Optimum chemical balance weighing designs for estimating the total weight

This deals with chemical balance weighing designs which attain a lower bound for the variance of the estimated total weight, The results extend those of Chacko Dey (1978).     

Optimum spring balance weighing designs for estimating the total weight

The problem of estimation of the total weight or objects using a spring balance weighing design has been deait with in this paper Based on a theorem by Dey and Gupta (1977) giving a lower bound for the variance of the estimated total weight, a necessary and sufficient condition for this lower bound to be attained is obtained. A few special cases where the lower bound is attained are enumerated.     

Scaled density models for binary response data and their D- and Ds-optimal designs

We introduce scaled density models for binary response data which can be much more reasonable than the traditional binary response models for particular types of binary response data. We show the maximum-likelihood estimates for the new models and it seems that the model works well with some sets of data. We also considered optimum designs for parameter estimation for the models and found that the D- and D_s-optimum designs are independent of parameters corresponding to the linear function of dose level, but the optimum designs are simple functions of a scale parameter only.     

Optimal designs for estimating the slope of a regression

In the common linear model with quantitative predictors we consider the problem of designing experiments for estimating the slope of the expected response in a regression. We discuss locally optimal designs, where the experimenter is only interested in the slope at a particular point, and standardized minimax optimal designs, which could be used if precise estimation of the slope over a given region is required. General results on the number of support points of locally optimal designs are derived if the regression functions form a Chebyshev system. For polynomial regression and Fourier regression models of arbitrary degree the optimal designs for estimating the slope of the regression are determined explicitly for many cases of practical interest.     

Wavelet designs for estimating nonparametric curves with heteroscedastic error

In this paper, we discuss the problem of constructing designs in order to maximize the accuracy of nonparametric curve estimation in the possible presence of heteroscedastic errors. Our approach is to exploit the flexibility of wavelet approximations to approximate the unknown response curve by its wavelet expansion thereby eliminating the mathematical difficulty associated with the unknown structure. It is expected that only finitely many parameters in the resulting wavelet response can be estimated by weighted least squares. The bias arising from this, compounds the natural variation of the estimates. Robust minimax designs and weights are then constructed to minimize mean-squared-error-based loss functions of the estimates. We find the periodic and symmetric properties of the Euclidean norm of the multiwavelet system useful in eliminating some of the mathematical difficulties involved. These properties lead us to restrict the search for robust minimax designs to a specific class of symmetric designs. We also construct minimum variance unbiased designs and weights which minimize the loss functions subject to a side condition of unbiasedness. We discuss an example from the nonparametric literature.     

D-optimal weighing designs and optimal chemical balance weighing designs for estimating the total weight

This paper provides D-optimal spring balance designs for estimating individual weights when the number of objects to be weighed in each weighing, B, is fixed. D-optimal chemical balance designs for estimating total weight under both homogeneous and nonhomogeneous error variances are found when the number of objects weighed in each weighing is ≥ B, a fixed number.

We indicate the restriction used in Chacko & Dey(1978) and Kageyama(1988), i.e. that chemical designs X be restricted to designs in which exactly “a” objects are replaced on the left pan and exactly “b” on the right pan in each of the weighings for a, b > 0, is unnecessary.     

Robust inference for estimating equations with nonignorably missing data based on SIR algorithm

Nonresponse is a very common phenomenon in survey sampling. Nonignorable nonresponse – that is, a response mechanism that depends on the values of the variable having nonresponse – is the most difficult type of nonresponse to handle. This article develops a robust estimation approach to estimating equations (EEs) by incorporating the modelling of nonignorably missing data, the generalized method of moments (GMM) method and the imputation of EEs via the observed data rather than the imputed missing values when some responses are subject to nonignorably missingness. Based on a particular semiparametric logistic model for nonignorable missing response, this paper proposes the modified EEs to calculate the conditional expectation under nonignorably missing data. We can apply the GMM to infer the parameters. The advantage of our method is that it replaces the non-parametric kernel-smoothing with a parametric sampling importance resampling (SIR) procedure to avoid nonparametric kernel-smoothing problems with high dimensional covariates. The proposed method is shown to be more robust than some current approaches by the simulations.     

