Log-hazard Function Research Articles

A Bayesian survival treed hazards model using latent Gaussian processes.

Survival models are used to analyze time-to-event data in a variety of disciplines. Proportional hazard models provide interpretable parameter estimates, but proportional hazard assumptions are not always appropriate. Non-parametric models are more flexible but often lack a clear inferential framework. We propose a Bayesian treed hazards partition model that is both flexible and inferential. Inference is obtained through the posterior tree structure and flexibility is preserved by modeling the log-hazard function in each partition using a latent Gaussian process. An efficient reversible jump Markov chain Monte Carlo algorithm is accomplished by marginalizing the parameters in each partition element via a Laplace approximation. Consistency properties for the estimator are established. The method can be used to help determine subgroups as well as prognostic and/or predictive biomarkers in time-to-event data. The method is compared with some existing methods on simulated data and a liver cirrhosis dataset.

Choline-to-N-acetyl aspartate and lipids-lactate-to-creatine ratios together with age assemble a significant Cox's proportional-hazards regression model for prediction of survival in high-grade gliomas.

A long-lasting concern has prevailed for the identification of predictive biomarkers for high-grade gliomas (HGGs) using MRI. However, a consensus of which imaging parameters assemble a significant survival model is still missing in the literature; we investigated the significant positive or negative contribution of several MR biomarkers in this tumour prognosis. A retrospective cohort of supratentorial HGGs [11 glioblastoma multiforme (GBM) and 17 anaplastic astrocytomas] included 28 patients (9 females and 19 males, respectively, with a mean age of 50.4 years, standard deviation: 16.28 years; range: 13-85 years). Oedema and viable tumour measurements were acquired using regions of interest in T1 weighted, T2 weighted, fluid-attenuated inversion recovery, apparent diffusion coefficient (ADC) and MR spectroscopy (MRS). We calculated Kaplan-Meier curves and obtained Cox's proportional hazards. During the follow-up period (3-98 months), 17 deaths were recorded. The median survival time was 1.73 years (range, 0.287-8.947 years). Only 3 out of 20 covariates (choline-to-N-acetyl aspartate and lipids-lactate-to-creatine ratios and age) showed significance in explaining the variability in the survival hazards model; score test: χ2 (3) = 9.098, p = 0.028. MRS metabolites overcome volumetric parameters of peritumoral oedema and viable tumour, as well as tumour region ADC measurements. Specific MRS ratios (Cho/Naa, L-L/Cr) might be considered in a regular follow-up for these tumours. Advances in knowledge: Cho/Naa ratio is the strongest survival predictor with a log-hazard function of 2.672 in GBM. Low levels of lipids-lactate/Cr ratio represent up to a 41.6% reduction in the risk of death in GBM.

Comparing Smoothing Techniques for Fitting the Nonlinear Effect of Covariate in Cox Models.

Background and Objective:Cox model is a popular model in survival analysis, which assumes linearity of the covariate on the log hazard function, While continuous covariates can affect the hazard through more complicated nonlinear functional forms and therefore, Cox models with continuous covariates are prone to misspecification due to not fitting the correct functional form for continuous covariates. In this study, a smooth nonlinear covariate effect would be approximated by different spline functions.Material and Methods:We applied three flexible nonparametric smoothing techniques for nonlinear covariate effect in the Cox models: penalized splines, restricted cubic splines and natural splines. Akaike information criterion (AIC) and degrees of freedom were used to smoothing parameter selection in penalized splines model. The ability of nonparametric methods was evaluated to recover the true functional form of linear, quadratic and nonlinear functions, using different simulated sample sizes. Data analysis was carried out using R 2.11.0 software and significant levels were considered 0.05.Results:Based on AIC, the penalized spline method had consistently lower mean square error compared to others to selection of smoothed parameter. The same result was obtained with real data.Conclusion:Penalized spline smoothing method, with AIC to smoothing parameter selection, was more accurate in evaluate of relation between covariate and log hazard function than other methods.

