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.     

A joint frailty model to estimate the recurrence process and the disease‐specific mortality process without needing the cause of death

In chronic diseases, such as cancer, recurrent events (such as relapses) are commonly observed; these could be interrupted by death. With such data, a joint analysis of recurrence and mortality processes is usually conducted with a frailty parameter shared by both processes. We examined a joint modeling of these processes considering death under two aspects: ‘death due to the disease under study' and ‘death due to other causes', which enables estimating the disease‐specific mortality hazard. The excess hazard model was used to overcome the difficulties in determining the causes of deaths (unavailability or unreliability); this model allows estimating the disease‐specific mortality hazard without needing the cause of death but using the mortality hazards observed in the general population. We propose an approach to model jointly recurrence and disease‐specific mortality processes within a parametric framework. A correlation between the two processes is taken into account through a shared frailty parameter. This approach allows estimating unbiased covariate effects on the hazards of recurrence and disease‐specific mortality. The performance of the approach was evaluated by simulations with different scenarios. The method is illustrated by an analysis of a population‐based dataset on colon cancer with observations of colon cancer recurrences and deaths. The benefits of the new approach are highlighted by comparison with the ‘classical' joint model of recurrence and overall mortality. Moreover, we assessed the goodness of fit of the proposed model. Comparisons between the conditional hazard and the marginal hazard of the disease‐specific mortality are shown, and differences in interpretation are discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

Additive and multiplicative covariate regression models for relative survival incorporating fractional polynomials for time-dependent effects

Relative survival is used to estimate patient survival excluding causes of death not related to the disease of interest. Rather than using cause of death information from death certificates, which is often poorly recorded, relative survival compares the observed survival to that expected in a matched group from the general population. Models for relative survival can be expressed on the hazard (mortality) rate scale as the sum of two components where the total mortality rate is the sum of the underlying baseline mortality rate and the excess mortality rate due to the disease of interest. Previous models for relative survival have assumed that covariate effects act multiplicatively and have thus provided relative effects of differences between groups using excess mortality rate ratios. In this paper we consider (i) the use of an additive covariate model, which provides estimates of the absolute difference in the excess mortality rate; and (ii) the use of fractional polynomials in relative survival models for the baseline excess mortality rate and time-dependent effects. The approaches are illustrated using data on 115 331 female breast cancer patients diagnosed between 1 January 1986 and 31 December 1990. The use of additive covariate relative survival models can be useful in situations when the excess mortality rate is zero or slightly less than zero and can provide useful information from a public health perspective. The use of fractional polynomials has advantages over the usual piecewise estimation by providing smooth estimates of the baseline excess mortality rate and time-dependent effects for both the multiplicative and additive covariate models. All models presented in this paper can be estimated within a generalized linear models framework and thus can be implemented using standard software.     

A multilevel excess hazard model to estimate net survival on hierarchical data allowing for non‐linear and non‐proportional effects of covariates

The excess hazard regression model is an approach developed for the analysis of cancer registry data to estimate net survival, that is, the survival of cancer patients that would be observed if cancer was the only cause of death. Cancer registry data typically possess a hierarchical structure: individuals from the same geographical unit share common characteristics such as proximity to a large hospital that may influence access to and quality of health care, so that their survival times might be correlated. As a consequence, correct statistical inference regarding the estimation of net survival and the effect of covariates should take this hierarchical structure into account. It becomes particularly important as many studies in cancer epidemiology aim at studying the effect on the excess mortality hazard of variables, such as deprivation indexes, often available only at the ecological level rather than at the individual level. We developed here an approach to fit a flexible excess hazard model including a random effect to describe the unobserved heterogeneity existing between different clusters of individuals, and with the possibility to estimate non‐linear and time‐dependent effects of covariates. We demonstrated the overall good performance of the proposed approach in a simulation study that assessed the impact on parameter estimates of the number of clusters, their size and their level of unbalance. We then used this multilevel model to describe the effect of a deprivation index defined at the geographical level on the excess mortality hazard of patients diagnosed with cancer of the oral cavity. Copyright © 2016 John Wiley & Sons, Ltd.     

Long term survival analysis: The curability of breast cancer

Methods of survival analysis for long-term follow-up studies are illustrated by a study of mortality in 3878 breast cancer patients in Edinburgh followed for up to 20 years. The problems of life tables, advantages of hazard plots and difficulties in statistical modelling are demonstrated by studying the relationship between survival and both clinical stage and initial menopausal status at diagnosis. To assess the ‘curability’ of breast cancer, mortality by year of follow-up is compared with expected mortality using Scottish age-specific death rates. Techniques for analysing such relative survival data include age-corrected life tables, ratio of observed to expected deaths and excess death rates. Finally, an additive hazard model is developed to incorporate covariates in the analysis of relative survival and curability.     

