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Response Surface Designs (RSDs) are widely used in process or product optimization studies to explore the input-response relationship. This paper is an attempt to provide a compilation of theoretic...

We contemplate an experimental situation in a 2 k -factorial experiment with acute resource crunch so that we need to conduct just a saturated design [SD] - with the understanding that precision of the estimates cannot be estimated from the data. It is known beforehand which effect(s)/interaction(s) are likely to be negligible. We examine the flexibility to the extent that an experimenter can make a choice of an SD in order to retain information on all the remaining [non-negligible] effects/interactions.

This research study is a continuation to Sapam et al. [2019. Latin square designs with neighbour effects. Journal of the Indian Society of Agricultural Statistics 73 (2):91–98 (online).] wherein we...

Estimation and inference are two key components toward the solution of any statistical problem; however, the inferential issues of statistical assessment of agreement among two or more raters have not been well developed as compared to the development of estimation procedures in this area. The fundamental reason for this gap is the complex expression of the concordance correlation coefficient (CCC) that is frequently used in assessing agreement among raters. Large sample-based statistical tests for CCC often fail to produce desired results for small samples. Hence, inferential procedures for small samples are urgently needed to evaluate agreement between raters. We argue that hypothesis testing of CCC has little value in practice due to the absence of a gold standard of agreement. In this article, we construct the generalized confidence interval (GCI) for CCC utilizing a bivariate normal distribution of measurements, and also develop a large sample-based confidence interval (LSCI). We establish satisfactory performance of GCI by providing the desired coverage probability (CP) via simulation. Results of GCI and LSCI are illustrated and compared with a data set of a recent study performed at U.S. Department of Veterans Affairs, Hines.

Functional Magnetic Resonance Imaging (fMRI) is a technology for studying how our brains respond to mental stimuli. At the design stage, one is interested in developing the best sequence of mental stimuli for collecting the most informative data in order to render the most precise inference about the ‘unknown parameters’ under an assumed statistical model. The simplest such model incorporates linear relation between mean response and the parameters describing the effects of the stimuli, applied at regularly spaced time points during the study period. In this paper, we introduce the linear model and discuss estimation issues and related concepts such as ‘orthogonality’ and ‘balance’.

This research study is devoted towards exploring the nature of error functions and estimable treatment contrasts in a model involving Graeco Latin square designs (GLSDs) of orders 4 and 5 in situations wherein there are left- and right-neighbor effects present in the underlying linear model. The model is assumed to be circular. It transpires from the current study that data analysis for GLSDs in the presence of neighbor effects has to be carefully handled.

The human brain is an amazingly complex network. Aberrant activities in this network can lead to various neurological disorders such as multiple sclerosis, Parkinson's disease, Alzheimer's disease, and autism. functional magnetic resonance imaging has emerged as an important tool to delineate the neural networks affected by such diseases, particularly autism. In this paper, we propose a special type of mixed-effects model together with an appropriate procedure for controlling false discoveries to detect disrupted connectivities for developing a neural network in whole brain studies. Results are illustrated with a large data set known as autism brain imaging data exchange which includes 361 subjects from eight medical centers.

In this paper we investigate an aspect of unbiased estimation of the reliability function, based on a single observation from a mixture of two exponential distributions with known mixing proportions. We introduce a few equivalent versions of unbiased estimators, and find that they give proper estimates only if negative weights on component mixture distributions are allowed.

With reference to “point estimation” of a real-valued parameter involved in the distribution of a real-valued random variable X, we consider a sample size n and an underlying unbiased estimator of for every where k is the minimum sample size needed for existence of unbiased estimator(s) of based on . We wish to investigate exact small-sample properties of the sequence of estimators considered here. This we study by considering what is termed “coverage probability” (CP) and defined as . For , we may redefine as since . When serves as a scale parameter, bounds to the ratio are more meaningful than bounds to the difference. In either case, it is desired that the sequence behaves like an increasing sequence for every . We may note in passing that we are asking for a property beyond “consistency” of a sequence of unbiased estimators. As is well known, consistency is a large-sample property. In this article we discuss several interesting features of the behavior of the in the exact sense.