Joint modelling of longitudinal and survival data: incorporating delayed entry and an assessment of model misspecification.

A now common goal in medical research is to investigate the inter‐relationships between a repeatedly measured biomarker, measured with error, and the time to an event of interest. This form of question can be tackled with a joint longitudinal‐survival model, with the most common approach combining a longitudinal mixed effects model with a proportional hazards survival model, where the models are linked through shared random effects. In this article, we look at incorporating delayed entry (left truncation), which has received relatively little attention. The extension to delayed entry requires a second set of numerical integration, beyond that required in a standard joint model. We therefore implement two sets of fully adaptive Gauss–Hermite quadrature with nested Gauss–Kronrod quadrature (to allow time‐dependent association structures), conducted simultaneously, to evaluate the likelihood. We evaluate fully adaptive quadrature compared with previously proposed non‐adaptive quadrature through a simulation study, showing substantial improvements, both in terms of minimising bias and reducing computation time. We further investigate, through simulation, the consequences of misspecifying the longitudinal trajectory and its impact on estimates of association. Our scenarios showed the current value association structure to be very robust, compared with the rate of change that we found to be highly sensitive showing that assuming a simpler trend when the truth is more complex can lead to substantial bias. With emphasis on flexible parametric approaches, we generalise previous models by proposing the use of polynomials or splines to capture the longitudinal trend and restricted cubic splines to model the baseline log hazard function. The methods are illustrated on a dataset of breast cancer patients, modelling mammographic density jointly with survival, where we show how to incorporate density measurements prior to the at‐risk period, to make use of all the available information. User‐friendly Stata software is provided. © 2015 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.

Modelling grouped survival times in toxicological studies using Generalized Additive Models

A method for combining a proportional-hazards survival time model with a bioassay model where the log-hazard function is modelled as a linear or smoothing spline function of log-concentration combined with a smoothing spline function of time is described. The combined model is fitted to mortality numbers, resulting from survival times that are grouped due to a common set of observation times, using Generalized Additive Models (GAMs). The GAM fits mortalities as conditional binomials using an approximation to the log of the integral of the hazard function and is implemented using freely-available, general software for fitting GAMs. Extensions of the GAM are described to allow random effects to be fitted and to allow for time-varying concentrations by replacing time with a calibrated cumulative exposure variable with calibration parameter estimated using profile likelihood. The models are demonstrated using data from a studies of a marine and a, previously published, freshwater taxa. The marine study involved two replicate bioassays of the effect of zinc exposure on survival of an Antarctic amphipod, Orchomenella pinguides. The other example modelled survival of the daphnid, Daphnia magna, exposed to potassium dichromate and was fitted by both the GAM and the process-based DEBtox model. The GAM fitted with a cubic regression spline in time gave a 61 % improvement in fit to the daphnid data compared to DEBtox due to a non-monotonic hazard function. A simulation study using each of these hazard functions as operating models demonstrated that the GAM is overall more accurate in recovering lethal concentration values across the range of forms of the underlying hazard function compared to DEBtox and standard multiple endpoint probit analyses.

Comparison of procedures to assess non‐linear and time‐varying effects in multivariable models for survival data

The focus of many medical applications is to model the impact of several factors on time to an event. A standard approach for such analyses is the Cox proportional hazards model. It assumes that the factors act linearly on the log hazard function (linearity assumption) and that their effects are constant over time (proportional hazards (PH) assumption). Variable selection is often required to specify a more parsimonious model aiming to include only variables with an influence on the outcome. As follow-up increases the effect of a variable often gets weaker, which means that it varies in time. However, spurious time-varying effects may also be introduced by mismodelling other parts of the multivariable model, such as omission of an important covariate or an incorrect functional form of a continuous covariate. These issues interact. To check whether the effect of a variable varies in time several tests for non-PH have been proposed. However, they are not sufficient to derive a model, as appropriate modelling of the shape of time-varying effects is required. In three examples we will compare five recently published strategies to assess whether and how the effects of covariates from a multivariable model vary in time. For practical use we will give some recommendations.

