Some New Results on the Multinomial Randomized Response Model

ABSTRACT

The randomized response technique is an effective survey method designed to elicit sensitive information while ensuring the privacy of the respondents. In this article, we present some new results on the randomization response model in situations wherein one or two response variables are assumed to follow a multinomial distribution. For a single sensitive question, we use the well-known Hopkins randomization device to derive estimates, both under the assumption of truthful and untruthful responses, and present a technique for making pairwise comparisons. When there are two sensitive questions of interest, we derive a Pearson product moment correlation estimator based on the multinomial model assumption. This estimator may be used to quantify the linear relationship between two variables when multinomial response data are observed according to a randomized-response protocol.     

Two-level minimum aberration designs in N =2 (mod 4) runs

For two-level factorials, we consider designs in N=2 (mod 4) runs as obtained by adding two runs, with a certain coincidence pattern, to an orthogonal array of strength two. These designs are known to be optimal main effect plans in a very broad sense in the absence of interactions. Among them, we explore the ones having minimum aberration, with a view to ensuring maximum model robustness even when interactions are possibly present. This is done by sequentially minimizing a measure of the bias caused by interactions of successively higher orders.     

Optimal Designs for Binary Logistic Regression with a Qualitative Classifier with Independent Levels

Dose response studies arise in many medical applications. Often, such studies are considered within the framework of binary-response experiments such as success-failure. In such cases, popular choices for modeling the probability of response are logistic or probit models. Design optimality has been well studied for the logistic model with a continuous covariate. A natural extension of the logistic model is to consider the presence of a qualitative classifier. In this work, we explore D-, A-, and E-optimal designs in a two-parameter, binary logistic regression model after introducing a binary, qualitative classifier with independent levels.     

Standard Exponential Cure Rate Model with Informative Censoring

In this article, we consider the standard cure rate model proposed by Boag (1949 Boag , J. W. ( 1949 ). Maximum likelihood estimates of the proportion of patients cured by cancer therapy . J. Roy. Statist. Soc. B 11 : 15 – 53 . [Google Scholar]) and Berkson and Gage (1952 Berkson , J. , Gage , R. ( 1952 ). Survival curve for cancer patients following treatment . J. Amer. Statist. Assoc. 47 : 501 – 515 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). We present a new definition of informative censoring similar to Lawless (1982 Lawless , J. F. ( 1982 ). Statistical Models and Methods for Lifetime Data . New York : Wiley .[Crossref] , [Google Scholar]) and the corresponding likelihood function. Under informative censoring, we obtain the Fisher information matrix of the exponential standard cure rate model. We verify, with simulated data, the impact caused by informative censoring in the coverage probabilities and in the lengths of asymptotic confidence intervals of the parameters of interest. An example with real data is analyzed.     

A new approximation to the distribution function of the studentized maximum root

The Studentized maximum root (SMR) distribution is useful for constructing simultaneous confidence intervals around product interaction contrasts in replicated two-way ANOVA. A three-moment approximation to the SMR distribution is proposed. The approximation requires the first three moments of the maximum root of a central Wishart matrix. These values are obtained by means of numerical integration. The accuracy of the approximation is compared to the accuracy of a two-moment approximation for selected two-way table sizes. Both approximations are reasonably accurate. The three-moment approximation is generally superior.     

Estimation of parameters of a pareto distribution by generalized order statistics

In this paper we address the problem of estimating the parameters of Pareto II distribution based on generalized order statistics. The estimators based on order statistics and record values are shown to be special cases of these estimators.     

A theorem on invariance of estimability of linear parametric functions in linear models

Consider the general linear model Y = Xβ + ? , where E[??'] = σ²I and rank of X is less than or equal to the number of columns of X. It is well known that the linear parametric function λ'β is estimable if and only if λ' is in the row space of X. This paper characterizes all orthogonal matrices P such that the row space of XP is equal to the row space of X, i.e. the estimability of λ'β is invariant under P. An additional property of these matrices is the invariance of the spectrum of the information matrix X'X. An application of the results is also given.     

Admissibility of Test Procedures for Linear Hypotheses in General Linear Model

This article shows that an F-test procedure is admissible for testing a linear hypothesis concerning one of the split mean vectors in a general linear model and an F-test procedure is also admissible for testing a linear hypothesis concerning another of the split mean vectors in the same model. These results are proved by showing that the critical functions of the tests are unique Bayes procedures with respect to proper prior distributions set in common for the null hypotheses and for the alternative ones, respectively.     

Accent on Teaching Materials

Estimation of covariance components in the multivariate random-effect model with nested covariance structure is discussed. There are two covariance matrices to be estimated, namely, the between-group and the within-group covariance matrices. These two covariance matrices are most often estimated by forming a multivariate analysis of variance and equating mean square matrices to their expectations. Such a procedure involves taking the difference between the between-group mean square and the within-group mean square matrices, and often produces an estimated between-group covariance matrix that is not nonnegative definite. We present estimators of the two covariance matrices that are always proper covariance matrices. The estimators are the restricted maximum likelihood estimators if the random effects are normally distributed. The estimation procedure is extended to more complicated models, including the twofold nested and the mixed-effect models. A numerical example is presented to illustrate the use of the estimation procedure.     

An Introduction to Ranking and Selection

This expository paper explains the philosophy of the indifference-zone approach and the subset-selection approach to ranking and selection procedures. It includes examples of operating characteristic curves and data applications for selection problems based on the binomial and normal distributions. A variety of different models and goals are provided with a list of references.