Multiple Hazards Model Research Articles

The Two-Way Proportional Hazards Model

SummarySurvival analysis problems often involve dual timescales, most commonly calendar date and lifetime, the latter being the elapsed time since an initiating event such as a heart transplant. In our main example attention is focused on the hazard rate of ‘death’ as a function of calendar date. Three different estimates are discussed, one each from proportional hazards analyses on the lifetime and the calendar date scales, and one from a symmetric approach called here the ‘two-way proportional hazards model’, a multiplicative hazards model going back to Lexis in the 1870s. The three are connected through a Poisson generalized linear model for the Lexis diagram. The two-way model is shown to combine the information from the two ‘one-way’ proportional hazards analyses efficiently, at the cost of more extensive parametric modelling.

Comparing nonnested Cox models

We derive the limiting distribution of the partial likelihood ratio under general conditions. The multiplicative hazards models being fitted may be nonnested and misspecified. The true model is not assumed to contain either model under consideration. The null hypothesis is that the models are equidistant in Kullback–Leibler metric applied to the rank likelihood. The statistic is consistent for the model which is closer to the truth. Its distribution depends on the unknown data‐generating mechanism. A sequential testing procedure is proposed for nonnested comparisons which is valid regardless of the true model. This involves a novel statistic for the equality of the fitted models which is separate from the partial likelihood. The methodology has important applications in model assessment. Simulations and a real example demonstrate its utility in selecting the functional forms of covariates and relative risks.

A flexible additive multiplicative hazard model

We present a new additive-multiplicative hazard model which consists of two components. The first component contains additive covariate effects through an additive Aalen model while the second component contains multiplicative covariate effects through a Cox regression model. The Aalen model allows for time-varying covariate effects, while the Cox model allows only a common time-dependence through the baseline. Approximate maximum likelihood estimators are derived by solving the simultaneous score equations for the nonparametric and parametric components of the model. The suggested estimators are provided with large-sample properties and are shown to be efficient. The efficient estimators depend, however, on some estimated weights. We therefore also consider unweighted estimators and describe their large-sample properties. We finally extend the model to allow for time-varying covariate effects in the multiplicative part of the model as well.

Fitting exponential family mixed models

The generalized linear model (McCullagh and Nelder, 1972) and the semiparametric multiplicative hazard model (Cox, 1972) have significantly influenced the way in which statistical modelling is taught and practiced. Common for the two model families is the assumption that conditionally on covariate information (including time) the observations are independent. The obvious difficulty in identifying and measuring all relevant covariates has pushed for methods that can jointly handle both mean and dependence structures. The early 1990s saw a myriad of approaches for dealing with multivariate generalized linear models. More recently, the hazard models have been extended to multivariate settings. Here we review (i) penalized likelihood, (ii) Monte Carlo EM, and (iii) Bayesian Markov chain Monte Carlo methods for fitting the generalized linear mixed models and the frailty models, and we discuss the rationale for choosing between the methods. The similarities of the toolboxes for these two multivariate model families open up for a new level of generality both in teaching and applied research. Two examples are used for illustration, involving censored failure time responses and Poisson responses, respectively.

Influence of Hepatitis C Virus Subtype on Hepatocellular Carcinogenesis: A Multivariate Analysis of a Retrospective Cohort of 593 Patients with Cirrhosis

Objective: The influence of hepatitis C virus (HCV) subtypes on hepatocellular carcinogenesis was prospectively investigated. Methods: A total of 593 patients with HCV-related cirrhosis were recruited and their HCV subtype was determined. Results: The carcinogenesis rates in the patients with HCV group 1 (genotype 1a +1b, n = 442) and group 2 (genotype 2a +2b, n = 136) were 32.0 and 26.6% at the end of the 5th year, 57.4 and 48.1% at the end of the 10th year and 71.8 and 71.0% at the end of the 15th year, respectively (p = 0.10). As to those patients without a history of regular drinking (i.e. less than 200 kg on a pure alcohol basis), the carcinogenesis rates in group 1 (n = 277) and group 2 (n = 90) were 30.8 and 16.3% at the end of the 5th year, 52.8 and 34.4% at the end of the 10th year and 70.6 and 67.1% at the end of the 15th year, respectively (p = 0.025). Although HCV subtype did not influence carcinogenesis in patients with a drinking history of 200 kg or more (p = 0.62), it significantly affected the carcinogenesis rate in patients without a history of regular drinking. Multivariate analysis showed that HCV subtype group 1 significantly increased the carcinogenesis in the group without a history of regular drinking after adjustment for age and gender (hazard ratio = 2.57, p = 0.0085). Conclusion: The interaction between HCV subtype and drinking history should be considered in the prediction of carcinogenesis using a multiplicative proportional hazard model.

