Article

# Exponential Survival with Covariance

Authors:
To read the full-text of this research, you can request a copy directly from the author.

## 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.

## No full-text available

... (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. ...
Article
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. ...
Full-text available
Article
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. ...
Article
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. ...
Article
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. ...
Full-text available
Book
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. ...
Full-text available
Article
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): ...
Article
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 ...
Article
The censored exponential regression model is commonly used for modeling lifetime data. In this paper, we derive a simple matrix formula for the second-order covariance matrix of the maximum likelihood estimators in this class of regression models. The general matrix formula covers many types of censoring commonly encountered in practice. Also, the formula only involves simple operations on matrices and hence is quite suitable for computer implementation. Monte Carlo simulations are provided to show that the second-order covariances can be quite pronounced in small to moderate sample sizes. Additionally, based on the second-order covariance matrix, we propose an alternative Wald statistic to test hypotheses in this class of regression models. Monte Carlo simulation experiments reveal that the alternative Wald test exhibits type I error probability closer to the chosen nominal level. We also present an empirical application for illustrative purposes.
Chapter
In both reliability and survival analyses, regression models are employed extensively for identifying factors associated with probability, hazard, risk, or survival of units being studied. This chapter introduces some of the regression models used in both reliability and survival analyses. The regression models include logistic regression, proportional hazards, accelerated failure time, and parametric regression models based on specific probability distributions.
Chapter
Chapter
Early developments in controlled clinical trials at the National Institutes of Health (NIH) took place mainly at the National Cancer Institute (NCI) and what was then the National Heart Institute (NHI) (subsequently the National Heart, Lung, and Blood Institute (NHLBI))beginning in the 1950s. This article reviews the developments from the early 1950s to the late 1960s at both institutes, summarizing the early efforts in clinical trials, the organizations set up to conduct and monitor the clinical trials, and the developments in statistical methodology that have formed the basis for conducting many of the present day randomized controlled trials. The early history of clinical trials at these institutes has been reviewed in more detail at NCI by Gehan & Schneiderman and at NHLBI by Halperin et al.
Chapter
Early developments in controlled clinical trials at the National Institutes of Health (NIH) took place mainly at the National Cancer Institute (NCI) and what was then the National Heart Institute (NHI) (subsequently the National Heart, Lung, and Blood Institute (NHLBI)) beginning in the 1950s. This chapter reviews the developments from the early 1950s to the late 1960s at both institutes, summarizing the early efforts in clinical trials, the organizations set up to conduct and monitor the clinical trials, and the developments in statistical methodology that have formed the basis for conducting many of the present day randomized controlled trials.
Chapter
Concomitant Variables The Role of Concomitant Variables in Planning Clinical Trials General Parametric Model of Hazard Function with Observed Concomitant Variables Additive Models of Hazard Rate Function Multiplicative Models Estimation in Multiplicative Models Assessment of the Adequacy of a Model: Tests of Goodness of Fit Selection of Concomitant Variables Treatment-Covariate Interaction Logistic Linear Models Time Dependent Concomitant Variables Concomitant Variables Regarded as Random Variables Posterior Distribution of Concomitant Variables Concomitant Variables in Competing Risk Models
Article
The “meningitis belt” is a region in sub-Saharan Africa where annual outbreaks of meningitis occur, with epidemics observed cyclically. While we know that meningitis is heavily dependent on seasonal trends, the exact pathways for contracting the disease are not fully understood and warrant further investigation. Most previous approaches have used large sample inference to assess impacts of weather on meningitis rates. However, in the case of rare events, the validity of such assumptions is uncertain. This work examines the meningitis trends in the context of rare events, with the specific objective of quantifying the underlying seasonal patterns in meningitis rates. We compare three main classes of models: the Poisson generalized linear model, the Poisson generalized additive model, and a Bayesian hazard model extended to accommodate count data and a changing at-risk population. We compare the accuracy and robustness of the models through the bias, RMSE, and standard deviation of the estimators, and also provide a detailed case study of meningitis patterns for data collected in Navrongo, Ghana.
Article
The presence of knowledge spillovers and shared human capital is at the heart of the Marhall–Arrow–Romer externalities hypothesis. Most of the earlier empirical contributions on knowledge externalities; however, considered data aggregated at a regional level so that conclusions are based on the arbitrary definition of jurisdictional spatial units: this is the essence of the so-called modifiable areal unit problem. A second limitation of these studies is constituted by the fact that, somewhat surprisingly, while concentrating on the effects of agglomeration on firm creation and growth, the literature has, conversely, largely ignored its effects on firm survival. The present paper aims at contributing to the existing literature by answering to some of the open methodological questions reconciling the literature of Cox proportional hazards model with that on point pattern and thus capturing the true nature of spatial information. We also present some empirical results based on Italian firm demography data collected and managed by the Italian National Institute of Statistics (ISTAT).
