Maximum likelihood estimation of variance components in a completely random bib design

ABSTRACT

The purposes of this paper are to abstract from a number of articles variance component estimation procedures which can be used for completely random balanced incomplete block designs, to develop an iterated least squares (ITLS) computing algorithm for calculating maximum likelihood estimates, and to compare these procedures by use of simulated experiments. Based on the simulated experiments, the estimated mean square errors of the ITLS estimates are generally less than*those for previously proposed analysis of variance and symmetric sums estimators.     

Three-factor symmetrical balanced designs with full efficiency for main effects

Some methods for constructing balanced design for 3-factor symmetrical factorial experiments in which all the main effects are completely unconfounded by using balanced arrays and BIB designs are proposed. The method is flexible in terms of selecting block size.     

Asymptotic relative efficiency of designs for factorial paired comparison experiments

A general approach for comparing designs of paired comparison experiments on the basis of the asymptotic relative efficiencies, in the Bahadur sense, of their respective likelihood ratio tests is discussed and extended to factorials. Explicit results for comparing five designs of 2^q factorial paired comparison experiments are obtained. These results indicate that some of the designs which require comparison of fewer distinct pairs of treatments than does the completely balanced design are, generally, more efficient for detecting main effects and/or certain interactions. The developments of this paper generalize the work of Littell and Boyett (1977) for comparing two designs of R x C factorial paired comparison experiments.     

Design considerations for small experiments and simple logistic regression

Inference for a generalized linear model is generally performed using asymptotic approximations for the bias and the covariance matrix of the parameter estimators. For small experiments, these approximations can be poor and result in estimators with considerable bias. We investigate the properties of designs for small experiments when the response is described by a simple logistic regression model and parameter estimators are to be obtained by the maximum penalized likelihood method of Firth [Firth, D., 1993, Bias reduction of maximum likelihood estimates. Biometrika, 80, 27–38]. Although this method achieves a reduction in bias, we illustrate that the remaining bias may be substantial for small experiments, and propose minimization of the integrated mean square error, based on Firth's estimates, as a suitable criterion for design selection. This approach is used to find locally optimal designs for two support points.     

A new extended generalized Gompertz distribution with statistical properties and simulations

Abstract

Statistical distributions are very useful in describing and predicting real world phenomena. In many applied areas there is a clear need for the extended forms of the well-known distributions. Generally, the new distributions are more flexible to model real data that present a high degree of skewness and kurtosis. The choice of the best-suited statistical distribution for modeling data is very important.

In this article, we proposed an extended generalized Gompertz (EGGo) family of EGGo. Certain statistical properties of EGGo family including distribution shapes, hazard function, skewness, limit behavior, moments and order statistics are discussed. The flexibility of this family is assessed by its application to real data sets and comparison with other competing distributions. The maximum likelihood equations for estimating the parameters based on real data are given. The performances of the estimators such as maximum likelihood estimators, least squares estimators, weighted least squares estimators, Cramer-von-Mises estimators, Anderson-Darling estimators and right tailed Anderson-Darling estimators are discussed. The likelihood ratio test is derived to illustrate that the EGGo distribution is better than other nested models in fitting data set or not. We use R software for simulation in order to perform applications and test the validity of this model.     

Fitting linear regression models to censored data by least squares and maximum likelihood methods

Several approaches have been suggested for fitting linear regression models to censored data. These include Cox's propor­tional hazard models based on quasi-likelihoods. Methods of fitting based on least squares and maximum likelihoods have also been proposed. The methods proposed so far all require special purpose optimization routines. We describe an approach here which requires only a modified standard least squares routine.

We present methods for fitting a linear regression model to censored data by least squares and method of maximum likelihood. In the least squares method, the censored values are replaced by their expectations, and the residual sum of squares is minimized. Several variants are suggested in the ways in which the expect­ation is calculated. A parametric (assuming a normal error model) and two non-parametric approaches are described. We also present a method for solving the maximum likelihood equations in the estimation of the regression parameters in the censored regression situation. It is shown that the solutions can be obtained by a recursive algorithm which needs only a least squares routine for optimization. The suggested procesures gain considerably in computational officiency. The Stanford Heart Transplant data is used to illustrate the various methods.     

On construction of constant block-sum partially balanced incomplete block designs

Abstract

Constant block-sum designs are of interest in repeated measures experimentation where the treatments levels are quantitative and it is desired that at the end of the experiments, all units have been exposed to the same constant cumulative dose. It has been earlier shown that the constant block-sum balanced incomplete block designs do not exist. As the next choice, we, in this article, explore and construct several constant block-sum partially balanced incomplete block designs. A natural choice is to first explore these designs via magic squares and Parshvanath yantram is found to be especially useful in generating designs for block size 4. Using other techniques such as pair-sums and, circular and radial arrangements, we generate a large number of constant block-sum partially balanced incomplete block designs. Their relationship with mixture designs is explored. Finally, we explore the optimization issues when constant block-sum may not be possible for the class of designs with a given set of parameters.     

