Survival estimation and the effects of dependency among animals

Survival models assume that fates of individuals are independent, yet the robustness of this assumption has been poorly quantified. We examine how empirically derived estimates of the variance of survival rates are affected by dependency in survival probability among individuals. We used Monte Carlo simulations to generate known amounts of dependency among pairs of individuals and analyzed these data with Kaplan-Meier and Cormack-Jolly-Seber models. Dependency significantly increased these empirical variances as compared to theoretically derived estimates of variance from the same populations. Using resighting data from 168 pairs of black brant ( Branta bernicla nigricans ), we used a resampling procedure and program RELEASE to estimate empirical and mean theoretical variances. We estimated that the relationship between paired individuals caused the empirical variance of the survival rate to be 155% larger than the empirical variance for unpaired individuals. Monte Carlo simulations and use of this resampling strategy can provide investigators with information on how robust their data are to this common assumption of independent survival probabilities.     

Applying the rescaling bootstrap under imputation: a simulation study

Resampling methods are a common measure to estimate the variance of a statistic of interest when data consist of nonresponse and imputation is used as compensation. Applying resampling methods usually means that subsamples are drawn from the original sample and that variance estimates are computed based on point estimators of several subsamples. However, newer resampling methods such as the rescaling bootstrap of Chipperfield and Preston [Efficient bootstrap for business surveys. Surv Methodol. 2007;33:167–172] include all elements of the original sample in the computation of its point estimator. Thus, procedures to consider imputation in resampling methods cannot be applied in the ordinary way. For such methods, modifications are necessary. This paper presents an approach applying newer resampling methods for imputed data. The Monte Carlo simulation study conducted in the paper shows that the proposed approach leads to reliable variance estimates in contrast to other modifications.     

Progressively Type-II censored competing risks data for exponential distributions based on sequential order statistics

We consider the progressively Type-II censored competing risks model based on sequential order statistics. It is assumed that the latent failure times are independent and the failure of each unit influences the lifetime distributions of the latent failure times of surviving units. We provide explicit expressions for the likelihood function of the available data under the conditional proportional hazard rate (CPHR) and the power trend conditional proportional hazard rate (PTCPHR) models. Under CPHR and PTCPHR models and assumption that the baseline distributions of the latent failure times are exponential, classical and Bayesian estimates of the unknown parameters are provided. Monte Carlo simulations are then performed for illustrative purposes. Finally, two datasets are analyzed.     

Nonparametric tests for right-censored data with biased sampling

Testing the equality of two survival distributions can be difficult in a prevalent cohort study when non random sampling of subjects is involved. Due to the biased sampling scheme, independent censoring assumption is often violated. Although the issues about biased inference caused by length-biased sampling have been widely recognized in statistical, epidemiological and economical literature, there is no satisfactory solution for efficient two-sample testing. We propose an asymptotic most efficient nonparametric test by properly adjusting for length-biased sampling. The test statistic is derived from a full likelihood function, and can be generalized from the two-sample test to a k-sample test. The asymptotic properties of the test statistic under the null hypothesis are derived using its asymptotic independent and identically distributed representation. We conduct extensive Monte Carlo simulations to evaluate the performance of the proposed test statistics and compare them with the conditional test and the standard logrank test for different biased sampling schemes and right-censoring mechanisms. For length-biased data, empirical studies demonstrated that the proposed test is substantially more powerful than the existing methods. For general left-truncated data, the proposed test is robust, still maintains accurate control of type I error rate, and is also more powerful than the existing methods, if the truncation patterns and right-censoring patterns are the same between the groups. We illustrate the methods using two real data examples.     

Bias in transition-specific survival and movement probabilities estimated using capture-recapture data

Transition probabilities can be estimated when capture-recapture data are available from each stratum on every capture occasion using a conditional likelihood approach with the Arnason-Schwarz model. To decompose the fundamental transition probabilities into derived parameters, all movement probabilities must sum to 1 and all individuals in stratum r at time i must have the same probability of survival regardless of which stratum the individual is in at time i + 1. If movement occurs among strata at the end of a sampling interval, survival rates of individuals from the same stratum are likely to be equal. However, if movement occurs between sampling periods and survival rates of individuals from the same stratum are not the same, estimates of stratum survival can be confounded with estimates of movement causing both estimates to be biased. Monte Carlo simulations were made of a three-sample model for a population with two strata using SURVIV. When differences were created in transition-specific survival rates for survival rates from the same stratum, relative bias was <2% in estimates of stratum survival and capture rates but relative bias in movement rates was much higher and varied. The magnitude of the relative bias in the movement estimate depended on the relative difference between the transition-specific survival rates and the corresponding stratum survival rate. The direction of the bias in movement rate estimates was opposite to the direction of this difference. Increases in relative bias due to increasing heterogeneity in probabilities of survival, movement and capture were small except when survival and capture probabilities were positively correlated within individuals.     