Conference Paper
This paper investigates several alternative methods of constructing confidence interval estimates based on the bootstrap and jackknife techniques for the parameter of the exponential regression model, when data is censored. Bootstrap confidence interval techniques, the bootstrap-t and bootstrap-percentile are compared with the confidence interval based on jackknife via coverage probability study using simulated data. The results clearly indicate that the jackknife technique works far better than any of the bootstrap techniques when dealing with censored data..
Article
The development of Statistical methodology for application to survival data has expanded rapidly in the last two decades. Increasing interest in the field of occupational health demands that the current state-of-the-art in modeling exposure-risk relationships be utilized in assessing potential dangers to worker health. It is the aim of this research to investigate the more sophisticated survivorship models in producing a quantitative risk assessment of lung cancer in U.S. uranium miners. The Cox proportional hazards model was chosen for this purpose. A variety of risk functions are examined, with a power function model providing the best fit. A number of risk factors influence the exposure-response relationship. Among these are a strong independent multiplicative effect of cigarette smoking t a positive effect for age at initial exposure, and a negative effect for time since last exposure. The nature of the temporal effects suggest that
Article
Consideration is given to the piecewise smooth estimation of hazard functions given by Cox's (1972) proportional hazards model. The smoothest estimate available hitherto is a step‐function which is not continuous. The method used is penalized maximum likelihood estimation introduced by Good and Gaskins (1971) and developed by Silverman (1978). This gives a quadratic spline with discontinuities in the slope at the times of death. The knot points are at times of death and censoring. The hazard function and corresponding survival curves are estimated for a group of 51 women with carcinoma of the vulva. The estimated survival curve is compared with those given by Breslow's (1972) and Kalbfleisch and Prentice's (1973) methods.
Article
In this paper I develop a new class of discrete-time, discrete-covariate models for modeling nonproportionality in event-history data within the log-multiplicative framework. The models specify nonproportionality in hazards to be a log-multiplicative product of two components: a nonproportionality pattern over time and a nonproportionality level per group. Illustrated with data from the U.S. National Longitudinal Mortality Study (Rogot et al. 1988) and from the 1980 June Current Population Survey (Wu and Tuma 1990), the log-multiplicative models are shown to be natural generalizations of proportional hazards models and should be applicable to a wide range of research areas.
Article
Regression models of the proportional hazards type are applied to the analysis of censored survival data. Methods of inference associated with the models and techniques for checking model assumptions are presented and applied to the analysis of some data arising from a clinical trial in medicine.
Article
A parametric mixture model provides a regression framework for analysing failure-time data that are subject to censoring and multiple modes of failure. The regression context allows us to adjust for concomitant variables and to assess their effects on the joint distribution of time and type of failure. The mixing parameters correspond to the marginal probabilities of the various failure types and are modelled as logistic functions of the covariates. The hazard rate for each conditional distribution of time to failure, given type of failure, is modelled as the product of a piece-wise exponential function of time and a log-linear function of the covariates. An EM algorithm facilitates the maximum likelihood analysis and illuminates the contributions of the censored observations. The methods are illustrated with data from a heart transplant study and are compared with a cause-specific hazard analysis. The proposed mixture model can also be used to analyse multivariate failure-time data.
Article
There are four regression techniques currently available for use with censored data which do not assume particular parametric families of survival distributions. They are due to (i) Cox (1972), (ii) Miller (1976), (iii) Buckley & James (1979), and (iv) Koul, Susarla & Van Ryzin (1981). These four methods are described, and their performances compared on the updated Stanford heart transplant data. Conclusions on the usefulness of the four procedures are drawn.
Article
A parametric model is proposed for the hazard function incorporating covariables. The model is flexible enough so that it does not unduly restrict the shape of the hazard function. This model does not require that the proportional hazards assumption be met, and it provides for testing whether the assumption is reasonable. The hazard rate is given as a polynomial in time with the coefficients of the various powers of time being possibly different functions of the vector of covariables. Methods for fitting the model to data and testing hypotheses about model parameters are presented. The degree of the polynomial for the hazard is chosen in a step-wise manner as part of the fitting of the model. Specification of the survival curve with covariables is straightforward as a result of the parametric nature of the model. Use of the model and methods for fitting and hypothesis testing are illustrated by application to two different cohort studies. For each analysis, a single covariate indicates in which of two treatment groups an individual belongs. It is found that a time-constant hazard and hence, the proportional hazards assumption are adequate for the first cohort examined. However, for the second cohort a time-varying hazard is required and the proportional hazards assumption is not suitable. Results obtained using other methods are compared with those of the proposed method for both of the cohorts. Good agreement among the different approaches is observed.
