Additive Rates Model Research Articles

A Family of Additive Teissier–Weibull Hazard Distributions for Modeling Bathtub-Shaped Failure Time Data

The paper introduces a new failure time distribution called additive Teissier–Weibull distribution. Among its features, it is capable of modeling increasing and bathtub hazard rates. It is thus useful in modeling heterogeneous populations and can be viewed as the failure time model of the system having two modes of failures that follow the Teissier and Weibull distributions, respectively. Some of its important properties are obtained, such as quantiles, moments and shapes of the probability density and hazard functions. Four different methods of estimation such as the maximum likelihood, least squares, weighted least squares and maximum product spacing are proposed to estimate the unknown parameters. To compare the performance of the estimates, an extensive simulation study is carried out with varying sample sizes. Finally, failure times of primary reactor pumps and power generators are fitted and analyzed to show that the new additive hazard rate model may give a better fit than many other existing models.

Augmented weighting estimators for the additive rates model under multivariate recurrent event data with missing event type.

Multivariate recurrent event data are frequently encountered in biomedical and epidemiological studies when subjects experience multiple types of recurrent events. In practice, the event type information may be missing due to a variety of reasons. In this article, we consider a semiparametric additive rates model for multivariate recurrent event data with missing event types. We develop the augmented inverse probability weighting technique to handle event types that are missing at random. The nonparametric kernel-assisted proposals for the missing mechanisms are studied. The resulting estimator is shown to be consistent and asymptotically normal. Extensive simulation studies and a real data application are provided to illustrate the validity and practical utility of the proposed method.

Additive rates model for recurrent event data with intermittently observed time-dependent covariates.

Various regression methods have been proposed for analyzing recurrent event data. Among them, the semiparametric additive rates model is particularly appealing because the regression coefficients quantify the absolute difference in the occurrence rate of the recurrent events between different groups. Estimation of the additive rates model requires the values of time-dependent covariates being observed throughout the entire follow-up period. In practice, however, the time-dependent covariates are usually only measured at intermittent follow-up visits. In this paper, we propose to kernel smooth functions involving time-dependent covariates across subjects in the estimating function, as opposed to imputing individual covariate trajectories. Simulation studies show that the proposed method outperforms simple imputation methods. The proposed method is illustrated with data from an epidemiologic study of the effect of streptococcal infections on recurrent pharyngitis episodes.

Sample size calculation for recurrent event data with additive rates models.

This paper discusses the design of clinical trials where the primary endpoint is a recurrent event with the focus on the sample size calculation. For the problem, a few methods have been proposed but most of them assume a multiplicative treatment effect on the rate or mean number of recurrent events. In practice, sometimes the additive treatment effect may be preferred or more appealing because of its intuitive clinical meaning and straightforward interpretation compared to a multiplicative relationship. In this paper, new methods are presented and investigated for the sample size calculation based on the additive rates model for superiority, non-inferiority, and equivalence trials. They allow for flexible baseline rate function, staggered entry, random dropout, and overdispersion in event numbers, and simulation studies show that the proposed methods perform well in a variety of settings. We also illustrate how to use the proposed methods to design a clinical trial based on real data.

Inverse probability weighted estimation for recurrent events data with missing category.

Modeling recurrent event data with multiple event types has drawn interest in recent biomedical studies due to its flexibility for understanding different risk factors for multiple recurrent event processes. However, in such data type, missing event type appears frequently because of various reasons such as recording ignorance or resource limitation. In this study, we aim to propose an inverse probability weighted estimation that is commonly used in the missing data literature to correct possibly biased estimation by a complete-case analysis. This approach is not limited to a specific form of the recurrent event model. We derive the large sample theory in a general form. We demonstrate that our approach can be applied to either multiplicative or additive rates model with practical sample size via comprehensive simulations. Nonmucoid and mucoid Pseudomonas aeruginosa infections of 14888 patients in 2016 Cystic Fibrosis Foundation Patient Registry data are analyzed to show that, without including 12% events with missing event type in the analysis, several factors may be misidentified as risk factors for the nonmucoid type of infections.

Additive-Multiplicative Rates Model for Recurrent Event Data with Intermittently Observed Time-Dependent Covariates.

Regression methods, including the proportional rates model and additive rates model, have been proposed to evaluate the effect of covariates on the risk of recurrent events. These two models have different assumptions on the form of the covariate effects. A more flexible model, the additive-multiplicative rates model, is considered to allow the covariates to have both additive and multiplicative effects on the marginal rate of recurrent event process. However, its use is limited to the cases where the time-dependent covariates are monitored continuously throughout the follow-up time. In practice, time-dependent covariates are often only measured intermittently, which renders the current estimation method for the additive-multiplicative rates model inapplicable. In this paper, we propose a semiparametric estimator for the regression coefficients of the additive-multiplicative rates model to allow intermittently observed time-dependent covariates. We present the simulation results for the comparison between the proposed method and the simple methods, including last covariate carried forward and linear interpolation, and apply the proposed method to an epidemiologic study aiming to evaluate the effect of time-varying streptococcal infections on the risk of pharyngitis among school children. The R package implementing the proposed method is available at www.github.com/TianmengL/rectime.

