Multiplicative Intensity Model Research Articles

A mixed effect model for clustered recurrent event data

For the clustered recurrent event data, a multiplicative intensity model with a mixed effect was proposed. The correlation within the same cluster was specified by a gamma frailty variable. The parameters of the model were estimated by maximum likelihood estimation. The inference procedure was carried out by EM algorithm. Some simulations were conducted to evaluate the proposed method, and a real data was illustrated.

Variational inference for multiplicative intensity models

We extend variational inference approximation of probability density functions to multiplicative intensity functions. For Bayesian nonparametrics, this provides a computationally efficient alternative to the blocked Gibbs sampler described in Ishwaran and James (2004). Simulation results are presented to demonstrate performance.

Goftte: A R package for assessing goodness-of-fit in proportional (sub) distributions hazards regression models

Background and objectiveIn this paper, we introduce a new R package goftte for goodness-of-fit assessment based on cumulative sums of model residuals useful for checking key assumptions in the Cox regression and Fine and Gray regression models. MethodsMonte-Carlo methods are used to approximate the null distribution of cumulative sums of model residuals. To limit the computational burden, the main routines used to approximate the null distributions are implemented in a parallel C++ programming environment. Numerical studies are carried out to evaluate the empirical type I error rates of the different testing procedures. The package and the documentation are available to users from CRAN R repositories. ResultsResults from simulation studies suggested that all statistical tests implemented in goftte yielded excellent control of the type I error rate even with modest sample sizes with high censoring rates. ConclusionsAs compared to other R packages goftte provides new useful method for testing functionals, such as Anderson-Darling type test statistics for checking assumptions about proportional (sub-) distribution hazards. Approximations for the null distributions of test statistics have been validated through simulation experiments. Future releases will provide similar tools for checking model assumptions in multiplicative intensity models for recurrent data. The package may help to spread the use of recent advocated goodness-of-fit techniques in semiparametric regression for time-to-event data.

The wild bootstrap for multivariate Nelson\u2013Aalen estimators

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

Oracle inequalities for the Lasso in the high-dimensional Aalen multiplicative intensity model

Dans le cadre general d’un processus de comptage, nous interessons a la facon d’obtenir un pronostic sur la duree de survie en fonction des covariables en grande dimension. Pour ce faire, nous construisons un estimateur de l’intensite conditionnelle. Nous l’estimons par le meilleur modele de Cox etant donne deux dictionnaires de fonctions. Le premier dictionnaire est utilise pour construire le logarithme du risque de base et le second, pour approximer le risque relatif. Nous introduisons une nouvelle procedure Lasso pondere avec une ponderation basee sur les donnees pour estimer les parametres inconnus du meilleur modele de Cox approximant l’intensite. Nous etablissons une inegalite oracle non-asymptotique en divergence de Kullback empirique, qui est la fonction de perte la plus appropriee a notre procedure. Nos resultats reposent sur une inegalite de Bernstein pour les martingales a sauts et sur des proprietes des fonctions self-concordantes.

Major adverse cardiac events in elderly patients with coronary artery disease undergoing noncardiac surgery: A multicenter prospective study in China

BackgroundMajor adverse cardiac events (MACEs) are important causes of perioperative morbidity and mortality for elderly patients undergoing non-cardiac surgery. Treatment and control rates for coronary artery disease (CAD) in Chinese patients are poorer than rates in western countries. However, no previous prospective study has focused on perioperative MACE in this population. Our aim was to ascertain the incidence and risk factors associated with MACEs in Chinese patients. MethodsConsecutive CAD patients, aged ≥60 years, who underwent non-cardiac surgery at five medical centers in China, were prospectively enrolled. Clinical variables, including electrocardiogram and troponin I levels, were evaluated to estimate MACEs. The main outcome was occurrence of at least one perioperative MACE from admittance to 30 days after surgery, defined as any of the following complications: cardiac death, nonfatal cardiac arrest, acute myocardial infarction (MI), congestive heart failure (CHF), and angina. MACE independent risk factors were based on the Andersen–Gill multiplicative intensity model. ResultsOf the 1422 patients recruited, 129 (9.1%) developed at least one MACE, and cardiac death occurred in 11 patients (0.8%). The independent risk factors contributing to postoperative MACE included age ≥75 years, female gender, history of MI, history of hypertension, high-risk surgery, intraoperative hypotension, and intraoperative hypoxemia. ConclusionsThe incidence of MACE in Chinese elderly patients with CAD who underwent non-cardiac surgery was 9.1%. Seven independent risk factors for a perioperative MACE were identified. Preventing intraoperative hypoxemia and hypotension may reduce the occurrence of MACE in these high risk patients.

