Additive risk models for survival data with high-dimensional covariates

As a useful alternative to Cox's proportional hazard model, the additive risk model assumes that the hazard function is the sum of the baseline hazard function and the regression function of covariates. This article is concerned with estimation and prediction for the additive risk models with right censored survival data, especially when the dimension of the covariates is comparable to or larger than the sample size. Principal component regression is proposed to give unique and numerically stable estimators. Asymptotic properties of the proposed estimators, component selection based on the weighted bootstrap, and model evaluation techniques are discussed. This approach is illustrated with analysis of the primary biliary cirrhosis clinical data and the diffuse large B-cell lymphoma genomic data. It is shown that this methodology is numerically stable and effective in dimension reduction, while still being able to provide satisfactory prediction and classification results.     

Attenuation caused by infrequently updated covariates in survival analysis

This paper deals with hazard regression models for survival data with time-dependent covariates consisting of updated quantitative measurements. The main emphasis is on the Cox proportional hazards model but also additive hazard models are discussed. Attenuation of regression coefficients caused by infrequent updating of covariates is evaluated using simulated data mimicking our main example, the CSL1 liver cirrhosis trial. We conclude that the degree of attenuation depends on the type of stochastic process describing the time-dependent covariate and that attenuation may be substantial for an Ornstein-Uhlenbeck process. Also trends in the covariate combined with non-synchronous updating may cause attenuation. Simple methods to adjust for infrequent updating of covariates are proposed and compared to existing techniques using both simulations and the CSL1 data. The comparison shows that while existing, more complicated methods may work well with frequent updating of covariates the simpler techniques may have advantages in larger data sets with infrequent updatings.     

Maximum likelihood estimation in the additive hazards model

The additive hazards model specifies the effect of covariates on the hazard in an additive way, in contrast to the popular Cox model, in which it is multiplicative. As the non-parametric model, additive hazards offer a very flexible way of modeling time-varying covariate effects. It is most commonly estimated by ordinary least squares. In this paper, we consider the case where covariates are bounded, and derive the maximum likelihood estimator under the constraint that the hazard is non-negative for all covariate values in their domain. We show that the maximum likelihood estimator may be obtained by separately maximizing the log-likelihood contribution of each event time point, and we show that the maximizing problem is equivalent to fitting a series of Poisson regression models with an identity link under non-negativity constraints. We derive an analytic solution to the maximum likelihood estimator. We contrast the maximum likelihood estimator with the ordinary least-squares estimator in a simulation study and show that the maximum likelihood estimator has smaller mean squared error than the ordinary least-squares estimator. An illustration with data on patients with carcinoma of the oropharynx is provided.     

Confidence bands for cumulative incidence curves under the additive risk model

In the context of competing risks, the cumulative incidence function is often used to summarize the cause-specific failure-time data. As an alternative to the proportional hazards model, the additive risk model is used to investigate covariate effects by specifying that the subject-specific hazard function is the sum of a baseline hazard function and a regression function of covariates. Based on such a formulation, we present an approach to constructing simultaneous confidence intervals for the cause-specific cumulative incidence function of patients with given risk factors. A melanoma data set is used for the purpose of illustration.     

Multiple Imputation Methods for Estimating Regression Coefficients in the Competing Risks Model with Missing Cause of Failure

We propose a method to estimate the regression coefficients in a competing risks model where the cause-specific hazard for the cause of interest is related to covariates through a proportional hazards relationship and when cause of failure is missing for some individuals. We use multiple imputation procedures to impute missing cause of failure, where the probability that a missing cause is the cause of interest may depend on auxiliary covariates, and combine the maximum partial likelihood estimators computed from several imputed data sets into an estimator that is consistent and asymptotically normal. A consistent estimator for the asymptotic variance is also derived. Simulation results suggest the relevance of the theory in finite samples. Results are also illustrated with data from a breast cancer study.     