Article
The problem of estimating regression parameters in an exponential model with censoring is considered when it is a priori suspected that the parameters may be restricted to a subspace. James-Stein (JS) type estimators are obtained which dominate the usual maximum likelihood (ml) estimators. The relative performance of the JS type estimators is compared to the ml estimators using quadratic distributional risk under local alternatives. It is demonstrated that the JS type estimators are asymptotically superior to the usual ml estimators. Further, it is shown that the JS type estimator is dominated by its truncated part.
Article
In the analysis of survival data, the statistician is often faced with the problem of competing risks, and a variety of approaches have been presented in the literature. One of the oldest and therefore most frequently used techniques involves the adjustment of rates appearing in a life table. This paper offers a general reformulation of the competing risks analysis described by Chiang (1961, 1968) in terms of matrices and matrix operations on an underlying contingency table. The method of weighted least squares is used to analyze estimates of survival rates adjusted for selected causes of failure as well as to fit probability distributions which effectively produce data reduction. Chi-square statistics are provided for testing hypotheses of interest in a large sample theory framework. Numerical examples illustrate the methods and the Introduction contains a succinct review of relevant literature. /// Pour analyser les nombres de survivants, le statisticien se heurte souvent au problème des risques en compétition; et on trouve dans la littérature diverses approches du problème. L'une des plus anciennes et, par suite, des plus fréquemment employées de ces techniques comporte l'ajustement de taux en forme de table de survie. Le présent article présente une formulation générale nouvelle de l'analyse des risques en compétition, décrite par Chiang (1961, 1968) en termes de matrices et d'opérateurs matriciels, sur une table de contingence sousjacente. On emploie la méthode des moindres carrés pondérés pour estimer les taux de survie ajustés relatifs à des causes de décès choisies, ainsi que pour ajuster des distributions de probabilités procurant une réduction efficace des données. Des Ki-carrés sont donnés pour tester des hypothèses intéressantes dans le cadre d'une théorie des grands échantillons. Des exemples numériques servent à illustrer les méthodes, tandis que l'introduction passe brièvement en revue la littérature relative à ce sujet.
Article
In this article, we consider the class of censored exponential regression models which is very useful for modeling lifetime data. Under a sequence of Pitman alternatives, the asymptotic expansions up to order n − 1/2 of the non null distribution functions of the likelihood ratio, Wald, Rao score, and gradient statistics are derive in this class of models. The non null asymptotic distribution functions of these statistics are obtained for testing a composite null hypothesis in the presence of nuisance parameters. The power of all four tests, which are equivalent to first order, are compared based on these non null asymptotic expansions. Furthermore, in order to compare the finite-sample performance of these tests in this class of models, we consider Monte Carlo simulations. We also present an empirical application for illustrative purposes.
Article
Log-linear models provide a flexible means of extending life table techniques for the analysis of censored survival data with categorical covariates, as discussed by Holford (1980) and Laird and Olivier (1981). We extend this methodology to incorporate cases in which one or more of the categorical covariates are sometimes missing. Maximum likelihood estimates of the parameters are calculated using data from all cases. This can result in large gains in efficiency over standard methods that require the exclusion of cases with incomplete data. With this approach, we assume that the hazard function, conditional on the covariates, is a stepwise function over disjoint intervals of time. The model has two parts: a log-linear model describing the hazard parameters, and a multinomial model describing the probabilities in the contingency table defined by the covariates. The main interest is in the model for the hazard parameters. We show how to calculate maximum likelihood estimates of parameters of the model either by an application of the EM algorithm in conjunction with one cycle of iterative proportional fitting in the M step or by using the Newton—Raphson algorithm. Estimates of standard errors are computed from the empirical information matrix. When using our proposed maximum likelihood approach, two additional assumptions are needed in addition to the usual assumptions of noninformative censoring. First, the mechanism causing missing covariates must be ignorable (Rubin 1976) in that the probability that a covariate is missing cannot depend on the covariate itself or on other covariates that are missing. The second assumption is that the distribution of the random censoring variable does not depend on any covariate that is missing. The first example, investigating the influence of several covariates on time to diagnosis of high blood pressure in a large cohort of men, shows clear gains in efficiency of our approach over analysis of complete cases and illustrates the flexibility of the log-linear approach. A second example of survival times of symptomatic and asymptomatic lymphoma patients shows interesting differences between the complete-case analysis and the maximum likelihood approach, which could be due to a nonrandom missing-value mechanism.
