Showing papers in "Technometrics in 1978"
TL;DR: In this article, the rank estimation of the rank A of the matrix Y, i.e., the estimation of how much of the data y ik is signal and how much is noise, is considered.
Abstract: By means of factor analysis (FA) or principal components analysis (PCA) a matrix Y with the elements y ik is approximated by the model Here the parameters α, β and θ express the systematic part of the data yik, “signal,” and the residuals ∊ ik express the “random” part, “noise.” When applying FA or PCA to a matrix of real data obtained, for example, by characterizing N chemical mixtures by M measured variables, one major problem is the estimation of the rank A of the matrix Y, i.e. the estimation of how much of the data y ik is “signal” and how much is “noise.” Cross validation can be used to approach this problem. The matrix Y is partitioned and the rank A is determined so as to maximize the predictive properties of model (I) when the parameters are estimated on one part of the matrix Y and the prediction tested on another part of the matrix Y.
2,468 citations
TL;DR: The performance of the new Gauss-Newton-like algorithm, called Dud for “doesn't use derivatives”, is evaluated on a number of standard test problems from the literature and it competes favorably with even the best derivative-based algorithms.
Abstract: Derivative-free nonlinear least squares algorithms which make efficient use of function evaluations are important for fitting models defined by systems of nonlinear differential equations. A new Gauss-Newton-like algorithm with these properties is developed. The performance of the new algorithm (called Dud for “doesn't use derivatives”) is evaluated on a number of standard test problems from the literature. On these problems Dud competes favorably with even the best derivative-based algorithms.
357 citations
TL;DR: The biplot graphical display of matrices is proposed as a data analytic tool for diagnosing the type of model to fit the data as discussed by the authors, where rows and columns of the data matrix are represented by markers and the diagnosis proceeds by examining these markers for collinearity.
Abstract: The biplot graphical display of matrices is proposed as a data analytic tool for diagnosing the type of model to fit the data. Rows and columns of the data matrix are represented by markers—the diagnosis proceeds by examining these markers for collinearity. Thus, for example. if set I of row markers are collinear and a set J of column markers are also collinear. a Tukey degree-of-freedom-for-non-additivity is diagnosed for the submatrix of these I-rows and J-columns: if the two lines are at 90°. an additive model is diagnosed. Mandel type models are similarly diagnosed when either the row or the column markers are collinear but not both. Illustrations from physical science data show how simple biplot diagnostic modeling is and how many features of the data are revealed. Of particular interest are biplots of three-way layouts which allow visual inspection of main effects and interactions.
189 citations
TL;DR: In this article, the authors present a maximum likelihood theory for large-sample optimum accelerated life test plans, which are used to estimate a simple linear relationship between (transformed) stress and product life.
Abstract: This paper presents maximum likelihood theory for large-sample optimum accelerated life test plans. The plans are used to estimate a simple linear relationship between (transformed) stress and product life, which has a Weibull or smallest extreme value distribution. Censored data are to be analyzed before all test units fail. The plans show that all test units need not run to failure and that more units should be tested at low test stresses than at high ones. The plans are illustrated with a voltage-accelerated life test of an electrical insulating fluid.
177 citations
TL;DR: In this article, a numerical procedure for the tabulation of average run lengths (ARL's) of geometric moving average charts is presented for various settings of the control limits, smoothing constant and shift in the nominal level of the process mean.
Abstract: A numerical procedure is presented for the tabulation of average run lengths (ARL's) of geometric moving average charts. Both one-and two-sided ARL's are given for various settings of the control limits, smoothing constant and shift in the nominal level of the process mean. Where comparison is possible. the tabulated ARL's are in agreement with those obtained by Roberts [3]. [4] using simulation techniques.
161 citations
TL;DR: In this article, Urn models and their application in modern discrete probability theory are discussed. But their application is not discussed in this paper, but instead in the context of modern Discrete Probability Theory.
Abstract: (1978). Urn Models and Their Application—An Approach to Modern Discrete Probability Theory. Technometrics: Vol. 20, No. 4, pp. 501-501.
142 citations
TL;DR: In this paper, the difference in favor of all-subsets can be arbitrarily large in examples where there are predictors which do poorly alone but do very well together, where the differences are measured on the data (sample values).
