Stochastic properties of the mixed accelerated hazard models

The accelerated hazard model in survival analysis assumes that the covariate effect acts the time scale of the baseline hazard rate. In this paper, we study the stochastic properties of the mixed accelerated hazard model since the covariate is considered basically unobservable. We build dependence structure between the population variable and the covariate, and also present some preservation properties. Using some well-known stochastic orders, we compare two mixed accelerated hazards models arising out of different choices of distributions for unobservable covariates or different baseline hazard rate functions.     

Weighted-bootstrap alignment of explanatory variables

Adjustment for covariates is a time-honored tool in statistical analysis and is often implemented by including the covariates that one intends to adjust as additional predictors in a model. This adjustment often does not work well when the underlying model is misspecified. We consider here the situation where we compare a response between two groups. This response may depend on a covariate for which the distribution differs between the two groups one intends to compare. This creates the potential that observed differences are due to differences in covariate levels rather than “genuine” population differences that cannot be explained by covariate differences. We propose a bootstrap-based adjustment method. Bootstrap weights are constructed with the aim of aligning bootstrap–weighted empirical distributions of the covariate between the two groups. Generally, the proposed weighted-bootstrap algorithm can be used to align or match the values of an explanatory variable as closely as desired to those of a given target distribution. We illustrate the proposed bootstrap adjustment method in simulations and in the analysis of data on the fecundity of historical cohorts of French-Canadian women.     

A Comparison of Analysis of Covariate-Adjusted Residuals and Analysis of Covariance

Various methods to control the influence of a covariate on a response variable are compared. These methods are ANOVA with or without homogeneity of variances (HOV) of errors and Kruskal–Wallis (K–W) tests on (covariate-adjusted) residuals and analysis of covariance (ANCOVA). Covariate-adjusted residuals are obtained from the overall regression line fit to the entire data set ignoring the treatment levels or factors. It is demonstrated that the methods on covariate-adjusted residuals are only appropriate when the regression lines are parallel and covariate means are equal for all treatments. Empirical size and power performance of the methods are compared by extensive Monte Carlo simulations. We manipulated the conditions such as assumptions of normality and HOV, sample size, and clustering of the covariates. The parametric methods on residuals and ANCOVA exhibited similar size and power when error terms have symmetric distributions with variances having the same functional form for each treatment, and covariates have uniform distributions within the same interval for each treatment. In such cases, parametric tests have higher power compared to the K–W test on residuals. When error terms have asymmetric distributions or have variances that are heterogeneous with different functional forms for each treatment, the tests are liberal with K–W test having higher power than others. The methods on covariate-adjusted residuals are severely affected by the clustering of the covariates relative to the treatment factors when covariate means are very different for treatments. For data clusters, ANCOVA method exhibits the appropriate level. However, such a clustering might suggest dependence between the covariates and the treatment factors, so makes ANCOVA less reliable as well.     

Bayesian methods for generalized linear models with covariates missing at random

The authors propose methods for Bayesian inference for generalized linear models with missing covariate data. They specify a parametric distribution for the covariates that is written as a sequence of one‐dimensional conditional distributions. They propose an informative class of joint prior distributions for the regression coefficients and the parameters arising from the covariate distributions. They examine the properties of the proposed prior and resulting posterior distributions. They also present a Bayesian criterion for comparing various models, and a calibration is derived for it. A detailed simulation is conducted and two real data sets are examined to demonstrate the methodology.     

Marginal and hazard ratio specific random data generation: Applications to semi-parametric bootstrapping

Cox's partial likelihood for censored time-to-event data can be interpreted as a permutation probability, whereby covariate values are permuted to the observed times-to-event and censoring times. This interpretation facilitates a simple method for jointly generating times-to-event and covariate tuples with considerable flexibility, including time dependence of the hazard ratio and specification of both the marginal time-to-event and covariate distributions. This interpretation also facilitates a method for semi-parametric bootstrapping of hazard ratio estimators.     

Covariate Order Tests for Covariate Effect

A new approach for constructing tests for association between a random right censored life time variable and a covariate is proposed. The basic idea is to first arrange the observations in increasing order of the covariate and then base the test on a certain point process defined by the observation times. Tests constructed by this approach are robust against outliers in the covariate values or misspecification of the covariate scale since they only use the ordering of the covariate. Of particular interest is a test based on the Anderson-Darling statistic. This test has good power properties both against monotonic and nonmonotonic dependencies between the covariate and the life time variable.     

