Competing risks models in discrete time with nominal or ordinal categories of response

The operation of digital systems are based on time slotting and it finds many applications particularly in communication networks. It is important to analyze a digital network on some digital concepts. In this paper we analyze the network buffer through discrete time queueing system in which queue is discrete in nature and it is used to analyze the performance of the communication networks buffer. Discrete time models can be used to represents such type of systems in terms of slotted time units to models any physical system. More than one state changes in discrete time models from one time unit to another. Discrete time markov chains are used to describes the behavior of the network under the specific conditions and various performance measures are evaluated such as queue length, mean waiting time and mean number in the system.

Many important factors affect the spread of childhood infectious disease. To better understand the fundamental drivers of infectious disease spread, several researchers have estimated seasonal transmission coefficients in discrete-time models. In this paper, we build upon this previous work and also develop a framework for efficient estimation using continuous differential equation models. We introduce nonlinear programming formulations to efficiently estimate model parameters and seasonal transmission profiles from existing case count data for the childhood disease, measles. We compare results from discrete time and continuous time models and address several shortcomings of the discrete-time method, including removing the need for the data reporting interval to match the time between successive cases in the chain of transmission or serial interval of the disease. Using a simultaneous approach for optimization of differential equation systems, we demonstrate that seasonal transmission parameters can be effectively estimated using continuous time models instead of discrete-time models.

Predictive performance of the competing risk model in screening for preeclampsia

Analysis of incomplete data in presence of competing risks

American Option Pricing with Discrete and Continuous Time Models: An Empirical Comparison

A competing-risks model refers to a situation where a system (or organism) is exposed to two or more causes of failure (or death) but its eventual failure (or death) can be attributed to exactly one of the causes of failure. The basic information available in the competing-risks situation is the time to failure of the system, and the corresponding cause of failure. In practice, the causes of failure are often statistically dependent (the latent failure time of an individual failing from one cause of failure is statistically correlated with the latent failure time of the same individual failing from a different cause of failure). This paper provides a simple nonparametric hypothesis test (SNPHT) for comparing the cumulative incidence functions of a competing-risks model when two causes of failure are possibly statistically dependent. The test statistic is the weighted sum of the differences of two cumulative incidence functions at system failure times. This paper: (1) proves that the test statistic has asymptotic normal distributions under both null and alternative hypotheses; and (2) derives an explicit formula for the power function of SNPHT. The simulation study for SNPHT, based on the absolutely continuous bivariate exponential model, shows that the simulated powers and the approximated powers calculated from the formula are consistent for a moderate sample size. SNPHT is very easy to use. The illustrative example involves the failure of small electrical appliances.

The hypothesis of independence between the failure time and the cause of failure is studied by using the conditional probabilities of failure due to a specific cause given that there is no failure up to certain fixed time. In practice, there are situations when the failure times are available for all units but the causes of failures might be missing for some units. We propose tests based on U-statistics to test for independence of the failure time and the cause of failure in the competing risks model when all the causes of failure cannot be observed. The asymptotic distribution is normal in each case. Simulation studies look at power comparisons for the proposed tests for two families of distributions. The one-sided and the two-sided tests based on Kendall type statistic perform exceedingly well in detecting departures from independence.

