Models for residual time to AIDS

The distributions of the time from Human Immunodeficiency Virus (HIV) infection to the onset of Acquired Immune Deficiency Syndrome (AIDS) and of the residual time to AIDS diagnosis are important for modeling the growth of the AIDS epidemic and for predicting onset of the disease in an individual. Markers such as CD4 counts carry valuable information about disease progression and therefore about the two survival distributions. Building on the framework set out by Jewell and Kalbfleisch (1992), we study these two survival distributions based on stochastic models for the marker process (X(t)) and a marker-dependent hazard (h()). We examine various plausible CD4 marker processes and marker-dependent hazard functions for AIDS proposed in recent literature. For a random effects plus Brownian motion marker process X(t)=(a+bt+BM(t))⁴, where a has a normal distribution, b<0 is an unknown parameter and BM(t) is Brownian motion, and marker-dependent hazard h(X(t)), we prove that, given CD4 cell count X(t), the residual time to AIDS distribution does not depend on the time since infection t. Using simulation and numerical integration, we find the marginal incubation period distribution, the marginal hazard and the residual time distribution for several combinations of marker processes and marker-dependent hazards. An example using data from the Multicenter AIDS Cohort Study is given. A simple regression model relating the cube root of residual time to AIDS to CD4 count is suggested.     

On computing estimates of a change-point in the Weibull regression hazard model

The hazard function describes the instantaneous rate of failure at a time t, given that the individual survives up to t. In applications, the effect of covariates produce changes in the hazard function. When dealing with survival analysis, it is of interest to identify where a change point in time has occurred. In this work, covariates and censored variables are considered in order to estimate a change-point in the Weibull regression hazard model, which is a generalization of the exponential model. For this more general model, it is possible to obtain maximum likelihood estimators for the change-point and for the parameters involved. A Monte Carlo simulation study shows that indeed, it is possible to implement this model in practice. An application with clinical trial data coming from a treatment of chronic granulomatous disease is also included.     

Mean Time to Failure of Weighted k-out-of-n: G Systems

The purpose of this article is to develop a Monte-Carlo simulation algorithm for computing mean time to failure (MTTF) of weighted-k-out-of-n:G and linear consecutive-weighted-k-out-of-n:G systems. Our algorithm is based on the use of appropriately defined stochastic process which represents the total weight of the system at time t. These stochastic processes are explicitly defined and used along with the ordered component lifetimes to simulate MTTF of the systems with weighted components.     

A Statistical Issue About a Phase III Multi-regional Trial for the Survival Data under Accelerated Failure Time Model with Unrecognized Heterogeneity

For the time-to-event outcome, current methods for sample size determination are based on the proportional hazard model. However, if the proportionality assumption fails to capture the relationship between the hazard time and covariates, the proportional hazard model is not suitable to analyze survival data. The accelerated failure time model is an alternative method to deal with survival data. In this article, we address the issue that the relationship between the hazard time and the treatment effect is satisfied with the accelerated failure time model to design a multi-regional trial for a phase III clinical trial. The log-rank test is employed to deal with the heterogeneous effect size among regions. The test statistic for the overall treatment effect is used to determine the total sample size for a multi-regional trial and the consistent trend is used to rationalize partition sample size to each region.     

Modeling Multilevel Survival Data Using Frailty Models

Often the dependence in multivariate survival data is modeled through an individual level effect called the frailty. Due to its mathematical simplicity, the gamma distribution is often used as the frailty distribution for hazard modeling. However, it is well known that the gamma frailty distribution has many drawbacks. For example, it weakens the effect of covariates. In addition, in the presence of a multilevel model, overall frailty comes from several levels. To overcome such drawbacks, more heavy-tailed distributions are needed to model the frailty distribution in order to incorporate extra variability. In this article, we develop a class of log-skew-t distributions for the frailty. This class includes the log-normal distribution along with many other heavy tailed distributions, e.g., log-Cauchy, log normal, and log-t as special cases.

Conditional on the frailty, the survival times are assumed to be independent with proportional hazard structure. The modeling process is then completed by assuming multilevel frailty-effects. Instead of tuning a strict parameterization of the baseline hazard function, we consider the partial likelihood approach and thus leave the baseline function unspecified. By eliminating the hazard, the pre-specification and computation are simplified considerably.     