Minimax designs for estimating slopes in a trigonometric regression model

Abstract

Designs for the first order trigonometric regression model over an interval on the real line are considered for the situation where estimation of the slope of the response surface at various points in the factor space is of primary interest. Minimization of the variance of the estimated slope at a point maximized over all points in the region of interest is taken as the design criterion. Optimal designs under the minimax criterion are derived for the situation where the design region and the region of interest are identical and a symmetric “partial cycle”. Some comparisons of the minimax designs with the traditional D- and A-optimal designs are provided. Efficiencies of some exact designs under the minimax criterion are also investigated.     

Optimum percentile estimating equations for nonlinear random coefficient models

In nonlinear random coefficients models, the means or variances of response variables may not exist. In such cases, commonly used estimation procedures, e.g., (extended) least-squares (LS) and quasi-likelihood methods, are not applicable. This article solves this problem by proposing an estimate based on percentile estimating equations (PEE). This method does not require full distribution assumptions and leads to efficient estimates within the class of unbiased estimating equations. By minimizing the asymptotic variance of the PEE estimates, the optimum percentile estimating equations (OPEE) are derived. Several examples including Weibull regression show the flexibility of the PEE estimates. Under certain regularity conditions, the PEE estimates are shown to be strongly consistent and asymptotic normal, and the OPEE estimates have the minimal asymptotic variance. Compared with the parametric maximum likelihood estimates (MLE), the asymptotic efficiency of the OPEE estimates is more than 98%, while the LS-type of procedures can have infinite variances. When the observations have outliers or do not follow the distributions considered in model assumptions, the article shows that OPEE is more robust than the MLE, and the asymptotic efficiency in the model misspecification cases can be above 150%.     

On d-optimal designs for binary data

We use the criterion of D-optimality of the Fisher information matrix to derive optimal vectors for binary data. Some concepts of totally positive functions and Polya functions of order II are used to derive properties of the determinant of the Fisher information matrix arising in quantal response bioassay and attribute life testing models. As is often the case in non-linear models the D-optimal vectors are functions of the unknown parameters. By using the criterion of D-optimality, general optimal vectors are characterized which could be used for constructing Bayesian or locally D-optimal designs.     

Optimum designs for estimation of optimum point under cost constraint

In this paper, we consider the estimation of the optimum factor combination in a response surface model. Assuming that the response function is quadratic concave and there is a linear cost constraint on the factor combination, we attempt to find the optimum design using the trace optimality criterion. As the criterion function involves the unknown parameters, we adopt a pseudo-Bayesian approach to resolve the problem.     

Optimum points of stratification for two or more stage designs

In this paper, an attempt is made to obtain optimum points of stratification for two or more stage designs with equal p.s.u.'s and the subsequent units. Stratification on the auxiliary variable when the study variable is closely related to the auxiliary variable has also been obtained. The determination of OPS in these cases have been illustrated with the help of some known specific distributions.     

Central composite designs for estimating the optimum conditions for a second-order model

Central composite designs which maximize both the precision and the accuracy of estimates of the extremal point of a second-order response surface for fixed values of the model parameters are constructed. Two optimality criteria are developed, the one relating to precision and based on the sum of the first-order approximations to the asymptotic variances and the other to accuracy and based on the sum of squares of the second-order approximations to the asymptotic biases of the estimates of the coordinates of the extremal point. Exact and continuous central composite designs are introduced and in particular designs which place no restriction on the pattern of the weights, termed benchmark designs, and designs which comprise equally weighted factorial and equally weighted axial points, termed axial-factorial designs, are explored. Algebraic results proved somewhat elusive and the requisite designs are obtained by a mix of algebra and numeric calculation or simply numerically. An illustrative example is presented and some interesting features which emerge from that example are discussed.     