Full-text available
Article
Many surveys include questions that attempt to measure the time of the most recent occurrence of some event, for example, last visit to a physician. Although it is tempting to apply survival (failure-time) methods to such data, the conditions under which such applications are appropriate have not been apparent. In this article it is shown that standard methods may be applied when the data arise from certain well-known stochastic processes. Special procedures may be necessary if the models include duration dependence, however. The methods are illustrated by the estimation of regression models for data on residential mobility.
Article
This paper presents a number of analyses to assess the effects of various covariates on the survival of patients in the Stanford Heart Transplantation Program. The data have been updated from previously published versions and include some additional covariates, such as measures of tissue typing. The methods used allow for simultaneous investigation of several covariates and provide estimates of the relative risk of transplantation as well as significance tests.
Article
The exact likelihood functions are examined for several examples of data chosen from the literature. These are compared with the likelihoods arising from the large sample approximations and with point estimates that were actually used in the literature. It is concluded that large sample approximations (application of standard maximum likelihood theory) can be misleading for inferences and should be checked against the actual likelihood functions. Similarly point estimates can be misleading or uninformative and their properties such as bias, variance, etc. are relatively unimportant. Because of the availability of high speed computers, exact methods and asymptotic comparisons are now feasible and this should be reflected where possible in the theory and application of statistical inference.
Article
A multicenter trial was carried out with patients with superficial Ta-T1 bladder tumors in 37 urology centers throughout Italy. After transurethral resection (TUR) patients were given intravesical doxorubicin instillations at the mean dose of 50 mg per instillation diluted in 50 ml of distilled water or physiologic saline. Chemoprophylaxis was performed at weekly intervals for the first 4 weeks and then monthly. Cystoscopies were taken every 3 months during the first year, every 4 months during the second year, and every 6 months thereafter. A total of 435 patients, with a median follow-up time of 436 days, were considered eligible for the evaluation of activity. The lowest recurrence rate and the longest disease-free interval was observed in the group of patients with primary or single tumor. Treatment was well tolerated; 119 patients (22.7%) complained of local adverse reactions, and 10 (1.9%) of systemic adverse reactions. In 34 of the patients (6.5%) treatment had to be discontinued.
Article
We consider some nonnormal regression situations in which there are many regressor varibles, and it is desired to determine good fitting models, according to the value of the likelihood ratio statistic for tests of submodels against the full model. Efficient computational algorithms for the normal linear model are adopted for use with nonnormal models. Even with as many as 10-15 regressor variables present, we find it is often possible to determine all of the better fitting models with relatively small amounts of computer time. The use of the procedures is illustrated on exponential, Poisson and binary regression models.
Article
This paper unites two different fields, survival and contingency table analysis, in a single analytical framework based on the log-linear model. We demonstrate that many currently popular approaches to modeling survival data, including the approaches of Glasser (1967), Cox (1972), Breslow (1972, 1974), and Holford (1976), can be handled by using existing computer packages developed for the log-linear analysis of contingency table data. More important, we demonstrate that the log-linear modeling system used to characterize counted data structures directly characterizes survival data as well. Counted data methodologies for testing and estimation are also applicable here. Much of the theoretical basis for this work has been independently derived by Holford (1980) and Aitkin and Clayton (1980). The emphasis in this paper is not to develop new methodologies, but rather to present new uses and interpretations for already familiar methodologies.
Full-text available
Article
A survey is given of techniques for covariance analysis of censored life data. Both parametric and nonpararnetric approaches are reviewed. An application is given to the evaluation of parolee followup data. We examine the effects of covariates, such as age, income, and drug use, on time to rearrest. One of these covariates varies with time. The records of two correctional institutions are compared after adjusting for non-homogeneity of covariate values.
Article
Four parametric survival models are compared. The constant hazard functions as a function of a covariate are: . The models were compared using as the distance between models the average over the covariate values of the sup norm between the two survival curves. It was found that 1) is a considerably different model than models 2), 3) and 4). For samples of size 100 and below the variability due to the estimation of parameters in models 2), 3) and 4) was greater than the distance between the models. The authors conclude that the choice between models 2), 3) and 4) is not crucial for small sample sizes.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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."
Article
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.