Model complexity and information in the data: Could it be a house built on sand?

Heisey et al. (2010), in an interesting paper, try to address a very difficult problem of analyzing spatially referenced, age specific prevalence data. The general goal of the analysis is to understand how force of infection changes as a function of age, time, and space. To further complicate matters, all the data considered in the paper are censored observations. Binary data are notoriously difficult to analyze, especially when latent processes are involved and prevalence is very low. Frankly, I was surprised by the complexity of the models they consider and the limited amount of information available to fit these models. I would like to congratulate them for trying to address such a difficult problem and in the process bringing to the attention of the ecologists some important statistical models in survival analysis. How does one generally deal with the conflicting issues of lack of information and desire to conduct inference about complex underlying processes? The standard approach is to compensate for lack of information by adding assumptions. This is done routinely in most statistical analyses by assuming a parametric model. For example, one can conduct inference in ANOVA without assuming any specific relationship between the treatment means if replicate observations are available at each treatment level. If such replicate data are not available, instead of giving up, we assume that there is a linear (or, some parametric) relationship between the covariates and the response, the regression approach. This is a smoothing assumption. Similarly, in one of the fundamental papers on statistical inference in the presence of nuisance parameters, Kiefer and Wolfowitz (1956) showed that simply assuming that the nuisance parameters arise from a distribution is enough of a smoothing assumption to estimate not only the parameters of interest but also the distribution function from which nuisance parameters are assumed to have arisen. Heisey et al. (2010) try to get away with the limited information available in the prevalence data, where all observations are censored, by imposing constraints on the log-hazard, a smoothing assumption of another sort. This is the easy part. The real questions are: (1) Given the limited amount of information in the data, what assumptions do we need until some inference is feasible? and (2) Are these inferences primarily driven by the data or by the assumptions? Technically, the answer to the first question is straightforward: add assumptions until the parameters in the model, at least the ones that are of scientific interest, are estimable given the data. The second question is qualitative. It is partially addressed by studying the sensitivity of the inferences on the parameters of interest (assuming they are identifiable) to the violations of the assumptions. I will discuss these issues in the remainder of the commentary. I assume readers are familiar with the basic descriptions in Heisey et al. (2010). Perhaps the easiest way out of the limited information in the prevalence data is to assume a specific parametric model for the log-hazard function. This does not guarantee that the parameters will be identifiable but it has the best chance. Heisey et al. (2010) do not take this easy way out. They aspire to assume less about the form of the log-hazard function. As they point out, the ‘‘nonparametric’’ MLE of the log-hazard is very choppy and unstable. It is generally not consistent, at least not at the usual ffiffiffi

Cox Models With Smooth Functional Effect of Covariates Measured With Error

We propose, develop, and implement a fully Bayesian inferential approach for the Cox model when the log hazard function contains unknown smooth functions of the variables measured with error. Our approach is to model nonparametrically both the log-baseline hazard and the smooth components of the log-hazard functions using low-rank penalized splines. Careful implementation of the Bayesian inferential machinery is shown to produce remarkably better results than the naive approach. Our methodology was motivated by and applied to the study of progression time to chronic kidney disease as a function of baseline kidney function and applied to the Atherosclerosis Risk in Communities study, a large epidemiological cohort study. This article has supplementary material online.

Polynomial spline estimation of partially linear single-index proportional hazards regression models

The Cox proportional hazards (PH) model usually assumes linearity of the covariates on the log hazard function, which may be violated because linearity cannot always be guaranteed. We propose a partially linear single-index proportional hazards regression model, which can model both linear and nonlinear covariate effects on the log hazard in the proportional hazards model. We adopt a polynomial spline smoothing technique to model the structured nonparametric single-index component for the nonlinear covariate effects. This method can reduce the dimensionality of the covariates being modeled, while, at the same time, can provide efficient estimates of the covariate effects. A two-step iterative algorithm to estimate the nonparametric component and the covariate effects is used for facilitating implementation. Asymptotic properties of the estimators are derived. Monte Carlo simulation studies are presented to compare the new method with the standard Cox linear PH model and some other comparable models. A case study with clinical trial data is presented for illustration.

