular outcomes were similar among users and non-users of NSAIDs, confirming that use of these drugs did not significantly affect the results from the Women's Health Initiative hormone trials.
Strengths and limitations: The original hormone trials were large, appropriately randomized studies that enrolled a diverse cohort of participants. Therefore, a large number of cardiovascular events occurred in the groups being compared, allowing this subsequent analysis to be done. One limitation is that use of COX-2 inhibitors in the trial was low; therefore, the investigators were not able to specifically test whether COX-2 inhibitor use (as opposed to NSAID use generally) might have affected their findings.
Contribution to the evidence: The investigators did not set out specifically to evaluate the cardiovascular safety of particular medications in this study. Rather, they wanted to see if these NSAIDs could have modified the effects of the hormone therapy. The secondary analysis done here shows that the main findings from the Women's Health Initiative hormone trials were not significantly affected by use of NSAIDs outside the trial.
PMCID: PMC1584256  PMID: 17016543
12.  Testing the proportional hazards assumption in case-cohort analysis 
Case-cohort studies have become common in epidemiological studies of rare disease, with Cox regression models the principal method used in their analysis. However, no appropriate procedures to assess the assumption of proportional hazards of case-cohort Cox models have been proposed.
We extended the correlation test based on Schoenfeld residuals, an approach used to evaluate the proportionality of hazards in standard Cox models. Specifically, pseudolikelihood functions were used to define “case-cohort Schoenfeld residuals”, and then the correlation of these residuals with each of three functions of event time (i.e., the event time itself, rank order, Kaplan-Meier estimates) was determined. The performances of the proposed tests were examined using simulation studies. We then applied these methods to data from a previously published case-cohort investigation of the insulin/IGF-axis and colorectal cancer.
Simulation studies showed that each of the three correlation tests accurately detected non-proportionality. Application of the proposed tests to the example case-cohort investigation dataset showed that the Cox proportional hazards assumption was not satisfied for certain exposure variables in that study, an issue we addressed through use of available, alternative analytical approaches.
The proposed correlation tests provide a simple and accurate approach for testing the proportional hazards assumption of Cox models in case-cohort analysis. Evaluation of the proportional hazards assumption is essential since its violation raises questions regarding the validity of Cox model results which, if unrecognized, could result in the publication of erroneous scientific findings.
PMCID: PMC3710085  PMID: 23834739
Proportional hazards; Schoenfeld residuals; Case-cohort studies; Cox models
13.  Modeling genome-wide replication kinetics reveals a mechanism for regulation of replication timing 
We developed analytical models of DNA replication that include probabilistic initiation of origins, fork progression, passive replication, and asynchrony.We fit the model to budding yeast genome-wide microarray data probing the replication fraction and found that initiation times correlate with the precision of timing.We extracted intrinsic origin properties, such as potential origin efficiency and firing-time distribution, which cannot be done using phenomenological approaches.We propose that origin timing is controlled by stochastically activated initiators bound to origin sites rather than explicit time-measuring mechanisms.
The kinetics of DNA replication must be controlled for cells to develop properly. Although the biochemical mechanisms of origin initiations are increasingly well understood, the organization of initiation timing as a genome-wide program is still a mystery. With the advance of technology, researchers have been able to generate large amounts of data revealing aspects of replication kinetics. In particular, the use of microarrays to probe the replication fraction of budding yeast genome wide has been a successful first step towards unraveling the details of the replication program (Raghuraman et al, 2001; Alvino et al, 2007; McCune et al, 2008). On the surface, the microarray data shows apparent patterns of early and late replicating regions and seems to support the prevailing picture of eukaryotic replication—origins are positioned at defined sites and initiated at defined, preprogrammed times (Donaldson, 2005). Molecular combing, a single-molecule technique, however, showed that the initiation of origins is stochastic (Czajkowsky et al, 2008). Motivated by these conflicting viewpoints, we developed a model that is flexible enough to describe both deterministic and stochastic initiation.
