Maximum Likelihood Inference for Multivariate Frailty Models Using an Automated Monte Carlo EM Algorithm

We present a maximum likelihood estimation procedure for the multivariate frailty model. The estimation is based on a Monte Carlo EM algorithm. The expectation step is approximated by averaging over random samples drawn from the posterior distribution of the frailties using rejection sampling. The maximization step reduces to a standard partial likelihood maximization. We also propose a simple rule based on the relative change in the parameter estimates to decide on sample size in each iteration and a stopping time for the algorithm. An important new concept is acquiring absolute convergence of the algorithm through sample size determination and an efficient sampling technique. The method is illustrated using a rat carcinogenesis dataset and data on vase lifetimes of cut roses. The estimation results are compared with approximate inference based on penalized partial likelihood using these two examples. Unlike the penalized partial likelihood estimation, the proposed full maximum likelihood estimation method accounts for all the uncertainty while estimating standard errors for the parameters.     

Statistical inference for Sobol pick-freeze Monte Carlo method

Many mathematical models involve input parameters, which are not precisely known. Global sensitivity analysis aims to identify the parameters whose uncertainty has the largest impact on the variability of a quantity of interest (output of the model). One of the statistical tools used to quantify the influence of each input variable on the output is the Sobol sensitivity index. We consider the statistical estimation of this index from a finite sample of model outputs. We study asymptotic and non-asymptotic properties of two estimators of Sobol indices. These properties are applied to significance tests and estimation by confidence intervals.     

Maximizing generalized linear mixed model likelihoods with an automated Monte Carlo EM algorithm

Two new implementations of the EM algorithm are proposed for maximum likelihood fitting of generalized linear mixed models. Both methods use random (independent and identically distributed) sampling to construct Monte Carlo approximations at the E-step. One approach involves generating random samples from the exact conditional distribution of the random effects (given the data) by rejection sampling, using the marginal distribution as a candidate. The second method uses a multivariate t importance sampling approximation. In many applications the two methods are complementary. Rejection sampling is more efficient when sample sizes are small, whereas importance sampling is better with larger sample sizes. Monte Carlo approximation using random samples allows the Monte Carlo error at each iteration to be assessed by using standard central limit theory combined with Taylor series methods. Specifically, we construct a sandwich variance estimate for the maximizer at each approximate E-step. This suggests a rule for automatically increasing the Monte Carlo sample size after iterations in which the true EM step is swamped by Monte Carlo error. In contrast, techniques for assessing Monte Carlo error have not been developed for use with alternative implementations of Monte Carlo EM algorithms utilizing Markov chain Monte Carlo E-step approximations. Three different data sets, including the infamous salamander data of McCullagh and Nelder, are used to illustrate the techniques and to compare them with the alternatives. The results show that the methods proposed can be considerably more efficient than those based on Markov chain Monte Carlo algorithms. However, the methods proposed may break down when the intractable integrals in the likelihood function are of high dimension.     

Design-based confidence limits for a distribution function

A method for constructing confidence limits for a distribution function is proposed. This method is a simple modification of the common method based on a normal approximation to the distribution of the estimated distribution function. The methods differ in how the estimated standard errors are used. The coverage properties of the two methods are compared in a simulation study. Coverage probabilities for the proposed method are found to be much closer to the nominal levels, particularly in the tails of the population distribution.     

关于多维指标抽样调查中删除指标的均值估计

针对多维度指标抽样调查情况,运用多变量幂函数逼近理论,对多指标中删除的指标构造出一类依赖于保留指标的估计量,通过对其性质的讨论,得出了该估计量的抽样误差与样本量的倒数同阶的结论。在抽样调查的实际应用中,可以根据给定的抽样误差,对部分指标进行调查,未被调查指标的均值通过调查指标值的估计去实现,从而达到了解全部指标信息的目的。     

Reliable error estimation for Sobol’ indices

In the field of sensitivity analysis, Sobol’ indices are sensitivity measures widely used to assess the importance of inputs of a model to its output. The estimation of these indices is often performed through Monte Carlo or quasi-Monte Carlo methods. A notable method is the replication procedure that estimates first-order indices at a reduced cost in terms of number of model evaluations. An inherent practical problem of this estimation is how to quantify the number of model evaluations needed to ensure that estimates satisfy a desired error tolerance. This article addresses this challenge by proposing a reliable error bound for first-order and total effect Sobol’ indices. Starting from the integral formula of the indices, the error bound is defined in terms of the discrete Walsh coefficients of the different integrands. We propose a sequential estimation procedure of Sobol’ indices using the error bound as a stopping criterion. The sequential procedure combines Sobol’ sequences with either Saltelli’s strategy to estimate both first-order and total effect indices, or the replication procedure to estimate only first-order indices.     