In the present work, thin films of carboxymethyl cellulose (CMC) and poly(vinyl alcohol) (PVA) were prepared by solution casting method and vanadium pentoxide (V2O5) was doped by in situ precipitation to yield CMC/PVA-V2O5 bionanocomposites. The materials were characterized by X-ray diffraction (XRD), atomic force microscopy (AFM), and Fourier transform infrared (FTIR) techniques and AC/DC conductivity, tensile strength, and microhardness measurements, and photo-luminescence (PL) and UV–Vis spectroscopic studies were conducted. The results of AFM and XRD studies provided information about morphology and crystalline features of the bionanocomposite film, respectively. The FTIR spectroscopy confirmed the presence of functional groups of the constituent polymers in the bionanocomposite and showed that V2O5 was successfully incorporated into CMC/PVA matrix. The AFM analysis of bionanocomposite films provided information about their morphology and roughness parameters. The PL spectroscopy indicated the appearance of blue shift after doping of V2O5 into the biopolymer film. The AC electrical conductivity was measured in the range 500–100,000 Hz and it was found that at room temperature the conductivity increases with frequency. Differential scanning calorimetry was used to look into the possible thermal transitions occurring in the bionanocomposites.

In large scale surveys, it is customary to accept unaltered the responses provided by the respondents, leaving no provision for investigators to pass on any circumstantial judgment on the responses. This restrictive practice sometimes vitiates the estimation of a parameter. Here we consider a real life scenario in which the data gathering process is compounded by possible interventions by investigators/supervisors who may provide thoughtful judgments on the quality/category of the response given by a respondent. In this modified scenario, we address the problem of estimating the unknown population proportion P. In the context of an illustrative example, we develop a possible randomization theory and computational formulas to estimate P under intervention effects.

We propose to discuss at length several examples from standard text books. All of these examples deal with analysis of covariance (ANCOVA) models and related analyses of data. We intend to capitalize on our understanding of optimal covariate designs (OCDs) in different ANCOVA models and re-visit these examples with a view to suggest optimal/nearly optimal designs for estimation of the covariate parameter(s). As we will see, for some examples our task is very much routine but for others, it is indeed a highly non trivial exercise. We intent to cover a total of six examples—divided in two parts. This is Part I—dealing with two examples.

Constructions of blocked mixture designs are considered in situations where BLUEs of the block effect contrasts are orthogonal to the BLUEs of the regression coefficients. Orthogonal arrays (OA), Balanced Arrays (BAs), incidence matrices of balanced incomplete block designs (BIBDs), and partially balanced incomplete block designs (PBIBDs) are used. Designs with equal and unequal block sizes are considered. Also both cases where the constants involved in the orthogonality conditions depend and do not depend on the factors have been taken into account. Some standard (already available) designs can be obtained as particular cases of the designs proposed here.

Pitman closeness criterion is a coverage probability-based criterion to examine the relative performances of estimators. Usually, the performance of the standard Graybill-Deal estimator of the common mean has been examined with respect to the mean squared error (variance). In this study we examine its performance with respect to the Pitman closeness criterion. Specifically, we compare a p-source based Graybill-Deal estimator against its q-source based competitors for q (< p)-source subsets of p-source data. The key references to this paper are [5] and [7].

In the present paper an attempt has been made to study the inter–and intra-block estimation of treatment effects contrasts with random block effects in the context of balanced ternary designs. The analysis is illustrated through examples.

We intend to discuss some standard and non-standard Data Integration Techniques with a hypothetical illustration from a study of environmental factors [to be called sources] and the extent of their toxic release in the surface areas of several locations in a closed region. The purpose is to rank the locations—by ’integrating’ the effects of toxic release from all the potential sources. From a practical point of view, it is highly unlikely that a single location can be the ’absolute best’ [i.e., the location possesses minimum toxic release for each of the toxicity sources] for any reasonable environmental study. Hence, there is a need for data integration to arrive at an overall ranking of the locations, keeping the differential toxic effects in mind. Naturally, rank 1 corresponds to the least ’compound’ effect of the toxicity sources.

We are given \(c\ge 2\) coins which are otherwise identical, except that there may be exactly (or at most) one fake coin among them which is known to be slightly lighter than the other genuine coins. Using only a two-pan weighing balance, we weigh subsets of coins sequentially in order to identify the counterfeit coin (or declare that all coins are genuine) using the fewest weighings on average. We find a formula for the smallest expected number of weighings, and another formula which determines an optimal number of coins to place on each pan during the first (and hence during each successive) weighing.

There is an impressive published literature on the statistical issues of assessment of agreement among two or more raters involving both qualitative and quantitative data. We propose to present basic ideas and relevant statistical methods toward assessment of agreement.