We modeled origin initiation as probabilistic events. We first propose a model where each origin is allowed to have its distinct ‘firing-time distribution.' Origins that have well-determined initiation times have narrow distributions, whereas more stochastic origins have wider distributions. Similar models based on simulations have previously been proposed (Lygeros et al, 2008; Blow and Ge, 2009; de Moura et al, 2010); however, our model is novel in that it is analytic. It is much faster than simulations and allowed us, for the first time, to fit genome-wide microarray data and extract parameters that describe the replication program in unprecedented detail (Figure 2).
Our main result is this: origins that fire early, on average, have precisely defined initiation times, whereas origins that fire late, on average, do not have a well-defined initiation time and initiate throughout S phase. What kind of global controlling mechanism can account for this trend? We propose a second model where an origin is composed of multiple initiators, each of which fires independently and identically. A good candidate for the initiator is the minichromosome maintenance (MCM) complex, as it is found to be associated with origin firing and loaded in abundance (Hyrien et al, 2003). We show that the aforementioned relationship can be explained quantitatively if the earlier-firing origins have more MCM complexes. This model offers a new view of replication: controlled origin timing can emerge from stochastic firing and does not need an explicit time-measuring mechanism, a ‘clock.' This model provides a new, detailed, plausible, and testable mechanism for replication timing control.
Our models also capture the effects of passive replication, which is often neglected in phenomenological approaches (Eshaghi et al, 2007). There are two ways an origin site can be replicated. The site can be replicated by the origin binding to it but can also be passively replicated by neighboring origins. This complication makes it difficult to extract the intrinsic properties of origins. By modeling passive replication, we can separate the contribution from each origin and extract the potential efficiency of origins, i.e., the efficiency of the origin given that there is no passive replication. We found that while most origins are potentially highly efficient, their observed efficiency varies greatly. This implies that many origins, though capable of initiating, are often passively replicated and appear dormant. Such a design makes the replication process robust against replication stress such as fork stalling (Blow and Ge, 2009). If two approaching forks stall, normally dormant origins in the region, not being passively replicated, will initiate to replicate the gap.
With the advance of the microarray and molecular-combing technology, experiments have been done to probe many different types of cells, and large amounts of replication fraction data have been generated. Our model can be applied to spatiotemporally resolved replication fraction data for any organism, as the model is flexible enough to capture a wide range of replication kinetics. The analytical model is also much faster than simulation-based models. For these reasons, we believe that the model is a powerful tool for analyzing these large datasets. This work opens the possibility for understanding the replication program across species in more rigor and detail (Goldar et al, 2009).
Microarrays are powerful tools to probe genome-wide replication kinetics. The rich data sets that result contain more information than has been extracted by current methods of analysis. In this paper, we present an analytical model that incorporates probabilistic initiation of origins and passive replication. Using the model, we performed least-squares fits to a set of recently published time course microarray data on Saccharomyces cerevisiae. We extracted the distribution of firing times for each origin and found that the later an origin fires on average, the greater the variation in firing times. To explain this trend, we propose a model where earlier-firing origins have more initiator complexes loaded and a more accessible chromatin environment. The model demonstrates how initiation can be stochastic and yet occur at defined times during S phase, without an explicit timing program. Furthermore, we hypothesize that the initiators in this model correspond to loaded minichromosome maintenance complexes. This model is the first to suggest a detailed, testable, biochemically plausible mechanism for the regulation of replication timing in eukaryotes.
PMCID: PMC2950085  PMID: 20739926
DNA replication program; genome-wide analysis; microarray data; replication-origin efficiency; stochastic modeling
14.  Gene–gene interaction analysis for the survival phenotype based on the Cox model 
Bioinformatics  2012;28(18):i582-i588.