Abstract: Although it is generally felt that the all-subsets, or “best” subset, approach is better than forward selection and backward elimination, the sequential procedures are still widely used. To see what advantage there is in doing all-subsets, this paper gtves both theoretical and empirical comparisons. It is shown that the difference in favor of all-subsets can be arbitrarily large in examples where there are predictors which do poorly alone but do very well together. Also. empirical comparisons on nine data sets show big dilrerences favoring all-subsets, when the differences are measured on the data (sample values). However, fairer comparisons based on known population values show very small differences favoring all-subsets. The only exception is the one data set which has predictors which do well together but poorly alone.
119 citations
TL;DR: The need for a multipurpose design strategy, which takes into account the need to design for estimation of the parameters of the correct model, is stressed.
Abstract: We review existing methods of experimental design for discriminating between rival regression models. Box and Hill's algorithm and its extensions. and the work of Atkinson and Fedorov and Atkinson and Cox are particularly considered. The relationship between the various methods is pointed out. Several unrelated and less formal algorithms are also mentioned. The need for a multipurpose design strategy, which takes into account the need to design for estimation of the parameters of the correct model, is stressed.
108 citations
TL;DR: In this article, the authors evaluate the relative performances of several mechanical selection rules, the ridge trace and the least squares procedure using computer simulation experiments and conclude that the ridge traces are more accurate than least squares estimates.
Abstract: Least squares estimates of the parameters in the usual linear regression model are likely to be too large in absolute value and possibly of the wrong sign when the vectors of explanatory variables are multicollinear. Hoer1 and Kennard have demonstrated that these undesirable effects of multicollinearity can be reduced by using “ridge” estimates in place of the least squares estimates. Unfortunately, ordinary ridge estimates depend on a value, k, which, in practice, is determined by the data. Several mechanical rules and a graphical procedure, known as the ridge trace, have been proposed for selecting k. In this paper we evaluate the relative performances of several mechanical selection rules, the ridge trace and the least squares procedure using computer simulation experiments.
107 citations
TL;DR: In this paper, minimum variance quadratic unbiased estimators (MIVQUE's) of variance components from unbalanced data are obtained for the one-way classification random model under normality.
Abstract: Minimum variance quadratic unbiased estimators (MIVQUE's) of variance components from unbalanced data are obtained for the one-way classification random model under normality. Explicit, computable expressions are given for the estimators, their variances, and their covariance. The variance expressions provide readily-calculated lower bounds for the variances of any quadratic unbiased estimators of the variance components. For unbalanced data, the estimators are functions of the data and of constants σ ao 2 and σ e 2, taken as a priori estimates of the variance components σ a 2 and σ e 2. The estimators are, for unbalanced data, only locally minimum variance, i.e., they are only minimum variance when σ ao 2 = σ a 2 and σ eo 2 = σ e 2. However, numerical results suggest that the “MIVQUE” of σ a 2 may have much smaller variance than the usual ANOVA estimator with unbalanced data, even when σ ao 2 and σ eo 2 deviate considerably from σ a 2 and σ e 2 respectively. In contrast, the ANOVA estimator of σ e 2 seem...
TL;DR: In this paper, the level and power of the usual F-test for the r-way layout in analysis of variance (ANOVA) are vitiated when the assumption of equality of error variances is violated.
Abstract: The level and power of the usual F-test for the r-way layout in analysis of variance (ANOVA) are vitiated when the assumption of equality of error variances is violated. We present procedures, with tables and approximations needed for implementation, which give exact tests with power and level completley independent of the unknown variances. Thus. the assumption of equal variances may be dropped.
TL;DR: In this paper, the authors present a review of methods of constructing confidence intervals for parameters or other characteristics of the Weibull or extreme value distribution, with emphasis on the flexibility of the method and on the computations which are necessary to use it.
Abstract: This paper reviews methods of constructing confidence intervals for parameters or other characteristics of the Weibull or extreme value distribution. The conditional method of obtaining confidence intervals is stressed, with emphasis on the flexibility of the method, and on the computations which are necessary to use it.
TL;DR: In this paper, an analogue of the Box-Hunter rotatability property for second order response surface designs in k independent variables is presented, which is called slope-rotatability by analogy with the corresponding property for the variance of the estimated response, ŷ.
Abstract: An analogue of the Box-Hunter rotatability property for second order response surface designs in k independent variables is presented. When such designs are used to estimate the first derivatives with respect to each independent variable the variance of the estimated derivative is a function of the coordinates of the point at which the derivative is evaluated and is also a function of the design. By choice of design it is possible to make this variance constant for all points equidistant from the design origin. This property is called slope-rotatability by analogy with the corresponding property for the variance of the estimated response, ŷ. For central composite designs slope-rotatability can be achieved simply by adjusting the axial point distances (α), so that the variance of the pure quadratic coefficients is one-fourth the variance of the mixed second order coefficients. Tables giving appropriate values of α have been constructed for 2 ≤ k ≤ 8. For 5 ≤ k ≤ 8 central composite designs involving fracti...