Hazard regression for interval-censored data with penalized spline.

This article introduces a new approach for estimating the hazard function for possibly interval- and right-censored survival data. We weakly parameterize the log-hazard function with a piecewise-linear spline and provide a smoothed estimate of the hazard function by maximizing the penalized likelihood through a mixed model-based approach. We also provide a method to estimate the amount of smoothing from the data. We illustrate our approach with two well-known interval-censored data sets. Extensive numerical studies are conducted to evaluate the efficacy of the new procedure.

Regression splines for threshold selection in survival data analysis.

The Cox proportional hazards model restricts the hazard ratio to be linear in the covariates. A survival model based on data from a clinical trial is developed using spline functions with variable knots to estimate the log hazard function. Moreover, the main point of the method is that a knot, seen as free parameters for a piecewise linear spline, represents a break point in the log hazard function which may be interpreted as a threshold value. The likelihood ratio test is used to select the final model and to determine the threshold number for a covariate. Confidence intervals for these threshold values are computed by bootstrapping the data. Two examples illustrate the method.

A population pharmacokinetic-pharmacodynamic analysis of repeated measures time-to-event pharmacodynamic responses: the antiemetic effect of ondansetron.

This paper presents and illustrates methodology for specifying, estimating, and evaluating a predictive model for repeated measures time-to-event responses. The illustrative example specifies a model of the antiemetic effect vs. concentration relationship for the 5-HT3 antagonist ondansetron in the human ipecac model for emesis. A key part of this model is a time-dependent log hazard function for emesis that is increased by ipecac administration and decreased by ondansetron concentration. The model is fit using an approximate maximum likelihood method. The data consist of the time free of emeses and, for those individuals with emetic episodes, the time(s) of the episode(s). Model evaluation is accomplished using residual plots adapted to time-to-event data and a "posterior predictive check" wherein observed data statistics are compared to those obtained from data simulated from the fitted model. The ondansetron concentration required to obtain a 50% reduction in the hazard of emesis is estimated to be 1.4 +/- 0.2 ng/ml, and the rate constant for elimination of ipecac-induced hazard is 1.5 +/- 0.2 hr-1.

Hazard Regression with Interval-Censored Data

In a recent paper, Kooperberg, Stone, and Truong (1995a) introduced hazard regression (HARE), in which linear splines and their tensor products are used to estimate the conditional log-hazard function based on possibly censored, positive response data and one or more covariates. Model selection is carried out in an adaptive fashion using maximum likelihood estimation of the unknown coefficients, Rao and Wald statistics to carry out stepwise addition and deletion of basis functions, and the Bayesian Information Criterion (BIC) to select the final model. In the present paper, the HARE methodology is extended to accommodate interval-censored data, time-dependent covariates, and cubic splines. The presence of interval-censored data means that the log-likelihood function may no longer be concave, presenting additional numerical challenges. The extended methodology is applied to a data set containing both interval-censoring and time-dependent covariates. The new software will be available in a future release of S-Plus.

Hazard Regression

Abstract Linear splines and their tensor products are used to estimate the conditional log-hazard function based on possibly censored, positive response data and one or more covariates. An automatic procedure involving the maximum likelihood method, stepwise addition, stepwise deletion, and the Bayes Information Criterion is used to select the final model. The possible models contain proportional hazards models as a subclass, which makes it possible to diagnose departures from proportionality. Cubic splines and two additional log terms are incorporated into a similar model for the unconditional log-hazard function to allow for greater flexibility in the extreme tails. A user interface based on S is described.