Analysis op failure time data by burr distribution

Sometimes it is appropriate to model the survival and failure time data by a non-monotonic failure rate distribution. This may be desirable when the course of disease is such that mortality reaches a peak after some finite period and then slowly declines.In this paper we study Burr, type XII model whose failure rate exhibits the above behavior. The location of the critical points (at which the monotonicity changes) for both the failure rate and the mean residual life function (MRLF) are studied. A procedure is described for estimating these critical points. Necessary and sufficient conditions for the existence and uniqueness of the maximum likelihood estimates are provided and it is shown that the conditions provided by Wingo (1993) are not sufficient. A data set pertaining to fibre failure strengths is analyzed and the maximum likelihood estimates of the critical points are obtained.     

Bayesian inferences given a type-2 censored sample from a burr distribution

A Bayesian approach is used to make inferences given a random sample of observations from a Burr distribution. Complete and type-2 censored samples are considered and inferences are made on the unknown parameters and the reliability function. In the case of a type-2 censored sample prediction bounds are derived for the unobserved sample values.     

Bayesian prediction with a random sample size for the burr lifetime distribution

This paper is concerned with the problem of deriving Bayesian prediction bounds for the Burr distribution when the sample size is a random variable. Prediction bounds for both the future observations (the case of two-sample prediction) and the remaining observations in the same sample (the case of one-sample prediction) will be derived. The analysis will depend mainly on assuming that the size of the sample is a random variable having the Poisson distribution. Finally, numerical examples are given to illustrate the results.     

Probabilistic tolerance design for a subsystem under burr distribution using taguchi's loss functions

Taguchi (1986) has derived tolerances for subcomponents, subsystems, parts and materials in which the relationship between a higher-level (Y) and a lower-level (X) quality characteristic is assumed to be deterministic and linear, namely, Y=_α+βX, without an error term. Tsai (1990) developed a probabilistic tolerance design for a subsystem in which a bivariate normal distribution between the above two quality characteristics as well as Taguchi's quadratic loss function were considered together to develop a closed form solution of the tolerance design for a subsystem. The Burr family is very rich for fitting sample data, and has positive domain. A bivariate Burr distribution can describe a nonlinear relationship between two quality characteristics, hence, it is adopted instead of a bivariate normal distribution and the simple solutions of three probabilistic tolerance desings for a subsystem are obtained for three cases of “nominal-is-best”, “smaller-is-berrer”, and “larger-is-beter” quality characteristics, by using Taguchi’ los functions, respectively.     

Recurrence relations for moments of progressive type-II right censored order statstics from burr distribution

In this paper some recurrence relations between moments of progressive Type-II right censored order statistics from doubly truncated Burr distribution are established. These recurrence relations would enable one to obtain all the single and product moments of Burr progressive Type-II right censored order statistics in a simple recursive manner.     

Bayesian design of experiment for quantal responses: What is promised versus what is delivered

This article considers a design problem in quantal response analysis, where an experimenter must choose a set of dose levels and number of independent observations to take at these levels, subject to some total sample size, in order to minimize the expected or predicted posterior variance of some characteristics ø of the tolerance distribution F_θ, with unknown parameters θ. An exact solution to this problem is demonstrated when ø is the unknown LD50 of the one parameter logistic tolerance distribution, under the restriction that an equal number of observations are taken at each of a set of equally spaced levels. The solution is based on a combination of simulated outcomes and Monte Carlo integration to evaluate the predicted variance. The numerical results are compared to those obtained previously by asymptotic approximations in Tsutakawa (1972), (J. Amer. Statist. Assoc. 67 584–590). The wide variability in the simulated posterior variance suggests that the expected posterior variance alone is not a good criterion for design selection.     

Prediction bounds for the burr model

Given a type 2 censored sample from the Burr life time distribution, Bayesian prediction bounds are derived for future observations. An approximate Bayesian method has been used to simplify the computation of the prediction bounds. Numerical examples are used to illustrate the procedures.     