There are situations when we need information on sensitive features/characteristics of the individuals in a population. Usual sampling techniques based on ‘direct’ questionnaire may not be fruitful and are likely to lead to ‘refusal’ or ‘untruthful’ reporting. To overcome such undesirable consequences, randomized response techniques [RRTs, in short] have been introduced and extensively used in real-life situations. We discuss some salient features of this technique.

From some basic considerations of formation of dyadic [pair-wise] binary [go/no-go] relations among a given number of units [individuals/households etc], we develop a social network. Basic parameters are then introduced and their quantifications are made for a given population network. Then a sampling problem is considered for formation of a sample network. Estimation of the population parameters based on data arising from a sample network are then discussed in details.

Social mobility is the movement of individuals, families, households, or other categories of people within or between social strata in a society. Societies organized by social class, rather than caste, usually allow greater social mobility; in such societies, one’s ability to achieve a higher social status can depend on factors such as social connections, wealth, effort, and education. In order to study the dynamics of social processes, it is natural to start by looking at the movement of people between social classes. Since such moves are largely unpredictable at the individual level, it is necessary for a model to describe mechanism of movement in probabilistic terms. A distinction has to be made between intergenerational mobility and intra-generational mobility. The former refers to social mobility which reflects changes of social class from one generation to another. Intra-generational mobility refers to changes of classes which take place during an individual’s life span. This type of movement is called occupational or labor mobility since it is usually more directly concerned with occupations. Many deterministic and stochastic models have been developed to study social and occupational mobility situations in the different parts of the world.

In pharmaceutical experiments, the rate of dissolution of a tablet is modeled in terms of the proportions of polymers and diluents used in the tablet. When more than one type of polymer and diluent are used, the total proportions of polymers and diluents in the tablet are generally subject to relational constraints, which give a range of acceptable values for each proportion. This paper considers two models for the mean dissolution rate subject to relational constraints on the polymers and diluents and attempts to find optimum designs for estimating the parameters in the models using the D-optimality criterion.

Clustering can be considered to be the most important unsupervised learning technique to find homogeneous groups in a collection of a moderately large number of data points. Clustering could be defined as the process of dividing items into unknown number of groups whose members are alike in some way. A cluster is therefore a collection of items those are similar among themselves and are dissimilar to the items belonging to other clusters. It can be shown that there is no absolute "best" criterion which would be independent of the final aim of the clustering. Hence, the structure of the clusters should be finalized by the user depending on the physical requirements. By depending on the nature of analysis, clustering is called an unsupervised learning method and classification is called a supervised learning method.

Principal component analysis (PCA) is a method for dimension reduction tool in order to reduce a large set of variables to a small set of components that still contains most of the information in the original data set. Under PCS, we transform a number of correlated variables into a smaller set of uncorrelated components called principal components.

The present study focuses on the problems of parameter estimation and growth prediction in a quadratic growth model based on repeated measurements of growth, where the parameters in the model are assumed to be functions of ‘treatments’ which are treated as mixtures. The study concentrates not only on the optimality aspects of designs for most efficient estimation of the parameters but also on optimal prediction of growth at designated time points.

Sudoku is a popular combinatorial puzzle. We give a brief overview of some mathematical features of a Sudoku square. Then we focus on interpreting Sudoku squares as experimental designs in order to meet a practical need.

In a mixture experiment, the response depends on the proportions of the mixing components. Canonical models of different degrees and also other models have been suggested to represent the mean response. Optimum designs for estimation of the parameters of the models have been investigated by different authors. In most cases, the optimum design includes the vertex points of the simplex as support points of the design, which are not mixture combinations in the true non-trivial sense. In this paper, optimum designs have been obtained when the experimental region is an ellipsoidal subspace of the entire factor space which does not cover the vertex points of the simplex.

Study of agreement between two or more comparable sets of measurements taken on each member of a population is needed in many areas. There is an impressive literature on the topic covering both the qualitative and quantitative features of study variables. In order to ascertain the extent of agreement between the quantitative ‘target’ variable X and the quantitative ‘test’ variable Y, coverage probability (CP) was introduced by Lin et al. (2002). CP is defined as the probability of (X, Y) falling into a strip along the direction of X = Y of a judiciously specified width 2d but symmetric with respect to X and Y. For a given d, the higher CP value is, the better agreement it indicates. This article dwells on the concept of agreement involving three or more comparable quantitative measurements taken on each member of a population. We contemplate on a situation wherein we have available a reference gold standard against which several competitors are to be compared with respect to their performance as judged by...

