The additive hazards model with high-dimensional regressors

The partly parametric and partly nonparametric additive risk model

Aalen's linear hazard rate regression model is a useful and increasingly popular alternative to Cox' multiplicative hazard rate model. It postulates that an individual has hazard rate function [Formula: see text] in terms of his covariate values [Formula: see text]. These are typically levels of various hazard factors, and may also be time-dependent. The hazard factor functions [Formula: see text] are the parameters of the model and are estimated from data. This is traditionally accomplished in a fully nonparametric way. This paper develops methodology for estimating the hazard factor functions when some of them are modelled parametrically while the others are left unspecified. Large-sample results are reached inside this partly parametric, partly nonparametric framework, which also enables us to assess the goodness of fit of the model's parametric components. In addition, these results are used to pinpoint how much precision is gained, using the parametric-nonparametric model, over the standard nonparametric method. A real-data application is included, along with a brief simulation study.

A boosting first-hitting-time model for survival analysis in high-dimensional settings

In this paper we propose a boosting algorithm to extend the applicability of a first hitting time model to high-dimensional frameworks. Based on an underlying stochastic process, first hitting time models do not require the proportional hazards assumption, hardly verifiable in the high-dimensional context, and represent a valid parametric alternative to the Cox model for modelling time-to-event responses. First hitting time models also offer a natural way to integrate low-dimensional clinical and high-dimensional molecular information in a prediction model, that avoids complicated weighting schemes typical of current methods. The performance of our novel boosting algorithm is illustrated in three real data examples.

ON ESTIMATION OF THE OPTIMAL TREATMENT REGIME WITH THE ADDITIVE HAZARDS MODEL

We propose a doubly robust estimation method for the optimal treatment regime based on an additive hazards model with censored survival data. Specifically, we introduce a new semiparametric additive hazard model which allows flexible baseline covariate effects in the control group and incorporates marginal treatment effect and its linear interaction with covariates. In addition, we propose a time-dependent propensity score to construct an A-learning type of estimating equations. The resulting estimator is shown to be consistent and asymptotically normal when either the baseline effect model for covariates or the propensity score is correctly specified. The asymptotic variance of the estimator is consistently estimated using a simple resampling method. Simulation studies are conducted to evaluate the finite-sample performance of the estimators and an application to AIDS clinical trial data is also given to illustrate the methodology.

Group Tests for High-dimensional Failure Time Data with the Additive Hazards Models

Statistical analysis of high-dimensional data has been attracting more and more attention due to the abundance of such data in various fields such as genetic studies or genomics and the existence of many interesting topics. Among them, one is the identification of a gene or genes that have significant effects on the occurrence of or are significantly related to a certain disease. In this paper, we will discuss such a problem that can be formulated as a group test or testing a group of variables or coefficients when one faces right-censored failure time response variable. For the problem, we develop a corrected variance reduced partial profiling (CVRPP) linear regression model and a likelihood ratio test procedure when the failure time of interest follows the additive hazards model. The numerical study suggests that the proposed method works well in practical situations and gives better performance than the existing one. An illustrative example is provided.

Quantification of Treatment Effect Modification on Both an Additive and Multiplicative Scale

Background

In both observational and randomized studies, associations with overall survival are by and large assessed on a multiplicative scale using the Cox model. However, clinicians and clinical researchers have an ardent interest in assessing absolute benefit associated with treatments. In older patients, some studies have reported lower relative treatment effect, which might translate into similar or even greater absolute treatment effect given their high baseline hazard for clinical events.

Methods

The effect of treatment and the effect modification of treatment were respectively assessed using a multiplicative and an additive hazard model in an analysis adjusted for propensity score in the context of coronary surgery.

Results

The multiplicative model yielded a lower relative hazard reduction with bilateral internal thoracic artery grafting in older patients (Hazard ratio for interaction/year = 1.03, 95%CI: 1.00 to 1.06, p = 0.05) whereas the additive model reported a similar absolute hazard reduction with increasing age (Delta for interaction/year = 0.10, 95%CI: -0.27 to 0.46, p = 0.61). The number needed to treat derived from the propensity score-adjusted multiplicative model was remarkably similar at the end of the follow-up in patients aged < = 60 and in patients >70.

Conclusions

The present example demonstrates that a lower treatment effect in older patients on a relative scale can conversely translate into a similar treatment effect on an additive scale due to large baseline hazard differences. Importantly, absolute risk reduction, either crude or adjusted, can be calculated from multiplicative survival models. We advocate for a wider use of the absolute scale, especially using additive hazard models, to assess treatment effect and treatment effect modification.