Motivation: For the past few decades, many statistical methods in genome-wide association studies (GWAS) have been developed to identify SNP–SNP interactions for case-control studies. However, there has been less work for prospective cohort studies, involving the survival time. Recently, Gui et al. (2011) proposed a novel method, called Surv-MDR, for detecting gene–gene interactions associated with survival time. Surv-MDR is an extension of the multifactor dimensionality reduction (MDR) method to the survival phenotype by using the log-rank test for defining a binary attribute. However, the Surv-MDR method has some drawbacks in the sense that it needs more intensive computations and does not allow for a covariate adjustment. In this article, we propose a new approach, called Cox-MDR, which is an extension of the generalized multifactor dimensionality reduction (GMDR) to the survival phenotype by using a martingale residual as a score to classify multi-level genotypes as high- and low-risk groups. The advantages of Cox-MDR over Surv-MDR are to allow for the effects of discrete and quantitative covariates in the frame of Cox regression model and to require less computation than Surv-MDR.
Results: Through simulation studies, we compared the power of Cox-MDR with those of Surv-MDR and Cox regression model for various heritability and minor allele frequency combinations without and with adjusting for covariate. We found that Cox-MDR and Cox regression model perform better than Surv-MDR for low minor allele frequency of 0.2, but Surv-MDR has high power for minor allele frequency of 0.4. However, when the effect of covariate is adjusted for, Cox-MDR and Cox regression model perform much better than Surv-MDR. We also compared the performance of Cox-MDR and Surv-MDR for a real data of leukemia patients to detect the gene–gene interactions with the survival time.
PMCID: PMC3436842  PMID: 22962485
15.  Accounting for individual differences and timing of events: estimating the effect of treatment on criminal convictions in heroin users 
The reduction of crime is an important outcome of opioid maintenance treatment (OMT). Criminal intensity and treatment regimes vary among OMT patients, but this is rarely adjusted for in statistical analyses, which tend to focus on cohort incidence rates and rate ratios. The purpose of this work was to estimate the relationship between treatment and criminal convictions among OMT patients, adjusting for individual covariate information and timing of events, fitting time-to-event regression models of increasing complexity.
National criminal records were cross linked with treatment data on 3221 patients starting OMT in Norway 1997–2003. In addition to calculating cohort incidence rates, criminal convictions was modelled as a recurrent event dependent variable, and treatment a time-dependent covariate, in Cox proportional hazards, Aalen’s additive hazards, and semi-parametric additive hazards regression models. Both fixed and dynamic covariates were included.
During OMT, the number of days with criminal convictions for the cohort as a whole was 61% lower than when not in treatment. OMT was associated with reduced number of days with criminal convictions in all time-to-event regression models, but the hazard ratio (95% CI) was strongly attenuated when adjusting for covariates; from 0.40 (0.35, 0.45) in a univariate model to 0.79 (0.72, 0.87) in a fully adjusted model. The hazard was lower for females and decreasing with older age, while increasing with high numbers of criminal convictions prior to application to OMT (all p < 0.001). The strongest predictors were level of criminal activity prior to entering into OMT, and having a recent criminal conviction (both p < 0.001). The effect of several predictors was significantly time-varying with their effects diminishing over time.
Analyzing complex observational data regarding to fixed factors only overlooks important temporal information, and naïve cohort level incidence rates might result in biased estimates of the effect of interventions. Applying time-to-event regression models, properly adjusting for individual covariate information and timing of various events, allows for more precise and reliable effect estimates, as well as painting a more nuanced picture that can aid health care professionals and policy makers.
PMCID: PMC4040473  PMID: 24886472
Maintenance treatment; Criminal activity; Time-to-event; Recurring event; Time-dependent covariate; Dynamic covariate
16.  Recurrent events and the exploding Cox model 
Lifetime data analysis  2010;16(4):525-546.
Counting process models have played an important role in survival and event history analysis for more than 30 years. Nevertheless, almost all models that are being used have a very simple structure. Analyzing recurrent events invites the application of more complex models with dynamic covariates. We discuss how to define valid models in such a setting. One has to check carefully that a suggested model is well defined as a stochastic process. We give conditions for this to hold. Some detailed discussion is presented in relation to a Cox type model, where the exponential structure combined with feedback lead to an exploding model. In general, counting process models with dynamic covariates can be formulated to avoid explosions. In particular, models with a linear feedback structure do not explode, making them useful tools in general modeling of recurrent events.