Randomized response multivariate designs for categorical data

Three approaches to multivariate estimation for categorical data using randomized response (RR) are described. In the first approach, practical only for 2×2 contingency tables, a multi-proportions design is used. In the second approach, a separate RR trial is used for each variate and it is noted that the multi­variate design matrix of conditional probabilities is given by the Kroneeker product of the univariate design matrices of each trial, provided that the trials are independent of each other in a certain sense. The third approach requires only a single randomization and thus may be viewed as the use of vector response. Finally, a special-purpose bivariate design is presented.     

Efficient initial designs for binary response data

In this paper we introduce a binary search algorithm that efficiently finds initial maximum likelihood estimates for sequential experiments where a binary response is modeled by a continuous factor. The problem is motivated by switching measurements on superconducting Josephson junctions. In this quantum mechanical experiment, the current is the factor controlled by the experimenter and a binary response indicating the presence or the absence of a voltage response is measured. The prior knowledge on the model parameters is typically poor, which may cause the common approaches of initial estimation to fail. The binary search algorithm is designed to work reliably even when the prior information is very poor. The properties of the algorithm are studied in simulations and an advantage over the initial estimation with equally spaced factor levels is demonstrated. We also study the cost-efficiency of the binary search algorithm and find the approximately optimal number of measurements per stage when there is a cost related to the number of stages in the experiment.     

Bayesian adaptive dose‐escalation designs for simultaneously estimating the optimal and maximum safe dose based on safety and efficacy

The main purpose of dose‐escalation trials is to identify the dose(s) that is/are safe and efficacious for further investigations in later studies. In this paper, we introduce dose‐escalation designs that incorporate both the dose‐limiting events and dose‐limiting toxicities (DLTs) and indicative responses of efficacy into the procedure. A flexible nonparametric model is used for modelling the continuous efficacy responses while a logistic model is used for the binary DLTs. Escalation decisions are based on the combination of the probabilities of DLTs and expected efficacy through a gain function. On the basis of this setup, we then introduce 2 types of Bayesian adaptive dose‐escalation strategies. The first type of procedures, called “single objective,” aims to identify and recommend a single dose, either the maximum tolerated dose, the highest dose that is considered as safe, or the optimal dose, a safe dose that gives optimum benefit risk. The second type, called “dual objective,” aims to jointly estimate both the maximum tolerated dose and the optimal dose accurately. The recommended doses obtained under these dose‐escalation procedures provide information about the safety and efficacy profile of the novel drug to facilitate later studies. We evaluate different strategies via simulations based on an example constructed from a real trial on patients with type 2 diabetes, and the use of stopping rules is assessed. We find that the nonparametric model estimates the efficacy responses well for different underlying true shapes. The dual‐objective designs give better results in terms of identifying the 2 real target doses compared to the single‐objective designs.     

Influence of missing data on compact designs for spacing experiments

Density optimization of a plantation is a classical task with important practical consequences. In this article, we present an adaptation of criss-cross design and an alternative analysis. If a tree is missing, the spacing of neighbouring trees is altered and considerable information is lost. We derive the estimate of the missing value that minimizes the residual sum of squares and obtain the analytical solution of the EM algorithm. The relationships between the two techniques are clarified. The method is applied to data from a plantation of Eucalyptus in the Congo.     

The performance of subset response surface designs for estimating third order terms

Response surface designs are widely used in industries like chemicals, foods, pharmaceuticals, bioprocessing, agrochemicals, biology, biomedicine, agriculture and medicine. One of the major objectives of these designs is to study the functional relationship between one or more responses and a number of quantitative input factors. However, biological materials have more run to run variation than in many other experiments, leading to the conclusion that smaller response surface designs are inappropriate. Thus designs to be used in these research areas should have greater replication. Gilmour (2006) introduced a wide class of designs called “subset designs” which are useful in situations in which run to run variation is high. These designs allow the experimenter to fit the second order response surface model. However, there are situations in which the second order model representation proves to be inadequate and unrealistic due to the presence of lack of fit caused by third or higher order terms in the true response surface model. In such situations it becomes necessary for the experimenter to estimate these higher order terms. In this study, the properties of subset designs, in the context of the third order response surface model, are explored.