Stress–strength reliability of a non-identical-component-strengths system based on upper record values from the family of Kumaraswamy generalized distributions

In an attempt to produce more realistic stress–strength models, this article considers the estimation of stress–strength reliability in a multi-component system with non-identical component strengths based on upper record values from the family of Kumaraswamy generalized distributions. The maximum likelihood estimator of the reliability, its asymptotic distribution and asymptotic confidence intervals are constructed. Bayes estimates under symmetric squared error loss function using conjugate prior distributions are computed and corresponding highest probability density credible intervals are also constructed. In Bayesian estimation, Lindley approximation and the Markov Chain Monte Carlo method are employed due to lack of explicit forms. For the first time using records, the uniformly minimum variance unbiased estimator and the closed form of Bayes estimator using conjugate and non-informative priors are derived for a common and known shape parameter of the stress and strength variates distributions. Comparisons of the performance of the estimators are carried out using Monte Carlo simulations, the mean squared error, bias and coverage probabilities. Finally, a demonstration is presented on how the proposed model may be utilized in materials science and engineering with the analysis of high-strength steel fatigue life data.     

Diagnostics for elliptical linear mixed models with first-order autoregressive errors

For longitudinal time series data, linear mixed models that contain both random effects across individuals and first-order autoregressive errors within individuals may be appropriate. Some statistical diagnostics based on the models under a proposed elliptical error structure are developed in this work. It is well known that the class of elliptical distributions offers a more flexible framework for modelling since it contains both light- and heavy-tailed distributions. Iterative procedures for the maximum-likelihood estimates of the model parameters are presented. Score tests for the presence of autocorrelation and the homogeneity of autocorrelation coefficients among individuals are constructed. The properties of test statistics are investigated through Monte Carlo simulations. The local influence method for the models is also given. The analysed results of a real data set illustrate the values of the models and diagnostic statistics.     

Robust estimation of dynamic fixed-effects panel data models

This paper extends an existing outlier-robust estimator of linear dynamic panel data models with fixed effects, which is based on the median ratio of two consecutive pairs of first-order differenced data. To improve its precision and robustness properties, a general procedure based on higher-order pairwise differences and their ratios is designed. The asymptotic distribution of this class of estimators is derived. Further, the breakdown point properties are obtained under contamination by independent additive outliers and by the patches of additive outliers, and are used to select the pairwise differences that do not compromise the robustness properties of the procedure. The proposed estimator is additionally compared with existing methods by means of Monte Carlo simulations.     

Approximate Methods for State-Space Models

State-space models provide an important body of techniques for analyzing time-series, but their use requires estimating unobserved states. The optimal estimate of the state is its conditional expectation given the observation histories, and computing this expectation is hard when there are nonlinearities. Existing filtering methods, including sequential Monte Carlo, tend to be either inaccurate or slow. In this paper, we study a nonlinear filter for nonlinear/non-Gaussian state-space models, which uses Laplace's method, an asymptotic series expansion, to approximate the state's conditional mean and variance, together with a Gaussian conditional distribution. This Laplace-Gaussian filter (LGF) gives fast, recursive, deterministic state estimates, with an error which is set by the stochastic characteristics of the model and is, we show, stable over time. We illustrate the estimation ability of the LGF by applying it to the problem of neural decoding and compare it to sequential Monte Carlo both in simulations and with real data. We find that the LGF can deliver superior results in a small fraction of the computing time.     

