Choice of Parametric Accelerated Life and Proportional Hazards Models for Survival Data: Asymptotic Results

We discuss the impact of misspecifying fully parametric proportional hazards and accelerated life models. For the uncensored case, misspecified accelerated life models give asymptotically unbiased estimates of covariate effect, but the shape and scale parameters depend on the misspecification. The covariate, shape and scale parameters differ in the censored case. Parametric proportional hazards models do not have a sound justification for general use: estimates from misspecified models can be very biased, and misleading results for the shape of the hazard function can arise. Misspecified survival functions are more biased at the extremes than the centre. Asymptotic and first order results are compared. If a model is misspecified, the size of Wald tests will be underestimated. Use of the sandwich estimator of standard error gives tests of the correct size, but misspecification leads to a loss of power. Accelerated life models are more robust to misspecification because of their log-linear form. In preliminary data analysis, practitioners should investigate proportional hazards and accelerated life models; software is readily available for several such models.     

Weighting methods for ties between event times and covariate change times

Recent work has shown that the presence of ties between an outcome event and the time that a binary covariate changes or jumps can lead to biased estimates of regression coefficients in the Cox proportional hazards model. One proposed solution is the Equally Weighted method. The coefficient estimate of the Equally Weighted method is defined to be the average of the coefficient estimates of the Jump Before Event method and the Jump After Event method, where these two methods assume that the jump always occurs before or after the event time, respectively. In previous work, the bootstrap method was used to estimate the standard error of the Equally Weighted coefficient estimate. However, the bootstrap approach was computationally intensive and resulted in overestimation. In this article, two new methods for the estimation of the Equally Weighted standard error are proposed. Three alternative methods for estimating both the regression coefficient and the corresponding standard error are also proposed. All the proposed methods are easy to implement. The five methods are investigated using a simulation study and are illustrated using two real datasets.     

Spatial measurement error in infectious disease models

Individual-level models (ILMs) for infectious disease can be used to model disease spread between individuals while taking into account important covariates. One important covariate in determining the risk of infection transfer can be spatial location. At the same time, measurement error is a concern in many areas of statistical analysis, and infectious disease modelling is no exception. In this paper, we are concerned with the issue of measurement error in the recorded location of individuals when using a simple spatial ILM to model the spread of disease within a population. An ILM that incorporates spatial location random effects is introduced within a hierarchical Bayesian framework. This model is tested upon both simulated data and data from the UK 2001 foot-and-mouth disease epidemic. The ability of the model to successfully identify both the spatial infection kernel and the basic reproduction number (R ₀) of the disease is tested.     

Approaches to regression analysis with multiple measurements from individual sampling units

Application of ordinary least-squares regression to data sets which contain multiple measurements from individual sampling units produces an unbiased estimator of the parameters but a biased estimator of the covariance matrix of the parameter estimates. The present work considers a random coefficient, linear model to deal with such data sets: this model permits many senses in which multiple measurements are taken from a sampling unit, not just when it is measured at several times. Three procedures to estimate the covariance matrix of the error term of the model are considered. Given these, three procedures to estimate the parameters of the model and their covariance matrix are considered; these are ordinary least-squares, generalized least-squares, and an adjusted ordinary least-squares procedure which produces an unbiased estimator of the covariance matrix of the parameters with small samples. These various procedures are compared in simulation studies using three examples from the biological literature. The possibility of testing hypotheses about the vector of parameters is also considered. It is found that all three procedures for regression estimation produce estimators of the parameters with bias of no practical consequence, Both generalized least-squares and adjusted ordinary least-squares generally produce estimators of the covariance matrix of the parameter estimates with bias of no practical consequence, while ordinary least-squares produces a negatively biased estimator. Neither ordinary nor generalized least-squares provide satisfactory hypothesis tests of the vector of parameter estimates. It is concluded that adjusted ordinary least-squares, when applied with either of two of the procedures used to estimate the error coveriance matrix, shows promise for practical application with data sets of the nature considered here.     

Structural Equation Models for Dealing With Spatial Confounding

In regression analyses of spatially structured data, it is common practice to introduce spatially correlated random effects into the regression model to reduce or even avoid unobserved variable bias in the estimation of other covariate effects. If besides the response the covariates are also spatially correlated, the spatial effects may confound the effect of the covariates or vice versa. In this case, the model fails to identify the true covariate effect due to multicollinearity. For highly collinear continuous covariates, path analysis and structural equation modeling techniques prove to be helpful to disentangle direct covariate effects from indirect covariate effects arising from correlation with other variables. This work discusses the applicability of these techniques in regression setups, where spatial and covariate effects coincide at least partly and classical geoadditive models fail to separate these effects. Supplementary materials for this article are available online.     