A semiparametric additive rates model for the weighted composite endpoint of recurrent and terminal events.

Recurrent event data with a terminal event commonly arise in longitudinal follow-up studies. We use a weighted composite endpoint of all recurrent and terminal events to assess the overall effects of covariates on the two types of events. A semiparametric additive rates model is proposed to analyze the weighted composite event process and the dependence structure among recurrent and terminal events is left unspecified. An estimating equation approach is developed for inference, and the asymptotic properties of the resulting estimators are established. The finite-sample behavior of the proposed estimators is evaluated through simulation studies, and an application to a bladder cancer study is illustrated.

Probabilistic mean value theorems for conditioned random variables with applications

AbstractIn this paper, we propose a probabilistic analogue of the mean value theorem for conditional nonnegative random variables ordered in the hazard rate and reversed hazard rate order, upon conditioning on intervals of the form (t,∞) and [0,t]. This result is then specialized within the proportional hazards model and the proportional reversed hazards model with applications to series systems in reliability theory and to absorption random times of linear birth‐death processes. We also study the comparison of residual entropies and discuss some connections to Wasserstein and stop‐loss distances of random variables. A treatment for the additive hazard rate model is finally provided, with an application to life annuities.

Semiparametric additive rates model for recurrent events data with intermittent gaps.

Statistical methods for analyzing recurrent events have attracted significant attention. The majority of existing works consider situations in which subjects are observed over time periods and events of interest that occurred during the course of follow-up are recorded. In some applications, a subject may leave the study for a period of time and then resume due to various reasons. During the absence, which is referred to as an intermittent gap in this study, it may be impossible to observe a recording of the event. A naive analysis disregards gaps and considers events to be a typical recurrent event dataset. However, this may result in biased estimations and misleading results. In this study, we build an additive rates model for recurrent event data considering intermittent gaps. We provide the asymptotic theories behind the proposed model, as well as the goodness of fit between observed and modeled values. Simulation studies reveal that the estimations perform well if intermittent gaps are taken into account. In addition, we utilized the longitudinal cohort of elderly patients who have type 2 diabetes and at least one record of a severe recurrent complication, hypoglycemia, from the National Health Insurance Research Database in Taiwan to demonstrate the proposed method.

Modeling Creep Deformation and Damage Behavior of Tempered Martensitic Steel in the Framework of Additive Creep Rate Formulation

Additive creep rate model has been developed to predict creep strain-time behavior of materials important to engineering creep design of components for high temperature applications. The model has two additive formulations: the first one is related to sine hyperbolic rate equation describing primary and secondary creep deformation based on the evolution of internal stress with strain/time, and the second defines the tertiary creep rate as a function of tertiary creep strain. In order to describe creep data accurately, tertiary creep rate relation based on MPC-Omega methodology has been appropriately modified. The applicability of the model has been demonstrated for tempered martensitic plain 9Cr-1Mo steel for different applied stresses at 873 K. Based on the observations, a power law relationship between internal stress and applied stress has been established for the steel. Further, a higher creep damage accumulation with increasing life fraction has been observed at low stresses than those obtained at high stresses.

A semiparametric additive rate model for a modulated renewal process.

Recurrent event data from a long single realization are widely encountered in point process applications. Modeling and analyzing such data are different from those for independent and identical short sequences, and the development of statistical methods requires careful consideration of the underlying dependence structure of the long single sequence. In this paper, we propose a semiparametric additive rate model for a modulated renewal process, and develop an estimating equation approach for the model parameters. The asymptotic properties of the resulting estimators are established by applying the limit theory for stationary mixing sequences. A block-based bootstrap procedure is presented for the variance estimation. Simulation studies are conducted to assess the finite-sample performance of the proposed estimators. An application to a data set from a cardiovascular mortality study is provided.

An additive marginal regression model for clustered recurrent event in the presence of a terminal event

ABSTRACTIn this article, an additive rate model is proposed for clustered recurrent event with a terminal event. The subjects are clustered by some property. For the clustered subjects, the recurrent event is precluded by the death. An estimating equation is developed for the model parameter and the baseline rate function. The asymptotic properties of the resulting estimators are established. In addition, a goodness-of-fit test is presented to assess the adequacy of the model. The finite-sample behavior of the proposed estimators is evaluated through simulation studies, and an application to a bladder cancer data is illustrated.

Partial sufficient dimension reduction on additive rates model for recurrent event data with high-dimensional covariates

Recurrent event data with an additive marginal rates function have been extensively studied in the literature. The existing statistical inference, however, faces the difficulty with high-dimensional covariates due to “curse of dimensionality”. Examples include gene expression and single nucleotide polymorphism data which have revolutionized our understanding of disease such as cancer recurrence. In this paper, a technique of partial sufficient dimension reduction is applied to an additive rates model for recurrent event data. A two-step procedure is proposed to estimate parameters. First, partial sufficient dimension reduction is used to estimate the basis of the partial central subspace and the structural dimension. Then the second step estimates the baseline and the regression function of covariates based on the estimated partial central subspace using the average surface method. Simulation is performed to confirm and assess the theoretical findings, and an application on a set of chronic granulomatous disease data is demonstrated.