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.

Predicting the survival time for diffuse large B-cell lymphoma using microarray data

The present study was conducted to predict survival time in patients with diffuse large B-cell lymphoma, DLBCL, based on microarray data using Cox regression model combined with seven dimension reduction methods. This historical cohort included 2042 gene expression measurements from 40 patients with DLBCL. In order to predict survival, a combination of Cox regression model was used with seven methods for dimension reduction or shrinkage including univariate selection, forward stepwise selection, principal component regression, supervised principal component regression, partial least squares regression, ridge regression and Losso. The capacity of predictions was examined by three different criteria including log rank test, prognostic index and deviance. MATLAB r2008a and RKWard software were used for data analysis. Based on our findings, performance of ridge regression was better than other methods. Based on ridge regression coefficients and a given cut point value, 16 genes were selected. By using forward stepwise selection method in Cox regression model, it was indicated that the expression of genes GENE3555X and GENE3807X decreased the survival time (P=0.008 and P=0.003, respectively), whereas the genes GENE3228X and GENE1551X increased survival time (P=0.002 and P<0.001, respectively). This study indicated that ridge regression method had higher capacity than other dimension reduction methods for the prediction of survival time in patients with DLBCL. Furthermore, a combination of statistical methods and microarray data could help to detect influential genes in survival.

Assessment of evaluation criteria for survival prediction from genomic data

Survival prediction from high-dimensional genomic data is dependent on a proper regularization method. With an increasing number of such methods proposed in the literature, comparative studies are called for and some have been performed. However, there is currently no consensus on which prediction assessment criterion should be used for time-to-event data. Without a firm knowledge about whether the choice of evaluation criterion may affect the conclusions made as to which regularization method performs best, these comparative studies may be of limited value. In this paper, four evaluation criteria are investigated: the log-rank test for two groups, the area under the time-dependent ROC curve (AUC), an R²-measure based on the Cox partial likelihood, and an R²-measure based on the Brier score. The criteria are compared according to how they rank six widely used regularization methods that are based on the Cox regression model, namely univariate selection, principal components regression (PCR), supervised PCR, partial least squares regression, ridge regression, and the lasso. Based on our application to three microarray gene expression data sets, we find that the results obtained from the widely used log-rank test deviate from the other three criteria studied. For future studies, where one also might want to include non-likelihood or non-model-based regularization methods, we argue in favor of AUC and the R²-measure based on the Brier score, as these do not suffer from the arbitrary splitting into two groups nor depend on the Cox partial likelihood.

Ranking prognosis markers in cancer genomic studies

In cancer research, high-throughput genomic studies have been extensively conducted, searching for markers associated with cancer diagnosis, prognosis and variation in response to treatment. In this article, we analyze cancer prognosis studies and investigate ranking markers based on their marginal prognosis power. To avoid ambiguity, we focus on microarray gene expression studies where genes are the markers, but note that the methodology and results are applicable to other high-throughput studies. The objectives of this study are 2-fold. First, we investigate ranking markers under three commonly adopted semiparametric models, namely the Cox, accelerated failure time and additive risk models. Data analysis shows that the ranking may vary significantly under different models. Second, we describe a nonparametric concordance measure, which has roots in the time-dependent ROC (receiver operating characteristic) framework and relies on much weaker assumptions than the semiparametric models. In simulation, it is shown that ranking using the concordance measure is not sensitive to model specification whereas ranking under the semiparametric models is. In data analysis, the concordance measure generates rankings significantly different from those under the semiparametric models.

Semiparametric prognosis models in genomic studies

Development of high-throughput technologies makes it possible to survey the whole genome. Genomic studies have been extensively conducted, searching for markers with predictive power for prognosis of complex diseases such as cancer, diabetes and obesity. Most existing statistical analyses are focused on developing marker selection techniques, while little attention is paid to the underlying prognosis models. In this article, we review three commonly used prognosis models, namely the Cox, additive risk and accelerated failure time models. We conduct simulation and show that gene identification can be unsatisfactory under model misspecification. We analyze three cancer prognosis studies under the three models, and show that the gene identification results, prediction performance of all identified genes combined, and reproducibility of each identified gene are model-dependent. We suggest that in practical data analysis, more attention should be paid to the model assumption, and multiple models may need to be considered.