Quasi-random resampling for the bootstrap

Quasi-random sequences are known to give efficient numerical integration rules in many Bayesian statistical problems where the posterior distribution can be transformed into periodic functions on then-dimensional hypercube. From this idea we develop a quasi-random approach to the generation of resamples used for Monte Carlo approximations to bootstrap estimates of bias, variance and distribution functions. We demonstrate a major difference between quasi-random bootstrap resamples, which are generated by deterministic algorithms and have no true randomness, and the usual pseudo-random bootstrap resamples generated by the classical bootstrap approach. Various quasi-random approaches are considered and are shown via a simulation study to result in approximants that are competitive in terms of efficiency when compared with other bootstrap Monte Carlo procedures such as balanced and antithetic resampling.     

Joint modelling of longitudinal binary data and survival data

The medical costs in an ageing society substantially increase when the incidences of chronic diseases, disabilities and inability to live independently are high. Healthy lifestyles not only affect elderly individuals but also influence the entire community. When assessing treatment efficacy, survival and quality of life should be considered simultaneously. This paper proposes the joint likelihood approach for modelling survival and longitudinal binary covariates simultaneously. Because some unobservable information is present in the model, the Monte Carlo EM algorithm and Metropolis-Hastings algorithm are used to find the estimators. Monte Carlo simulations are performed to evaluate the performance of the proposed model based on the accuracy and precision of the estimates. Real data are used to demonstrate the feasibility of the proposed model.     

Modeling multiple‐response categorical data from complex surveys

Although “choose all that apply” questions are common in modern surveys, methods for analyzing associations among responses to such questions have only recently been developed. These methods are generally valid only for simple random sampling, but these types of questions often appear in surveys conducted under more complex sampling plans. The purpose of this article is to provide statistical analysis methods that can be applied to “choose all that apply” questions in complex survey sampling situations. Loglinear models are developed to incorporate the multiple responses inherent in these types of questions. Statistics to compare models and to measure association are proposed and their asymptotic distributions are derived. Monte Carlo simulations show that tests based on adjusted Pearson statistics generally hold their correct size when comparing models. These simulations also show that confidence intervals for odds ratios estimated from loglinear models have good coverage properties, while being shorter than those constructed using empirical estimates. Furthermore, the methods are shown to be applicable to more general problems of modeling associations between elements of two or more binary vectors. The proposed analysis methods are applied to data from the National Health and Nutrition Examination Survey. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

Estimation and prediction of the Burr type XII distribution based on record values and inter-record times

The maximum likelihood and Bayesian approaches for parameter estimations and prediction of future record values have been considered for the two-parameter Burr Type XII distribution based on record values with the number of trials following the record values (inter-record times). Firstly, the Bayes estimates are obtained based on a joint bivariate prior for the shape parameters. In this case, the Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo (MCMC) method due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The MCMC method has been also used to construct the highest posterior density credible intervals. Secondly, the Bayes estimates are obtained with respect to a discrete prior for the first shape parameter and a conjugate prior for other shape parameter. The Bayes and the maximum likelihood estimates are compared in terms of the estimated risk by the Monte Carlo simulations. We further consider the non-Bayesian and Bayesian prediction for future lower record arising from the Burr Type XII distribution based on record data. The comparison of the derived predictors is carried out by using Monte Carlo simulations. A real data are analysed for illustration purposes.     

Normality Test Based on a Truncated Mean Characterization

This article generalizes a characterization based on a truncated mean to include higher truncated moments, and introduces a new normality goodness-of-fit test based on the truncated mean. The test is a weighted integral of the squared distance between the empirical truncated mean and its expectation. A closed form for the test statistic is derived. Assuming known parameters, the mean and the variance of the test are derived under the normality assumption. Moreover, a limiting distribution for the proposed test as well as an approximation are obtained. Also, based on Monte Carlo simulations, the power of the test is evaluated against stable, symmetric, and skewed classes of distributions. The test proves compatibility with prominent tests and shows higher power for a wide range of alternatives.     

Robust estimation of variance components

New robust estimates for variance components are introduced. Two simple models are considered: the balanced one-way classification model with a random factor and the balanced mixed model with one random factor and one fixed factor. However, the method of estimation proposed can be extended to more complex models. The new method of estimation we propose is based on the relationship between the variance components and the coefficients of the least-mean-squared-error predictor between two observations of the same group. This relationship enables us to transform the problem of estimating the variance components into the problem of estimating the coefficients of a simple linear regression model. The variance-component estimators derived from the least-squares regression estimates are shown to coincide with the maximum-likelihood estimates. Robust estimates of the variance components can be obtained by replacing the least-squares estimates by robust regression estimates. In particular, a Monte Carlo study shows that for outlier-contaminated normal samples, the estimates of variance components derived from GM regression estimates and the derived test outperform other robust procedures.     

