Dynamic regression hazards models for relative survival

A natural way of modelling relative survival through regression analysis is to assume an additive form between the expected population hazard and the excess hazard due to the presence of an additional cause of mortality. Within this context, the existing approaches in the parametric, semiparametric and non-parametric setting are compared and discussed. We study the additive excess hazards models, where the excess hazard is on additive form. This makes it possible to assess the importance of time-varying effects for regression models in the relative survival framework. We show how recent developments can be used to make inferential statements about the non-parametric version of the model. This makes it possible to test the key hypothesis that an excess risk effect is time varying in contrast to being constant over time. In case some covariate effects are constant, we show how the semiparametric additive risk model can be considered in the excess risk setting, providing a better and more useful summary of the data. Estimators have explicit form and inference based on a resampling scheme is presented for both the non-parametric and semiparametric models. We also describe a new suggestion for goodness of fit of relative survival models, which consists on statistical and graphical tests based on cumulative martingale residuals. This is illustrated on the semiparametric model with proportional excess hazards. We analyze data from the TRACE study using different approaches and show the need for more flexible models in relative survival.     

Prediction of short-term survival with an application in primary biliary cirrhosis.

Many long-term follow-up studies for survival accumulate repeated measurements of prognostic factors. Survival models which include only covariate values at baseline do not use all available information, and do not relate to survival predictions for times other than at that baseline. Time-dependent covariate models (which update covariate values as measurements occur through time) might be used, though limitations of software for estimating the underlying hazard functions and difficulty in relating hazard function changes to survival prediction present serious drawbacks. By dividing each patient's follow-up into successive intervals of equal length (using a length of interest for prediction) and with measurements available at the start of each, we describe how an analysis taking person-intervals as the observation units can be undertaken using readily available software to produce short-term survival models. We show that this approach is related to both the baseline and time-dependent covariate models. The method is illustrated using data from a long-term study of patients with primary biliary cirrhosis, where interest is in short-term survival predictions to aid the decision when to undertake liver transplantation.     

Regression models for relative survival

Four approaches to estimating a regression model for relative survival using the method of maximum likelihood are described and compared. The underlying model is an additive hazards model where the total hazard is written as the sum of the known baseline hazard and the excess hazard associated with a diagnosis of cancer. The excess hazards are assumed to be constant within pre-specified bands of follow-up. The likelihood can be maximized directly or in the framework of generalized linear models. Minor differences exist due to, for example, the way the data are presented (individual, aggregated or grouped), and in some assumptions (e.g. distributional assumptions). The four approaches are applied to two real data sets and produce very similar estimates even when the assumption of proportional excess hazards is violated. The choice of approach to use in practice can, therefore, be guided by ease of use and availability of software. We recommend using a generalized linear model with a Poisson error structure based on collapsed data using exact survival times. The model can be estimated in any software package that estimates GLMs with user-defined link functions (including SAS, Stata, S-plus, and R) and utilizes the theory of generalized linear models for assessing goodness-of-fit and studying regression diagnostics.     

Estimating net survival: the importance of allowing for informative censoring

Net survival, the one that would be observed if cancer were the only cause of death, is the most appropriate indicator to compare cancer mortality between areas or countries. Several parametric and non-parametric methods have been developed to estimate net survival, particularly when the cause of death is unknown. These methods are based either on the relative survival ratio or on the additive excess hazard model, the latter using the general population mortality hazard to estimate the excess mortality hazard (the hazard related to net survival). The present work used simulations to compare estimator abilities to estimate net survival in different settings such as the presence/absence of an age effect on the excess mortality hazard or on the potential time of follow-up, knowing that this covariate has an effect on the general population mortality hazard too. It showed that when age affected the excess mortality hazard, most estimators, including specific survival, were biased. Only two estimators were appropriate to estimate net survival. The first is based on a multivariable excess hazard model that includes age as covariate. The second is non-parametric and is based on the inverse probability weighting. These estimators take differently into account the informative censoring induced by the expected mortality process. The former offers great flexibility whereas the latter requires neither the assumption of a specific distribution nor a model-building strategy. Because of its simplicity and availability in commonly used software, the nonparametric estimator should be considered by cancer registries for population-based studies.     