Box‐Cox transformations in linear models: Large sample theory and tests of normality

The authors provide a rigorous large sample theory for linear models whose response variable has been subjected to the Box‐Cox transformation. They provide a continuous asymptotic approximation to the distribution of estimators of natural parameters of the model. They show, in particular, that the maximum likelihood estimator of the ratio of slope to residual standard deviation is consistent and relatively stable. The authors further show the importance for inference of normality of the errors and give tests for normality based on the estimated residuals. For non‐normal errors, they give adjustments to the log‐likelihood and to asymptotic standard errors.     

Prediction Error of Small Area Predictors Shrinking Both Means and Variances

The article considers a new approach for small area estimation based on a joint modelling of mean and variances. Model parameters are estimated via expectation–maximization algorithm. The conditional mean squared error is used to evaluate the prediction error. Analytical expressions are obtained for the conditional mean squared error and its estimator. Our approximations are second‐order correct, an unwritten standardization in the small area literature. Simulation studies indicate that the proposed method outperforms the existing methods in terms of prediction errors and their estimated values.     

Improved generalized Fourier amplitude sensitivity test (FAST) for model assessment

The Fourier amplitude sensitivity test (FAST) can be used to calculate the relative variance contribution of model input parameters to the variance of predictions made with functional models. It is widely used in the analyses of complicated process modeling systems. This study provides an improved transformation procedure of the Fourier amplitude sensitivity test (FAST) for non-uniform distributions that can be used to represent the input parameters. Here it is proposed that the cumulative probability be used instead of probability density when transforming non-uniform distributions for FAST. This improvement will increase the accuracy of transformation by reducing errors, and makes the transformation more convenient to be used in practice. In an evaluation of the procedure, the improved procedure was demonstrated to have very high accuracy in comparison to the procedure that is currently widely in use.     

Variance Estimation under Two‐Phase Sampling

We consider the variance estimation of the weighted likelihood estimator (WLE) under two‐phase stratified sampling without replacement. Asymptotic variance of the WLE in many semiparametric models contains unknown functions or does not have a closed form. The standard method of the inverse probability weighted (IPW) sample variances of an estimated influence function is then not available in these models. To address this issue, we develop the variance estimation procedure for the WLE in a general semiparametric model. The phase I variance is estimated by taking a numerical derivative of the IPW log likelihood. The phase II variance is estimated based on the bootstrap for a stratified sample in a finite population. Despite a theoretical difficulty of dependent observations due to sampling without replacement, we establish the (bootstrap) consistency of our estimators. Finite sample properties of our method are illustrated in a simulation study.     

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.     

基于分层随机抽样的季节指数的抽样估计研究

由于传统的季节指数分析方法是一种描述统计,本文提出了采用分层随机抽样的季节指数估计量,给出了估计量的偏误和均方误差以及均方误差的估计,并在此基础上分析了季节指数的假设检验以及最优估计量的确定。     

Designing clinical trials with uncertain estimates of variability

An estimated sample size is a function of three components: the required power, the predetermined Type I error rate, and the specified effect size. For Normal data the standardized effect size is taken as the difference between two means divided by an estimate of the population standard deviation. However, in early phase trials one may not have a good estimate of the population variance as it is often based on the results of a few relatively small trials. The imprecision of this estimate should be taken into account in sample size calculations. When estimating a trial sample size this paper recommends that one should investigate the sensitivity of the trial to the assumptions made about the variance and consider being adaptive in one's trial design. Copyright © 2004 John Wiley & Sons Ltd.     

Multiple-bias modelling for analysis of observational data

Summary.  Conventional analytic results do not reflect any source of uncertainty other than random error, and as a result readers must rely on informal judgments regarding the effect of possible biases. When standard errors are small these judgments often fail to capture sources of uncertainty and their interactions adequately. Multiple-bias models provide alternatives that allow one systematically to integrate major sources of uncertainty, and thus to provide better input to research planning and policy analysis. Typically, the bias parameters in the model are not identified by the analysis data and so the results depend completely on priors for those parameters. A Bayesian analysis is then natural, but several alternatives based on sensitivity analysis have appeared in the risk assessment and epidemiologic literature. Under some circumstances these methods approximate a Bayesian analysis and can be modified to do so even better. These points are illustrated with a pooled analysis of case–control studies of residential magnetic field exposure and childhood leukaemia, which highlights the diminishing value of conventional studies conducted after the early 1990s. It is argued that multiple-bias modelling should become part of the core training of anyone who will be entrusted with the analysis of observational data, and should become standard procedure when random error is not the only important source of uncertainty (as in meta-analysis and pooled analysis).     

