Article

Exponential Survival with Covariance

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Abstract

In applied problems one often wishes to compare the mortality experience of two or more groups of individuals who are known not to be comparable with respect to a covariable. This paper presents an approach to this problem by assuming that the force of mortality for each individual is a function of the covariable. Extension to the case where more than one covariable is present is indicated. It is also suggested that the present method is adaptable to an actuarial type analysis.

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... (ii) To correct for lack of comparability with respect to the concomitant variables among groups whose survival is being compared (Feigl and Zelen, 1965;Glasser, 1967). ...
... Let z be the observed value of the random variable corresponding to white blood cell (WBC) count, and 6 0 be the parameter for effect due to membership in a specific population. Concurrently Glasser (1967) suggested an exponential model to adjust censored survival data following surgery for lung cancer in two populations which differed with respect to their age distributions. ...
... The dat~are some preliminary results from a clinical trial reported byGlasser (1967) of survival following surgery for lung cancer. The two populations are. ...
... A snowball sampling approach was also utilized. Snowball sampling is a non-probability sampling technique used to discover uncommon characteristics in samples (Glasser, 1967). Because it was difficult to get participants on the phenomena of interest, a snowball sampling technique was employed (see Table 2). ...
... Similarly, all interviews were examined on line by line basis. Breaking down the data into numerous separate concepts, incidences and occurrences is part of open coding (Glasser, 1967). The analysts then gave these incidents/ideas/etc. a name or a code to symbolize them. ...
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Purpose – This paper explores the steps/countermeasures taken by firms to address supply chain disruptions in the wake of COVID-19. Design/methodology/approach – This study employs a case study methodology and employs 46 semi-structured interviews with senior managers of the three buying firms, four distribution centres and four supplying firms based in four countries (Pakistan, Sri Lanka, China and India). Findings – Results reveal that manufacturers are refining production schedules to meet the production challenges. Distributors are working with secondary suppliers to meet the inventory shortage. Finally, supplying firms are evaluating the impact of demand, focusing on short-term demand-supply strategy, preparing for channel shifts, opening up additional channels of communication with key customers, understanding immediate customer’s demand and priorities and finally becoming more agile. Research limitations/implications – There are some limitations to this study. First, the results of this study cannot be generalized to a wider population. Second, this study explores the interpretations of senior managers based in four Asian countries only. Practical implications – Supply chain firms can use these findings to understand how COVID-19 is affecting firms. Firms can also use the suggestions provided in this study to mitigate the impact of COVID-19 and make the best out of this pandemic. Originality/value – This study contributes to the supply chain disruption literature by exploring the robust countermeasure taken by supply chain firms amid COVID-19 outbreak. In particular, it explores such countermeasures from the perspective of three different entities (buyer, supplier and distributor) based in four different countries in the South Asian region.
... Basu and Mawaziny [3] studied a system with m independent components and found that the parameter estimators perform well when the component lifetimes are identically distributed. Arasan and Daud [1] studied log-linear-exponential regression model of Glasser [9] incorporated in a parallel system model with two covariates and Type II censoring and found that the parallel model with two covariates work well, especially when the data has low censoring proportion, high number of components in the system and large sample size. ...
... Feigl and Zelen [8] proposed that concomitant variates are important as within a treatment group, the patients will vary in disease status hence respond differently to treatment. Glasser [9] proposed that covariates should be incorporated in the model to get a better model, and it can be extended by adding additional covariable between groups with large differences. Breslow [4] claimed that incorporation of covariance is important for prognosis and that the adjustment for covariance had a marked effect on the treatment comparison. ...
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This research aims to model the lifetime of parallel components system with covariates, right, and interval censored data. The lifetimes of the components are assumed to follow the exponential distribution, with constant failure rates. A simulation study is conducted to assess the performance of the maximum likelihood estimates, without and with midpoint imputation method at various sample sizes, censoring proportions, and number of components in the system. The combination which produces the best parameter estimates is then identified by comparing the bias, standard error and root mean square error of these estimates. The simulation results indicate that the midpoint imputation method produces more efficient and accurate parameter estimates with smaller bias, standard error and root mean square error. Also, in general, better estimates are obtained at low censoring levels, large sample sizes, and a high number of parallel components in the system. The proposed model is then fitted to a modified real data of diabetic retinopathy patients. Following that, the non-parametric log-rank test and Wald hypothesis test are carried out to check the significance of the covariate, age in the model. The results show that the model fits the data rather well and the age of patients has no significant effect on the survival time of the patients’ eyes.