Regression analysis with multiplicative and time-varying additive regression coefficients with examples from breast and colon cancer

Regression analysis may be used to simplify the representation of mortality rates when there are many significant prognostic covariates or to adjust for confounding effects. The principal request of the regression model in this range of use is to have unbiased parameter estimates. A model with constant multiplicative and time-varying additive regression coefficients is discussed. The model allows some covariate effects to be multiplicative while allowing others to have a time-varying additive effect. Thus, it is a mix of classical Cox regression and Aalen's additive risk model. A major characteristic of cancer mortality rates, in contrast to general mortality rates, is that hazard rates, after a potentially initial increase, decrease, although not always tending to zero. Cancer diseases, like breast and colon cancer, have significantly increased cause-specific mortality rates even 20 years after diagnosis. Another major feature in cancer survival analysis is that many covariate effects are time-varying. Some covariate effects, like age at diagnosis, may only be significant for a limited time after diagnosis. Furthermore, some treatment procedures may initially decrease the mortality, while the long-term effect may be opposite. A third issue is that average covariate effects are very often not multiplicative. Estimation is carried out iteratively; the cumulative additive regression functions are estimated non-parametrically using a least-squares method and the multiplicative parameters are estimated from the partial likelihood. The method is applied on 3201 female breast cancer and 1372 male colon cancer patients.     

Survival extrapolation in the presence of cause specific hazards

Health economic evaluations require estimates of expected survival from patients receiving different interventions, often over a lifetime. However, data on the patients of interest are typically only available for a much shorter follow‐up time, from randomised trials or cohorts. Previous work showed how to use general population mortality to improve extrapolations of the short‐term data, assuming a constant additive or multiplicative effect on the hazards for all‐cause mortality for study patients relative to the general population. A more plausible assumption may be a constant effect on the hazard for the specific cause of death targeted by the treatments. To address this problem, we use independent parametric survival models for cause‐specific mortality among the general population. Because causes of death are unobserved for the patients of interest, a polyhazard model is used to express their all‐cause mortality as a sum of latent cause‐specific hazards. Assuming proportional cause‐specific hazards between the general and study populations then allows us to extrapolate mortality of the patients of interest to the long term. A Bayesian framework is used to jointly model all sources of data. By simulation, we show that ignoring cause‐specific hazards leads to biased estimates of mean survival when the proportion of deaths due to the cause of interest changes through time. The methods are applied to an evaluation of implantable cardioverter defibrillators for the prevention of sudden cardiac death among patients with cardiac arrhythmia. After accounting for cause‐specific mortality, substantial differences are seen in estimates of life years gained from implantable cardioverter defibrillators. © 2014 The Authors Statistics in Medicine Published by John Wiley & Sons Ltd.     

Flexible parametric models for relative survival, with application in coronary heart disease

Relative survival is frequently used in population-based studies as a method for estimating disease-related mortality without the need for information on cause of death. We propose an extension to relative survival of a flexible parametric model proposed by Royston and Parmar for censored survival data. The model provides smooth estimates of the relative survival and excess mortality rates by using restricted cubic splines on the log cumulative excess hazard scale. The approach has several advantages over some of the more standard relative survival models, which adopt a piecewise approach, the main being the ability to model time on a continuous scale, the survival and hazard functions are obtained analytically and it does not use split-time data.     

Relative survival multistate Markov model

Prognostic studies often have to deal with two important challenges: (i) separating effects of predictions on different 'competing' events and (ii) uncertainty about cause of death. Multistate Markov models permit multivariable analyses of competing risks of, for example, mortality versus disease recurrence. On the other hand, relative survival methods help estimate disease-specific mortality risks even in the absence of data on causes of death. In this paper, we propose a new Markov relative survival (MRS) model that attempts to combine these two methodologies. Our MRS model extends the existing multistate Markov piecewise constant intensities model to relative survival modeling. The intensity of transitions leading to death in the MRS model is modeled as the sum of an estimable excess hazard of mortality from the disease of interest and an 'offset' defined as the expected hazard of all-cause 'natural' mortality obtained from relevant life-tables. We evaluate the new MRS model through simulations, with a design based on registry-based prognostic studies of colon cancer. Simulation results show almost unbiased estimates of prognostic factor effects for the MRS model. We also applied the new MRS model to reassess the role of prognostic factors for mortality in a study of colorectal cancer. The MRS model considerably reduces the bias observed with the conventional Markov model that does not permit accounting for unknown causes of death, especially if the 'true' effects of a prognostic factor on the two types of mortality differ substantially.     

