Small sample study on estimators of life distributions and of mean survival times using randomly censored data

Whereas large-sample properties of the estimators of survival distributions using censored data have been studied by many authors, exact results for small samples have been difficult to obtain. In this paper we obtain the exact expression for the ath moment (a > 0) of the Bayes estimator of survival distribution using the censored data under proportional hazard model. Using the exact expression we compute the exact mean, variance and MSE of the Bayes estimator. Also two estimators ofthe mean survival time based on the Kaplan-Meier estimator and the Bayes estimator are compared for small samples under proportional hazards.     

Confidence intervals centred on bootstrap smoothed estimators

Bootstrap smoothed (bagged) parameter estimators have been proposed as an improvement on estimators found after preliminary data‐based model selection. A result of Efron in 2014 is a very convenient and widely applicable formula for a delta method approximation to the standard deviation of the bootstrap smoothed estimator. This approximation provides an easily computed guide to the accuracy of this estimator. In addition, Efron considered a confidence interval centred on the bootstrap smoothed estimator, with width proportional to the estimate of this approximation to the standard deviation. We evaluate this confidence interval in the scenario of two nested linear regression models, the full model and a simpler model, and a preliminary test of the null hypothesis that the simpler model is correct. We derive computationally convenient expressions for the ideal bootstrap smoothed estimator and the coverage probability and expected length of this confidence interval. In terms of coverage probability, this confidence interval outperforms the post‐model‐selection confidence interval with the same nominal coverage and based on the same preliminary test. We also compare the performance of the confidence interval centred on the bootstrap smoothed estimator, in terms of expected length, to the usual confidence interval, with the same minimum coverage probability, based on the full model.     

On mantel-haenszel type estimators in simple nested case-control studies

We are concerned with nested case-control studies in this article. For proportional hazards model, a class of over-all estimators of hazard ratios is presented when simple samples are drawn from risk sets. These estimators have the form of the Mantel-Haenszel estimator of odds ratio, and are consistent not only for large strata, but also for sparse data. Consistent estimators of the variances of the proposed hazard ratio estimators are also developed. An example is given to illustrate the proposed estimators.     

A New Estimator for Cox Proportional Hazard Regression Model in Presence of Collinearity

We propose a new approach to estimate the parameters of the Cox proportional hazards model in the presence of collinearity. Generally, a maximum partial likelihood estimator is used to estimate parameters for the Cox proportional hazards model. However, the maximum partial likelihood estimators can be seriously affected by the presence of collinearity since the parameter estimates result in large variances.

In this study, we develop a Liu-type estimator for Cox proportional hazards model parameters and compare it with a ridge regression estimator based on the scalar mean squared error (MSE). Finally, we evaluate its performance through a simulation study.     

On Local Polynomial Modelling of the Additive Risk Model

The additive risk model provides an alternative modelling technique for failure time data to the proportional hazards model. In this article, we consider the additive risk model with a nonparametric risk effect. We study estimation of the risk function and its derivatives with a parametric and an unspecified baseline hazard function respectively. The resulting estimators are the local likelihood and the local score estimators. We establish the asymptotic normality of the estimators and show that both methods have the same formula for asymptotic bias but different formula for variance. It is found that, in some special cases, the local score estimator is of the same efficiency as the local likelihood estimator though it does not use the information about the baseline hazard function. Another advantage of the local score estimator is that it has a closed form and is easy to implement. Some simulation studies are conducted to evaluate and compare the performance of the two estimators. A numerical example is used for illustration.     

Comparison of preliminary test estimators based on generalized order statistics from proportional hazard family using Pitman measure of closeness

In this article, based on generalized order statistics from a family of proportional hazard rate model, we use a statistical test to generate a class of preliminary test estimators and shrinkage preliminary test estimators for the proportionality parameter. These estimators are compared under Pitman measure of closeness (PMC) as well as MSE criteria. Although the PMC suffers from non transitivity, in the first class of estimators, it has the transitivity property and we obtain the Pitman-closest estimator. Analytical and graphical methods are used to show the range of parameter in which preliminary test and shrinkage preliminary test estimators perform better than their competitor estimators. Results reveal that when the prior information is not too far from its real value, the proposed estimators are superior based on both mentioned criteria.     