This thesis is motivated by the limitations of the existing discrete time competing risks models vis-a-vis the treatment of data that comes with missing failure causes or a sizableproportions of cured subjects. The discrete time models that have been suggested to date (Davis and Lawrance, 1989; Tutz and Schmid, 2016; Ambrogi et al., 2009; Lee et al., 2018) are cause-specific-hazard denominated. Clearly, this fact summarily disqualifies these models from consideration if data comes with missing failure causes. It is also a well documented fact that naive application of the cause-specific-hazards to data that has a sizable proportion of cured subjects may produce downward biased estimates for these quantities. The existing models can be considered within the multiple imputation framework (Rubin, 1987) for handling missing failure causes, but the prospects of scaling them up for handling cured subjects are minimal, if not nil. In this thesis we address these issues concerning the treatment of missing failure causes and cured subjects in discrete time settings. Towards that end, we focus on the mixture model (Larson and Dinse, 1985) and the vertical model (Nicolaie et al., 2010) because these models possess certain properties which dovetail with the objectives of this thesis. The mixture model has been upgraded into a model that can handle cured subjects. Nicolaie et al. (2015) have demonstrated that the vertical model can also handle missing failure causes as is. Nicolaie et al. (2018) have also extended the vertical model to deal with cured subjects. Our strategy in this thesis is to exploit both the mixture model and the vertical model as a launching pad to advance discrete time models for handling data that comes with missing failure causes or cured subjects. Iqoqa. Isisusa salo mqingo wocwaningo silethwe ukugqoza kwezindlela ezibheka ubungozi bento kanye nendlela okubhekwa ngayo imininingo ekhuluma ngokwehluleka ukuhlonza imbangela yenkinga noma ukuqhamuka nekhambi lokwelapha lokho osekuyisisulu. Izindlela ezibheka isikhathi yizona ezisaphakanyiswayo kuze kube yimanje (Davis & Lawrance, 1989; Tutz & Schmid, 2016; Ambrogi et al., 2009; Lee et al., 2018) uma kubhekwa imbangela eqondene ngqo nengozi leyo. Kuyacaca ukuthi lezi zindlela ngeke zithathwe njengezisebenza kahle uma imininingo yocwaningo iveza ukuthi ziyahluleka ukuhlonza izimbangela zokungasebenzi kahle kokucwaningwayo. Kunobufakazi obusekwa wucwaningo futhi ukuthi ukusetshenziswa ngokungacopheleli izindlela ezibheka ngqo imbangela yenkinga ngemininingo yocwaningo eveza isibalo esikhulu sezisulu ezingalashwayo kuwukuhlehlela emuva kwezokwelapha ngoba kungaletha imiphumela echemile. Izindlela ezisebenzayo okwamanje zingahlolisiswa kusetshenziswa indlela exubile yokuhlola, imultiple imputation framework (Rubin, 1987) ukuhlonza izimbangela zokungasebenzi kahle kokucwaningwayo, kodwa amathuba okukhuphula ukusebenza kwazo ukulawula isibalo sokulashwa kwezisulu ezicwaningwayo mancane kakhulu, noma nje awekho. Kulo mqingo wocwaningo, sibheka lezi zinto nokuhluleka ukuhlonza izimbangela zokungasebenzi kahle kokucwaningwayo kanye nokwelashwa kwezisulu ngesikhathi esithize esibekiwe. Ekugcineni, sigxile ekusetshenzisweni kwendlela exubile (Nicolaie et al., 2010) ngoba le ndlela iyahambelana nezinhlosongqangi zalolu cwaningo. Indlela exubile ikwazile ukwenziwa kangcono ibe yindlela ekwaziyo ukulawula izisulu esezilaphekile. UNicolaie et al. (2015) bavezile ukuthi indlela ebheka phezulu iyakwazi ukulawula ukuhlonza izimbangela zokungasebenzi kahle kokucwaningwayo. UNicolaie et al. (2018) baphinde basebenzisa indlela eya phezulu ukubheka ukwelashwa kwezisulu. Isu lethu kulo mqingo wocwaningo ukusebenzisa kokubili, indlela yocwaningo exubile kanye nendlela eya phezulu njengezindlela ezikwazi ukubheka ukubaluleka kwesikhathi kanye nokucubungula imininingo mayelana nezimbangela zokungasebenzi kahle kokucwaningwayo.