An overall strategy based on regression models to estimate relative survival and model the effects of prognostic factors in cancer survival studies

Relative survival provides a measure of the proportion of patients dying from the disease under study without requiring the knowledge of the cause of death. We propose an overall strategy based on regression models to estimate the relative survival and model the effects of potential prognostic factors. The baseline hazard was modelled until 10 years follow-up using parametric continuous functions. Six models including cubic regression splines were considered and the Akaike Information Criterion was used to select the final model. This approach yielded smooth and reliable estimates of mortality hazard and allowed us to deal with sparse data taking into account all the available information. Splines were also used to model simultaneously non-linear effects of continuous covariates and time-dependent hazard ratios. This led to a graphical representation of the hazard ratio that can be useful for clinical interpretation. Estimates of these models were obtained by likelihood maximization. We showed that these estimates could be also obtained using standard algorithms for Poisson regression.     

Non-parametric estimation for baseline hazards function and covariate effects with time-dependent covariates

Often in many biomedical and epidemiologic studies, estimating hazards function is of interest. The Breslow's estimator is commonly used for estimating the integrated baseline hazard, but this estimator requires the functional form of covariate effects to be correctly specified. It is generally difficult to identify the true functional form of covariate effects in the presence of time-dependent covariates. To provide a complementary method to the traditional proportional hazard model, we propose a tree-type method which enables simultaneously estimating both baseline hazards function and the effects of time-dependent covariates. Our interest will be focused on exploring the potential data structures rather than formal hypothesis testing. The proposed method approximates the baseline hazards and covariate effects with step-functions. The jump points in time and in covariate space are searched via an algorithm based on the improvement of the full log-likelihood function. In contrast to most other estimating methods, the proposed method estimates the hazards function rather than integrated hazards. The method is applied to model the risk of withdrawal in a clinical trial that evaluates the anti-depression treatment in preventing the development of clinical depression. Finally, the performance of the method is evaluated by several simulation studies.     

Estimation of the adjusted cause-specific cumulative probability using flexible regression models for the cause-specific hazards

In competing risks setting, we account for death according to a specific cause and the quantities of interest are usually the cause-specific hazards (CSHs) and the cause-specific cumulative probabilities. A cause-specific cumulative probability can be obtained with a combination of the CSHs or via the subdistribution hazard. Here, we modeled the CSH with flexible hazard-based regression models using B-splines for the baseline hazard and time-dependent (TD) effects. We derived the variance of the cause-specific cumulative probabilities at the population level using the multivariate delta method and showed how we could easily quantify the impact of a covariate on the cumulative probability scale using covariate-adjusted cause-specific cumulative probabilities and their difference. We conducted a simulation study to evaluate the performance of this approach in its ability to estimate the cumulative probabilities using different functions for the cause-specific log baseline hazard and with or without a TD effect. In the scenario with TD effect, we tested both well-specified and misspecified models. We showed that the flexible regression models perform nearly as well as the nonparametric method, if we allow enough flexibility for the baseline hazards. Moreover, neglecting the TD effect hardly affects the cumulative probabilities estimates of the whole population but impacts them in the various subgroups. We illustrated our approach using data from people diagnosed with monoclonal gammopathy of undetermined significance and provided the R-code to derive those quantities, as an extension of the R-package mexhaz .     

Marginal structural models to estimate the causal effect of zidovudine on the survival of HIV-positive men

Standard methods for survival analysis, such as the time-dependent Cox model, may produce biased effect estimates when there exist time-dependent confounders that are themselves affected by previous treatment or exposure. Marginal structural models are a new class of causal models the parameters of which are estimated through inverse-probability-of-treatment weighting; these models allow for appropriate adjustment for confounding. We describe the marginal structural Cox proportional hazards model and use it to estimate the causal effect of zidovudine on the survival of human immunodeficiency virus-positive men participating in the Multicenter AIDS Cohort Study. In this study, CD4 lymphocyte count is both a time-dependent confounder of the causal effect of zidovudine on survival and is affected by past zidovudine treatment. The crude mortality rate ratio (95% confidence interval) for zidovudine was 3.6 (3.0-4.3), which reflects the presence of confounding. After controlling for baseline CD4 count and other baseline covariates using standard methods, the mortality rate ratio decreased to 2.3 (1.9-2.8). Using a marginal structural Cox model to control further for time-dependent confounding due to CD4 count and other time-dependent covariates, the mortality rate ratio was 0.7 (95% conservative confidence interval = 0.6-1.0). We compare marginal structural models with previously proposed causal methods.     