A Bayesian Adjustment for Covariate Misclassification with Correlated Binary Outcome Data

Estimated associations between an outcome variable and misclassified covariates tend to be biased when the methods of estimation that ignore the classification error are applied. Available methods to account for misclassification often require the use of a validation sample (i.e. a gold standard). In practice, however, such a gold standard may be unavailable or impractical. We propose a Bayesian approach to adjust for misclassification in a binary covariate in the random effect logistic model when a gold standard is not available. This Markov Chain Monte Carlo (MCMC) approach uses two imperfect measures of a dichotomous exposure under the assumptions of conditional independence and non-differential misclassification. A simulated numerical example and a real clinical example are given to illustrate the proposed approach. Our results suggest that the estimated log odds of inpatient care and the corresponding standard deviation are much larger in our proposed method compared with the models ignoring misclassification. Ignoring misclassification produces downwardly biased estimates and underestimate uncertainty.     

On Confounding, Prediction and Efficiency in the Analysis of Longitudinal and Cross-sectional Clustered Data

Abstract.  In the analysis of clustered and/or longitudinal data, it is usually desirable to ignore covariate information for other cluster members as well as future covariate information when predicting outcome for a given subject at a given time. This can be accomplished through con-ditional mean models which merely condition on the considered subject's covariate history at each time. Pepe & Anderson (Commun. Stat. Simul. Comput. 23, 1994 , 939) have shown that ordinary generalized estimating equations may yield biased estimates for the parameters in such models, but that valid inferences can be guaranteed by using a diagonal working covariance matrix in the equations. In this paper, we provide insight into the nature of this problem by uncovering substantive data-generating mechanisms under which such biases will result. We then propose a class of asymptotically unbiased estimators for the parameters indexing the suggested conditional mean models. In addition, we provide a representation for the efficient estimator in our class, which attains the semi-parametric efficiency bound under the model, along with an efficient algorithm for calculating it. This algorithm is easy to apply and may realize major efficiency improvements as demonstrated through simulation studies. The results suggest ways to improve the efficiency of inverse-probability-of-treatment estimators which adjust for time-varying confounding, and are used to estimate the effect of discontinuing highly active anti-retroviral therapy (HAART) on viral load in HIV-infected patients.     

Estimation following selection of the largest of two normal means

This paper considers the problem of estimation when one of a number of populations, assumed normal with known common variance, is selected on the basis of it having the largest observed mean. Conditional on selection of the population, the observed mean is a biased estimate of the true mean. This problem arises in the analysis of clinical trials in which selection is made between a number of experimental treatments that are compared with each other either with or without an additional control treatment. Attempts to obtain approximately unbiased estimates in this setting have been proposed by Shen [2001. An improved method of evaluating drug effect in a multiple dose clinical trial. Statist. Medicine 20, 1913–1929] and Stallard and Todd [2005. Point estimates and confidence regions for sequential trials involving selection. J. Statist. Plann. Inference 135, 402–419]. This paper explores the problem in the simple setting in which two experimental treatments are compared in a single analysis. It is shown that in this case the estimate of Stallard and Todd is the maximum-likelihood estimate (m.l.e.), and this is compared with the estimate proposed by Shen. In particular, it is shown that the m.l.e. has infinite expectation whatever the true value of the mean being estimated. We show that there is no conditionally unbiased estimator, and propose a new family of approximately conditionally unbiased estimators, comparing these with the estimators suggested by Shen.     

Bootstrap procedures in a spatial-temporal model

Two bootstrap procedures are introduced into the hybrid of the backfitting algorithm and the Cochrane–Orcutt procedure in the estimation of a spatial-temporal model. The use of time blocks of consecutive observations in resampling steps proved to be optimal in terms of stability and efficiency of estimates. Between iterations, there were minimal changes in the empirical distributions of the parameter estimates associated with the covariate and temporal effects indicating convergence of the algorithm. Crop yield data are used to illustrate the proposed methods.

The simulation study indicated that prediction error from the fitted model (estimated from either Method 1 or Method 2) is very low. Also, the prediction error is relatively robust to the number of spatial units and the number of time points.     

Estimation of dependence between paired correlated failure times in the presence of covariate measurement error

Summary. In many biomedical studies, covariates are subject to measurement error. Although it is well known that the regression coefficients estimators can be substantially biased if the measurement error is not accommodated, there has been little study of the effect of covariate measurement error on the estimation of the dependence between bivariate failure times. We show that the dependence parameter estimator in the Clayton–Oakes model can be considerably biased if the measurement error in the covariate is not accommodated. In contrast with the typical bias towards the null for marginal regression coefficients, the dependence parameter can be biased in either direction. We introduce a bias reduction technique for the bivariate survival function in copula models while assuming an additive measurement error model and replicated measurement for the covariates, and we study the large and small sample properties of the dependence parameter estimator proposed.     