Over the years, few attempts have been made to use survival analysis techniques in dental research. Moreover, the use of survival analysis was restricted to a simple method of drawing a survival curve using the Kaplan-Meier technique, but later prognostic factors were included utilizing the Cox Proportional Hazard (CPH) model. More recently, the CPH model has been incorporated with a frailty parameter to account for the correlation of observations within a subject. A recent publication has utilized a subdistribution hazard model with time-dependent covariates. In actual dental practice, tooth failure could be due to a variety of reasons other than the cause of failure of interest. To have a realistic estimate of the risk of failure for cause i, one should also include the risk of failure due to causes j, j = 1 , 2 , … n , j ≠ i , which could occur prior to cause i. This competing risk scenario has been attempted in the last ten to fifteen years or so to estimate the probability of failure in the presence of competing events. Precise estimate of survival (failure) probabilities using competing risk model has implications in the dental cost-effectiveness and cost-utility studies. Periodontal loss increases with age and higher in MM class III as compared to MM class II, a two-and-half times increase to a three-and-a-half times depending on the chosen model. The CPH model overestimated the hazard rate compared to the Fine-Grey subdistribution model. Simulation results support the robustness of parametric models and Fine-Grey subdistribution hazard model.

SUMMARY We consider the problem of estimating the regression coefficients in a competing risks model, where the relationship between the cause-specific hazard for the cause of interest and covariates is described using linear transformation models and when cause of failure is missing at random for a subset of individuals. Using the theory of Robins et al. (1994) for missing data problems and the approach of Chen et al. (2002) for estimating regression coefficients for linear transformation models, we derive augmented inverse probability weighted complete-case estimators for the regression coefficients that are doubly robust. Simulations demonstrate the relevance of the theory in finite samples.

Consider the competing risks model where the individuals are subjected to k failures. Tests for independence between time to failure and cause of failure are widely discussed in literature. We propose a quantile-based test for testing independence of failure time and cause of failure. A simulation study looks at the performance of the proposed test. The proposed test is applied to a few real data sets to illustrate its use.

Discrete competing risk model with application to modeling bus-motor failure data

We discuss various limits of a simple random exchange model that can be used for the distribution of wealth. We start from a discrete state space — discrete time version of this model and, under suitable scaling, we show its functional convergence to a continuous space — discrete time model. Then, we show a thermodynamic limit of the empirical distribution to the solution of a kinetic equation of Boltzmann type. We solve this equation and we show that the solutions coincide with the appropriate limits of the invariant measure for the Markov chain. In this way we complete Boltzmann’s program of deriving kinetic equations from random dynamics for this simple model. Three families of invariant measures for the mean field limit are discovered and we show that only two of those families can be obtained as limits of the discrete system while the third is extraneous.

The survival competing risks model in discrete time based on multinomial logistic regression, proposed by Luo et al. (2016), models the non-linear and irregular shape of hazard functions by incorporating a time-dependent spline into the multinomial logistic regression. This model also directly includes longitudinal variables in the regression. Due to the issues arising from including both baseline and longitudinal covariates in the extended form as proposed, and considering that the latter may be subject to error, this article suggests an extension of the existing model. The proposed extension utilizes the concept of joint models for longitudinal and survival data, which is an effective approach for integrating simultaneousness both baseline and time-dependent covariates into the survival model.,

For several specific queueing models with a vacation policy, the stationary system occupancy at the beginning of a random slot is distributed as the sum of two independent random variables. One of these variables is the stationary number of customers in an equivalent queueing system with no vacations. For models in continuous time with Poissonian arrivals, this result is well-known, and referred to as stochastic decomposition, with proof provided by Fuhrmann and Cooper. For models in discrete time, this result received less attention, with no proof available to date. In this paper, we first establish a proof of the decomposition result in discrete time. When compared to the proof in continuous time, conditions for the proof in discrete time are somewhat more general. Second, we explore four different examples: non-preemptive priority systems, slot-bound priority systems, polling systems, and fiber delay line (FDL) buffer systems. The first two examples are known results from literature that are given here as an illustration. The third is a new example, and the last one (FDL buffer systems) shows new results. It is shown that in some cases the queueing analysis can be considerably simplified using this decomposition property.