TL;DR: In this article, the Weibull process is used to model reliability growth and the problem of inference and prediction is solved using the maximum likelihood estimators, and integral expressions for the distributions of these functions are presented which make it possible to compute, using numerical integration, the probabilities needed for inference.
Abstract: Some problems of inference and prediction arise when the Weibull process is used to model reliability growth. Their solution can be based on certain functions involving the maximum likelihood estimators. Integral expressions for the distributions of these functions are presented which make it possible to compute, using numerical integration, the probabilities needed for inference.
TL;DR: In this article, the use of moving sums (MOSUMS) of recursively calculated residuals is demonstrated for testing the constancy of the expected value of a series of T independent observations normally distributed with equal variances.
Abstract: The use of moving sums (MOSUMS) of recursively calculated residuals (Hack1 [9]) is demonstrated for testing the constancy of the expected value of a series of T independent observations normally distributed with equal variances. The cases of known or unknown expected value are treated separately; in the situation of unknown variance the recursive estimation of the variance is suggested in the course of the sequential application of the test procedure. Conservative probability limits are constructed and an assessment of the effective error probabilities and the power of the method is given in simulation studies. Finally the use of the procedure is demonstrated in an example.
TL;DR: Tolerance limits and confidence limits on reliability are proposed for the two-parameter exponential distribution in this paper, with the advantage that solutions to both the tolerance limit problem and the confidence limit problem can be written explicitly.
Abstract: Tolerance limits and confidence limits on reliability, which closely approximate exact limits. are proposed for the two-parameter exponential distribution. These approximations have the advantage that solutions to both the tolerance limit problem and the confidence limit problem can be written explicitly.
TL;DR: In this paper, minimum cost p-charts are presented for 36 sets of cost data and three degrees of shift in the process mean, which are expressed as 0.1, 0.5 and 1.0 multiples of the initial process standard deviation.
Abstract: For two initial process levels, po ′ = 0.01 and po ′ = 0.05, minimum cost p-charts are presented for 36 sets of cost data and three degrees of shift in the process mean. Minimum cost designs are also given for other special cases. Actually minimum cost designs are derived for np-charts and converted to p-charts. The degrees of shift are expressed as 0.1, 0.5 and 1.0 multiples of the initial process standard deviation. This has some but not a full stabilizing effect. In general it is found that small shifts require larger samples taken less frequently, but generally with a lower overall intensity of sampling. It is also found that with rare exceptions charts designed to detect small shifts have acceptance limits of zero. The effects of variations in the cost parameters and in the rate of occurrence of the assignable cause are also presented in the paper and a brief comparison is made with a paper on p-charts by W. K. Chiu and another by D. C. Montgomery, J. F. Mance and R. G. Herkes.
TL;DR: In this article, a method based on minimizing estimators of the criteria E (L 1 2 ) and E(L 2 2 ) is given for selecting the k 1, in generalized ridge regression; an explicit, closed form solution is developed for the resulting estimator β* of the vector of parameters,β, in the linear regression model.
Abstract: A method, based upon minimizing estimators of the criteria E(L 1 2) and E(L 2 2), is given for selecting the k 1, in generalized ridge regression; an explicit, closed form solution is developed for the resulting estimator β* of the vector of parameters,β, in the linear regression model. A method is also proposed for obtaining constrained generalized ridge estimators, with constraints placed upon β*, to utilize a priori information concerning the signs of the model parameters along with ridge estimation. These and other ridge estimation procedures are then examined and compared with least squares or constrained least squares estimators in a Monte Carlo study which demonstrates their effectiveness. An example is given to indicate their behavior and use on an applied problem.
TL;DR: In this article, the posterior distribution of β is discussed, when it is assumed a priori that any given set of k observations has uniform probability I/( n k ) of being spurious.
Abstract: We deal with the situation covered by the univariate general linear model, that is, it is intended that n observations be generated in accordance with the usual model y = Xβ + e however, it is feared that k of the observations are spurious, that is, not generated in the manner intended, so that for an unknown set of k distinct integers, say (i 1, … ik ), a subset of the first n integers, we have, specifically, that ytj , = x tj ′, β + a j , + ∊ tj , where in general x t ′ denotes the t-th row of X, and where (a 1, …, a k ), so called shift parameters, are such that – ∞ < a j , < ∞. In this paper, we discuss the posterior distribution of β, when indeed it is assumed a priori that any given set of k observations has uniform probability I/( n k ) of being spurious. The properties of the posterior of β are discussed, and the results used in an example using data generated from a response surface design. Ad hoc procedures are discussed for gaining information on k, when k is unknown. These ad hoc procedures ar...