Two-component mixture cure rate model with spline estimated nonparametric components

Summary In some survival analysis of medical studies, there are often long-term survivors who can be considered as permanently cured. The goals in these studies are to estimate the noncured probability of the whole population and the hazard rate of the susceptible subpopulation. When covariates are present as often happens in practice, to understand covariate effects on the noncured probability and hazard rate is of equal importance. The existing methods are limited to parametric and semiparametric models. We propose a two-component mixture cure rate model with nonparametric forms for both the cure probability and the hazard rate function. Identifiability of the model is guaranteed by an additive assumption that allows no time-covariate interactions in the logarithm of hazard rate. Estimation is carried out by an expectation-maximization algorithm on maximizing a penalized likelihood. For inferential purpose, we apply the Louis formula to obtain point-wise confidence intervals for noncured probability and hazard rate. Asymptotic convergence rates of our function estimates are established. We then evaluate the proposed method by extensive simulations. We analyze the survival data from a melanoma study and find interesting patterns for this study.     

Generalized additive models with interval-censored data and time-varying covariates: application to human immunodeficiency virus infection in hemophiliacs

We describe a method for extending smooth nonparametric modeling methods to time-to-event data where the event may be known only to lie within a window of time. Maximum penalized likelihood is used to fit a discrete proportional hazards model that also models the baseline hazard, and left-truncation and time-varying covariates are accommodated. The implementation follows generalized additive modeling conventions, allowing both parametric and smooth terms and specifying the amount of smoothness in terms of the effective degrees of freedom. We illustrate the method on a well-known interval-censored data set on time of human immunodeficiency virus infection in a multicenter study of hemophiliacs. The ability to examine time-varying covariates, not available with previous methods, allows detection and modeling of nonproportional hazards and use of a time-varying covariate that fits the data better and is more plausible than a fixed alternative.     

Maximum likelihood analysis of logistic regression models with incomplete covariate data and auxiliary information

This article presents a new method for maximum likelihood estimation of logistic regression models with incomplete covariate data where auxiliary information is available. This auxiliary information is extraneous to the regression model of interest but predictive of the covariate with missing data. Ibrahim (1990, Journal of the American Statistical Association 85, 765-769) provides a general method for estimating generalized linear regression models with missing covariates using the EM algorithm that is easily implemented when there is no auxiliary data. Vach (1997, Statistics in Medicine 16, 57-72) describes how the method can be extended when the outcome and auxiliary data are conditionally independent given the covariates in the model. The method allows the incorporation of auxiliary data without making the conditional independence assumption. We suggest tests of conditional independence and compare the performance of several estimators in an example concerning mental health service utilization in children. Using an artificial dataset, we compare the performance of several estimators when auxiliary data are available.     

Functional additive models for optimizing individualized treatment rules

A novel functional additive model is proposed, which is uniquely modified and constrained to model nonlinear interactions between a treatment indicator and a potentially large number of functional and/or scalar pretreatment covariates. The primary motivation for this approach is to optimize individualized treatment rules based on data from a randomized clinical trial. We generalize functional additive regression models by incorporating treatment-specific components into additive effect components. A structural constraint is imposed on the treatment-specific components in order to provide a class of additive models with main effects and interaction effects that are orthogonal to each other. If primary interest is in the interaction between treatment and the covariates, as is generally the case when optimizing individualized treatment rules, we can thereby circumvent the need to estimate the main effects of the covariates, obviating the need to specify their form and thus avoiding the issue of model misspecification. The methods are illustrated with data from a depression clinical trial with electroencephalogram functional data as patients' pretreatment covariates.     

Fitting semiparametric additive hazards models using standard statistical software

The Cox proportional hazards model has become the standard in biomedical studies, particularly for settings in which the estimation covariate effects (as opposed to prediction) is the primary objective. In spite of the obvious flexibility of this approach and its wide applicability, the model is not usually chosen for its fit to the data, but by convention and for reasons of convenience. It is quite possible that the covariates add to, rather than multiply the baseline hazard, making an additive hazards model a more suitable choice. Typically, proportionality is assumed, with the potential for additive covariate effects not evaluated or even seriously considered. Contributing to this phenomenon is the fact that many popular software packages (e.g., SAS, S-PLUS/R) have standard procedures to fit the Cox model (e.g., proc phreg, coxph), but as of yet no analogous procedures to fit its additive analog, the Lin and Ying (1994) semiparametric additive hazards model. In this article, we establish the connections between the Lin and Ying (1994) model and both Cox and least squares regression. We demonstrate how SAS's phreg and reg procedures may be used to fit the additive hazards model, after some straightforward data manipulations. We then apply the additive hazards model to examine the relationship between Model for End-stage Liver Disease (MELD) score and mortality among patients wait-listed for liver transplantation.     