Non‐Parametric Estimation of the Conditional Distribution of the Interjumping Times for Piecewise‐Deterministic Markov Processes

ABSTRACTThis paper presents a non‐parametric method for estimating the conditional density associated to the jump rate of a piecewise‐deterministic Markov process. In our framework, the estimation needs only one observation of the process within a long time interval. Our method relies on a generalization of Aalen's multiplicative intensity model. We prove the uniform consistency of our estimator, under some reasonable assumptions related to the primitive characteristics of the process. A simulation study illustrates the behaviour of our estimator.

Nonparametric estimation of the jump rate for non-homogeneous marked renewal processes

This paper is devoted to the nonparametric estimation of the jump rate and the cumulative rate for a general class of non-homogeneous marked renewal processes, defined on a separable metric space. In our framework, the estimation needs only one observation of the process within a long time. Our approach is based on a generalization of the multiplicative intensity model, introduced by Aalen in the seventies. We provide consistent estimators of these two functions, under some assumptions related to the ergodicity of an embedded chain and the characteristics of the process. The paper is illustrated by a numerical example.

Survival Data Analysis using Additive and Multiplicative Gamma Polygonal Hazards Function

Proportional hazards model (PHM) is commonly used in survival analysis for estimating the effects of different covariates influencing the survival data. The hazard function in proportional hazards model (PHM) is commonly defined as a product of the baseline hazard function and a non-negative function of covariates. However, the hazard function may also be presented as the sum of the baseline hazard function and a nonnegative function of covariates. We propose the new additive and multiplicative Gamma Polygonal in the hazards function using OpenBugs Statistical Packages. Both models are an alternative to the existing additive and multiplicative models but the new additive Gamma polygonal intensity model is quite complex compared to the new multiplicative Gamma polygonal intensity model. The application for right-censored data from Bayesian perspective will be discussed and the Markov Chain Monte Carlo (MCMC) method will be used to compute the Bayesian estimator using Leukemia data and DSR data. The results obtained show that the propose model is as good as the existing models in analyzing paired survival data. Keywords: Additive model; Multiplicative model; Baseline hazards function; MCMC 2010 Mathematics Subject Classification: 46N60; 92B99

Point Process Modelling for Directed Interaction Networks

SummaryNetwork data often take the form of repeated interactions between senders and receivers tabulated over time. A primary question to ask of such data is which traits and behaviours are predictive of interaction. To answer this question, a model is introduced for treating directed interactions as a multivariate point process: a Cox multiplicative intensity model using covariates that depend on the history of the process. Consistency and asymptotic normality are proved for the resulting partial-likelihood-based estimators under suitable regularity conditions, and an efficient fitting procedure is described. Multicast interactions—those involving a single sender but multiple receivers—are treated explicitly. The resulting inferential framework is then employed to model message sending behaviour in a corporate e-mail network. The analysis gives a precise quantification of which static shared traits and dynamic network effects are predictive of message recipient selection.

Additive Intensity Regression Models in Corporate Default Analysis

We consider additive intensity (Aalen) models as an alternative to the multiplicative intensity (Cox) models for analyzing the default risk of a sample of rated, nonfinancial U.S. firms. The setting allows for estimating and testing the significance of time-varying effects. We use a variety of model checking techniques to identify misspecifications. In our final model, we find evidence of time-variation in the effects of distance-to-default and short-to-long term debt. Also we identify interactions between distance-to-default and other covariates, and the quick ratio covariate is significant. None of our macroeconomic covariates are significant.