Analysis of a fixed center effect additive rates model for recurrent event data

A center effect additive rates model is suggested to analyze recurrent event data. The proposed model is a useful alternative to the center effect proportional rates model and provides a direct interpretation of parameters. The traditional estimation methods treat the centers as categorical variables, and they comprise many parameters when the number of centers is large and thus may not be feasible in many situations. An estimation method based on the difference in the observed to the expected number of recurrent events is recommended to address the deficiency of the traditional method. The asymptotic properties of the proposed estimator are established. We develop a goodness-of-fit test for model checking. Simulations are conducted to evaluate the small sample performance and show the computational advantage of the suggested method. The proposed methodology is applied to the Childhood Cancer Survivor Study.

Additive rate model with auxiliary covariates

In this paper, we consider an inference method for recurrent event data in which the primary exposure covariate is assessed only in a validation set, while as an auxiliary covariate for the main exposure is available for the full cohort. Additive rate model is considered. The existing estimating equations in the absence of primary exposure are corrected by taking use of the validation data and auxiliary information, which yield consistent and asymptotically normal estimators of the regression parameters. The estimated baseline mean process is shown to converge weakly to a zero-mean Gaussian process. Extensive simulations are conducted to evaluate finite sample performance.

MEAN RESIDUAL LIFE FUNCTION FOR ADDITIVE AND MULTIPLICATIVE HAZARD RATE MODELS

This paper deals with the mean residual life function (MRLF) and its monotonicity in the case of additive and multiplicative hazard rate models. It is shown that additive (multiplicative) hazard rate does not imply reduced (proportional) MRLF and vice versa. Necessary and sufficient conditions are obtained for the two models to hold simultaneously. In the case of non-monotonic failure rates, the location of the turning points of the MRLF is investigated in both the cases. The case of random additive and multiplicative hazard rate is also studied. The monotonicity of the mean residual life is studied along with the location of the turning points. Examples are provided to illustrate the results.

A semiparametric additive rates model for multivariate recurrent events with missing event categories

Multivariate recurrent event data arise in many clinical and observational studies, in which subjects may experience multiple types of recurrent events. In some applications, event times can be always observed, but types for some events may be missing. In this article, a semiparametric additive rates model is proposed for analyzing multivariate recurrent event data when event categories are missing at random. A weighted estimating equation approach is developed to estimate parameters of interest, and the resulting estimators are shown to be consistent and asymptotically normal. In addition, a lack-of-fit test is presented to assess the adequacy of the model. Simulation studies demonstrate that the proposed method performs well for practical settings. An application to a platelet transfusion reaction study is provided.

A competing risks approach to "biologic" interaction.

In epidemiology, the concepts of "biologic" and "statistical" interactions have been the subject of extensive debate. We present a new approach to biologic interaction based on Rothman's original (Am J Epidemiol, 104:587-592, 1976) discussion of sufficient causes. We do this in a probabilistic framework using competing risks and argue that sufficient cause interaction between two factors can be evaluated via the parameters in a particular statistical model, the additive hazard rate model. We present empirical conditions for presence of sufficient cause interaction and an example based on data from a liver cirrhosis trial illustrates the ideas.

Regression analysis of mixed recurrent-event and panel-count data with additive rate models.

Event-history studies of recurrent events are often conducted in fields such as demography, epidemiology, medicine, and social sciences (Cook and Lawless, 2007, The Statistical Analysis of Recurrent Events. New York: Springer-Verlag; Zhao et al., 2011, Test 20, 1-42). For such analysis, two types of data have been extensively investigated: recurrent-event data and panel-count data. However, in practice, one may face a third type of data, mixed recurrent-event and panel-count data or mixed event-history data. Such data occur if some study subjects are monitored or observed continuously and thus provide recurrent-event data, while the others are observed only at discrete times and hence give only panel-count data. A more general situation is that each subject is observed continuously over certain time periods but only at discrete times over other time periods. There exists little literature on the analysis of such mixed data except that published by Zhu et al. (2013, Statistics in Medicine 32, 1954-1963). In this article, we consider the regression analysis of mixed data using the additive rate model and develop some estimating equation-based approaches to estimate the regression parameters of interest. Both finite sample and asymptotic properties of the resulting estimators are established, and the numerical studies suggest that the proposed methodology works well for practical situations. The approach is applied to a Childhood Cancer Survivor Study that motivated this study.

Further Results on Dynamic Additive Hazard Rate Model

In the past, the proportional and additive hazard rate models have been investigated in the works. Nanda and Das (2011) introduced and studied the dynamic proportional (reversed) hazard rate model. In this paper we study the dynamic additive hazard rate model, and investigate its aging properties for different aging classes. The closure of the model under some stochastic orders has also been investigated. Some examples are also given to illustrate different aging properties and stochastic comparisons of the model.