1,721,073 research outputs found
J. N. Srivastava and experimental design
No full textJ. N. Srivastava was a tremendously productive statistical researcher for five decades. He made significant contributions in many areas of statistics, including multivariate analysis and sampling theory. A constant throughout his career was the attention he gave to problems in discrete experimental design, where many of his best known publications are found. This paper focuses on his design work, tracing its progression, recounting his key contributions and ideas, and assessing its continuing impact. A synopsis of his design-related editorial and organizational roles is also included
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Search for the Optimum Variance Components Estimates in Mixed Effects Models
Full text linkThis dissertation aims at searching for the optimum variance components estimates in the mixed-effects model. Traditional estimation methods of the variance components include the analysis of variance/method of moment (ANOVA/MoM) estimation, which is the optimum estimation (OPE) when the data are balanced, the maximum likelihood estimation (MLE) and the restricted maximum likelihood estimation (REMLE). However, when the data have small sample sizes and unbalanced structures, the optimum estimates do not exist, ML estimates are biased, MLE and REMLE cannot provide the closed-form expressions of the estimates to study their small-sample statistical properties. To solve those problems, we proposed the near optimum estimation (NOPE) method and the average optimum estimation (AOPE) method when the data are unbalanced in DOE. When estimating the variance components and a linear function of variance components in SAE, we proposed methods of finding the unbiased quadratic estimators with smaller variances than the corresponding MoM estimators. We presented simulation studies to evaluate the estimation performance of our proposed methods and compare them with MoM, ML and REML. All of our proposed estimators have closed-form expressions and do not require the functional form in the distributional assumptions
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Robust and Optimum Fractional Factorial Designs
Full text linkThis thesis is devoted to the study of robust and optimum fractional factorial designs. We consider models that contain the general mean, main effects, and k two-factor interactions for 2m fractional factorial experiments. We define Si to be the set of all (1 × m) vectors, with elements 1 and -1 of weight i, where the weight of a vector is the number of nonzero elements in it. We present the robustness property of two classes of designs D={S0, S1, Sm-1, Sm} and D1={S0, S1, S2, Sm} with respect to any t runs as well as a specific set of t runs in the sense that the full estimation capacity of the designs remain when we delete any t runs as well as specific t runs from the designs D and D1. The number of runs are (2+m) and [2+m+.5(m)(m-1)] in D and D1 respectively. We introduce a general structure M for the information matrices of a class of models possibly describing the data from a fractional factorial experiment with m factors each at two levels and n runs. We characterize all the eigenvalues and eigenvectors for such matrices M. For m=4 we establish the robustness property of the design D7={S0, S1, S3}. The runs of D7 are contained in design D when m=4. We show all the information matrices from design D7 and designs obtained from D7 by deleting some runs are special cases of M. Let DT be the class of designs with n runs for estimating the main effects only and let FT be the class of foldover designs with 2n runs, n runs from T in DT and another n runs from -T, having full estimation capacity for k=1. We prove that if T* in DT is E-optimum, the foldover design [T*, -T*] is optimum design with respect to AMCR and GMCR in FT. Furthermore, if T* is D- and A- optimum with a special structure for X'1T*X1T* we prove [T*, -T*] is GD, AD, GT, and AT optimal in FT
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Characterization of Special Variance Structures for Designs in Model Identification and Discrimination
Full text linkModels containing the general mean, main effects, and all possible k two-factor interaction effects are considered for factorial experiments with m factors observed at two levels each. Specifically, fractional factorial designs consisting of n runs which permit the identification and discrimination of the models of interest are evaluated and classified. The classifications are dependent on a new property introduced in this dissertation, denoted PV, g≥1, which relies on the structure of the variance-covariance matrix for the estimates of the model parameters. Designs with the property, PV, g≥1, permit to divide the models in the class considered into g groups so that all models in a group have equal variances for the least squares estimates of the k two-factor interaction effects. Ghosh-Tian optimum designs [Ghosh and Tian (2006)] for m=4, n=6,...,11, k=1, and m=5, n=7,...,16, k=1 are classified with respect to g values, g≥1, in the property PV and presented through illustrative examples. Several characterizations of designs with PV, g≥1, are provided for the case when k=1. Designs such as balanced designs, isomorphic designs and complementary designs with the property PV are proven to have P<1V for k=1. Tables identifying all such balanced designs are provided. It is noted that although D9.2 and D14 are not balanced, these designs in fact take a special form resulting in P<1V for k=1. These special forms are investigated in depth. In addition, the construction of all designs giving P<1V for m=3, n=5, 6, 7, 8, k=1 and m=4, n=6,...,16, k=1 are described. Further, occurrences of P<1V are presented for fractional factorial designs when m=5, n=7, 8, 9, k=1.Finally, additional characterizations of PV, g≥1, when P>1 are given and illustrated through various examples. Special designs are presented with the property P<1V for maxk which is the largest value of k in the models of interest that the design has the ability to identify and discriminate. It is shown that balanced designs will have P<1V for k=mC2 . Tables identifying all such balanced designs are provided
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Near Uniformly Minimum Variance Quadratic Unbiased Estimation of Variance Components in Mixed Effects Models
Full text linkSeveral methods are available in literature for estimating the variance components in mixed effects models. In this thesis we consider the general mixed effects model without making any distributional assumptions. The quadratic unbiased estimators are considered for estimating the variance components. The uniformly minimum variance quadratic unbiased estimation (UMVQUE) of variance components is investigated for the data obtained from both balanced and unbalanced designs. In spite of its attractive properties, the UMVQUE may not always be possible. When the UMVQUE is not possible, we propose two alternative methods for estimating the variance components. We first introduce a method of near uniformly minimum variance quadratic unbiased estimation (NUMVQUE) for an unbalanced incomplete block design. When the UMVQUE of variance components is not possible for a design with replicated blocks but it is possible with a single replication of blocks, we propose another method of average uniformly minimum variance quadratic unbiased estimation (AUMVQUE). The maximum likelihood estimation (MLE) and restricted maximum likelihood estimation (REMLE) are likelihood based procedures and therefore require the distributional assumptions to estimate the variance components. We present a simulation study to evaluate the performance of our proposed estimation methods and compare them with MLE and REMLE