Modeling rating transitions

The time-continuous discrete-state Markov process is a model for rating transitions. One parameter, namely the intensity to migrate to an adjacent rating state, implies an ordinal rating to have an intuitive metric. State-specific intensities generalize such state-stationarity. Observing Markov processes from a multiplicative intensity model, the maximum likelihood parameter estimators for both models can be studied with the score statistic, written as a martingale transform of the processes that count transitions between the rating states. A Taylor expansion reveals consistency and asymptotic normality of the parameter estimates, resulting in a χ 2 -distributed likelihood ratio of state-stationarity against the state-specific model. This extends to time-stationarity. Simulations contrast the asymptotic results with finite samples. An application to a sufficiently large set of credit rating histories shows that the one-parameter model can be a good starting point.

Early detection of a change in Poisson rate after accounting for population size effects

Motivated by applications in bio and syndromic surveillance, this arti- cle is concerned with the problem of detecting a change in the mean of Poisson distributions after taking into account the effects of population size. The family of generalized likelihood ratio (GLR) schemes is proposed and its asymptotic opti- mality properties are established under the classical asymptotic setting. However, numerical simulation studies illustrate that the GLR schemes are at times not as efficient as two families of ad-hoc schemes based on either the weighted likelihood ratios or the adaptive threshold method that adjust the effects of population sizes. To explain this, a further asymptotic optimality analysis is developed under a new asymptotic setting that is more suitable to our finite-sample numerical simulations. In addition, we extend our approaches to a general setting with arbitrary probability distributions, as well as to the continuous-time setting involving the multiplicative intensity models for Poisson processes, but further research is needed.

A Bootstrap Algorithm for Data from a Periodic Multiplicative Intensity Function

The periodic multiplicative intensity model is considered. A new bootstrap method for non stationary counting processes which intensity function has some periodicity properties is presented. Its main advantage is that it does not destroy the temporal order and the original periodicity of the underlying counting process. The proposed algorithm is used to construct a bootstrap version of the maximum likelihood hazard function estimator. The consistency of the bootstrap method is shown. A possible modification of the proposed bootstrap method is discussed. The bootstrap simultaneous confidence intervals for the hazard function are presented. The telecommunication network traffic real data example is discussed.

Anaemia in a phase 2 study of a blood stage falciparum malaria vaccine

BackgroundA Phase 1-2b study of the blood stage malaria vaccine AMA1-C1/Alhydrogel was conducted in 336 children in Donéguébougou and Bancoumana, Mali. In the Phase 2 portion of the study (n = 300), no impact on parasite density or clinical malaria was seen; however, children who received the study vaccine had a higher frequency of anaemia (defined as haemoglobin < 8.5 g/dL) compared to those who received the comparator vaccine (Hiberix). This effect was one of many tested and was not significant after adjusting for multiple comparisons.MethodsTo further investigate the possible impact of vaccination on anaemia, additional analyses were conducted including patients from the Phase 1 portion of the study and controlling for baseline haemoglobin, haemoglobin types S or C, alpha-thalassaemia, G6PD deficiency, and age. A multiplicative intensity model was used, which generalizes Cox regression to allow for multiple events. Frailty effects for each subject were used to account for correlation of multiple anaemia events within the same subject. Intensity rates were calculated with reference to calendar time instead of time after randomization in order to account for staggered enrollment and seasonal effects of malaria incidence. Associations of anaemia with anti-AMA1 antibody were further explored using a similar analysis.ResultsA strong effect of vaccine on the incidence of anaemia (risk ratio [AMA1-C1 to comparator (Hiberix)]= 2.01, 95% confidence interval [1.26,3.20]) was demonstrated even after adjusting for baseline haemoglobin, haemoglobinopathies, and age, and using more sophisticated statistical models. Anti-AMA1 antibody levels were not associated with this effect.ConclusionsWhile these additional analyses show a robust effect of vaccination on anaemia, this is an intensive exploration of secondary results and should, therefore, be interpreted with caution. Possible mechanisms of the apparent adverse effect on haemoglobin of vaccination with AMA1-C1/Alhydrogel and implications for blood stage vaccine development are discussed. The potential impact on malaria-associated anaemia should be closely evaluated in clinical trials of AMA1 and other blood stage vaccines in malaria-exposed populations.

