Aalen's Additive Model Research Articles

Survival Regression Model Allowing for Exposure-Mediator Interaction: Analysis of Kenya Demographic Health Survey (KDHS) 2014 Data

Although infant and under-five mortality rates have decreased over the past decade, Kenya, like many other African nations, did not achieve the Millennium Development Goal (MDG) 4 target. To accelerate progress towards reaching Sustainable Development Goal 3 by 2030, it is essential to understand the factors influencing under-five child mortality (UFCM), taking into account all potential confounding variables and effect modifiers. The child mortality rate serves as a key health indicator for any country. This study used data from the 2014 Kenya Demographic and Health Survey (KHDS). This paper introduces the concept of decomposing the total effect of an independent variable into three components: a direct effect, an indirect effect, and an interactive effect. We attempt to account for the direct effect of an independent variable on the outcome, then proceed to check the effect of the presence of a possible mediator and, furthermore, the possible interactions between an exposure variable and a mediator variable. The outcome variable was UFCM, and the study was to determine the effect of mother’s education on UFCM, in the presence of mediators such as mother’s income and the effect via the interaction between mother’s education and maternal income was estimated. To capture the effect of mediation and interactions in the context of survival analysis, an Aalen additive model, including a product term for the exposure and mediator term, was developed. The methods were further illustrated with practical approach to KDHS data. KDHS has data on a broad scope of risk factors for UFCM. Computations for all data sets was implemented using the freely available R-software package. This analysis suggests that while a significant portion of the impact of maternal education on UFCM is mediated by increasing maternal income, it is the interaction between maternal education and maternal income that leads to a reduction in UFCM. Interventions targeting an increase in income among mothers with no education level would yield a greater reduction in UFCM than interventions targeting mothers with a higher education level. The total effect was ascribed to an interaction between the mother’s education level and maternal income, and part of it is attributable to pure indirect effect, and a given proportion is attributable to pure direct effect. The majority of the total effect (70%)is attributed to the interaction between change from no education level to primary education level and maternal income. 22% is associated with the pure direct effect, while 8% is linked to the pure indirect effect. Interventions with a given increase in income among those with no education level would yield a greater reduction in UFCM than interventions targeting mothers with a higher education level.

A causal inference approach to compare leukaemia treatment outcome in the absence of randomization and with dependent censoring

One cause of poor outcomes in children of low-income countries affected by acute lymphoblastic leukaemia (ALL) is loss to follow-up due to abandonment of treatment. Assuming this type of loss to follow-up as independent censoring, as in standard Kaplan-Meier estimates, ignores the likely association of abandonment with biologic and socio-economic factors related to outcome. Moreover, when comparing treatment protocols adopted in different time periods, possible imbalances in patients' characteristics must be considered. We aim to compare the outcome of children enrolled in two subsequent protocols for ALL treatment (2000-2007 and 2008-2015) in Honduras, taking both dependent censoring due to abandonment of treatment and imbalances between patient characteristics into account. Marginal structural models based on inverse probability of treatment and censoring (IPTC) weighting allow the estimation of potential event-free survival (EFS) as if no abandonment of treatment occurred and the whole cohort was exposed, or not, to both protocols. An Aalen additive model and a logistic-regression model were used to build abandonment and treatment weights, respectively. The two protocols recruited 514 and 717 patients. Measured baseline covariates in both protocols were gender, age, white blood cell count, central nervous system involvement, tumour histology and socio-economic status. The potential EFS is slightly higher under the more recent protocol in the first 3 years but no difference is estimated in the long period [survival difference at 5 years (95% confidence interval) = 0.1% (-0.97%; 1.13%)]. Both protocols would allow reducing the event rate by 12-13% if there was no abandonment of treatment. Using IPTC weighting, we found a similar potential effect of the two treatment protocols if the imbalance due to the different distribution of potential confounders and to abandonment of therapy was removed.

On goodness-of-fit tests for Aalen's additive model based on left-truncated right-censored or doubly censored data

ABSTRACTGandy and Jensen (2005) proposed goodness-of-fit tests for Aalen's additive risk model. In this article, we demonstrate that the approach of Gandy and Jensen (2005) can be applied to left-truncated right-censored (LTRC) data and doubly censored data. A simulation study is conducted to investigate the performance of the proposed tests. The proposed tests are illustrated using heart transplant data.

The partly Aalen's model for recurrent event data with a dependent terminal event.

Recurrent event data are commonly observed in biomedical longitudinal studies. In many instances, there exists a terminal event, which precludes the occurrence of additional repeated events, and usually there is also a nonignorable correlation between the terminal event and recurrent events. In this article, we propose a partly Aalen's additive model with a multiplicative frailty for the rate function of recurrent event process and assume a Cox frailty model for terminal event time. A shared gamma frailty is used to describe the correlation between the two types of events. Consequently, this joint model can provide the information of temporal influence of absolute covariate effects on the rate of recurrent event process, which is usually helpful in the decision-making process for physicians. An estimating equation approach is developed to estimate marginal and association parameters in the joint model. The consistency of the proposed estimator is established. Simulation studies demonstrate that the proposed approach is appropriate for practical use. We apply the proposed method to a peritonitis cohort data set for illustration.

Penalised logistic regression and dynamic prediction for discrete-time recurrent event data.