A model combining excess and relative mortality for population‐based studies

Considering expected mortality provides an attractive approach to analyse mortality of population‐based cohorts of patients presenting with a chronic disease. Two classes of methods are available: either modelling the excess mortality using an additive hazard model or modelling the relative mortality using a multiplicative hazard model. Because these two models are informative to look for factors associated with mortality related to a chronic disease, we developed an alternative model modelling both the excess and the relative mortality. We generalised Andersen and Vaeth's model to fit covariates and obtain directly an estimation of the Excess Mortality Ratio and Relative Mortality Ratio for each covariate. We assessed the performances of the combined model by using simulations, and it appeared satisfactorily. We illustrate the combined model by data collected in patients presenting with end‐stage renal disease and treated by dialysis. The combined model offers the possibility of performing pure additive and multiplicative models and thus to compare their log‐likelihoods. The combined model appears useful to select one of these pure models or to conclude to the need of modelling both excess and relative mortality. In this latter case, our model enabled better describing the effect of covariates on the excess and relative mortality. Copyright © 2013 John Wiley & Sons, Ltd.     

Hazard regression model and cure rate model in colon cancer relative survival trends: are they telling the same story?

Hazard regression models and cure rate models can be advantageously used in cancer relative survival analysis. We explored the advantages and limits of these two models in colon cancer and focused on the prognostic impact of the year of diagnosis on survival according to the TNM stage at diagnosis. The analysis concerned 9,998 patients from three French registries. In the hazard regression model, the baseline excess death hazard and the time-dependent effects of covariates were modelled using regression splines. The cure rate model estimated the proportion of ‘cured’ patients and the excess death hazard in ‘non-cured’ patients. The effects of year of diagnosis on these parameters were estimated for each TNM cancer stage. With the hazard regression model, the excess death hazard decreased significantly with more recent years of diagnoses (hazard ratio, HR 0.97 in stage III and 0.98 in stage IV, P < 0.001). In these advanced stages, this favourable effect was limited to the first years of follow-up. With the cure rate model, recent years of diagnoses were significantly associated with longer survivals in ‘non-cured’ patients with advanced stages (HR 0.95 in stage III and 0.97 in stage IV, P < 0.001) but had no significant effect on cure (odds ratio, OR 0.99 in stages III and IV, P > 0.5). The two models were complementary and concordant in estimating colon cancer survival and the effects of covariates. They provided two different points of view of the same phenomenon: recent years of diagnosis had a favourable effect on survival, but not on cure.     

A proportional regression model for 20 year survival of colon cancer in Norway

Twenty year survival of all Norwegians with colon cancer registered in a period of 10 years is estimated by both relative survival rates, and with a proportional regression model for the excess intensity. Male colon cancer patients have a significant positive excess mortality at least 20 years after diagnosis, while the excess mortality for females is about zero after 10 years. Stratified analyses for men indicate non-proportionality throughout the follow-up period, and when this information is included in the regression model, there are significant effects of age between 60 and 70 years and for pelvic cancer. The use of proportional regression models is also discussed when excess intensities are close to zero or negative.     

Multiple regression of cost data: use of generalised linear models

OBJECTIVE: Choosing an appropriate method for regression analyses of cost data is problematic because it must focus on population means while taking into account the typically skewed distribution of the data. In this paper we illustrate the use of generalised linear models for regression analysis of cost data. METHODS: We consider generalised linear models with either an identity link function (providing additive covariate effects) or log link function (providing multiplicative effects), and with gaussian (normal), overdispersed poisson, gamma, or inverse gaussian distributions. These are applied to estimate the treatment effects in two randomised trials adjusted for baseline covariates. Criteria for choosing an appropriate model are presented. RESULTS: In both examples considered, the gaussian model fits poorly and other distributions are to be preferred. When there are variables of prognostic importance in the model, using different distributions can materially affect the estimates obtained; it may also be possible to discriminate between additive and multiplicative covariate effects. CONCLUSIONS: Generalised linear models are attractive for the regression of cost data because they provide parametric methods of analysis where a variety of non-normal distributions can be specified and the way covariates act can be altered. Unlike the use of data transformation in ordinary least-squares regression, generalised linear models make inferences about the mean cost directly.     