Characterization of singular balanced fractional sm factorial designs derivable from balanced arrays with maximum number of constraints

An upper bound on the maximum number of constraints for s-symbol balanced arrays is derived. It is shown that balanced fractional s^m factorial designs derived from some balanced arrays with the maximum possible number of constraints become singular.     

Some Recent Developments in Econometric Inference

Abstract

Recent results in information theory, see Soofi (1996; 2001) for a review, include derivations of optimal information processing rules, including Bayes' theorem, for learning from data based on minimizing a criterion functional, namely output information minus input information as shown in Zellner (1988; 1991; 1997; 2002). Herein, solution post data densities for parameters are obtained and studied for cases in which the input information is that in (1) a likelihood function and a prior density; (2) only a likelihood function; and (3) neither a prior nor a likelihood function but only input information in the form of post data moments of parameters, as in the Bayesian method of moments approach. Then it is shown how optimal output densities can be employed to obtain predictive densities and optimal, finite sample structural coefficient estimates using three alternative loss functions. Such optimal estimates are compared with usual estimates, e.g., maximum likelihood, two‐stage least squares, ordinary least squares, etc. Some Monte Carlo experimental results in the literature are discussed and implications for the future are provided.     

Optimal designs for statistical analysis with Zernike polynomials

The Zernike polynomials arise in several applications such as optical metrology or image analysis on a circular domain. In the present paper, we determine optimal designs for regression models which are represented by expansions in terms of Zernike polynomials. We consider two estimation methods for the coefficients in these models and determine the corresponding optimal designs. The first one is the classical least squares method and Φ _p -optimal designs in the sense of Kiefer [Kiefer, J., 1974, General equivalence theory for optimum designs (approximate theory). Annals of Statistics, 2 849–879.] are derived, which minimize an appropriate functional of the covariance matrix of the least squares estimator. It is demonstrated that optimal designs with respect to Kiefer's Φ _p -criteria (p>?∞) are essentially unique and concentrate observations on certain circles in the experimental domain. E-optimal designs have the same structure but it is shown in several examples that these optimal designs are not necessarily uniquely determined. The second method is based on the direct estimation of the Fourier coefficients in the expansion of the expected response in terms of Zernike polynomials and optimal designs minimizing the trace of the covariance matrix of the corresponding estimator are determined. The designs are also compared with the uniform designs on a grid, which is commonly used in this context.     

New Section on Social Statistics

The analysis of a general k-factor factorial experiment having unequal numbers of observations per cell is complex. For the special case of a 2 ^k experiment with unequal numbers of observations per cell, the method of unweighted means provides a simple vehicle for analysis that requires no matrix inversion and can be used with existing software programs for the analysis of balanced data. All numerator sums of squares for testing main effects and interactions are χ² with one degree of freedom. In addition, for tests having one degree of freedom in any factorial experiment, the method of unweighted means may be modified to yield exact tests.     

Constructions of Nested Pairwise Efficiency and Variance Balanced Designs

ABSTRACT

Nested pairwise efficiency and variance balanced designs form a new class of block designs. In this article, two methods of constructing such designs from a symmetric balanced incomplete block design are proposed with some illustrations.     

Robust 2k factorial design with Weibull error distributions

It is well known that the least squares method is optimal only if the error distributions are normally distributed. However, in practice, non-normal distributions are more prevalent. If the error terms have a non-normal distribution, then the efficiency of least squares estimates and tests is very low. In this paper, we consider the 2^k factorial design when the distribution of error terms are Weibull W(p,σ). From the methodology of modified likelihood, we develop robust and efficient estimators for the parameters in 2^k factorial design. F statistics based on modified maximum likelihood estimators (MMLE) for testing the main effects and interaction are defined. They are shown to have high powers and better robustness properties as compared to the normal theory solutions. A real data set is analysed.     

The first-order random coefficient integer valued autoregressive process with the occasional level shift random noise based on dual empirical likelihood

Abstract

This paper investigates the first-order random coefficient integer valued autoregressive process with the occasional level shift random noise based on dual empirical likelihood. The limiting distribution of log empirical likelihood ratio statistic is constructed. Asymptotic convergence and confidence region results of empirical likelihood ratio are given. Hypothesis testing is considering, and maximum empirical likelihood estimation for parameter is acquired. Simulations are given to show that the maximum empirical likelihood estimation is more efficient than the conditional least squares estimation.     

