Model Assisted Statistics and Applications

It is widely recognized that many cancer therapies are effective only for a subset of patients. However clinical studies are most often powered to detect an overall treatment effect. To address this issue, classification methods are increasingly being used to predict a subset of patients which respond differently to treatment. This study begins with a brief history of classification methods with an emphasis on applications involving melanoma. Nonparametric methods suitable for predicting subsets of patients responding differently to treatment are then reviewed. Each method has different ways of incorporating continuous, categorical, clinical and high-throughput covariates. More recent methods have built-in dimension reduction methods for high throughput data. Pre-validation is one method of assessing the added value of high-throughput data to clinical covariates. The way in which treatment interactions are incorporated is important if the goal is to predict a subset of patients which respond differently to treatment. For nonparametric methods, distance measures specific to the method are used to make classification decisions. Approaches are outlined which employ these distances to measure treatment interactions. It is hoped that this study will stimulate more development of nonparametric methods to predict subsets of patients responding differently to treatment.

The shared frailty models allow for the unbiased heterogeneity or statistical dependence between the observed survival data. The most common shared frailty model is a model in which hazard function is a product of a random factor (frailty) and the baseline hazard function which is common to all individuals. There are certain assumptions about the baseline distribution and the distribution of frailty. In this paper, we consider shared gamma frailty model with three different baseline distributions namely, the generalized Rayleigh, the weighted exponential and the extended Weibull distributions. With these three baseline distributions we propose three different shared frailty models. We also compare these models with the models where the above mentioned distributions are considered without frailty. We develop the Bayesian estimation procedure using Markov Chain Monte Carlo (MCMC) technique to estimate the parameters involved in these models. We present a simulation study to compare the true values of the parameters with the estimated values. A search of the literature suggests that currently no work has been done for these three baseline distributions with a shared gamma frailty so far. We also apply these three models by using a real life bivariate survival data set of McGilchrist and Aisbett [15] related to the kidney infection data and a better model is suggested for the data.

Experimental designs under two-way blocking structure are used to separate out two cross-classified non-interacting sources of variation in the experimental material. However, there may exist situations wherein the experimental area contain some stony patches or other features that tend to clump in compact areas where these designs cannot capture the variability due to these compact areas. For such situations, experimental designs that are capable of removing three sources of variability can be more advantageously used. In this paper, some methods of constructing designs under three-way blocking structure have been developed. Joint information matrix has been derived and the efficiency factor of these designs has been computed as compared to an orthogonal design. List of parameters of the designs obtained has been prepared for number of treatments <25 along with the efficiency factor for each design.

In this paper we investigated the asymptotic distribution of the bootstrap parameter estimator of a first order autoregressive AR(1) model. We described the asymptotic distribution of such estimator by applying the delta method and employing two different approaches, and concluded that the two approaches lead to the same conclusion, viz. both results converge in distribution to a normal distribution. We also presented the Monte Carlo simulation of the residuals bootstrap and application with real data was carried out in order to yield apparent conclusions.

Bayes estimators are obtained in case of Pareto distribution for its shape parameter, mean income, Gini index and a Poverty measure for both censored and complete setup. The said estimators are obtained using Jeffreys' non-informative invariant prior and the extension of Jeffreys' prior information. Using simulation techniques, the relative efficiency of proposed estimators with the existing estimators using two-parameter exponential prior is obtained. It turns out that the Bayesian method with Jeffreys' non-informative invariant prior results in smaller expected loss function as compared to existing estimators using two-parameter exponential prior.

This article considers generalized maximum likelihood and Bayesian estimations of reliability parameters under some balanced loss functions when the data are progressively Type II censored from a compound Rayleigh distribution. This is done with respect to a conjugate prior on scale parameter and a discrete prior on shape parameter of the distribution. Posterior risks of generalized maximum likelihood and Bayes estimates are also obtained under balanced loss functions. A real life application to the survival times of patients is also described for the developed estimation methods. A Monte Carlo simulation study has been carried out to examine and compare the performance of estimates on the basis of posterior risks. The simulation study indicates that Bayesian estimation should be preferred over generalized maximum likelihood estimation for estimation of the said parameters. This study will be very useful to researchers, practitioners, and statisticians where such type of life test is needed and especially where a compound Rayleigh model is used.