... An alternative representation of the hazard that remains non-negative regardless of parameter values is given by which has frequently been called a log linear model. In effect, this model adjusts the hazard rate by a multiplicative factor which is a function of the covariates z where:' = (zl'zZ""'zp)' Glasser (1967) average of all the ij individual ages. There were two groups considered. ...
... were adjusted so that they were treated as occurring at t i . This model was compared to the linear exponential model of Feigl and Zelen (1965) and the log linear exponential model of Glasser (1967) for the comparison of survival curves related to a clinical trial of maintenance therapy for childhood leukemia. Recently, Holford (1976) has considered a model similar to that presented by Breslow (1974), where the under- Other authors have investigated topics related to linear and log linear exponential models. ...
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Methods introduced by Lagakos for incorporating infonnation from a time-dependent covariable (an intervening event) into the analysis of failure times are generalized. The research was motivated by two follow-up studies--one involving industrial workers where dis-I ability retirement is the intervening event and the other, patients with coronary artery disease where the first non-fatal myocardial infarctiOn after diagnosis for the disease is the intervening event. The generalized model
... In this case, y has an extreme value distribution with PDF f ðyÞ ¼ exp ðy À lÞ exp fÀ exp ðy À lÞg, where y 2 R and l ¼ log ðhÞ 2 R: This regression model is one of the most important parametric regression models for the analysis of lifetime data and it is employed in several applications. It has been studied by many researchers such as Feigl and Zelen (1965), Glasser (1967), Cox and Snell (1968), Prentice (1973), Lawless (1976), Ahn (1994) and Colosimo (1997, 1999), among others. ...
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This paper deals with the issue of testing hypotheses in the censored exponential regression model in small and moderate-sized samples. We focus on four tests, namely the Wald, likelihood ratio, score, and gradient tests. These tests rely on asymptotic results and are unreliable when the sample size is not large enough to guarantee a good agreement between the exact distribution of the test statistic under a null hypothesis and the corresponding reference chi-squared asymptotic distribution. Bartlett and Bartlett-type corrections typically attenuate the size distortion of the tests. These corrections are available in the literature for the likelihood ratio and score tests in the class of censored exponential regression models. A Bartlett-type correction for the gradient test is derived in this paper in this class of models. Additionally, we also propose bootstrap-based inferential improvements to the four tests mentioned. We numerically compare the tests through extensive Monte Carlo simulation experiments. The numerical results reveal that the corrected and bootstrapped tests exhibit type I error probability closer to the chosen nominal level with virtually no power loss. We also present an empirical application for illustrative purposes.
... 13 Klein et al. 1990Piantadosi, Crowley 1995;Zelterman, Curtsinger 1995;Mudholkar et al. 1995;Rossa 2002. 14 Feigl, Zelen 1965Zippin, Armitage 1966;Glasser 1967. 15 Aalen 1989Huffer, McKeague 1991;Lin, Ying 1994;Kim, Lee 1998. ...
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The monograph introduces non-parametric methods of unbiased estimation for right-censored data.
... Parametric models are obtained when a functional form is specified for the standard hazard. A constant hazard leads to the exponential model proposed by Glasser (1967) and further studied by Prentice (1973) and Breslow(1974), .but many other functional forms have been proposed in the literature, see e.g. ...
... While there are a plethora of methods for estimation of the hazard rate (for examples, see [30,32,31]), a simple and convenient method for describing the time-varying pattern is with a piecewise constant function. In general theory, this is known as the piecewise exponential (PE) hazards model [19,47,29,28,33], which easily accommodates time-varying covariates (such as monthly weather measurements) as well as a changing number of subjects at risk (such as the increase in the population of Navrongo between 1998 and 2008). We are interested in examining how this model (represented through the Poisson generalized linear model [43]) compares to the modern, Bayesian MRH model, which can also be expressed as a PE model but has a prior structure that lends itself to estimation in the instance of small observed failures. ...
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The "meningitis belt" is a region in sub-Saharan Africa where annual outbreaks of meningitis occur, with large epidemics observed cyclically. While we know that meningitis is heavily dependent on seasonal trends (in particular, weather), the exact pathways for contracting the disease are not fully understood and warrant further investigation. This manuscript examines meningitis trends in the context of survival analysis, quantifying underlying seasonal patterns in meningitis rates through the hazard rate for the population of Navrongo, Ghana. We compare three candidate models: the commonly used Poisson generalized linear model, the Bayesian multi-resolution hazard model, and the Poisson generalized additive model. We compare the accuracy and robustness of the models through the bias, RMSE, and the standard deviation. We provide a detailed case study of meningitis patterns for data collected in Navrongo, Ghana.