PMCID: PMC4066394  PMID: 20625827
Recurrent events; Cox regression; Explosion; Honest process; Birth process; Aalen regression; Stochastic differential equation; Lipschitz condition; Feller criterion; Martingale problem
17.  Efficient Semiparametric Estimation of Short-term and Long-term Hazard Ratios with Right-Censored Data 
Biometrics  2013;69(4):10.1111/biom.12097.
The proportional hazards assumption in the commonly used Cox model for censored failure time data is often violated in scientific studies. Yang and Prentice (2005) proposed a novel semiparametric two-sample model that includes the proportional hazards model and the proportional odds model as sub-models, and accommodates crossing survival curves. The model leaves the baseline hazard unspecified and the two model parameters can be interpreted as the short-term and long-term hazard ratios. Inference procedures were developed based on a pseudo score approach. Although extension to accommodate covariates was mentioned, no formal procedures have been provided or proved. Furthermore, the pseudo score approach may not be asymptotically efficient. We study the extension of the short-term and long-term hazard ratio model of Yang and Prentice (2005) to accommodate potentially time-dependent covariates. We develop efficient likelihood-based estimation and inference procedures. The nonparametric maximum likelihood estimators are shown to be consistent, asymptotically normal, and asymptotically efficient. Extensive simulation studies demonstrate that the proposed methods perform well in practical settings. The proposed method successfully captured the phenomenon of crossing hazards in a cancer clinical trial and identified a genetic marker with significant long-term effect missed by using the proportional hazards model on age-at-onset of alcoholism in a genetic study.
PMCID: PMC3868993  PMID: 24328712
Semiparametric hazards rate model; Non-parametric likelihood; Proportional hazards model; Proportional odds model; Semiparametric efficiency
18.  Survival Analysis of Irish Amyotrophic Lateral Sclerosis Patients Diagnosed from 1995–2010 
PLoS ONE  2013;8(9):e74733.
The Irish ALS register is a valuable resource for examining survival factors in Irish ALS patients. Cox regression has become the default tool for survival analysis, but recently new classes of flexible parametric survival analysis tools known as Royston-Parmar models have become available.
We employed Cox proportional hazards and Royston-Parmar flexible parametric modeling to examine factors affecting survival in Irish ALS patients. We further examined the effect of choice of timescale on Cox models and the proportional hazards assumption, and extended both Cox and Royston-Parmar models with time varying components.
On comparison of models we chose a Royston-Parmar proportional hazards model without time varying covariates as the best fit. Using this model we confirmed the association of known survival markers in ALS including age at diagnosis (Hazard Ratio (HR) 1.34 per 10 year increase; 95% CI 1.26–1.42), diagnostic delay (HR 0.96 per 12 weeks delay; 95% CI 0.94–0.97), Definite ALS (HR 1.47 95% CI 1.17–1.84), bulbar onset disease (HR 1.58 95% CI 1.33–1.87), riluzole use (HR 0.72 95% CI 0.61–0.85) and attendance at an ALS clinic (HR 0.74 95% CI 0.64–0.86).
Our analysis explored the strengths and weaknesses of Cox proportional hazard and Royston-Parmar flexible parametric methods. By including time varying components we were able to gain deeper understanding of the dataset. Variation in survival between time periods appears to be due to missing data in the first time period. The use of age as timescale to account for confounding by age resolved breaches of the proportional hazards assumption, but in doing so may have obscured deficiencies in the data. Our study demonstrates the need to test for, and fully explore, breaches of the Cox proportional hazards assumption. Royston-Parmar flexible parametric modeling proved a powerful method for achieving this.
PMCID: PMC3786977  PMID: 24098664
19.  The relative efficiency of time-to-threshold and rate of change in longitudinal data 
Contemporary clinical trials  2011;32(5):685-693.