298 Cd3+Cd4+-Lymphocytes as Biomarker Predicting the Outcome of Pediatric Haematopoietic Stem Cell Transplantation Recipients

Introduction: Haematopoietic stem cell transplantation (HSCT) may be complicated by severe infectious complications, graft-versus-host disease (GvHD) and relapse/graft failures depending on the lymphocyte immune reconstitution. The goal of this study was to identify biomarkers of immune-reconstitution that may predict outcome in the first three months after transplant.Methods: All children transplanted between 2005-2007 within the UMCU were prospectively included in this study. Immunophenotyping was performed every two weeks after HSCT. In this study we analyzed several parameters of immunereconstitution: AUC, the time and maximum CD3+CD4+, CD3+CD8+ T-cell and CD19+B-cell numbers, the ratio between CD4/CD8 and the relationship between naive and memory cells in relation with survival, acute- and chronic-GvHD. We focused on the age-related differences of the immune system and donor sources. We used a multiplicative intensity model in R.Results: 56 recipients received bone marrow, 12 received a sibling donor, while 26 patients received cord blood derived stem cells. The median age at HSCT was 5.9 years (range 0.11 - 18.1). The log-AUC of CD4+ cells in the first 3 months after HSCT positively predicted survival (P < 0.001, HR 0.68, CI95% 0.55-0.85.Conclusion: The AUC of the number of CD4+ cells, 0-90 days after HSCT, predicted long term outcome after HSCT. A low number of CD4 cells (an AUC of approximately 2000 cells*days) in the first period (< 3mths) after HSCT predicted poor survival. Both in future studies and in clinical practice this biomarker may be used to predict outcome early after transplantation.

Asymptotic results for fitting marginal hazard models from stratified case-cohort studies with multiple disease outcomes

In stratified case-cohort designs, samplings of case-cohort samples are conducted via a stratified random sampling based on covariate information available on the entire cohort members. In this paper, we extended the work of Kang and Cai (2009) to a generalized stratified case-cohort study design for failure time data with multiple disease outcomes. Under this study design, we developed weighted estimating procedures for model parameters in marginal multiplicative intensity models and for the cumulative baseline hazard function. The asymptotic properties of the estimators are studied using martingales, modern empirical process theory, and results for finite population sampling.

A likelihood ratio test for stationarity of rating transitions

We study the time-stationarity of rating transitions, modelled by a time-continuous discrete-state Markov process and derive a likelihood ratio test. For multiple Markov processes from a multiplicative intensity model, maximum likelihood parameter estimates can be written as martingale transform of the processes, counting transitions between the rating states, so that the profile partial likelihood ratio is asymptotically χ 2 -distributed. An application to an internal rating data set reveals highly significant instationarity.

Local linear density estimation for filtered survival data, with bias correction

A class of local linear kernel density estimators based on weighted least-squares kernel estimation is considered within the framework of Aalen's multiplicative intensity model. This model includes the filtered data model that, in turn, allows for truncation and/or censoring in addition to accommodating unusual patterns of exposure as well as occurrence. It is shown that the local linear estimators corresponding to all different weightings have the same pointwise asymptotic properties. However, the weighting previously used in the literature in the i.i.d. case is seen to be far from optimal when it comes to exposure robustness, and a simple alternative weighting is to be preferred. Indeed, this weighting has, effectively, to be well chosen in a ‘pilot’ estimator of the survival function as well as in the main estimator itself. We also investigate multiplicative and additive bias-correction methods within our framework. The multiplicative bias-correction method proves to be the best in a simulation study comparing the performance of the considered estimators. An example concerning old-age mortality demonstrates the importance of the improvements provided.

Smoothed estimator of the periodic hazard function

A smoothed estimator of the periodic hazard function is considered and its asymptotic probability distribution and bootstrap simultaneous confidence intervals are derived. Moreover, consistency of the bootstrap method is proved and some applications of the developed theory are presented. The bootstrap method is based on the phase-consistent resampling scheme developed in Dudek and Leśkow [A. Dudek, J. Leśkow, Bootstrap algorithm in periodic multiplicative intensity model, to appear].