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Optimum Designs for Identification and Discrimination within a Class of Competing Linear Regression Models
Full text linkWe consider the problem of finding optimum designs for model identification and discrimination where the dependence of the response variable Y on an explanatory variable X can be described by at most a third order model. We therefore consider a class that includes all the models up to a maximum of third order with linear, quadratic, and cubic coefficients present. In addition all models have an intercept parameter. A general class of designs with 4 distinct points x_1, x_2, x_3, and x_4 is considered with replications n1; n2; n3, and n4 respectively, satisfying n_1 + n_2 + n_3 + n_4 = n where n is known in advance. While discriminating between two models from the class of models considered, the true model may or may not be one of them. We define the predictive criterion function I and the fitting criterion function J. When the functions I and J are dependent on more than one model parameters, we define the additional criterion functions K_I and K_J. We use the proposed optimality criterion functions to obtain the optimal designs for the model identification and discrimination
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Estimation of the Parameters of Skew Normal Distribution by Approximating the Ratio of the Normal Density and Distribution Functions
Full text linkThe normal distribution is symmetric and enjoys many important properties. That is why it is widely used in practice. Asymmetry in data is a situation where the normality assumption is not valid. Azzalini (1985) introduces the skew normal distribution reflecting varying degrees of skewness. The skew normal distribution is mathematically tractable and includes the normal distribution as a special case. It has three parameters: location, scale and shape. In this thesis we attempt to respond to the complexity and challenges in the maximum likelihood estimates of the three parameters of the skew normal distribution. The complexity is traced to the ratio of the normal density and distribution function in the likelihood equations in the presence of the skewness parameter. Solution to this problem is obtained by approximating this ratio by linear and non-linear functions. We observe that the linear approximation performs quite satisfactorily. In this thesis, we present a method of estimation of the parameters of the skew normal distribution based on this linear approximation. We define a performance measure to evaluate our approximation and estimation method based on it. We present the simulation studies to illustrate the methods and evaluate their performances
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Evaluation of the Design Effect for Optimizing the Model Discrimination Strength to Detect Non-Zero Interactions in Factorial Experiments
Full text linkIt is important for a design to be able to discriminate between all pairwise model comparisons when searching for non-zero interactions. If a design does not have this capability, finding non-zero interactions may not be possible. We propose a procedure at analyzing the model discrimination strength when searching for non-zero interactions by understanding the pairwise differenced error sum of squares for a given design. This is done by calculating eigenvalues and eigenvectors of differenced projection matrices which are completely dependent on the design and not the observed values for the response variable. Using this procedure, we have compared two balanced designs and two Placket-Burman (1946) designs which both have n=12 runs, m=5 factors each with 2 levels, and searching over (5¦2)=10 two-factor interaction effects. Additionally, ridge regression and LASSO were used for a model selection approach which both use tuning parameters to calculate the model parameter estimates. All three model selection approaches were used on all four example designs to compare the performance when searching for non-zero interactions
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New Methods for Solving Maximum Likelihood Estimating Equations of Logistic and Probit Regression Models
Full text linkSeveral iterative methods are available in literature for solving the Maximum Likelihood Estimating Equations (MLEEs) of logistic and probit regression models. Generalized Self Consistency (GSC) method is such an existing iterative method. We introduce a new idea using the paired observations and combine it with the GSC method for both logistic and probit regression models and propose several new methods for solving MLEEs. For probit regression model, we introduce a linear approximation method for finding the exact solution of MLEEs. We illustrate the proposed methods with a real data as well as a simulated data and compare their performances with the existing methods. We investigate some theoretical properties of our estimates. We also present a meaningful method of choosing the initial values of parameters for the iterative methods
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Estimation of Parameters for Logistic Regression Model in Dose Response Study with A Single Compound or Mixture of Compounds
Full text linkWe investigate the estimation issues for count data in dose response model. Inthis thesis, we are considering logistic dose response model for a mixture experimentwith two drugs. We propose two new methods of estimation of parameters for thismodel by forming the observation pairs. The standard maximum likelihood estima-tion method uses the numerical methods for solving the estimating equations. Thismethod requires an initial set of values for the parameters in the model. The standardprocedure normally uses the initial values as zero or some convenient numbers withoutany justication. We present two very systematic methods of nding the initial valuesof parameters of the maximum likelihood estimating equations (MLEE). Our methodsare based on two criterion functions, the log-likelihood and the other function . Wethen use the initial values and the corresponding criterion function to obtain the nalsolution of MLEE. We demonstrate that when we consider only two doses from thedata, we do have an exact analytic expression for the solution of estimating equations.We use that fact to obtain the initial values of parameters in these models. Then wehave used the search algorithm for performing the optimization to nd the nal esti-mates. The proposed methods are transparent in the selection of the initial values ofparameters. The proposed methods are computer intensive like bootstrap and jack-knife methods popular among statisticians. We have also compared our estimates withthe estimates obtained by SAS and R. The proposed methods compare favorably with SAS and R in terms numerical values of the estimates and the performance time of theestimates. We illustrate our methods with a data set (Giltinan, 1998). We present alsosome simulated data to illustrate our methods
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