Estimation for semi-Markov models with partial observations via self-consistency equations

Nonparametric estimators of the survival function S(t) = P(T ≥ t) for a partially observed time variable T have been defined by several methods, in particular, by integral self-consistency equations. The author establishes explicit expressions of the estimators in an additive form and extend this approach to several cases: a left-truncated and right-censored variable and the left-censored or left-truncated sojourn times of a right-censored semi-Markov process. These estimators are always identical to the product-limit estimators if hazard functions may be defined.     

Detecting change in a hazard regression model with right-censoring

The hazard function plays an important role in survival analysis and reliability, since it quantifies the instantaneous failure rate of an individual at a given time point t, given that this individual has not failed before t. In some applications, abrupt changes in the hazard function are observed, and it is of interest to detect the location of such a change. In this paper, we consider testing of existence of a change in the parameters of an exponential regression model, based on a sample of right-censored survival times and the corresponding covariates. Likelihood ratio type tests are proposed and non-asymptotic bounds for the type II error probability are obtained. When the tests lead to acceptance of a change, estimators for the location of the change are proposed. Non-asymptotic upper bounds of the underestimation and overestimation probabilities are obtained. A short simulation study illustrates these results.     

Graphical representation of survival curves in the presence of time-dependent categorical covariates with application to liver transplantation

Graphical representation of survival curves is often used to illustrate associations between exposures and time-to-event outcomes. However, when exposures are time-dependent, calculation of survival probabilities is not straightforward. Our aim was to develop a method to estimate time-dependent survival probabilities and represent them graphically. Cox models with time-dependent indicators to represent state changes were fitted, and survival probabilities were plotted using pre-specified times of state changes. Time-varying hazard ratios for the state change were also explored. The method was applied to data from the Adult-to-Adult Living Donor Liver Transplantation Cohort Study (A2ALL). Survival curves showing a ‘split’ at a pre-specified time t allow for the qualitative comparison of survival probabilities between patients with similar baseline covariates who do and do not experience a state change at time t. Time since state change interactions can be visually represented to reflect changing hazard ratios over time. A2ALL study results showed differences in survival probabilities among those who did not receive a transplant, received a living donor transplant, and received a deceased donor transplant. These graphical representations of survival curves with time-dependent indicators improve upon previous methods and allow for clinically meaningful interpretation.     

A model for markers and latent health status

We extend the bivariate Wiener process considered by Whitmore and co-workers and model the joint process of a marker and health status. The health status process is assumed to be latent or unobservable. The time to reach the primary end point or failure (death, onset of disease, etc.) is the time when the latent health status process first crosses a failure threshold level. Inferences for the model are based on two kinds of data: censored survival data and marker measurements. Covariates, such as treatment variables, risk factors and base-line conditions, are related to the model parameters through generalized linear regression functions. The model offers a much richer potential for the study of treatment efficacy than do conventional models. Treatment effects can be assessed in terms of their influence on both the failure threshold and the health status process parameters. We derive an explicit formula for the prediction of residual failure times given the current marker level. Also we discuss model validation. This model does not require the proportional hazards assumption and hence can be widely used. To demonstrate the usefulness of the model, we apply the methods in analysing data from the protocol 116a of the AIDS Clinical Trials Group.     

Forecasting Vector ARMA Processes With Systematically Missing Observations

The following two predictors are compared for time series with systematically missing observations: (a) A time series model is fitted to the full series X_t , and forecasts are based on this model, (b) A time series model is fitted to the series with systematically missing observations Y _τ, and forecasts are based on the resulting model. If the data generation processes are known vector autoregressive moving average (ARMA) processes, the first predictor is at least as efficient as the second one in a mean squared error sense. Conditions are given for the two predictors to be identical. If only the ARMA orders of the generation processes are known and the coefficients are estimated, or if the process orders and coefficients are estimated, the first predictor is again, in general, superior. There are, however, exceptions in which the second predictor, using seemingly less information, may be better. These results are discussed, using both asymptotic theory and small sample simulations. Some economic time series are used as illustrative examples.     