Practical recommendations for reporting Fine‐Gray model analyses for competing risk data

In survival analysis, a competing risk is an event whose occurrence precludes the occurrence of the primary event of interest. Outcomes in medical research are frequently subject to competing risks. In survival analysis, there are 2 key questions that can be addressed using competing risk regression models: first, which covariates affect the rate at which events occur, and second, which covariates affect the probability of an event occurring over time. The cause‐specific hazard model estimates the effect of covariates on the rate at which events occur in subjects who are currently event‐free. Subdistribution hazard ratios obtained from the Fine‐Gray model describe the relative effect of covariates on the subdistribution hazard function. Hence, the covariates in this model can also be interpreted as having an effect on the cumulative incidence function or on the probability of events occurring over time. We conducted a review of the use and interpretation of the Fine‐Gray subdistribution hazard model in articles published in the medical literature in 2015. We found that many authors provided an unclear or incorrect interpretation of the regression coefficients associated with this model. An incorrect and inconsistent interpretation of regression coefficients may lead to confusion when comparing results across different studies. Furthermore, an incorrect interpretation of estimated regression coefficients can result in an incorrect understanding about the magnitude of the association between exposure and the incidence of the outcome. The objective of this article is to clarify how these regression coefficients should be reported and to propose suggestions for interpreting these coefficients.     

Comparison of additive and multiplicative models for reproductive risk factors and post-menopausal breast cancer

The mathematical multistage model of carcinogenesis proposed by Breslow and Day indicates that if hormonal risk factors for breast cancer are late-stage carcinogens then the use of an additive model (excess absolute risk) should be preferrable to the present standard use of the multiplicative model (excess relative risk). To test different models, information from a large prospective study of 431,604 women aged 45–74 years in 1970 with follow-up to 1985 on reproductive factors and post-menopausal breast cancer mortality was used to compute goodness-of-fit statistics. The relative risk function, ranging from multiplicative to additive, was explored by changing the exponent in a power transformation. The analysis found evidence of better fit by the additive compared with the multiplicative models, consistent with the proposed mathematical model.     

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.     

How to control for unmeasured confounding in an observational time‐to‐event study with exposure incidence information: the treatment choice Cox model

In an observational study of the effect of a treatment on a time‐to‐event outcome, a major problem is accounting for confounding because of unknown or unmeasured factors. We propose including covariates in a Cox model that can partially account for an unknown time‐independent frailty that is related to starting or stopping treatment as well as the outcome of interest. These covariates capture the times at which treatment is started or stopped and so are called treatment choice (TC) covariates. Three such models are developed: first, an interval TC model that assumes a very general form for the respective hazard functions of starting treatment, stopping treatment, and the outcome of interest and second, a parametric TC model that assumes that the log hazard functions for starting treatment, stopping treatment, and the outcome event include frailty as an additive term. Finally, a hybrid TC model that combines attributes from the parametric and interval TC models. As compared with an ordinary Cox model, the TC models are shown to substantially reduce the bias of the estimated hazard ratio for treatment when data are simulated from a realistic Cox model with residual confounding due to the unobserved frailty. The simulations also indicate that the bias decreases or levels off as the sample size increases. A TC model is illustrated by analyzing the Women's Health Initiative Observational Study of hormone replacement for post‐menopausal women. Published 2017. This article has been contributed to by US Government employees and their work is in the public domain in the USA.     

Test for additive interaction in proportional hazards models

PURPOSE: We describe a method for testing and estimating a two-way additive interaction between two categorical variables, each of which has greater than or equal to two levels. METHODS: We test additive and multiplicative interactions in the same proportional hazards model and measure additivity by relative excess risk due to interaction (RERI), proportion of disease attributable to interaction (AP), and synergy index (S). A simulation study was used to compare the performance of these measures of additivity. Data from the Atherosclerosis Risk in Communities cohort study with a total of 15,792 subjects were used to exemplify the methods. RESULTS: The test and measures of departure from additivity depend neither on follow-up time nor on the covariates. The simulation study indicates that RERI is the best choice of measures of additivity using a proportional hazards model. The examples indicated that an interaction between two variables can be statistically significant on additive measure (RERI=1.14, p=0.04) but not on multiplicative measure (beta3=0.59, p=0.12) and that additive and multiplicative interactions can be in opposite directions (RERI=0.08, beta3=-0.08). CONCLUSIONS: The method has broader application for any regression models with a rate as the dependent variable. In the case that both additive and multiplicative interactions are statistically significant and in the opposite direction, the interpretation needs caution.     

Net time‐dependent ROC curves: a solution for evaluating the accuracy of a marker to predict disease‐related mortality

Developing prognostic markers of mortality for patients with chronic disease is important for identifying subjects at high risk of death and optimizing medical management. The usual approach in this regard is the use of time‐dependent ROC curves, which are well adapted for censored data. Nevertheless, an important part of the mortality may not be due to the chronic disease, and it is often impossible to individually determine whether or not the deaths are related to the disease itself. In survival regression, one solution is to distinguish between the expected mortality of one general population (from life tables) and the excess mortality related to the disease, by using an additive relative survival model. In this paper, we propose a new estimator of time‐dependent ROC curves, which includes this concept of net survival, in order to evaluate the capacity of a marker to predict disease‐specific mortality. We performed simulations in order to validate this estimator. We also illustrate this method using two different applications: (i) predicting mortality related to primary biliary cirrhosis of the liver and (ii) predicting mortality related to kidney transplantation in end‐stage renal disease patients. For each application, we evaluated a scoring system already established. The results demonstrate the utility of the proposed estimator of net time‐dependent ROC curves. Copyright © 2014 John Wiley & Sons, Ltd.     