Nonparametric estimators for shift in quantal bioassay

Let F(x) and F(x+θ) be log dose-response curves for a standard preparation and a test preparation, respectively, in a parallel quantal bioassay designed to test the relative potency of a drug, toxicant, or some other substance, and suppose the form of F is unknown. Several estimators of the shift parameter θ or relative potency, are compared, including some generalized and trimmed Spearman-Kärber estimators and a non parametric maximum likelihood estimator. Both point and interval estimation are discussed. Some recommendations concerning the choices of estimators are offered.     

On the influence curve for quantal bioassay

Influence curves are defined for estimators of the ED50 in quantal bioassay, assuming a symmetic tolerance distribution. A definition of influence-curve robustness is suggested. Influence curves are obtained for L-,M- and R-estimators, and their robustness is investigated. The influence curve of the logistic maximum likelihood estimator is studied.     

Inference and prediction for the burr type x model based on records

Based on record values, the maximum likelihood, minimum variance unbiased and Bayes estimators of the one parameter of the Burr type X distribution are computed and compared. The Bayesian and non-Bayesian confidence intervals for this parameter are also presented. A Bayesian prediction interval for the sth future record is obtained in a closed form. Based on simulated record values, numerical computations and comparisons between the different estimators are given     

Sampling based approach for one-hit and multi-hit models in quantal bioassay

Bayesian methods for estimating the dose response curves with the one-hit model, the gamma multi-hit model, and their modified versions with Abbott's correction are studied. The Gibbs sampling approach with data augmentation and with the Metropolis algorithm is employed to compute the Bayes estimates of the potency curves. In addition, estimation of the relative additional risk and the virtually safe dose is studied. Model selection based on conditional predictive ordinates from cross-validated data is developed.     

Empirical likelihood confidence bands for distribution functions with missing responses

Distribution function estimation plays a significant role of foundation in statistics since the population distribution is always involved in statistical inference and is usually unknown. In this paper, we consider the estimation of the distribution function of a response variable Y with missing responses in the regression problems. It is proved that the augmented inverse probability weighted estimator converges weakly to a zero mean Gaussian process. A augmented inverse probability weighted empirical log-likelihood function is also defined. It is shown that the empirical log-likelihood converges weakly to the square of a Gaussian process with mean zero and variance one. We apply these results to the construction of Gaussian process approximation based confidence bands and empirical likelihood based confidence bands of the distribution function of Y. A simulation is conducted to evaluate the confidence bands.     

Analysis of a factorial quantal response assay using inverse regression

An inverse regression approach to analyzing quantal response assays with one quanritative independent- variable and any number of qualitative independent variables is presented. A general factorial model ror LQC- inverted relationship and methods of testing hypotheses estimating qulitative Interest are developed. This naper provides a modified method of analysis which is easier to apply than previously existing methods.     

Benchmark profile and inferences for joint-exposure quantal data in quantitative risk assessment

Abstract

In risk assessment, it is often desired to make inferences on the minimum dose levels (benchmark doses or BMDs) at which a specific benchmark risk (BMR) is attained. The estimation and inferences of BMDs are well understood in the case of an adverse response to a single-exposure agent. However, the theory of finding BMDs and making inferences on the BMDs is much less developed for cases where the adverse effect of two hazardous agents is studied simultaneously. Deutsch and Piegorsch [2012. Benchmark dose profiles for joint-action quantal data in quantitative risk assessment. Biometrics 68(4):1313–22] proposed a benchmark modeling paradigm in dual exposure setting—adapted from the single-exposure setting—and developed a strategy for conducting full benchmark analysis with joint-action quantal data, and they further extended the proposed benchmark paradigm to continuous response outcomes [Deutsch, R. C., and W. W. Piegorsch. 2013. Benchmark dose profiles for joint-action continuous data in quantitative risk assessment. Biometrical Journal 55(5):741–54]. In their 2012 article, Deutsch and Piegorsch worked exclusively with the complementary log link for modeling the risk with quantal data. The focus of the current paper is on the logit link; particularly, we consider an Abbott-adjusted [A method of computing the effectiveness of an insecticide. Journal of Economic Entomology 18(2):265–7] log-logistic model for the analysis of quantal data with nonzero background response. We discuss the estimation of the benchmark profile (BMP)—a collection of benchmark points which induce the prespecified BMR—and propose different methods for building benchmark inferences in studies involving two hazardous agents. We perform Monte Carlo simulation studies to evaluate the characteristics of the confidence limits. An example is given to illustrate the use of the proposed methods.     