Graphical Tests for the Assumption of Gamma and Inverse Gaussian Frailty Distributions

The common choices of frailty distribution in lifetime data models include the Gamma and Inverse Gaussian distributions. We present diagnostic plots for these distributions when frailty operates in a proportional hazards framework. Firstly, we present plots based on the form of the unconditional survival function when the baseline hazard is assumed to be Weibull. Secondly, we base a plot on a closure property that applies for any baseline hazard, namely, that the frailty distribution among survivors at time t has the same form as the original distribution, with the same shape parameter but different scale parameter. We estimate the shape parameter at different values of t and examine whether it is constant, that is, whether plotted values form a straight line parallel to the time axis. We provide simulation results assuming Weibull baseline hazard and an example to illustrate the methods.     

OPTIMAL DESIGNS FOR FITTING A PROPORTIONAL HAZARDS REGRESSION MODEL TO DATA SUBJECT TO CENSORING

Suppose the probability model for failure time data, subject to censoring, is specified by the hazard function λ(t)exp(β_T x), where x is a vector of covariates. Analytical difficulties involved in finding the optimal design are avoided by assuming that λ is completely specified and by using D-optimality based on the information matrix for β Optimal designs are found to depend on β, but some results of practical consequence are obtained. It is found that censoring does not affect the choice of design appreciably when β^T x ≥ 0 for all points of the feasible region, but may have an appreciable effect when β_ix_i 0, for all i and all points in the feasible experimental region. The nature of the effect is discussed in detail for the cases of one and two parameters. It is argued that in practical biomedical situations the optimal design is almost always the same as for uncensored data.     

Bayesian one-sample prediction of future observations under Pareto distribution

Suppose that the length of time in years for which a business operates until failure has a Pareto distribution. Let t ₁?t ₂?t _r denote the survival lifetimes of the first r of a random sample of n businesses. Bayesian predictions are to be made on the ordered failure times of the remaining (n???r) businesses, using the conditional probability function. Numerical examples are given to illustrate our results.     

Estimation of Survival Function Subject to Time-Dependent Covariates and Left Truncation

Abstract

Satten et al. [Satten, G. A., Datta, S., Robins, J. M. (2001). Estimating the marginal survival function in the presence of time dependent covariates. Statis. Prob. Lett. 54: 397--403] proposed an estimator [denoted by ?(t)] of survival function of failure times that is in the class of survival function estimators proposed by Robins [Robins, J. M. (1993). Information recovery and bias adjustment in proportional hazards regression analysis of randomized trials using surrogate markers. In: Proceedings of the American Statistical Association-Biopharmaceutical Section. Alexandria, VA: ASA, pp. 24--33]. The estimator is appropriate when data are subject to dependent censoring. In this article, it is demonstrated that the estimator ?(t) can be extended to estimate the survival function when data are subject to dependent censoring and left truncation. In addition, we propose an alternative estimator of survival function [denoted by ? _w (t)] that is represented as an inverse-probability-weighted average Satten and Datta [Satten, G. A., Datta, S. (2001). The Kaplan–Meier estimator as an inverse-probability-of-censoring weighted average. Amer. Statist. Ass. 55: 207--210]. Simulation results show that when truncation is not severe the mean squared error of ?(t) is smaller than that of ? _w (t), except for the case when censoring is light. However, when truncation is severe, ? _w (t) has the advantage of less bias and the situation can be reversed.     

The analysis of incomplete data using stochastic covariates

In the development of many diseases there are often associated random variables which continuously reflect the progress of a subject towards the final expression of the disease (failure). At any given time these processes, which we call stochastic covariates, may provide information about the current hazard and the remaining time to failure. Likewise, in situations when the specific times of key prior events are not known, such as the time of onset of an occult tumour or the time of infection with HIV-1, it may be possible to identify a stochastic covariate which reveals, indirectly, when the event of interest occurred. The analysis of carcinogenicity trials which involve occult tumours is usually based on the time of death or sacrifice and an indicator of tumour presence for each animal in the experiment. However, the size of an occult tumour observed at the endpoint represents data concerning tumour development which may convey additional information concerning both the tumour incidence rate and the rate of death to which tumour-bearing animals are subject. We develop a stochastic model for tumour growth and suggest different ways in which the effect of this growth on the hazard of failure might be modelled. Using a combined model for tumour growth and additive competing risks of death, we show that if this tumour size information is used, assumptions concerning tumour lethality, the context of observation or multiple sacrifice times are no longer necessary in order to estimate the tumour incidence rate. Parametric estimation based on the method of maximum likelihood is outlined and is applied to simulated data from the combined model. The results of this limited study confirm that use of the stochastic covariate tumour size results in more precise estimation of the incidence rate for occult tumours.     