THE EXAMINATION AND ANALYSIS OF RESIDUALS FOR SOME BIASED ESTIMATORS IN LINEAR REGRESSION

In the presence of collinearity certain biased estimation procedures like ridge regression, generalized inverse estimator, principal component regression, Liu estimator, or improved ridge and Liu estimators are used to improve the ordinary least squares (OLS) estimates in the linear regression model. In this paper new biased estimator (Liu estimator), almost unbiased (improved) Liu estimator and their residuals will be analyzed and compared with OLS residuals in terms of mean-squared error.     

Bayesian inference for pairwise interacting point processes

Pairwise interacting point processes are commonly used to model spatial point patterns. To perform inference, the established frequentist methods can produce good point estimates when the interaction in the data is moderate, but some methods may produce severely biased estimates when the interaction in strong. Furthermore, because the sampling distributions of the estimates are unclear, interval estimates are typically obtained by parametric bootstrap methods. In the current setting however, the behavior of such estimates is not well understood. In this article we propose Bayesian methods for obtaining inferences in pairwise interacting point processes. The requisite application of Markov chain Monte Carlo (MCMC) techniques is complicated by an intractable function of the parameters in the likelihood. The acceptance probability in a Metropolis-Hastings algorithm involves the ratio of two likelihoods evaluated at differing parameter values. The intractable functions do not cancel, and hence an intractable ratio r must be estimated within each iteration of a Metropolis-Hastings sampler. We propose the use of importance sampling techniques within MCMC to address this problem. While r may be estimated by other methods, these, in general, are not readily applied in a Bayesian setting. We demonstrate the validity of our importance sampling approach with a small simulation study. Finally, we analyze the Swedish pine sapling dataset (Strand 1972) and contrast the results with those in the literature.     

A flexible and efficient spatial interpolator for radar rainfall estimation

A key challenge in rainfall estimation is spatio-temporal variablility. Weather radars are used to estimate precipitation with high spatial and temporal resolution. Due to the inherent errors in radar estimates, spatial interpolation has been often employed to calibrate the estimates. Kriging is a simple and popular spatial interpolation method, but the method has several shortcomings. In particular, the prediction is quite unstable and often fails to be performed when sample size is small. In this paper, we proposed a flexible and efficient spatial interpolator for radar rainfall estimation, with several advantages over kriging. The method is illustrated using a real-world data set.     

Using Auxiliary Time-Dependent Covariates to Recover Information in Nonparametric Testing with Censored Data

Murrayand Tsiatis (1996) described a weighted survival estimate thatincorporates prognostic time-dependent covariate informationto increase the efficiency of estimation. We propose a test statisticbased on the statistic of Pepe and Fleming (1989, 1991) thatincorporates these weighted survival estimates. As in Pepe andFleming, the test is an integrated weighted difference of twoestimated survival curves. This test has been shown to be effectiveat detecting survival differences in crossing hazards settingswhere the logrank test performs poorly. This method uses stratifiedlongitudinal covariate information to get more precise estimatesof the underlying survival curves when there is censored informationand this leads to more powerful tests. Another important featureof the test is that it remains valid when informative censoringis captured by the incorporated covariate. In this case, thePepe-Fleming statistic is known to be biased and should not beused. These methods could be useful in clinical trials with heavycensoring that include collection over time of covariates, suchas laboratory measurements, that are prognostic of subsequentsurvival or capture information related to censoring.     

Estimation Bias in the First-Order Autoregressive Model and Its Impact on Predictions and Prediction Intervals

The least squares estimate of the autoregressive coefficient in the AR(1) model is known to be biased towards zero, especially for parameters close to the stationarity boundary. Several methods for correcting the autoregressive parameter estimate for the bias have been suggested. Using simulations, we study the bias and the mean square error of the least squares estimate and the bias-corrections proposed by Kendall and Quenouille.

We also study the mean square forecast error and the coverage of the 95% prediction interval when using the biased least squares estimate or one of its bias-corrected versions. We find that the estimation bias matters little for point forecasts, but that it affects the coverage of the prediction intervals. Prediction intervals for forecasts more than one step ahead, when calculated with the biased least squares estimate, are too narrow.     