A note on tests for interaction in quantal response data

There are few distribution-free methods for detecting interaction in fixed-dose trials involving quantal response data, despite the fact that such trials are common. We present three new tests to address this issue, including a simple bootstrap procedure. We examine the power of the likelihood ratio test and our new bootstrap test statistic using an innovative linear extrapolation power-estimation technique described in Boos, D. D. and Zhang, J. (2000) in Monte Carlo evaluation of resampling-based hypothesis tests. Journal of the American Statistical Association, 95, 486–492.     

A survival-adjusted quantal response test for comparing tumour incidence rates

Summary.  The paper presents a case-study of skin fibromas among male rats in the 2-year cancer bioassay of methyleugenol that was conducted by the US National Toxicology Program (NTP). In animal carcinogenicity experiments such as this one, tumour rates are often compared with the Cochran–Armitage (CA) trend test. The operating characteristics of the CA test, however, can be adversely affected by survival differences across groups and by the assumed dose metric. Survival-adjusted generalizations of the CA test have been proposed, but they are still sensitive to the choice of scores that are assigned to the dose groups. We present an alternative test, which outperforms the survival-adjusted CA test which is currently used by the NTP to compare incidence rates. Simulated data from a wide range of realistic situations show that the operating characteristics of the test proposed are superior to those of the NTP's survival-adjusted CA test, especially for rare tumours and wide logarithmic spacings of the dose metric.     

Multivariate measurement error models based on Student-t distribution under censored responses

Measurement error models constitute a wide class of models that include linear and nonlinear regression models. They are very useful to model many real-life phenomena, particularly in the medical and biological areas. The great advantage of these models is that, in some sense, they can be represented as mixed effects models, allowing us to implement well-known techniques, like the EM-algorithm for the parameter estimation. In this paper, we consider a class of multivariate measurement error models where the observed response and/or covariate are not fully observed, i.e., the observations are subject to certain threshold values below or above which the measurements are not quantifiable. Consequently, these observations are considered censored. We assume a Student-t distribution for the unobserved true values of the mismeasured covariate and the error term of the model, providing a robust alternative for parameter estimation. Our approach relies on a likelihood-based inference using an EM-type algorithm. The proposed method is illustrated through some simulation studies and the analysis of an AIDS clinical trial dataset.     

A comparison of using weighted distribution and joint modeling for analyzing non-ignorable missing responses

In this study, we reconsider weighted distribution from the perspective of missing mechanism since weighted distribution instead of being the distribution of the whole population of interest is only the distribution of respondents (sub-population). After defining some weighted distributions by different mechanisms for indicator of response, we show, by some simulation studies, that using weighted distributions may lead to biased estimates of parameters under the non-ignorable missing mechanism. On the other hand, joint modeling of the response and selection mechanism could result in more efficient and valid estimates of parameters. The lower root of mean squared errors of estimates from the joint modeling approach than those of the weighted distribution is a warranty to the statement that the joint modeling method is more efficient than weighted distribution; this is proved by diverse simulation studies along the article. However, these two methods of the weighted approach and joint modeling give similar results if the selection mechanism is at random. Finally, the methods are applied and compared in the analysis of one well-used real dataset.