Strong approximations for time-varying infinite-server queues with non-renewal arrival and service processes

In real stochastic systems, the arrival and service processes may not be renewal processes. For example, in many telecommunication systems such as internet traffic where data traffic is bursty, the sequence of inter-arrival times and service times are often correlated and dependent. One way to model this non-renewal behavior is to use Markovian Arrival Processes (MAPs) and Markovian Service Processes (MSPs). MAPs and MSPs allow for inter-arrival and service times to be dependent, while providing the analytical tractability of simple Markov processes. To this end, we prove fluid and diffusion limits for MAP_t/MSP_t/∞ queues by constructing a new Poisson process representation for the queueing dynamics and leveraging strong approximations for Poisson processes. As a result, the fluid and diffusion limit theorems illuminate how the dependence structure of the arrival or service processes can affect the sample path behavior of the queueing process. Finally, our Poisson representation for MAPs and MSPs is useful for simulation purposes and may be of independent interest.     

Using categorical markers as auxiliary variables in log‐rank tests and hazard ratio estimation

Markers, which are prognostic longitudinal variables, can be used to replace some of the information lost due to right censoring. They may also be used to remove or reduce bias due to informative censoring. In this paper, the authors propose novel methods for using markers to increase the efficiency of log‐rank tests and hazard ratio estimation, as well as parametric estimation. They propose a «plug‐in» methodology that consists of writing the test statistic or estimate of interest as a functional of Kaplan–Meier estimators. The latter are then replaced by an efficient estimator of the survival curve that incorporates information from markers. Using simulations, the authors show that the resulting estimators and tests can be up to 30% more efficient than the usual procedures, provided that the marker is highly prognostic and that the frequency of censoring is high.     

The covariance of the number of renewals in a fixed time and the ensuing excess life

The covariance of the number of renewals in a fixed time N ^t and the ensuing excess life time Y ^t is derived using matrix-analytic methods for the stationary PH-renewal process. Specific results for the Erlang and hyperexponential processes are provided to illustrate the ease of computation. Properties concerning the sign and the behavior of the covariance as t→∞ are provided throughout. Parameter estimation for renewal processes which cannot be fully observed serves as the motivation for our derivations. These statistical applications as well as links to estimation for service time distributions in queues shed light on the type of problems for which the covariance is useful.     

Simple bounds for terminating Poisson and renewal shock processes

A system subject to a point process of shocks is considered. The shocks occur in accordance with a renewal process or a nonhomogeneous Poisson process. Each shock independently of the previous history leads to a system failure with probability θ and is survived with a complimentary probability $\text{θ}\text{̄}$. A number of problems in reliability and safety analysis can be interpreted by means of this model. The exact solution for the probability of survival $\text{W}\text{̄}\text{(t,θ)}$ can be obtained only in the form of infinite series (renewal process of shocks). Approximate solutions and new simple bounds for the probability of survival are obtained. The introduced method is based on the notion of a stochastic hazard rate process. A supplementary characteristic in this analysis is the mean of the hazard rate process. This method makes it possible to consider a generalization important in practical applications when the probability of a system failure under the effect of a current shock depends on the time since the previous one.     

Robust multivariate mixture regression models with incomplete data

Multivariate mixture regression models can be used to investigate the relationships between two or more response variables and a set of predictor variables by taking into consideration unobserved population heterogeneity. It is common to take multivariate normal distributions as mixing components, but this mixing model is sensitive to heavy-tailed errors and outliers. Although normal mixture models can approximate any distribution in principle, the number of components needed to account for heavy-tailed distributions can be very large. Mixture regression models based on the multivariate t distributions can be considered as a robust alternative approach. Missing data are inevitable in many situations and parameter estimates could be biased if the missing values are not handled properly. In this paper, we propose a multivariate t mixture regression model with missing information to model heterogeneity in regression function in the presence of outliers and missing values. Along with the robust parameter estimation, our proposed method can be used for (i) visualization of the partial correlation between response variables across latent classes and heterogeneous regressions, and (ii) outlier detection and robust clustering even under the presence of missing values. We also propose a multivariate t mixture regression model using MM-estimation with missing information that is robust to high-leverage outliers. The proposed methodologies are illustrated through simulation studies and real data analysis.