Analysis of ordered covariate effects among groups with repeated measurements

Testing procedures for ordered covariate effects are developed in the repeated measures experiment. The maximum likelihood estimators of covariate effects under the ordered hypothesis are approximated by the isotonic regression of their unconstrained estimators. The asymptotic null distributions of the test statistics are chi-bar-square distributions which are mixtures of chi-square distributions. A Monte-Carlo simulation reveals that the proposed test for ordered covariate effects is seriously more powerful than the usual chi-square test that neglects the information on the order restriction. These testing methods are applied for analyzing the effect of vitamin E diet supplement on growth rate of animals.     

Bayesian Methods for Missing Covariates in Cure Rate Models

We propose methods for Bayesian inference for missing covariate data with a novel class of semi-parametric survival models with a cure fraction. We allow the missing covariates to be either categorical or continuous and specify a parametric distribution for the covariates that is written as a sequence of one dimensional conditional distributions. We assume that the missing covariates are missing at random (MAR) throughout. We propose an informative class of joint prior distributions for the regression coefficients and the parameters arising from the covariate distributions. The proposed class of priors are shown to be useful in recovering information on the missing covariates especially in situations where the missing data fraction is large. Properties of the proposed prior and resulting posterior distributions are examined. Also, model checking techniques are proposed for sensitivity analyses and for checking the goodness of fit of a particular model. Specifically, we extend the Conditional Predictive Ordinate (CPO) statistic to assess goodness of fit in the presence of missing covariate data. Computational techniques using the Gibbs sampler are implemented. A real data set involving a melanoma cancer clinical trial is examined to demonstrate the methodology.     

New tests for stochastic order with application to case control studies

It is shown that if a binary regression function is increasing then retrospective sampling induces a stochastic ordering of the covariate distributions among the responders, which we call cases, and the non-responders, which we call controls. We also show that if the covariate distributions are stochastically ordered then the regression function must be increasing. This means that testing whether the regression function is monotone is equivalent to testing whether the covariate distributions are stochastically ordered. Capitalizing on these new probabilistic observations we proceed to develop two new non-parametric tests for stochastic order. The new tests are based on either the maximally selected, or integrated, chi-bar statistic of order one. The tests are easy to compute and interpret and their large sampling distributions are easily found. Numerical comparisons show that they compare favorably with existing methods in both small and large samples. We emphasize that the new tests are applicable to any testing problem involving two stochastically ordered distributions.     

Bayesian latent variable models for clustered mixed outcomes

A general framework is proposed for modelling clustered mixed outcomes. A mixture of generalized linear models is used to describe the joint distribution of a set of underlying variables, and an arbitrary function relates the underlying variables to be observed outcomes. The model accommodates multilevel data structures, general covariate effects and distinct link functions and error distributions for each underlying variable. Within the framework proposed, novel models are developed for clustered multiple binary, unordered categorical and joint discrete and continuous outcomes. A Markov chain Monte Carlo sampling algorithm is described for estimating the posterior distributions of the parameters and latent variables. Because of the flexibility of the modelling framework and estimation procedure, extensions to ordered categorical outcomes and more complex data structures are straightforward. The methods are illustrated by using data from a reproductive toxicity study.     

Missing covariates in generalized linear models when the missing data mechanism is non-ignorable

We propose a method for estimating parameters in generalized linear models with missing covariates and a non-ignorable missing data mechanism. We use a multinomial model for the missing data indicators and propose a joint distribution for them which can be written as a sequence of one-dimensional conditional distributions, with each one-dimensional conditional distribution consisting of a logistic regression. We allow the covariates to be either categorical or continuous. The joint covariate distribution is also modelled via a sequence of one-dimensional conditional distributions, and the response variable is assumed to be completely observed. We derive the E- and M-steps of the EM algorithm with non-ignorable missing covariate data. For categorical covariates, we derive a closed form expression for the E- and M-steps of the EM algorithm for obtaining the maximum likelihood estimates (MLEs). For continuous covariates, we use a Monte Carlo version of the EM algorithm to obtain the MLEs via the Gibbs sampler. Computational techniques for Gibbs sampling are proposed and implemented. The parametric form of the assumed missing data mechanism itself is not `testable' from the data, and thus the non-ignorable modelling considered here can be viewed as a sensitivity analysis concerning a more complicated model. Therefore, although a model may have `passed' the tests for a certain missing data mechanism, this does not mean that we have captured, even approximately, the correct missing data mechanism. Hence, model checking for the missing data mechanism and sensitivity analyses play an important role in this problem and are discussed in detail. Several simulations are given to demonstrate the methodology. In addition, a real data set from a melanoma cancer clinical trial is presented to illustrate the methods proposed.     