Business failure of new firms: an empirical analysis using a multiplicative hazards model

This paper investigates business failure of new firms. Using a multiplicative hazards model, we estimate the determinants of business failure among new manufacturing firms in Tokyo during the period 1986 to 1994. It is found that a new firm without sufficient capital or a sufficient size has a higher risk of business failure. It is also found that the new firm tends to have more difficulty surviving in an industry characterized by a high entry rate. With respect to the timing of entry, the new firm that entered just before or after the collapse of the so-called bubble economy has been more likely to fail. Furthermore, we estimate a regression model based not only on age but also on calendar time. By using a regression model based on calendar time, it is found that the firm's age has been related to business failure.

Analysis of failure times for multiple infections following bone marrow transplantation: an application of the multiple failure time proportional hazards model.

We report an application of the proportional hazards model for multiple failure times in a study arising from the Bone Marrow Transplant Database at the University of Minnesota. The study compared the risk of infections after transplantation for patients who received allogeneic bone marrow transplants from unrelated donors (URD) versus related donors (RD). In 249 patients there was a total of 365 infections in 2.5 years of follow-up. The multiple failure time model uses all the data and provides empirical estimates of standard errors that incorporate the within-person dependencies in the data. The estimate of relative risk associated with URD was 1.4 (naive 95 per cent confidence interval 1.14 to 1.73, empirical 1.08 to 1.79), compared to the estimate 1.6 (naive or empirical, 1.1 to 2.1) from the proportional hazards model on 165 first infections only. The multivariate model gives great flexibility in modelling, for example, in accommodating a separate base hazard function for each type of failure and in allowing analysis of intervals between infections as an alternative to analysis of time from a marker event, here transplantation.

Analysis of failure times for multiple infections following bone marrow transplantation: an application of the multiple failure time proportional hazards model

Analysis of failure times for multiple infections following bone marrow transplantation: an application of the multiple failure time proportional hazards model

A nonparametric multiplicative hazard model for event history analysis

A major issue in exploring and analyzing life history data with multiple states and events is the development and availability of flexible methods that allow simultaneous incorporation and estimation of baseline hazards, detection and modelling of nonlinear functional forms of covariates and time-varying effects, and the possibility to include time-dependent covariates. In this paper we consider a nonparametric multiplicative hazard model that takes into account these aspects. Embedded in the counting process approach, estimation is based on penalized likelihoods and splines. The methods are illustrated by two real data applications, one to a more conventional survival data set with two absorbing states, and one to more complex sleep-electroencephalography data with multiple recurrent states of sleep.

Radiation enterocolitis: overview of the past 15 years.

From April 1980 to April 1995 a total of 54 patients (53 women, 1 man) were hospitalized in our department for the surgical treatment of radiation enterocolitis. Two surgical protocols were applied for these patients: intestinal decompression procedures alone (intestinal bypass, colostomy, or both; n = 18) or an intestinal resection in addition to decompression (n = 36). The clinical factors contributing to survival after irradiation were retrospectively reviewed by a multiple variate proportional hazards model. As a result, patients treated with decompression procedures alone had an 11 times higher risk for death than those treated with the addition of intestinal resection. In the former group, 5 of 18 patients died of bleeding from the remaining intestine after operation. We concluded that surgical resection of the diseased intestine is a useful procedure for treating radiation enterocolitis to reduce intestinal bleeding from the irradiated intestine.

The attractiveness of an additive hazard model: an example from medical demography.

With the intention of studying sociodemographic differentials in theelevated mortality from cancer, a mixed additive-multiplicativecontinuous-time hazard model with categorical covariates is suggested. Thismodel is a simple and plausible extension of the multiplicative hazard modeldemographers are well acquainted with. As an empirical example, survival fromprostate cancer among Norwegian men is examined on the basis of individualregister and census data. The model has some advantages compared to themodels based on an additive mortality structure that have previously beenused in studies of cancer survival. In other demographic research areas, itcould be an interesting alternative to the commonly employed multiplicativehazard model.

Weighted empiricals and the product-limit estimator in the multiplicative hazard and time transfer regression model

In a model that combines the usual multiplicative hazard regression model and the accelerated life model or its log transform, weighted empirical processes are used to define estimators of the underlying hazard and survival functions with a specified multiplicative factor in the hazard rate. The covariates as well as the weights are allowed to be time-dependent. The weak convergence of the estimators are obtained. These estimators can be used to define and analyze parametric estimators in the multiplicative hazard model, the censored regression or the accelerated life model.