An Estimator of the Survival Function Basedon the Semi-Markov Model Under Dependent Censorship

Lee and Wolfe (Biometrics vol. 54 pp. 1176–1178, 1998) proposed the two-stage sampling design for testing the assumption of independent censoring, which involves further follow-up of a subset of lost-to-follow-up censored subjects. They also proposed an adjusted estimator for the survivor function for a proportional hazards model under the dependent censoring model. In this paper, a new estimator for the survivor function is proposed for the semi-Markov model under the dependent censorship on the basis of the two-stage sampling data. The consistency and the asymptotic distribution of the proposed estimator are derived. The estimation procedure is illustrated with an example of lung cancer clinical trial and simulation results are reported of the mean squared errors of estimators under a proportional hazards and two different nonproportional hazards models.     

Small area estimation of mean price of habitation transaction using time-series and cross-sectional area-level models

In this paper, a new small domain estimator for area-level data is proposed. The proposed estimator is driven by a real problem of estimating the mean price of habitation transaction at a regional level in a European country, using data collected from a longitudinal survey conducted by a national statistical office. At the desired level of inference, it is not possible to provide accurate direct estimates because the sample sizes in these domains are very small. An area-level model with a heterogeneous covariance structure of random effects assists the proposed combined estimator. This model is an extension of a model due to Fay and Herriot [5], but it integrates information across domains and over several periods of time. In addition, a modified method of estimation of variance components for time-series and cross-sectional area-level models is proposed by including the design weights. A Monte Carlo simulation, based on real data, is conducted to investigate the performance of the proposed estimators in comparison with other estimators frequently used in small area estimation problems. In particular, we compare the performance of these estimators with the estimator based on the Rao–Yu model [23]. The simulation study also accesses the performance of the modified variance component estimators in comparison with the traditional ANOVA method. Simulation results show that the estimators proposed perform better than the other estimators in terms of both precision and bias.     

Nonparametric estimation of the survival function from censored data based on relative risk function

We present a nonparametric estimator for distribution function under random censorship from the right. Our approach is based on estimating the relative risk function, and the resulting estimator is closely related with the Kaplan-Meier’s product-limit and Breslow’s exponential hazard estimators. We also consider the general proportional hazards model when the competing risks are variable censored from the left.     

k-Step shape estimators based on spatial signs and ranks

In this paper, the shape matrix estimators based on spatial sign and rank vectors are considered. The estimators considered here are slight modifications of the estimators introduced in Dümbgen (1998) and Oja and Randles (2004) and further studied for example in Sirkiä et al. (2009). The shape estimators are computed using pairwise differences of the observed data, therefore there is no need to estimate the location center of the data. When the estimator is based on signs, the use of differences also implies that the estimators have the so called independence property if the estimator, that is used as an initial estimator, has it. The influence functions and limiting distributions of the estimators are derived at the multivariate elliptical case. The estimators are shown to be highly efficient in the multinormal case, and for heavy-tailed distributions they outperform the shape estimator based on sample covariance matrix.     

Penalized and Shrinkage Estimation in the Cox Proportional Hazards Model

This article considers the shrinkage estimation procedure in the Cox's proportional hazards regression model when it is suspected that some of the parameters may be restricted to a subspace. We have developed the statistical properties of the shrinkage estimators including asymptotic distributional biases and risks. The shrinkage estimators have much higher relative efficiency than the classical estimator, furthermore, we consider two penalty estimators—the LASSO and adaptive LASSO—and compare their relative performance with that of the shrinkage estimators numerically. A Monte Carlo simulation experiment is conducted for different combinations of irrelevant predictors and the performance of each estimator is evaluated in terms of simulated mean squared error. Simulation study shows that the shrinkage estimators are comparable to the penalty estimators when the number of irrelevant predictors in the model is relatively large. The shrinkage and penalty methods are applied to two real data sets to illustrate the usefulness of the procedures in practice.     