... En efecto, supongamos que para cada individuo j, además del tiempo de sobrevivencia, se tiene también información sobre p covariables zj = ( z j j, z Pj). En este caso, un modelo en donde los riesgos h(t) dependan de las covariables z puede formularse como (Glasser, 1967): ...
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Introducción La mayor parte de los eventos de que se ocupa la demografía pueden ser vistos -y de hecho, es una dimensión que siempre subyace en el análisis-como riesgos o probabilidades en competencia. El caso que a modo ilustrativo hemos decidido analizar aquí, la legalización de uniones consensúales, es un buen ejemplo de esto: una unión con-sensual puede terminar como tal, primero, por alguna forma de lega-lización o, segundo, por alguna forma de separación. Entre las legali-zaciones podemos considerar las diversas combinaciones de formas de matrimonio, y entre las separaciones debemos considerar la viudez o la disolución de la unión. Puesto que todas estas opciones son excluyen-tes se dice que están "en competencia". Así, para estimar las probabilida-des de legalización (libres del efecto perturbador que introduce, por ejemplo, la viudez) es necesario considerar el fenómeno como ex-puesto a dos procesos de decremento. Otros fenómenos demográfi-cos como la mortalidad vista por causas suponen la acción de múlti-ples procesos (independientes) de decremento. * Coordinador nacional del Programa de Educación, Salud y Alimentación (Pro-gresa).
... (1989) e Lindsey (1997), o modelo complementar log-log para ensaios de diluição (Fisher, 1922), os modelos probit (Bliss, 1935) e logit (Berkson, 1944;Dyke and Patterson, 1952;Rasch, 1960) para proporções, os modelos log-lineares para dados de contagens (Birch, 1963), os modelos de regressão para análise de sobrevivência (Feigl and Zelen, 1965;Zippin and Armitage, 1966;Glasser, 1967). ...
... (g) os modelos de testes de vida, envolvendo a distribuição exponencial (Feigl e Zelen, 1965;Zippin e Armitage, 1966;Gasser, 1967); ...
... -regressão linear múltipla, envolvendo distribuição normal (Legendre, Gauss, início do século XIX); -análise de variância para experimentos planejados, envolvendo distribuição normal (Fisher, 1920(Fisher, a 1935; -função de verossimilhança, um procedimento geral para inferência a respeito de qualquer modelo estatístico (Fisher, 1922); -modelo complemento log-log para ensaios de diluição, envolvendo distribuição binomial (Fisher, 1922); -família exponencial, uma classe de distribuições com propriedades ótimas (estatísticas suficientes) para a estimação dos parâmetros (Fisher, 1934); -modelo probit para proporções, envolvendo distribuição binomial (Bliss, 1935); -modelo logístico para proporções, envolvendo distribuição binomial (Berkson, 1944;Dyke & Patterson, 1952); -modelo logístico para análise de itens, envolvendo distribuição Bernoulli (Rasch, 1960); -modelos log-lineares para contagens, envolvendo distribuição Poisson e multinomial (Birch, 1963); -modelos de regressão para dados de sobrevivência, envolvendo distribuição exponencial (Feigl & Zelen, 1965;Zippin & Armitage, 1966;Gasser, 1967); -polinômios inversos para ensaios de adubação, envolvendo distribuição gama (Nelder, 1966). Nelder & Wedderburn (1972) mostraram, então, que a maioria dos problemas estatísticos, que surgem nas áreas de agricultura, demografia, ecologia, economia, geografia, geologia, história, medicina, ciência política, psicologia, sociologia, zootecnia etc, podem ser formulados, de uma maneira unificada, como modelos de regressão. ...
... • modelos de regressão para dados de sobrevivência -distribuição exponencial com ligação recíproca ou logarítmica, 1 (Feigl & Zelen, 1965, Zippin & Armitage, 1966, Gasser, 1967; ...
... When a vector of covariates, X, is present, the exponential model may be extended by assuming an underlying constant hazard rate in each group, with the proportional hazards model assumed for the effect of X, so that AJX) = Ajexp(,BX) (Glasser, 1967). Employing ...