We consider methods for the analysis of discrete-time recurrent event data, when interest is mainly in prediction. The Aalen additive model provides an extremely simple and effective method for the determination of covariate effects for this type of data, especially in the presence of time-varying effects and time varying covariates, including dynamic summaries of prior event history. The method is weakened for predictive purposes by the presence of negative estimates. The obvious alternative of a standard logistic regression analysis at each time point can have problems of stability when event frequency is low and maximum likelihood estimation is used. The Firth penalised likelihood approach is stable but in removing bias in regression coefficients it introduces bias into predicted event probabilities. We propose an alterative modified penalised likelihood, intermediate between Firth and no penalty, as a pragmatic compromise between stability and bias. Illustration on two data sets is provided.

A penalized algorithm for event-specific rate models for recurrent events.

We introduce a covariate-specific total variation penalty in two semiparametric models for the rate function of recurrent event process. The two models are a stratified Cox model, introduced in Prentice and others (1981. On the regression analysis of multivariate failure time data. Biometrika 68: , 373-379.), and a stratified Aalen's additive model. We show the consistency and asymptotic normality of our penalized estimators. We demonstrate, through a simulation study, that our estimators outperform classical estimators for small-to-moderate sample sizes. Finally, an application to the bladder tumor data of Byar (1980. The veterans administration study of chemoprophylaxis for recurrent stage 1 bladder tumors: comparison of placebo, pyridoxine, and topical thiotepa. In Pavone-Macaluso, M. Smith, P. H. and F. Edsmyn (editors), Bladder Tumors and Others Topics in Urological Oncology, pp. 363-370.) is presented.

Estimation of Causal Odds of Concordance using the Aalen Additive Model

AbstractA simple summary of a treatment effect is attractive, which is part of the explanation of the success of the Cox model when analysing time‐to‐event data since the relative risk measure is such a convenient summary measure. In practice, however, the Cox model may fail to give a reasonable fit, very often because of time‐changing treatment effect. The Aalen additive hazards model may be a good alternative as time‐changing effects are easily modelled within this model, but results are then evidently more complicated to communicate. In such situations, the odds of concordance measure (OC) is a convenient way of communicating results, and recently Martinussen & Pipper (2012) showed how a variant of the OC measure may be estimated based on the Aalen additive hazards model. In this study, we propose an estimator that should be preferred in observational studies as it always estimates the causal effect on the chosen scale, only assuming that there are no un‐measured confounders. The resulting estimator is shown to be consistent and asymptotically normal, and an estimator of its limiting variance is provided. Two real applications are provided.

Estimation of odds of concordance based on the Aalen additive model

The Cox regression model is often used when analyzing survival data as it provides a convenient way of summarizing covariate effects in terms of relative risks. The proportional hazards assumption may not hold, however. A typical violation of the assumption is time-changing covariate effects. Under such scenarios one may use more flexible models but the results from such models may be complicated to communicate and it is desirable to have simple measures of a treatment effect, say. In this paper we focus on the odds-of-concordance measure that was recently studied by Schemper et al. (Stat Med 28:2473-2489, 2009). They suggested to estimate this measure using weighted Cox regression (WCR). Although WCR may work in many scenarios no formal proof can be established. We suggest an alternative estimator of the odds-of-concordance measure based on the Aalen additive hazards model. In contrast to the WCR, one may derive the large sample properties for this estimator making formal inference possible. The estimator also allows for additional covariate effects.

A SAS macro for direct adjusted survival curves based on Aalen's additive model

Aalen's additive model defines the covariate effects on the survival outcome to be time-varying. The aim of this paper is to explore the application of Aalen's model in assessing treatment effects, adjusting for possibly different patient profiles between the treatment groups. We treat Aalen's model as the underlying model and compute the direct adjusted survival curves for different treatments. The treatment comparison can be performed by finding a confidence interval for the difference between two direct adjusted survivals. We developed a SAS macro that calculates the direct adjusted survival estimates and the standard errors, based on Aalen's model. In this paper, we give two examples to explain the syntax for using the macro.

Sampling Adjusted Analysis of Dynamic Additive Regression Models for Longitudinal Data

We consider a modelling approach to longitudinal data that aims at estimating flexible covariate effects in a model where the sampling probabilities are modelled explicitly. The joint modelling yields simple estimators that are easy to compute and analyse, even if the sampling of the longitudinal responses interacts with the response level. An incorrect model for the sampling probabilities results in biased estimates. Non‐representative sampling occurs, for example, if patients with an extreme development (based on extreme values of the response) are called in for additional examinations and measurements. We allow covariate effects to be time‐varying or time‐constant. Estimates of covariate effects are obtained by solving martingale equations locally for the cumulative regression functions. Using Aalen's additive model for the sampling probabilities, we obtain simple expressions for the estimators and their asymptotic variances. The asymptotic distributions for the estimators of the non‐parametric components as well as the parametric components of the model are derived drawing on general martingale results. Two applications are presented. We consider the growth of cystic fibrosis patients and the prothrombin index for liver cirrhosis patients. The conclusion about the growth of the cystic fibrosis patients is not altered when adjusting for a possible non‐representativeness in the sampling, whereas we reach substantively different conclusions about the treatment effect for the liver cirrhosis patients.