INTERVAL CENSORED SURVIVAL DATA: A GENERALIZED LINEAR MODELLING APPROACH

A method is described for weak parametric modelling of arbitrarily interval censored survival data using generalized linear models. The method makes use of an associated Bernoulli model, with standard errors based on the observed information matrix. Three types of models are discussed: additive and multiplicative hazard models with piecewise constant baseline hazard, and a proportional hazards model with discrete baseline survivor function. These models may be fitted in the statistical package GLIM.

Trichomonas vaginalis and cervical cancer : A prospective study in China

The relationship between Trichomonas vaginalis infection and cervical cancer was investigated prospectively in a cohort of 16,797 women aged 25 years or more who were followed from 1974 to 1985 within the framework of a cervical screening program in Jingan, China. Personal interviews were conducted by trained interviewers when the women first entered the screening program. At initial screening, 421 (2.51%) women had a positive cytologic diagnosis of T. vaginalis infection. Ninety-nine incident cases of pathologically confirmed squamaus cell carcinoma were identified from the cohort, with a total of 140,018 person-years of observation. T. vaginalis infection was found to contribute to the risk of cervical cancer, as determined by crude estimates and after adjustment for potential confounding effects. In a multiple proportional hazards model, the relative risk for cervical cancer was 3.3 (95% confidence interval: 1.5 to 7.4) among women with T. vaginalis infection. Furthermore, in the multivariate analysis, increased risk of cervical cancer was associated with the following factors: number of extramarital sexual partners of both the subjects and their spouses, cigarette smoking, and irregular menstruation. Having a large number of negative Pap smears was associated with lower risk. This study suggests that there might be an association between T. vaginalis infection and the risk of cervical cancer, but only 4 to 5% of cervical cancer in Chinese women may be attributable to T. vaginalis infection.

Assessing the Losses Caused by an Industrial Intervention: A Hierarchical Bayesian Approach

A nonparametric multiplicative hazard model is proposed, and applied to the estimation of losses caused by a strike at a Quebec aluminium smelter in 1967. The Bayesian approach is adopted, and Gibbs sampling for the numerical computations. We obtain a posterior distribution of the hazard rate and the corresponding predictive distribution of the projected losses. The main merit of the approach proposed is that the uncertainties concerning the actual losses, both in terms of the total number of operating days lost and the excess number of damaged aluminium reduction cells, can be quantified with probabilities. The compensation of the losses is discussed, and the results are compared with some point estimates from earlier analyses of the same data.

Simple Parametric and Nonparametric Models for Excess and Relative Mortality

This paper studies two classes of hazard-rate-based models for the mortality in a group of individuals taking normal life expectancy into account. In a multiplicative hazard model, the estimate for the relative mortality generalises the standardised mortality ratio, and the adequacy of a model with constant relative mortality can be tested using a type of total time on test statistic. In an additive hazard model, continuous-time generalisations of a "corrected" survival curve and a "normal" survival curve are obtained, and the adequacy of a model with constant excess mortality can again be tested using a type of total time on test statistic. A model including both the multiplicative hazard model and the additive hazard model is briefly considered. The use of the models is illustrated on a set of data concerning survival after operation for malignant melanoma.

Cohabitation and social background: Trends observed for Swedish women born between 1936 and 1960

This paper throws some light on changing family-building patterns in Sweden during the 1960s and 1970s. The family-building events analyzed are first informal cohabitation, first marriage, birth of first child (both outside a union and within a consensual union), and separations from childless informal cohabitation. Results from three previous studies are first summarized. These analyze the patterns by socio-economic and regional origin, applying multiple-decrement techniques separately for each background factor. The new results presented here are based on analyses that use multiplicative hazard models, permitting isolation of the effects of socio-economic background and region of childhood and adolescence respectively. It is generally found that socio-economic background is more important than the region where the woman grew up. Marriage intensities, however, which exhibit a clear regional pattern, are exceptions to this.

Multiplicative and additive models with external controls in a cohort study of cancer mortality.

The use of additive and multiplicative hazard models is examined for a cohort study of 2696 women followed up for 12 years. The multiplicative model implied that women with a haemoglobin level less than 12 g/dl were at higher risk from cancer, and the additive model showed that this risk was confined to women after the menopause. Despite difficulties in fitting and in interpretation, additive model can be useful in the analysis of cohort studies.