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We discuss the optimal allocation problem in a multi-level stress test with Type-II censoring and Weibull (extreme value) regression model. We derive the maximum-likelihood estimators and their asymptotic variance–covariance matrix through the Fisher information. Four optimality criteria are used to discuss the optimal allocation problem. Optimal allocation of units, both exactly for small sample sizes and asymptotically for large sample sizes, for two- and four-stress-level situations are determined numerically. Conclusions and discussions are provided based on the numerical studies.
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The Poisson regression model for the analysis of life table and follow-up data with covariates is presented. An example is presented to show how this technique can be used to construct a parsimonious model which describes a set of survival data. All parameters in the model, the hazard and survival functions are estimated by maximum likelihood.
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The variance of the estimated hazard ratio between two groups when there is one categorized continuous gamma-distributed covariate is derived using exponential and Weibull regression models and asymptotic theory. Categorizing a continuous covariate increases the variance of the estimated hazard ratio and decreases the efficiency of the analysis. The efficiency of categorization is studied as a function of the strength of the relation between survival time and the covariate, the choice of cut points used in categorizing, and the number of categories. An application of the results to an advanced lung cancer clinical trial is given.
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Assuming the time-dependent component of Cox's form of the hazard function to be specified by the constant, Weibull, and Gompertz, produces three failure density models. Using simulation techniques, size and power comparisons between tests of hypotheses (concerning the regression parameters) arising from Cox's nonparametric method and tests arising from the three parametric models are made. The test statistics arise from the likelihood ratio criterion and the asymptotic normality property of maximum likelihood estimators.
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In the present work, we find a set of reliability functionals to fix up an allocation strategy among K(≥2) treatments when the response distributions, conditionally dependent on some continuous prognostic variable, are exponential with unknown linear regression functions as the means of the respective conditional distributions. Targeting such reliability functionals, we propose a covariate-adjusted response-adaptive randomization procedure for the multi-treatment single-period clinical trial under the Koziol–Green model for informative censoring. We compare the proposed procedure with its competitive covariate-eliminated procedure.
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Gaddum, the author of the first of the four articles included in this study, points out that logarithmic transmutations of data or of some function of the raw data may be normally distributed and homoscedastic in cases where the original data were not normally distributed or variances of arrays were not equal. He therefore describes and advocates the use of such transformation (which he calls lognormal distributions) to increase the accuracy and scope of the conclusions that may be drawn from experimental observations. Allen elaborates the method in applications to geological data. Pearce investigated empirically the properties of two types of logarithmic transformation [$X = log(2x + 1)$ and $X = log(x + 1)$] and found that the transformations did allow the assumption of homoscedasticity (as Gaddum suggested they would) even though the raw data did not. The original author emphasizes again in the last article that: "The choice of the appropriate technique for any given problem must always be based, if possible, on direct evidence of its suitability."
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In lifetesting, medical follow-up, and other fields the observation of the time of occurrence of the event of interest (called a death) may be prevented for some of the items of the sample by the previous occurrence of some other event (called a loss). Losses may be either accidental or controlled, the latter resulting from a decision to terminate certain observations. In either case it is usually assumed in this paper that the lifetime (age at death) is independent of the potential loss time; in practice this assumption deserves careful scrutiny. Despite the resulting incompleteness of the data, it is desired to estimate the proportion P(t) of items in the population whose lifetimes would exceed t (in the absence of such losses), without making any assumption about the form of the function P(t). The observation for each item of a suitable initial event, marking the beginning of its lifetime, is presupposed. For random samples of size N the product-limit (PL) estimate can be defined as follows: List and label the N observed lifetimes (whether to death or loss) in order of increasing magnitude, so that one has \(0 \leqslant t_1^\prime \leqslant t_2^\prime \leqslant \cdots \leqslant t_N^\prime .\) Then \(\hat P\left( t \right) = \Pi r\left[ {\left( {N - r} \right)/\left( {N - r + 1} \right)} \right]\), where r assumes those values for which \(t_r^\prime \leqslant t\) and for which \(t_r^\prime\) measures the time to death. This estimate is the distribution, unrestricted as to form, which maximizes the likelihood of the observations. Other estimates that are discussed are the actuarial estimates (which are also products, but with the number of factors usually reduced by grouping); and reduced-sample (RS) estimates, which require that losses not be accidental, so that the limits of observation (potential loss times) are known even for those items whose deaths are observed. When no losses occur at ages less than t the estimate of P(t) in all cases reduces to the usual binomial estimate, namely, the observed proportion of survivors.