Issues for stratified randomization based on a factor derived from a continuous baseline variable

Stratified randomization based on the baseline value of the primary analysis variable is common in clinical trial design. We illustrate from a theoretical viewpoint the advantage of such a stratified randomization to achieve balance of the baseline covariate. We also conclude that the estimator for the treatment effect is consistent when including both the continuous baseline covariate and the stratification factor derived from the baseline covariate. In addition, the analysis of covariance model including both the continuous covariate and the stratification factor is asymptotically no less efficient than including either only the continuous baseline value or only the stratification factor. We recommend that the continuous baseline covariate should generally be included in the analysis model. The corresponding stratification factor may also be included in the analysis model if one is not confident that the relationship between the baseline covariate and the response variable is linear. In spite of the above recommendation, one should always carefully examine relevant historical data to pre-specify the most appropriate analysis model for a perspective study.     

A distribution free test to detect general dependence between a response variable and a covariate in the presence of heteroscedastic treatment effects

In this paper, we present a test of independence between the response variable, which can be discrete or continuous, and a continuous covariate after adjusting for heteroscedastic treatment effects. The method involves first augmenting each pair of the data for all treatments with a fixed number of nearest neighbours as pseudo‐replicates. Then a test statistic is constructed by taking the difference of two quadratic forms. The statistic is equivalent to the average lagged correlations between the response and nearest neighbour local estimates of the conditional mean of response given the covariate for each treatment group. This approach effectively eliminates the need to estimate the nonlinear regression function. The asymptotic distribution of the proposed test statistic is obtained under the null and local alternatives. Although using a fixed number of nearest neighbours pose significant difficulty in the inference compared to that allowing the number of nearest neighbours to go to infinity, the parametric standardizing rate for our test statistics is obtained. Numerical studies show that the new test procedure has robust power to detect nonlinear dependency in the presence of outliers that might result from highly skewed distributions. The Canadian Journal of Statistics 38: 408–433; 2010 © 2010 Statistical Society of Canada     

Asymptotic Theory for the QMLE in GARCH-X Models With Stationary and Nonstationary Covariates

This article investigates the asymptotic properties of the Gaussian quasi-maximum-likelihood estimators (QMLE’s) of the GARCH model augmented by including an additional explanatory variable—the so-called GARCH-X model. The additional covariate is allowed to exhibit any degree of persistence as captured by its long-memory parameter d_x; in particular, we allow for both stationary and nonstationary covariates. We show that the QMLE’s of the parameters entering the volatility equation are consistent and mixed-normally distributed in large samples. The convergence rates and limiting distributions of the QMLE’s depend on whether the regressor is stationary or not. However, standard inferential tools for the parameters are robust to the level of persistence of the regressor with t-statistics following standard Normal distributions in large sample irrespective of whether the regressor is stationary or not. Supplementary materials for this article are available online.     

THE POTENTIAL FOR BIAS IN PRINCIPAL CAUSAL EFFECT ESTIMATION WHEN TREATMENT RECEIVED DEPENDS ON A KEY COVARIATE

Motivated by a potential-outcomes perspective, the idea of principal stratification has been widely recognized for its relevance in settings susceptible to posttreatment selection bias such as randomized clinical trials where treatment received can differ from treatment assigned. In one such setting, we address subtleties involved in inference for causal effects when using a key covariate to predict membership in latent principal strata. We show that when treatment received can differ from treatment assigned in both study arms, incorporating a stratum-predictive covariate can make estimates of the "complier average causal effect" (CACE) derive from observations in the two treatment arms with different covariate distributions. Adopting a Bayesian perspective and using Markov chain Monte Carlo for computation, we develop posterior checks that characterize the extent to which incorporating the pretreatment covariate endangers estimation of the CACE. We apply the method to analyze a clinical trial comparing two treatments for jaw fractures in which the study protocol allowed surgeons to overrule both possible randomized treatment assignments based on their clinical judgment and the data contained a key covariate (injury severity) predictive of treatment received.     