Relative efficiency of three estimators in a polynomial regression with measurement errors

In a polynomial regression with measurement errors in the covariate, the latter being supposed to be normally distributed, one has (at least) three ways to estimate the unknown regression parameters: one can apply ordinary least squares (OLS) to the model without regard to the measurement error or one can correct for the measurement error, either by correcting the estimating equation (ALS) or by correcting the mean and variance functions of the dependent variable, which is done by conditioning on the observable, error ridden, counter part of the covariate (SLS). While OLS is biased, the other two estimators are consistent. Their asymptotic covariance matrices and thus their relative efficiencies can be compared to each other, in particular for the case of a small measurement error variance. In this case, it appears that ALS and SLS become almost equally efficient, even when they differ noticeably from OLS.     

A new class of measurement-error models,with applications to dietary data

Measurement-error modelling occurs when one cannot observe a covariate, but instead has possibly replicated surrogate versions of this covariate measured with error. The vast majority of the literature in measurement-error modelling assumes (typically with good reason) that given the value of the true but unobserved (latent) covariate, the replicated surrogates are unbiased for latent covariate and conditionally independent. In the area of nutritional epidemiology, there is some evidence from biomarker studies that this simple conditional independence model may break down due to two causes: (a) systematic biases depending on a person's body mass index, and (b) an additional random component of bias, so that the error structure is the same as a one-way random-effects model. We investigate this problem in the context of (1) estimating distribution of usual nutrient intake, (2) estimating the correlation between a nutrient instrument and usual nutrient intake, and (3) estimating the true relative risk from an estimated relative risk using the error-prone covariate. While systematic bias due to body mass index appears to have little effect, the additional random effect in the variance structure is shown to have a potentially important effect on overall results, both on corrections for relative risk estimates and in estimating the distribution of usual nutrient intake. However, the effect of dietary measurement error on both factors is shown via examples to depend strongly on the data set being used. Indeed, one of our data sets suggests that dietary measurement error may be masking a strong risk of fat on breast cancer, while for a second data set this masking is not so clear. Until further understanding of dietary measurement is available, measurement-error corrections must be done on a study-specific basis, sensitivity analyses should be conducted, and even then results of nutritional epidemiology studies relating diet to disease risk should be interpreted cautiously.     

Multiple imputation for continuous variables using a Bayesian principal component analysis

ABSTRACT

We propose a multiple imputation method based on principal component analysis (PCA) to deal with incomplete continuous data. To reflect the uncertainty of the parameters from one imputation to the next, we use a Bayesian treatment of the PCA model. Using a simulation study and real data sets, the method is compared to two classical approaches: multiple imputation based on joint modelling and on fully conditional modelling. Contrary to the others, the proposed method can be easily used on data sets where the number of individuals is less than the number of variables and when the variables are highly correlated. In addition, it provides unbiased point estimates of quantities of interest, such as an expectation, a regression coefficient or a correlation coefficient, with a smaller mean squared error. Furthermore, the widths of the confidence intervals built for the quantities of interest are often smaller whilst ensuring a valid coverage.     

Robust inference for bivariate point processes

In the analysis of recurrent events where the primary interest lies in studying covariate effects on the expected number of events occurring over a period of time, it is appealing to base models on the cumulative mean function (CMF) of the processes (Lawless & Nadeau 1995). In many chronic diseases, however, more than one type of event is manifested. Here we develop a robust inference procedure for joint regression models for the CMFs arising from a bivariate point process. Consistent parameter estimates with robust variance estimates are obtained via unbiased estimating functions for the CMFs. In most situations, the covariance structure of the bivariate point processes is difficult to specify correctly, but when it is known, an optimal estimating function for the CMFs can be obtained. As a convenient model for more general settings, we suggest the use of the estimating functions arising from bivariate mixed Poisson processes. Simulation studies demonstrate that the estimators based on this working model are practically unbiased with robust variance estimates. Furthermore, hypothesis tests may be based on the generalized Wald or generalized score tests. Data from a trial of patients with bronchial asthma are analyzed to illustrate the estimation and inference procedures.     

空间回归模型选择的反思

空间计量经济学存在两种最基本的模型:空间滞后模型和空间误差模型,这里旨在重新思考和探讨这两种空间回归模型的选择,结论为:Moran’s I指数可以用来判断回归模型后的残差是否存在空间依赖性;在实证分析中,采用拉格朗日乘子检验判断两种模型优劣是最常见的做法。然而,该检验仅仅是基于统计推断而忽略了理论基础,因此,可能导致选择错误的模型;在实证分析中,空间误差模型经常被选择性遗忘,而该模型的适用性较空间滞后模型更为广泛;实证分析大多缺乏空间回归模型设定的探讨,Anselin提出三个统计量,并且,如果模型设定正确,应该遵从Wald统计量>Log likelihood统计量>LM统计量的排列顺序。