Adjustment for Missingness Using Auxiliary Information in Semiparametric Regression

Summary .  In this article, we study the estimation of mean response and regression coefficient in semiparametric regression problems when response variable is subject to nonrandom missingness. When the missingness is independent of the response conditional on high-dimensional auxiliary information, the parametric approach may misspecify the relationship between covariates and response while the nonparametric approach is infeasible because of the curse of dimensionality. To overcome this, we study a model-based approach to condense the auxiliary information and estimate the parameters of interest nonparametrically on the condensed covariate space. Our estimators possess the double robustness property, i.e., they are consistent whenever the model for the response given auxiliary covariates or the model for the missingness given auxiliary covariate is correct. We conduct a number of simulations to compare the numerical performance between our estimators and other existing estimators in the current missing data literature, including the propensity score approach and the inverse probability weighted estimating equation. A set of real data is used to illustrate our approach.     

Some interrelationships among the regression coefficient estimates arising in a class of models appropriate to response-time data.

Interrelationships among three response-time models which incorporate covariate information are explored. The most general of these models is the logistic-exponential in which the log odds of the probability of responding in a fixed interval is assumed to be a linear function of the covariates; this model includes a parameter W for the width of discrete time intervals in which responses occur. As W leads to O this model is equivalent to a continuous time exponential model in which the log hazard is linear in the covariates. As W leads to infininity it is equivalent to a continuous time exponential model in which the hazard itself is a linear function of the covariates. This second model was fitted to the data used in an earlier publication describing the logistic exponential model, and very close agreement of the estimates of the regression coefficients is demonstrated.     

Penalized partial likelihood for frailties and smoothing splines in time to first insemination models for dairy cows

In many epidemiological studies time to event data are clustered and the physiological relationship between (time-dependent) covariates and the log hazard is often not linear as assumed in the Cox model. Introducing frailties in the Cox model can account for the clustering of the data and smoothing splines can be used to describe nonlinear relations. These two extensions of the Cox model are introduced jointly and it is shown how penalized partial likelihood techniques can be used to fit the extended model. We demonstrate the need for such a model to study the relation between the physiological covariates milk ureum and protein concentration and the log hazard of first insemination in dairy cows, with the farms as clusters.     

A partly parametric additive risk model

Aalen's additive risk model allows the influence of each covariateto vary separately over time. Although allowing greater flexibilityof temporal structure than a Cox model, Aalen's model is morelimited in the number of covariates it can handle. We introducea partly parametric version of Aalen's model in which the influenceof only a few covariates varies nonparametrically over time,and that of the remaining covariates is constant. Efficientprocedures for fitting this new model are developed and studied.The approach is applied to data from the Medical Research Council'smyelomatosis trials.     

Use of Two-Part Regression Calibration Model to Correct for Measurement Error in Episodically Consumed Foods in a Single-Replicate Study Design: EPIC Case Study