TL;DR: In this article, a double-edged jack knife procedure for constructing confidence regions in nonlinear regression is examined using Monte Carlo simulation, and it is shown that for moderate sample sizes the jack knife cannot be trusted in establishing joint confidence regions.
Abstract: A jack knife procedure for constructing confidence regions in nonlinear regression is examined using Monte Carlo simulation. The jack knife promises to be asymptotically double-edged, being both independent of linearizing approximations to the regression surface and insensitive to specification of the error distribution. For moderate sample sizes the jack knife cannot be trusted in establishing joint confidence regions.
TL;DR: In this article, a stepwise procedure based on likelihood ratio statistics was used to determine the number of multiplicative components in a fixed two-way model and an approximation was presented to obtain percentage points for these likelihood ratios statistics.
Abstract: Stepwise procedures based upon likelihood ratio statistics are used to determine the number of multiplicative components in a fixed two-way model. An approximation is presented to obtain percentage points for these likelihood ratio statistics.
TL;DR: In this article, a modification for the WE statistic for testing that a sample comes from the exponential distribution is given. The modification is for the case where the origin is known, and has been devised to make use of the tables already provided by Shapiro and Wilk.
Abstract: A modification is given for the WE , statistic for testing that a sample comes from the exponential distribution. The modification is for the case where the origin is known, and has been devised to make use of the tables already provided by Shapiro and Wilk. Power studies are given to show that it is worthwhile using the modification when possible.
TL;DR: In this paper, D-optimal fractions of three-level factorial designs for p factors are constructed for factorial effects models (2 ≤ p ≤ 4) and quadratic response surface models ( 2 ≤ p ≥ 5) using an exchange algorithm for maximizing |X′X| and an algorithm which produces D-optimally balanced array designs.
Abstract: D-optimal fractions of three-level factorial designs for p factors are constructed for factorial effects models (2 ≤ p ≤ 4) and quadratic response surface models (2 ≤ p ≤ 5). These designs are generated using an exchange algorithm for maximizing |X′X| and an algorithm which produces D-optimal balanced array designs. The design properties for the DETMAX designs and the balanced array designs are tabulated. An example is given to illustrate the use of such designs.
TL;DR: In this article, the exact prediction intervals for the kth future observation of a Weibull process were derived based on maximum likelihood estimation, and an exact, closed-form prediction limit was given.
Abstract: Exact prediction intervals, based upon maximum likelihood estimation, for the kth future observation of a Weibull process are derived. For k = 1, an exact, closed-form prediction limit is given. For k > 1, the exact solution, and a simpler approximate prediction limit are both given.
TL;DR: In this article, the distribution of the one sample Kolmogorov-Smirnov statistics for truncated or censored samples is presented for sample sizes exceeding 25; they are based on the asymptotic distribution derived by Koziol and Byar and the exact power of these tests for finite sample sizes can be calculated.
Abstract: Formulas and tables of significance points are presented for the distribution of the one sample Kolmogorov-Smirnov statistics for truncated or censored samples as proposed by Barr and Davidson [1]. Approximate formulas for the significance points are given for sample sizes exceeding 25; they are based on the asymptotic distribution derived by Koziol and Byar [2]. In addition, it is shown how the exact power of these tests for finite sample sizes can be calculated.
TL;DR: In this article, the extent of multicollinearity in a set of regression data with respect to parameter confidence regions and tests of hypotheses, the effective sample size, and predictability is evaluated.
Abstract: Problems of multicollinearity in regression analysis are considered and numerical measures established to assess the extent of multicollinearity in a set of regression data with respect to parameter confidence regions and tests of hypotheses, the effective sample size, and predictability.
TL;DR: The authors proposed a two-stage test for the presence of two outliers or one outlier in two-way tables, where the test statistics are estimated by Monte Carlo generations, and approximations to the percentage points are suggested.
Abstract: Previous work by Gentleman and Wilk on outliers in two-way tables is summarized, and their statistic QK is discussed. Instead of a plot of QK values, we propose a two-stage test for the presence of two outliers or one outlier. Percentage points for the test statistics are estimated by Monte Carlo generations, and approximations to the percentage points are suggested. The 8 × 12 case is examined in detail and extensions to other cases are briefly discussed.