Properties of Designs in Experiments on Spatial Lattices

ABSTRACT

When spatial variation is present in experiments, it is clearly sensible to use designs with favorable properties under both generalized and ordinary least squares. This will make the statistical analysis more robust to misspecification of the spatial model than would be the case if designs were based solely on generalized least squares. In this article, treatment information is introduced as a way of studying the ordinary least squares properties of designs. The treatment information is separated into orthogonal frequency or polynomial components which are assumed to be independent under the spatial model. The well-known trend-resistant designs are those with no treatment information at the very low order frequency or polynomial components which tend to have the higher variances under the spatial model. Ideally, designs would be chosen with all the treatment information distributed at the higher-order components. However, the results in this article show that there are limits on how much trend resistance can be achieved as there are many constraints on the treatment information. In addition, appropriately chosen Williams squares designs are shown to have favorable properties under both ordinary and generalized least squares. At all times, the ordinary least squares properties of the designs are balanced against the generalized least squares objectives of optimizing neighbor balance.     

On the analysis of unbalanced two-level supersaturated designs via generalized linear models

Supersaturated designs (SSDs) are factorial designs in which the number of experimental runs is smaller than the number of parameters to be estimated in the model. While most of the literature on SSDs has focused on balanced designs, the construction and analysis of unbalanced designs has not been developed to a great extent. Recent studies discuss the possible advantages of relaxing the balance requirement in construction or data analysis of SSDs, and that unbalanced designs compare favorably to balanced designs for several optimality criteria and for the way in which the data are analyzed. Moreover, the effect analysis framework of unbalanced SSDs until now is restricted to the central assumption that experimental data come from a linear model. In this article, we consider unbalanced SSDs for data analysis under the assumption of generalized linear models (GLMs), revealing that unbalanced SSDs perform well despite the unbalance property. The examination of Type I and Type II error rates through an extensive simulation study indicates that the proposed method works satisfactorily.     

Comparisons of search designs using search probabilities

Search designs are considered for searching and estimating one nonzero interaction from the two and three factor interactions under the search linear model. We compare three 12-run search designs D1, D2, and D3, and three 11-run search designs D4, D5, and D6, for a 2⁴ factorial experiment. Designs D2 and D3 are orthogonal arrays of strength 2, D1 and D4 are balanced arrays of full strength, D5 is a balanced array of strength 2, and D6 is obtained from D3 by deleting the duplicate run. Designs D4 and D5 are also obtained by deleting a run from D1 and D2, respectively. Balanced arrays and orthogonal arrays are commonly used factorial designs in scientific experiments. “Search probabilities” are calculated for the comparison of search designs. Three criteria based on search probabilities are presented to determine the design which is most likely to identify the nonzero interaction. The calculation of these search probabilities depends on an unknown parameter ρ which has a signal-to-noise ratio form. For a given value of ρ, Criteria I and II are newly proposed in this paper and Criteria III is given in Shirakura et al. (Ann. Statist. 24 (6) (1996) 2560). We generalize Criteria I–III for all values of ρ so that the comparison of search designs can be made without requiring a specific value of ρ. We have developed simplified methods for comparing designs under these three criteria for all values of ρ. We demonstrate, under all three criteria, that the balanced array D1 is more likely to identify the nonzero interaction than the orthogonal arrays D2 and D3, and the design D4 is more likely to identify the nonzero interaction than the designs D5 and D6.The methods of comparing designs developed in this paper are applicable to other factorial experiments for searching one nonzero interaction of any order.     

Optimal design results for 2n factorial paired comparison experiments

Four general classes of partially balanced designs for 2ⁿ factorials, corresponding to four different forms of a general null hypothesis H on factorial effects, are presented. For the typical design in each class, the simplified form of the non-centrality parameter λ² of the asymptotic chi-square distribution of the likelihood ratio statistic for testing the corresponding form of H₀ is derived under defined local alternatives. Optimal designs d₁ maximizing λ² in the i-th class and minimizing the trace, determinant and largest eigenvalue of a defined covariance matrix, i =1,…,4, are determined.     

Minimum aberration blocking schemes for 128-run designs

Several criteria have been proposed for ranking blocked fractional factorial designs. For large fractional factorial designs, the most appropriate minimum aberration criterion was one proposed by Cheng and Wu (2002). We justify this assertion and propose a novel construction method to overcome the computational challenge encountered in large fractional factorial designs. Tables of minimum aberration blocked designs are presented for N=128 runs and n=8–64 factors.     

Local influence of nonlinear mixed effects model based on M-estimation

Abstract

In this work we mainly study the local influence in nonlinear mixed effects model with M-estimation. A robust method to obtain maximum likelihood estimates for parameters is presented, and the local influence of nonlinear mixed models based on robust estimation (M-estimation) by use of the curvature method is systematically discussed. The counting formulas of curvature for case weights perturbation, response variable perturbation and random error covariance perturbation are derived. Simulation studies are carried to access performance of the methods we proposed. We illustrate the diagnostics by an example presented in Davidian and Giltinan, which was analyzed under the non-robust situation.