In epidemiologic studies, measurement error in dietary variables often attenuates association between dietary intake and disease occurrence. To adjust for the attenuation caused by error in dietary intake, regression calibration is commonly used. To apply regression calibration, unbiased reference measurements are required. Short-term reference measurements for foods that are not consumed daily contain excess zeroes that pose challenges in the calibration model. We adapted two-part regression calibration model, initially developed for multiple replicates of reference measurements per individual to a single-replicate setting. We showed how to handle excess zero reference measurements by two-step modeling approach, how to explore heteroscedasticity in the consumed amount with variance-mean graph, how to explore nonlinearity with the generalized additive modeling (GAM) and the empirical logit approaches, and how to select covariates in the calibration model. The performance of two-part calibration model was compared with the one-part counterpart. We used vegetable intake and mortality data from European Prospective Investigation on Cancer and Nutrition (EPIC) study. In the EPIC, reference measurements were taken with 24-hour recalls. For each of the three vegetable subgroups assessed separately, correcting for error with an appropriately specified two-part calibration model resulted in about three fold increase in the strength of association with all-cause mortality, as measured by the log hazard ratio. Further found is that the standard way of including covariates in the calibration model can lead to over fitting the two-part calibration model. Moreover, the extent of adjusting for error is influenced by the number and forms of covariates in the calibration model. For episodically consumed foods, we advise researchers to pay special attention to response distribution, nonlinearity, and covariate inclusion in specifying the calibration model.     

Analysis of covariance with incomplete data via semiparametric model transformations

We propose a method for fitting semiparametric models such as the proportional hazards (PH), additive risks (AR), and proportional odds (PO) models. Each of these semiparametric models implies that some transformation of the conditional cumulative hazard function (at each t) depends linearly on the covariates. The proposed method is based on nonparametric estimation of the conditional cumulative hazard function, forming a weighted average over a range of t-values, and subsequent use of least squares to estimate the parameters suggested by each model. An approximation to the optimal weight function is given. This allows semiparametric models to be fitted even in incomplete data cases where the partial likelihood fails (e.g., left censoring, right truncation). However, the main advantage of this method rests in the fact that neither the interpretation of the parameters nor the validity of the analysis depend on the appropriateness of the PH or any of the other semiparametric models. In fact, we propose an integrated method for data analysis where the role of the various semiparametric models is to suggest the best fitting transformation. A single continuous covariate and several categorical covariates (factors) are allowed. Simulation studies indicate that the test statistics and confidence intervals have good small-sample performance. A real data set is analyzed.     

Modeling clustered, discrete, or grouped time survival data with covariates

We have developed methods for modeling discrete or grouped time, right-censored survival data collected from correlated groups or clusters. We assume that the marginal hazard of failure for individual items within a cluster is specified by a linear log odds survival model and the dependence structure is based on a gamma frailty model. The dependence can be modeled as a function of cluster-level covariates. Likelihood equations for estimating the model parameters are provided. Generalized estimating equations for the marginal hazard regression parameters and pseudolikelihood methods for estimating the dependence parameters are also described. Data from two clinical trials are used for illustration purposes.     

The Analysis of Life Table and Follow-up Data with Covariates Using Poisson Regression Model

The Poisson regression model for the analysis of life table and follow-up data with covariates is presented. An example is presented to show how this technique can be used to construct a parsimonious model which describes a set of survival data. All parameters in the model, the hazard and survival functions are estimated by maximum likelihood.     

Multivariate survival models induced by genetic frailties,with application to linkage analysis

We derive a multivariate survival model for age of onset data of a sibship from an additive genetic gamma frailty model constructed basing on the inheritance vectors, and investigate the properties of this model. Based on this model, we propose a retrospective likelihood approach for genetic linkage analysis using sibship data. This test is an allele-sharing-based test, and does not require specification of genetic models or the penetrance functions. This new approach can incorporate both affected and unaffected sibs, environmental covariates and age of onset or age at censoring information and, therefore, provides a practical solution for mapping genes for complex diseases with variable age of onset. Small simulation study indicates that the proposed method performs better than the commonly used allele-sharing-based methods for linkage analysis, especially when the population disease rate is high. We applied this method to a type 1 diabetes sib pair data set and a small breast cancer data set. Both simulated and real data sets also indicate that the method is relatively robust to the misspecification to the baseline hazard function.     

Mixture Hazard Models for Lifetime Data

We propose a general family of mixture hazard models to analyze lifetime data associated with bathtub and multimodal hazard functions. With this model we have a great flexibility for fitting lifetime data. Its version with covariates has the proportional hazard and the accelerated failure time models as special cases. A Bayesian analysis is presented for the model using informative priors, using sampling‐based approaches to perform the Bayesian computations. A real example with a medical data illustrates the methodology.