A comparison of mean squared error approximations for a small estimated state space model*

Summary: This paper investigates mean squared errors for unobserved states in state space models when estimation uncertainty of hyperparameters is taken into account. Three alternative approximations to mean squared errors with estimation uncertainty are compared in a Monte Carlo experiment, where the random walk with noise model serves as DGP: A naive method which neglects estimation uncertainty completely, an approximation based on an expansion around the true state with respect to the estimated parameters, and a bootstrap approach. Overall, the bootstrap method performs best in the simulations. However, the gains are not systematic, and the computationally burden of this method is relatively high.*This paper represents the authors personal opinions and does not necessarily reflect the views of the Deutsche Bundesbank. I am grateful to Malte Knüppel, Jeong-Ryeol Kurz-Kim, Karl-Heinz Tödter and a referee for helpful comments. The computer programs for this paper were written in Ox and SsfPack, see Doornik (1998) and Koopman et al. (1999). The used SsfPack version is 2.2.     

Global sensitivity analysis with dependence measures

Global sensitivity analysis with variance-based measures suffers from several theoretical and practical limitations, since they focus only on the variance of the output and handle multivariate variables in a limited way. In this paper, we introduce a new class of sensitivity indices based on dependence measures which overcomes these insufficiencies. Our approach originates from the idea to compare the output distribution with its conditional counterpart when one of the input variables is fixed. We establish that this comparison yields previously proposed indices when it is performed with Csiszár f-divergences, as well as sensitivity indices which are well-known dependence measures between random variables. This leads us to investigate completely new sensitivity indices based on recent state-of-the-art dependence measures, such as distance correlation and the Hilbert–Schmidt independence criterion. We also emphasize the potential of feature selection techniques relying on such dependence measures as alternatives to screening in high dimension.     

Uncertainty and sensitivity analysis in the presence of stochastic and subjective uncertainty

Uncertainty and sensitivity analyses for systems that involve both stochastic (i.e., aleatory) and subjective (i.e., epistemic) uncertainty are discussed. In such analyses, the dependent variable is usually a complementary cumulative distribution function (CCDF) that arises from stochastic uncertainty; uncertainty analysis involves the determination of a distribution of CCDFs that results from subjective uncertainty, and sensitivity analysis involves the determination of the effects of subjective uncertainty in individual variables on this distribution of CCDFs. Uncertainty analysis is presented as an integration problem involving probability spaces for stochastic and subjective uncertainty. Approximation procedures for the underlying integrals are described that provide an assessment of the effects of stochastic uncertainty, an assessment of the effects of subjective uncertainty, and a basis for performing sensitivity studies. Extensive use is made of Latin hypercube sampling, importance sampling and regression-based sensitivity analysis techniques. The underlying ideas, which are initially presented in an abstract form, are central to the design and performance of real analyses. To emphasize the connection between concept and computational practice, these ideas are illustrated with an analysis involving the MACCS reactor accident consequence model a, performance assessment for the Waste Isolation Pilot Plant, and a probabilistic risk assessment for a nuclear power station.     

An extension of the placebo‐based pattern‐mixture model

Pattern‐mixture models provide a general and flexible framework for sensitivity analyses of nonignorable missing data in longitudinal studies. The placebo‐based pattern‐mixture model handles missing data in a transparent and clinically interpretable manner. We extend this model to include a sensitivity parameter that characterizes the gradual departure of the missing data mechanism from being missing at random toward being missing not at random under the standard placebo‐based pattern‐mixture model. We derive the treatment effect implied by the extended model. We propose to utilize the primary analysis based on a mixed‐effects model for repeated measures to draw inference about the treatment effect under the extended placebo‐based pattern‐mixture model. We use simulation studies to confirm the validity of the proposed method. We apply the proposed method to a clinical study of major depressive disorders. Copyright © 2013 John Wiley & Sons, Ltd.     

Computing first-order sensitivity indices with contribution to the sample mean plot

In this paper, we investigate the use of the contribution to the sample mean plot (CSM plot) as a graphical tool for sensitivity analysis (SA) of computational models. We first provide an exact formula that links, for each uncertain model input X_j, the CSM plot C_j(·) with the first-order variance-based sensitivity index S_j. We then build a new estimate for S_j using polynomial regression of the CSM plot. This estimation procedure allows the computation of S_j from given data, without any SA-specific design of experiment. Numerical results show that this new S_j estimate is efficient for large sample sizes, but that at small sample sizes it does not compare well with other S_j estimation techniques based on given data, such as the effective algorithm for computing global sensitivity indices method or metamodel-based approaches.     

On the Bayesian estimation of the weighted Lindley distribution

The weighted distributions provide a comprehensive understanding by adding flexibility in the existing standard distributions. In this article, we considered the weighted Lindley distribution which belongs to the class of the weighted distributions and investigated various its properties. Although, our main focus is the Bayesian analysis however, stochastic ordering, the Bonferroni and the Lorenz curves, various entropies and order statistics derivations are obtained first time for the said distribution. Different types of loss functions are considered; the Bayes estimators and their respective posterior risks are computed and compared. The different reliability characteristics including hazard function, stress and strength analysis, and mean residual life function are also analysed. The Lindley approximation and the importance sampling are described for estimation of parameters. A simulation study is designed to inspect the effect of sample size on the estimated parameters. A real-life application is also presented for the illustration purpose.