Semi-parametric additive risk models: application to injury duration study

A strategy for optimal fitting of multiplicative and additive hazards regression models

Background

In survival analysis, data can be modeled using either a multiplicative hazards regression model (such as the Cox model) or an additive hazards regression model (such as Lin’s or Aalen’s model). While several diagnostic tools are available to check the assumptions underpinning each type of model, there is no defined procedure to fit these models optimally. Moreover, the two types of models are rarely combined in survival analysis. Here, we propose a strategy for optimal fitting of multiplicative and additive hazards regression models in survival analysis.

Methods

This section details our proposed strategy for optimal fitting of multiplicative and additive hazards regression models, with a focus on the assumptions underpinning each type of model, the diagnostic tools used to check these assumptions, and the steps followed to fit the data. The proposed strategy draws on classical diagnostic tools (Schoenfeld and martingale residuals) and less common tools (pseudo-observations, martingale residual processes, and Arjas plots).

Results

The proposed strategy is applied to a dataset of patients with myocardial infarction (TRACE data frame). The effects of 5 covariates (age, sex, diabetes, ventricular fibrillation, and clinical heart failure) on the hazard of death are analyzed using multiplicative and additive hazards regression models. The proposed strategy is shown to fit the data optimally.

Conclusions

Survival analysis is improved by using multiplicative and additive hazards regression models together, but specific steps must be followed to fit the data optimally. By providing different measures of the same effect, our proposed strategy allows for better interpretation of the data.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12874-021-01273-2.

A gap time model based on a multiplicative marginal rate function that accounts for zero-recurrence units

In this article, we propose an alternative gap time model based on a multiplicative marginal rate function, which is formulated considering each gap time conditional on the previous recurrence times. In this formulation, the gap times are treated equally and the relation between successive events is no longer a problem. Furthermore, this article considers the inclusion of a proportion of zero-recurrence units (for which the event of interest will not occur) into the model to analyze recurrent event data. Inference aspects of the proposed model are discussed through maximum likelihood approach. A simulation study is carried out to examine the performance of the estimation procedure. The model is applied to hospital readmission data among colorectal cancer patients.

Power and sample size calculation for the additive hazard model

Sample size determination is essential during the planning phases of a study. When the study endpoint is the time to an event, Cox proportional hazard model is the traditional technique to analyze the effects of covariates on survival time. In contrast to the proportional hazard model, the additive hazard (AH) model specifies that the effect of covariates additively increases or decreases the hazard function. The popularity of this model is that it gives a more intuitive interpretation without the proportional hazard assumption. Because there is no literature estimating the required sample size based on the AH model, we provide a flexible formula for calculating the required sample size. The proposed formula incorporates time-independent and time-dependent effects of covariates without complicated mathematical calculations. The performance of the method was evaluated by extensive simulations. Finally, some pilot studies are shown to illustrate the applications.

The exponential-Poisson model for recurrent event data: an application to a set of data on malaria in Brazil

In this paper, we introduce a new model for recurrent event data characterized by a baseline rate function fully parametric, which is based on the exponential-Poisson distribution. The model arises from a latent competing risk scenario, in the sense that there is no information about which cause was responsible for the event occurrence. Then, the time of each recurrence is given by the minimum lifetime value among all latent causes. The new model has a particular case, which is the classical homogeneous Poisson process. The properties of the proposed model are discussed, including its hazard rate function, survival function, and ordinary moments. The inferential procedure is based on the maximum likelihood approach. We consider an important issue of model selection between the proposed model and its particular case by the likelihood ratio test and score test. Goodness of fit of the recurrent event models is assessed using Cox-Snell residuals. A simulation study evaluates the performance of the estimation procedure in the presence of a small and moderate sample sizes. Applications on two real data sets are provided to illustrate the proposed methodology. One of them, first analyzed by our team of researchers, considers the data concerning the recurrence of malaria, which is an infectious disease caused by a protozoan parasite that infects red blood cells.

Additive and multiplicative hazards modeling for recurrent event data analysis

BACKGROUND

Sequentially ordered multivariate failure time or recurrent event duration data are commonly observed in biomedical longitudinal studies. In general, standard hazard regression methods cannot be applied because of correlation between recurrent failure times within a subject and induced dependent censoring. Multiplicative and additive hazards models provide the two principal frameworks for studying the association between risk factors and recurrent event durations for the analysis of multivariate failure time data.

METHODS

Using emergency department visits data, we illustrated and compared the additive and multiplicative hazards models for analysis of recurrent event durations under (i) a varying baseline with a common coefficient effect and (ii) a varying baseline with an order-specific coefficient effect.

RESULTS

The analysis showed that both additive and multiplicative hazards models, with varying baseline and common coefficient effects, gave similar results with regard to covariates selected to remain in the model of our real dataset. The confidence intervals of the multiplicative hazards model were wider than the additive hazards model for each of the recurrent events. In addition, in both models, the confidence interval gets wider as the revisit order increased because the risk set decreased as the order of visit increased.

CONCLUSIONS

Due to the frequency of multiple failure times or recurrent event duration data in clinical and epidemiologic studies, the multiplicative and additive hazards models are widely applicable and present different information. Hence, it seems desirable to use them, not as alternatives to each other, but together as complementary methods, to provide a more comprehensive understanding of data.