Visualizing covariates in proportional hazards model

We present a graphical method called the rank‐hazard plot that visualizes the relative importance of covariates in a proportional hazards model. The key idea is to rank the covariate values and plot the relative hazard as a function of ranks scaled to interval [0, 1]. The relative hazard is plotted with respect to the reference hazard, which can be, for example, the hazard related to the median of the covariate. Transformation to scaled ranks allows plotting of covariates measured in different units in the same graph, which helps in the interpretation of the epidemiological relevance of the covariates. Rank‐hazard plots show the difference of hazards between the extremes of the covariate values present in the data and can be used as a tool to check if the proportional hazards assumption leads to reasonable estimates for individuals with extreme covariate values. Alternative covariate definitions or different transformations applied to covariates can be also compared using rank‐hazard plots. We apply rank‐hazard plots to the data from the FINRISK study where population‐based cohorts have been followed up for events of cardiovascular diseases and compare the relative importance of the covariates cholesterol, smoking, blood pressure and body mass index. The data from the Study to Understand Prognoses Preferences Outcomes and Risks of Treatment (SUPPORT) are used to visualize nonlinear covariate effects. The proposed graphics work in other regression models with different interpretations of the y‐axis. Copyright © 2009 John Wiley & Sons, Ltd.     

Estimating survival probabilities by exposure levels: utilizing vital statistics and complex survey data with mortality follow‐up

We present a two‐step approach for estimating hazard rates and, consequently, survival probabilities, by levels of general categorical exposure. The resulting estimator utilizes three sources of data: vital statistics data and census data are used at the first step to estimate the overall hazard rate for a given combination of gender and age group, and cohort data constructed from a nationally representative complex survey with linked mortality records, are used at the second step to divide the overall hazard rate by exposure levels. We present an explicit expression for the resulting estimator and consider two methods for variance estimation that account for complex multistage sample design: (1) the leaving‐one‐out jackknife method, and (2) the Taylor linearization method, which provides an analytic formula for the variance estimator. The methods are illustrated with smoking and all‐cause mortality data from the US National Health Interview Survey Linked Mortality Files, and the proposed estimator is compared with a previously studied crude hazard rate estimator that uses survey data only. The advantages of a two‐step approach and possible extensions of the proposed estimator are discussed. Copyright © 2015 John Wiley & Sons, Ltd.     

Methods for maternal age-standardization of the incidence of congenital abnormalities

In order to compare the birth incidences of particular congenital abnormalities in different populations, it is often necessary to allow for the effects of maternal age. Three age-adjusted indices are defined, based on analogous indices from the literature on mortality studies, namely, the Standardized Mortality Ratio, the Comparative Mortality Figure and Kerridge's Inverse Method. In most practical situations the differences between them are likely to be trival. However, the first index is maximally efficient under multiplicative risk models and is easily adjusted for incomplete data. The second is the only one to provide valid comparisons under additive, as well as multiplicative, models. The third has the advantage that it does not require a knowledge of the maternal age distribution of all births in the population. The use of the three indices is illustrated with published data on Down's syndrome.     

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.     

Assessing cumulative incidence functions under the semiparametric additive risk model

In analyzing competing risks data, a quantity of considerable interest is the cumulative incidence function. Often, the effect of covariates on the cumulative incidence function is modeled via the proportional hazards model for the cause‐specific hazard function. As the proportionality assumption may be too restrictive in practice, we consider an alternative more flexible semiparametric additive hazards model of (Biometrika 1994; 81 :501–514) for the cause‐specific hazard. This model specifies the effect of covariates on the cause‐specific hazard to be additive as well as allows the effect of some covariates to be fixed and that of others to be time varying. We present an approach for constructing confidence intervals as well as confidence bands for the cause‐specific cumulative incidence function of subjects with given values of the covariates. Furthermore, we also present an approach for constructing confidence intervals and confidence bands for comparing two cumulative incidence functions given values of the covariates. The finite sample property of the proposed estimators is investigated through simulations. We conclude our paper with an analysis of the well‐known malignant melanoma data using our method. Published in 2009 by John Wiley & Sons, Ltd.     

Multiplicative interaction in network meta‐analysis

Meta‐analysis of a set of clinical trials is usually conducted using a linear predictor with additive effects representing treatments and trials. Additivity is a strong assumption. In this paper, we consider models for two or more treatments that involve multiplicative terms for interaction between treatment and trial. Multiplicative models provide information on the sensitivity of each treatment effect relative to the trial effect. In developing these models, we make use of a two‐way analysis‐of‐variance approach to meta‐analysis and consider fixed or random trial effects. It is shown using two examples that models with multiplicative terms may fit better than purely additive models and provide insight into the nature of the trial effect. We also show how to model inconsistency using multiplicative terms. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

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.