Randomized, placebo-controlled trials often use time-to-event as the primary endpoint, even when a continuous measure of disease severity is available. We compare the power to detect a treatment effect using either rate of change, as estimated by linear models of longitudinal continuous data, or time-to-event estimated by Cox proportional hazards models. We propose an analytic inflation factor for comparing the two types of analyses assuming that the time-to-event can be expressed as a time-to-threshold of the continuous measure. We conduct simulations based on a publicly available Alzheimer's disease data set in which the time-to-event is algorithmically defined based on a battery of assessments. A Cox proportional hazards model of the time-to-event endpoint is compared to a linear model of a single assessment from the battery. The simulations also explore the impact of baseline covariates in either analysis.
PMCID: PMC3148349  PMID: 21554992
longitudinal data; survival analysis; linear mixed models; marginal linear models; power
20.  Polifeprosan 20, 3.85% carmustine slow-release wafer in malignant glioma: evidence for role in era of standard adjuvant temozolomide 
Core Evidence  2012;7:115-130.
The Polifeprosan 20 with carmustine (BCNU, bis-chloroethylnitrosourea, Gliadel®) polymer implant wafer is a biodegradable compound containing 3.85% carmustine which slowly degrades to release carmustine and protects it from exposure to water with resultant hydrolysis until the time of release. The carmustine implant wafer was demonstrated to improve survival in blinded placebo-controlled trials in selected patients with newly diagnosed or recurrent malignant glioma, with little increased risk of adverse events. Based on these trials and other supporting data, US and European regulatory authorities granted approval for its use in recurrent and newly diagnosed malignant glioma, and it remains the only approved local treatment. The preclinical and clinical data suggest that it is optimally utilized primarily in the proportion of patients who may have total or near total removal of gross tumor. The aim of this work was to review the evidence for the use of carmustine implants in the management of malignant astrocytoma (World Health Organization grades III and IV), including newly diagnosed and recurrent disease, especially in the setting of a standard of care that has changed since the randomized trials were completed. Therapy has evolved such that patients now generally receive temozolomide chemotherapy during and after radiotherapy treatment. For patients undergoing repeat resection for malignant glioma, a randomized, blinded, placebo-controlled trial demonstrated a median survival for 110 patients who received carmustine polymers of 31 weeks compared with 23 weeks for 122 patients who only received placebo polymers. The benefit achieved statistical significance only on analysis adjusting for prognostic factors rather than for the randomized groups as a whole (hazard ratio = 0.67, P = 0.006). A blinded, placebo-controlled trial has also been performed for carmustine implant placement in newly diagnosed patients prior to standard radiotherapy. Median survival was improved from 11.6 to 13.9 months (P = 0.03), with a 29% reduction in the risk of death. When patients with glioblastoma multiforme alone were analyzed, the median survival improved from 11.4 to 13.5 months, but this improvement was not statistically significant. When a Cox’s proportional hazard model was utilized to account for other potential prognostic factors, there was a significant 31% reduction in the risk of death (P = 0.04) in this subgroup. Data from other small reports support these results and confirm that the incidence of adverse events does not appear to be increased meaningfully. Given the poor prognosis without possibility of cure, these benefits from a treatment with a favorable safety profile were considered meaningful. There is randomized evidence to support the use of carmustine wafers placed during resection of recurrent disease. Therefore, although there is limited specific evidence, this treatment is likely to be efficacious in an environment when nearly all patients receive temozolomide as part of initial management. Given that half of the patients in the randomized trial assessing the value of carmustine implants in recurrent disease had received prior chemotherapy, it is likely that this remains a valuable treatment at the time of repeat resection, even after temozolomide. There are data from multiple reports to support safety. Although there is randomized evidence to support the use of this therapy in newly diagnosed patients who will receive radiotherapy alone, it is now standard to administer both adjuvant temozolomide and radiotherapy. There are survival outcome reports for small cohorts of patients receiving temozolomide with radiotherapy, but this information is not sufficient to support firm recommendations. Based on the rationale and evidence of safety, this approach appears to be a reasonable option as more information is acquired. Available data support the safety of using carmustine wafers in this circumstance, although special attention to surgical guidelines for implanting the wafers is warranted.