Effect of assay measurement error on parameter estimation in concentration–QTc interval modeling

Linear mixed‐effects models (LMEMs) of concentration–double‐delta QTc intervals (QTc intervals corrected for placebo and baseline effects) assume that the concentration measurement error is negligible, which is an incorrect assumption. Previous studies have shown in linear models that independent variable error can attenuate the slope estimate with a corresponding increase in the intercept. Monte Carlo simulation was used to examine the impact of assay measurement error (AME) on the parameter estimates of an LMEM and nonlinear MEM (NMEM) concentration–ddQTc interval model from a ‘typical’ thorough QT study. For the LMEM, the type I error rate was unaffected by assay measurement error. Significant slope attenuation ( > 10%) occurred when the AME exceeded > 40% independent of the sample size. Increasing AME also decreased the between‐subject variance of the slope, increased the residual variance, and had no effect on the between‐subject variance of the intercept. For a typical analytical assay having an assay measurement error of less than 15%, the relative bias in the estimates of the model parameters and variance components was less than 15% in all cases. The NMEM appeared to be more robust to AME error as most parameters were unaffected by measurement error. Monte Carlo simulation was then used to determine whether the simulation–extrapolation method of parameter bias correction could be applied to cases of large AME in LMEMs. For analytical assays with large AME ( > 30%), the simulation–extrapolation method could correct biased model parameter estimates to near‐unbiased levels. Copyright © 2013 John Wiley & Sons, Ltd.     

Estimation and prediction of the Kumaraswamy distribution based on record values and inter-record times

ABSTRACT

The maximum likelihood and Bayesian approaches for estimating the parameters and the prediction of future record values for the Kumaraswamy distribution has been considered when the lower record values along with the number of observations following the record values (inter-record-times) have been observed. The Bayes estimates are obtained based on a joint bivariate prior for the shape parameters. In this case, Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo (MCMC) method due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The MCMC method has been also used to construct the highest posterior density credible intervals. The Bayes and the maximum likelihood estimates are compared by using the estimated risk through Monte Carlo simulations. We further consider the non-Bayesian and Bayesian prediction for future lower record values arising from the Kumaraswamy distribution based on record values with their corresponding inter-record times and only record values. The comparison of the derived predictors are carried out by using Monte Carlo simulations. Real data are analysed for an illustration of the findings.     

Volatility and GMM — Monte Carlo studies and empirical estimations

In this article we examine small sample properties of a generalized method of moments (GMM) estimation using Monte Carlo simulations. We assume that the generated time series describe the stochastic variance rate of a stock index. we use a mean reverting square-root process to simulate the dynamics of this instantaneous variance rate. The time series obtained are used to estimate the parameters of the assumed variance rate process by applying GMM. Our results are described and compared to estimates from empirical data which consist of volatility as well as daily volume data of the German stock market. One of our main findings is that estimates of the mean reverting parameter that are not significantly different from zero do not necessarily imply a rejection of the hypothesis of a mean reverting behavior of the underlying stochastic process.     

Bootstrapping moving average models

Summary In recent years, the bootstrap method has been extended to time series analysis where the observations are serially correlated. Contributions have focused on the autoregressive model producing alternative resampling procedures. In contrast, apart from some empirical applications, very little attention has been paid to the possibility of extending the use of the bootstrap method to pure moving average (MA) or mixed ARMA models. In this paper, we present a new bootstrap procedure which can be applied to assess the distributional properties of the moving average parameters estimates obtained by a least square approach. We discuss the methodology and the limits of its usage. Finally, the performance of the bootstrap approach is compared with that of the competing alternative given by the Monte Carlo simulation. Research partially supported by CNR and MURST.     

Bayesian inference for random coefficient dynamic panel data models

We develop a hierarchical Bayesian approach for inference in random coefficient dynamic panel data models. Our approach allows for the initial values of each unit's process to be correlated with the unit-specific coefficients. We impose a stationarity assumption for each unit's process by assuming that the unit-specific autoregressive coefficient is drawn from a logitnormal distribution. Our method is shown to have favorable properties compared to the mean group estimator in a Monte Carlo study. We apply our approach to analyze energy and protein intakes among individuals from the Philippines.