Information in the sample covariate distribution in prevalent cohorts

Methods of estimation and inference about survival distributions based on length-biased samples are well-established. Comparatively little attention has been given to the assessment of covariate effects in the context of length-biased samples, but prevalent cohort studies often have this objective. We show that, like the survival distribution, the covariate distribution from a prevalent cohort study is length-biased, and that this distribution may contain parametric information about covariate effects on the survival time. As a result, a likelihood based on the joint distribution of the survival time and the covariates yields estimates of covariate effects which are at least as efficient as estimates arising from a traditional likelihood which conditions on covariate values in the length-biased sample. We also investigate the empirical bias of estimators arising from a joint likelihood when the population covariate distribution is misspecified. The asymptotic relative efficiencies and empirical biases under model misspecification are assessed for both proportional hazards and accelerated failure time models. The various methods considered are applied in an illustrative analysis of risk factors for death following onset of dementia using data collected in the Canadian Study of Health and Aging.     

A comparison of propensity score methods: a case-study estimating the effectiveness of post-AMI statin use

There is an increasing interest in the use of propensity score methods to estimate causal effects in observational studies. However, recent systematic reviews have demonstrated that propensity score methods are inconsistently used and frequently poorly applied in the medical literature. In this study, we compared the following propensity score methods for estimating the reduction in all-cause mortality due to statin therapy for patients hospitalized with acute myocardial infarction: propensity-score matching, stratification using the propensity score, covariate adjustment using the propensity score, and weighting using the propensity score. We used propensity score methods to estimate both adjusted treated effects and the absolute and relative risk reduction in all-cause mortality. We also examined the use of statistical hypothesis testing, standardized differences, box plots, non-parametric density estimates, and quantile-quantile plots to assess residual confounding that remained after stratification or matching on the propensity score. Estimates of the absolute reduction in 3-year mortality ranged from 2.1 to 4.5 per cent, while estimates of the relative risk reduction ranged from 13.3 to 17.0 per cent. Adjusted estimates of the reduction in the odds of 3-year death varied from 15 to 24 per cent across the different propensity score methods.     

A marginalized two‐part model for semicontinuous data

In health services research, it is common to encounter semicontinuous data characterized by a point mass at zero followed by a right‐skewed continuous distribution with positive support. Examples include health expenditures, in which the zeros represent a subpopulation of patients who do not use health services, while the continuous distribution describes the level of expenditures among health services users. Semicontinuous data are typically analyzed using two‐part mixture models that separately model the probability of health services use and the distribution of positive expenditures among users. However, because the second part conditions on a non‐zero response, conventional two‐part models do not provide a marginal interpretation of covariate effects on the overall population of health service users and non‐users, even though this is often of greatest interest to investigators. Here, we propose a marginalized two‐part model that yields more interpretable effect estimates in two‐part models by parameterizing the model in terms of the marginal mean. This model maintains many of the important features of conventional two‐part models, such as capturing zero‐inflation and skewness, but allows investigators to examine covariate effects on the overall marginal mean, a target of primary interest in many applications. Using a simulation study, we examine properties of the maximum likelihood estimates from this model. We illustrate the approach by evaluating the effect of a behavioral weight loss intervention on health‐care expenditures in the Veterans Affairs health‐care system. Copyright © 2014 John Wiley & Sons, Ltd.     

Cumulative cause-specific mortality for cancer patients in the presence of other causes: a crude analogue of relative survival

A common population-based cancer progress measure for net survival (survival in the absence of other causes) of cancer patients is relative survival. Relative survival is defined as the ratio of a population of observed survivors in a cohort of cancer patients to the proportion of expected survivors in a comparable set of cancer-free individuals in the general public, thus giving a measure of excess mortality due to cancer. Relative survival was originally designed to address the question of whether or not there is evidence that patients have been cured. It has proven to be a useful survival measure in several areas, including the evaluation of cancer control efforts and the application of cure models. However, it is not representative of the actual survival patterns observed in a cohort of cancer patients. This paper suggests a measure for cumulative crude (in the presence of other causes) cause-specific probability of death for a population diagnosed with cancer. The measure does not use cause of death information which can be unreliable for population cancer registries. Point estimates and variances are derived for crude cause-specific probability of death using relative survival instead of cause of death information. Examples are given for men diagnosed with localized prostate cancer over the age of 70 and women diagnosed with regional breast cancer using Surveillance, Epidemiology and End Results (SEER) Program data. The examples emphasize the differences in crude and net mortality measures and suggest areas where a crude measure is more informative. Estimates of this type are especially important for older patients as new screening modalities detect cancers earlier and choice of treatment or even 'watchful waiting' become viable options. Published in 2000 by John Wiley & Sons, Ltd.