PMCID: PMC3484478  PMID: 23118709
carmustine; Polifeprosan 20; malignant glioma
21.  Multi-state model for studying an intermediate event using time-dependent covariates: application to breast cancer 
The aim of this article is to propose several methods that allow to investigate how and whether the shape of the hazard ratio after an intermediate event depends on the waiting time to occurrence of this event and/or the sojourn time in this state.
A simple multi-state model, the illness-death model, is used as a framework to investigate the occurrence of this intermediate event. Several approaches are shown and their advantages and disadvantages are discussed. All these approaches are based on Cox regression. As different time-scales are used, these models go beyond Markov models. Different estimation methods for the transition hazards are presented. Additionally, time-varying covariates are included into the model using an approach based on fractional polynomials. The different methods of this article are then applied to a dataset consisting of four studies conducted by the German Breast Cancer Study Group (GBSG). The occurrence of the first isolated locoregional recurrence (ILRR) is studied. The results contribute to the debate on the role of the ILRR with respect to the course of the breast cancer disease and the resulting prognosis.
We have investigated different modelling strategies for the transition hazard after ILRR or in general after an intermediate event. Including time-dependent structures altered the resulting hazard functions considerably and it was shown that this time-dependent structure has to be taken into account in the case of our breast cancer dataset. The results indicate that an early recurrence increases the risk of death. A late ILRR increases the hazard function much less and after the successful removal of the second tumour the risk of death is almost the same as before the recurrence. With respect to distant disease, the appearance of the ILRR only slightly increases the risk of death if the recurrence was treated successfully.
It is important to realize that there are several modelling strategies for the intermediate event and that each of these strategies has restrictions and may lead to different results. Especially in the medical literature considering breast cancer development, the time-dependency is often neglected in the statistical analyses. We show that the time-varying variables cannot be neglected in the case of ILRR and that fractional polynomials are a useful tool for finding the functional form of these time-varying variables.
PMCID: PMC3700854  PMID: 23786493
22.  Aspirin and recurrent intracerebral hemorrhage in cerebral amyloid angiopathy 
Neurology  2010;75(8):693-698.
To identify and compare clinical and neuroimaging predictors of primary lobar intracerebral hemorrhage (ICH) recurrence, assessing their relative contributions to recurrent ICH.
Subjects were consecutive survivors of primary ICH drawn from a single-center prospective cohort study. Baseline clinical, imaging, and laboratory data were collected. Survivors were followed prospectively for recurrent ICH and intercurrent aspirin and warfarin use, including duration of exposure. Cox proportional hazards models were used to identify predictors of recurrence stratified by ICH location, with aspirin and warfarin exposures as time-dependent variables adjusting for potential confounders.
A total of 104 primary lobar ICH survivors were enrolled. Recurrence of lobar ICH was associated with previous ICH before index event (hazard ratio [HR] 7.7, 95% confidence interval [CI] 1.4–15.7), number of lobar microbleeds (HR 2.93 with 2–4 microbleeds present, 95% CI 1.3–4.0; HR = 4.12 when ≥5 microbleeds present, 95% CI 1.6–9.3), and presence of CT-defined white matter hypodensity in the posterior region (HR 4.11, 95% CI 1.01–12.2). Although aspirin after ICH was not associated with lobar ICH recurrence in univariate analyses, in multivariate analyses adjusting for baseline clinical predictors, it independently increased the risk of ICH recurrence (HR 3.95, 95% CI 1.6–8.3, p = 0.021).
Recurrence of lobar ICH is associated with previous microbleeds or macrobleeds and posterior CT white matter hypodensity, which may be markers of severity for underlying cerebral amyloid angiopathy. Use of an antiplatelet agent following lobar ICH may also increase recurrence risk.
= cerebral amyloid angiopathy;
= confidence interval;
= CT-defined white matter hypodensity;
= hazard ratio;
= intracerebral hemorrhage;
= variance inflation factor.