Reducing bias in parameter estimates from stepwise regression in proportional hazards regression with right-censored data

When variable selection with stepwise regression and model fitting are conducted on the same data set, competition for inclusion in the model induces a selection bias in coefficient estimators away from zero. In proportional hazards regression with right-censored data, selection bias inflates the absolute value of parameter estimate of selected parameters, while the omission of other variables may shrink coefficients toward zero. This paper explores the extent of the bias in parameter estimates from stepwise proportional hazards regression and proposes a bootstrap method, similar to those proposed by Miller (Subset Selection in Regression, 2nd edn. Chapman & Hall/CRC, 2002) for linear regression, to correct for selection bias. We also use bootstrap methods to estimate the standard error of the adjusted estimators. Simulation results show that substantial biases could be present in uncorrected stepwise estimators and, for binary covariates, could exceed 250% of the true parameter value. The simulations also show that the conditional mean of the proposed bootstrap bias-corrected parameter estimator, given that a variable is selected, is moved closer to the unconditional mean of the standard partial likelihood estimator in the chosen model, and to the population value of the parameter. We also explore the effect of the adjustment on estimates of log relative risk, given the values of the covariates in a selected model. The proposed method is illustrated with data sets in primary biliary cirrhosis and in multiple myeloma from the Eastern Cooperative Oncology Group.     

An additive–multiplicative mean model for panel count data with dependent observation and dropout processes

This paper discusses regression analysis of panel count data with dependent observation and dropout processes. For the problem, a general mean model is presented that can allow both additive and multiplicative effects of covariates on the underlying point process. In addition, the proportional rates model and the accelerated failure time model are employed to describe possible covariate effects on the observation process and the dropout or follow‐up process, respectively. For estimation of regression parameters, some estimating equation‐based procedures are developed and the asymptotic properties of the proposed estimators are established. In addition, a resampling approach is proposed for estimating a covariance matrix of the proposed estimator and a model checking procedure is also provided. Results from an extensive simulation study indicate that the proposed methodology works well for practical situations, and it is applied to a motivating set of real data.     

Lifetime estimation from doubly censored data and dirichlet process prior knowledge on the observable random vector

The purpose of this paper is to present a nonparametric Bayesian procedure for estimating a survival curve in a double censoring situation. Assuming a proportional hazard rates model, we propose a consistent estimation of lifetime, based on a Dirichlet process prior knowledge on the observable random vector. Some large sample properties of this estimator are also derived, We prove strong consistency and asymptotic weak convergence to a Gaussian pro cess. Finally, a simulation study is presented in order to analyze the behavior of the proposed estimator, and establish some comparisons to other estimators.     

Three modifications of the principle of the minque

Nonnegative estimators for the variance components of a linear model are obtained by ignoring the condition for unbiasedness in the principle of the MINQUE. An estimator is derived when the priori weights are proportional to the variance components. The ordinary sample variance is shown to be the nonnegative MINQUE. Efficiencies of the three estimators are examined for some special cases of the model.     

Confidence regions based on inefficient estimators

We consider an unbiased estimator of a function of mean value parameters, which is not efficient. This inefficient estimator is correlated with a residual vector. Thus, if a unit dispersion is unknown, it is impossible to determine the correct confidence region for a function of mean value parameters via a standard estimator of an unknown dispersion with the exception of the case when the ordinary least squares (OLS) estimator is considered in a model with a special covariance structure such that the OLS and the generalized least squares (GLS) estimator are the same, that is the OLS estimator is efficient. Two different estimators of a unit dispersion independent of an inefficient estimator are derived in a singular linear statistical model. Their quality was verified by simulations for several types of experimental designs. Two new estimators of the unit dispersion were compared with the standard estimators based on the GLS and the OLS estimators of the function of the mean value parameters. The OLS estimator was considered in the incorrect model with a different covariance matrix such that the originally inefficient estimator became efficient. The numerical examples led to a slightly surprising result which seems to be due to data behaviour. An example from geodetic practice is presented in the paper.     