The problem of unbiased estimation of the mean life and the reliability for an exponential life distribution using time censored sample data are considered. We prove that there does not exist an unbiased estimator of the mean life based on a time censored sample. On the other hand, for reliability estimation at an arbitrarily specified time point, we give necessary and sufficient conditions for the existence of (i) an unbiased estimator (ii) the uniformly minimum variance unbiased estimator. We also provide a characterization of the class of unbiased estimators of the reliability, based on a sufficient statistic in situations where unbiased estimators exist.

In this article, an attempt has been made to settle the question of existence of unbiased estimator of the key parameter p of the quasi-binomial distributions of Type I (QBD I) and of Type II (QBD II), with/without any knowledge of the other parameter φ appearing in the expressions for probability functions of the QBD's. This is studied with reference to a single observation, a random sample of finite size m as also with samples drawn by suitably defined sequential sampling rules.

Surveillance is a major issue in the today's world. The need to develop adequate surveillance strategies is genuine in many spheres of human activity. In this article we focus on a specific problem of surveillance and discuss some related statistical issues. This specific problem deals with a possible change in the incidence rate of an event (to a higher value) when a system is studied at discrete time points. We apply the exponentially weighted moving average methods for detecting an increased incidence rate per exposure unit of an event. Different measures of evaluation, suitable in different types of applications, such as the expected delay, out-of-control average run length, and probability of successful detection, are studied. Analytical bounds are provided for those measures of evaluation. Results from intensive simulations indicate that the analytical bounds perform fairly well when the weight parameter is large.

In this article we investigate the problem of ascertaining A- and D-optimal designs in a cubic regression model with random coefficients. Our interest lies in estimation of all the parameters or in only those except the intercept term. Assuming the variance ratios to be known, we tabulate D-optimal designs for various combinations of the variance ratios. A-optimality does not pose any new problem in the random coefficients situation.

In this chapter, optimum covariate designs (OCDs) have been considered for the set-up of the balanced treatment incomplete block (BTIB) designs, which form an important class of test-control designs. It is seen that the OCDs depend much on the methods of construction of the basic BTIB designs. The presentation in this chapter closely follows the work of Dutta, Das, J Stat Plan Inference 143:1203–1214, 2013, Dutta and Das (2013) who considered the problem of construction of OCDs on the series of BTIB designs mainly described in Bechhofer, Tamhane, Technometrics 23:45–57, 1981, Bechhofer and Tamhane (1981) and Das, Dey, Kageyama, Sinha, Australas J Combinatorics 32:243–252, 2005, Das et al. (2005).

Two motivating examples are presented in order to appreciate the importance of optimal choice of covariates. Basics in linear models are discussed. Chapter-wise summary of the work covered and choice of various experimental design settings are discussed.

In this chapter, we consider the problem of finding optimum designs for the estimation of covariate parameters in a BIBD with parameters \(v,~b,~r,~k,~\lambda \). When \(n\ne \)0 (mod 4), it is not possible to find designs attaining minimum variance (\(\frac{\sigma ^2}{n}\)) for the estimated covariate parameters. Conditions for D-optimum designs for the estimation of covariate parameters are established with the usual assumption that each of the values of the csovariates belongs to the interval [\(-1,1\)]. Some constructions of D-optimal designs have been provided for symmetric balanced incomplete block design (SBIBD) with parameters \(b=v,~r=k=v-1,~\lambda =v-2\) when \(k\equiv 2\) (mod 4) and b is an odd integer.

Optimum covariate designs in simple set-ups such as CRD, RBD and some series of BIBD have already been considered. BIBDs do not exist for many parametric combinations. So there is a need to consider other block designs different from the BIBDs suitable for desired parametric combinations. In this chapter, OCDs have been considered for the less-restrictive set-ups of partially balanced incomplete block designs (PBIBD), which are popular among practitioners. As in the case of BIBDs, the OCDs depend much on the methods of construction of the basic PBIBDs. We focus on one well-known subclass of two-associate class PBIBDs, viz. group divisible designs (GDDs) and develop OCDs based on the GDDs. As before, different combinatorial arrangements and tools such as orthogonal arrays, Hadamard matrices and different kinds of matrix-products, viz. Khatri-Rao product, Kronecker product, etc., have been conveniently used to construct OCDs with as many covariates as possible. A list of OCDs based on some classes of GDDs available in Clatworthy, Tables of two-associate class partially balanced designs, (1973) is given in the appendix for ready reference.