PMCID: PMC2931649  PMID: 20733144
23.  Competing risk models to estimate the excess mortality and the first recurrent-event hazards 
In medical research, one common competing risks situation is the study of different types of events, such as disease recurrence and death. We focused on that situation but considered death under two aspects: "expected death" and "excess death", the latter could be directly or indirectly associated with the disease.
The excess hazard method allows estimating an excess mortality hazard using the population (expected) mortality hazard. We propose models combining the competing risks approach and the excess hazard method. These models are based on a joint modelling of each event-specific hazard, including the event-free excess death hazard. The proposed models are parsimonious, allow time-dependent hazard ratios, and facilitate comparisons between event-specific hazards and between covariate effects on different events. In a simulation study, we assessed the performance of the estimators and showed their good properties with different drop-out censoring rates and different sample sizes.
We analyzed a population-based dataset on French colon cancer patients who have undergone curative surgery. Considering three competing events (local recurrence, distant metastasis, and death), we showed that the recurrence-free excess mortality hazard reached zero six months after treatment. Covariates sex, age, and cancer stage had the same effects on local recurrence and distant metastasis but a different effect on excess mortality.
The proposed models consider the excess mortality within the framework of competing risks. Moreover, the joint estimation of the parameters allow (i) direct comparisons between covariate effects, and (ii) fitting models with common parameters to obtain more parsimonious models and more efficient parameter estimators.
PMCID: PMC3123657  PMID: 21612632
Excess hazard; Competing risks; Time-dependent hazard ratio; Regression splines; Cancer; Population-based study
24.  Alternative statistical methods for estimating efficacy of interferon beta-1b for multiple sclerosis clinical trials 
In the randomized study of interferon beta-1b (IFN beta-1b) for multiple sclerosis (MS), it has usually been evaluated the simple annual relapse rate as the study endpoint. This study aimed to investigate the performance of various regression models using information regarding the time to each recurrent event and considering the MS specific data generation process, and to estimate the treatment effect of a MS clinical trial data.
We conducted a simulation study with consideration of the pathological characteristics of MS, and applied alternative efficacy estimation methods to real clinical trial data, including 5 extended Cox regression models for time-to-event analysis, a Poisson regression model and a Poisson regression model with Generalized Estimating Equations (GEE). We adjusted for other important covariates that may have affected the outcome.
We compared the simulation results for each model. The hazard ratios of real data were estimated for each model including the effects of other covariates. The results (hazard ratios of high-dose to low-dose) of all models were approximately 0.7 (range, 0.613 - 0.769), whereas the annual relapse rate ratio was 0.714.
The precision of the treatment estimation was increased by application of the alternative models. This suggests that the use of alternative models that include recurrence event data may provide better analyses.
PMCID: PMC3118202  PMID: 21612661
25.  Joint Modeling and Estimation for Recurrent Event Processes and Failure Time Data 
Recurrent event data are commonly encountered in longitudinal follow-up studies related to biomedical science, econometrics, reliability, and demography. In many studies, recurrent events serve as important measurements for evaluating disease progression, health deterioration, or insurance risk. When analyzing recurrent event data, an independent censoring condition is typically required for the construction of statistical methods. In some situations, however, the terminating time for observing recurrent events could be correlated with the recurrent event process, thus violating the assumption of independent censoring. In this article, we consider joint modeling of a recurrent event process and a failure time in which a common subject-specific latent variable is used to model the association between the intensity of the recurrent event process and the hazard of the failure time. The proposed joint model is flexible in that no parametric assumptions on the distributions of censoring times and latent variables are made, and under the model, informative censoring is allowed for observing both the recurrent events and failure times. We propose a “borrow-strength estimation procedure” by first estimating the value of the latent variable from recurrent event data, then using the estimated value in the failure time model. Some interesting implications and trajectories of the proposed model are presented. Properties of the regression parameter estimates and the estimated baseline cumulative hazard functions are also studied.
PMCID: PMC3780991  PMID: 24068850
Borrow-strength method; Frailty; Informative censoring; Joint model; Nonstationary Poisson process

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