Proportional hazards model for competing risks data with missing cause of failure

We consider the semiparametric proportional hazards model for the cause-specific hazard function in analysis of competing risks data with missing cause of failure. The inverse probability weighted equation and augmented inverse probability weighted equation are proposed for estimating the regression parameters in the model, and their theoretical properties are established for inference. Simulation studies demonstrate that the augmented inverse probability weighted estimator is doubly robust and the proposed method is appropriate for practical use. The simulations also compare the proposed estimators with the multiple imputation estimator of Lu and Tsiatis (2001). The application of the proposed method is illustrated using data from a bone marrow transplant study.     

An empirical investigation of different operating characteristics of several estimators of the intraclass correlation in the analysis of binary data

This paper is concerned with properties (bias, standard deviation, mean square error and efficiency) of twenty six estimators of the intraclass correlation in the analysis of binary data. Our main interest is to study these properties when data are generated from different distributions. For data generation we considered three over-dispersed binomial distributions, namely, the beta-binomial distribution, the probit normal binomial distribution and a mixture of two binomial distributions. The findings regarding bias, standard deviation and mean squared error of all these estimators, are that (a) in general, the distributions of biases of most of the estimators are negatively skewed. The biases are smallest when data are generated from the beta-binomial distribution and largest when data are generated from the mixture distribution; (b) the standard deviations are smallest when data are generated from the beta-binomial distribution; and (c) the mean squared errors are smallest when data are generated from the beta-binomial distribution and largest when data are generated from the mixture distribution. Of the 26, nine estimators including the maximum likelihood estimator, an estimator based on the optimal quadratic estimating equations of Crowder (1987), and an analysis of variance type estimator is found to have least amount of bias, standard deviation and mean squared error. Also, the distributions of the bias, standard deviation and mean squared error for each of these estimators are, in general, more symmetric than those of the other estimators. Our findings regarding efficiency are that the estimator based on the optimal quadratic estimating equations has consistently high efficiency and least variability in the efficiency results. In the important range in which the intraclass correlation is small (≤0 5), on the average, this estimator shows best efficiency performance. The analysis of variance type estimator seems to do well for larger values of the intraclass correlation. In general, the estimator based on the optimal quadratic estimating equations seems to show best efficiency performance for data from the beta-binomial distribution and the probit normal binomial distribution, and the analysis of variance type estimator seems to do well for data from the mixture distribution.     

Nonparametric Shrinkage Estimation for Aalen's Additive Hazards Model

Aalen's nonparametric additive model in which the regression coefficients are assumed to be unspecified functions of time is a flexible alternative to Cox's proportional hazards model when the proportionality assumption is in doubt. In this paper, we incorporate a general linear hypothesis into the estimation of the time‐varying regression coefficients. We combine unrestricted least squares estimators and estimators that are restricted by the linear hypothesis and produce James‐Stein‐type shrinkage estimators of the regression coefficients. We develop the asymptotic joint distribution of such restricted and unrestricted estimators and use this to study the relative performance of the proposed estimators via their integrated asymptotic distributional risks. We conduct Monte Carlo simulations to examine the relative performance of the estimators in terms of their integrated mean square errors. We also compare the performance of the proposed estimators with a recently devised LASSO estimator as well as with ridge‐type estimators both via simulations and data on the survival of primary billiary cirhosis patients.     

The Statistical Properties of the Equity Estimator

In this article we consider the Equity estimator proposed by Krishnamurthi and Rangaswamy. We show that this estimator is inconsistent and does not necessarily improve on the mean squared error (MSE) of the least squares (LS) estimator. We perform a Monte Carlo experiment based on the price-promotion model used in marketing research, with marketing data, comparing the MSE of the Equity estimator to that of two empirical Bayes estimators and the LS estimator. We find that the empirical Bayes estimators have substantially smaller MSE than the Equity estimator in almost every case.