Bootstrapping the conditional copula

This paper is concerned with inference about the dependence or association between two random variables conditionally upon the given value of a covariate. A way to describe such a conditional dependence is via a conditional copula function. Nonparametric estimators for a conditional copula then lead to nonparametric estimates of conditional association measures such as a conditional Kendall's tau. The limiting distributions of nonparametric conditional copula estimators are rather involved. In this paper we propose a bootstrap procedure for approximating these distributions and their characteristics, and establish its consistency. We apply the proposed bootstrap procedure for constructing confidence intervals for conditional association measures, such as a conditional Blomqvist beta and a conditional Kendall's tau. The performances of the proposed methods are investigated via a simulation study involving a variety of models, ranging from models in which the dependence (weak or strong) on the covariate is only through the copula and not through the marginals, to models in which this dependence appears in both the copula and the marginal distributions. As a conclusion we provide practical recommendations for constructing bootstrap-based confidence intervals for the discussed conditional association measures.     

Conditions for consistent estimation in mixed-effects models for binary matched-pairs data

Parametric mixed-effects logistic models can provide effective analysis of binary matched-pairs data. Responses are assumed to follow a logistic model within pairs, with an intercept which varies across pairs according to a specified family of probability distributions G. In this paper we give necessary and sufficient conditions for consistent covariate effect estimation and present a geometric view of estimation which shows that when the assumed family of mixture distributions is rich enough, estimates of the effect of the binary covariate are typically consistent. The geometric view also shows that under the conditions for consistent estimation, the mixed-model estimator is identical to the familar conditional-likelihood estimator for matched pairs. We illustrate the findings with some examples.     

Unmasking the Trend Sought by Jonckheere-Type Tests for Right-Censored Data

Medical and epidemiological studies often involve groups of subjects associated with increasing levels of exposure to a risk factor. Survival of the groups is expected to follow the same order as the level of exposure. Formal tests for this trend fall into the regression framework if one knows what function of exposure to use as a covariate. When unknown, a linear function of exposure level is often used. Jonckheere-type tests for trend have generated continued interest largely because they do not require specification of a covariate. This paper shows that the Jonckheere-type test statistics are special cases of a generalized linear rank statistic with time-dependent covariates which unfortunately depend on the initial group sizes and censoring distributions. Using asymptotic relative efficiency calculations, the Jonckheere tests are compared to standard linear rank tests based on a linear covariate over a spectrum of shapes for the true trend.     

Generalized symmetrical partial linear model

In this work, we propose a new model called generalized symmetrical partial linear model, based on the theory of generalized linear models and symmetrical distributions. In our model the response variable follows a symmetrical distribution such a normal, Student-t, power exponential, among others. Following the context of generalized linear models we consider replacing the traditional linear predictors by the more general predictors in whose case one covariate is related with the response variable in a non-parametric fashion, that we do not specified the parametric function. As an example, we could imagine a regression model in which the intercept term is believed to vary in time or geographical location. The backfitting algorithm is used for estimating the parameters of the proposed model. We perform a simulation study for assessing the behavior of the penalized maximum likelihood estimators. We use the quantile residuals for checking the assumption of the model. Finally, we analyzed real data set related with pH rivers in Ireland.     

Testing for Covariate Effect in the Cox Proportional Hazards Regression Model

This article presents methods for testing covariate effect in the Cox proportional hazards model based on Kullback–Leibler divergence and Renyi's information measure. Renyi's measure is referred to as the information divergence of order γ (γ ≠ 1) between two distributions. In the limiting case γ → 1, Renyi's measure becomes Kullback–Leibler divergence. In our case, the distributions correspond to the baseline and one possibly due to a covariate effect. Our proposed statistics are simple transformations of the parameter vector in the Cox proportional hazards model, and are compared with the Wald, likelihood ratio and score tests that are widely used in practice. Finally, the methods are illustrated using two real-life data sets.