Showing papers in "Technometrics in 1968"
TL;DR: In this article, several methods of estimating error rates in discriminant analysis are evaluated by sampling methods, and two methods in most common use are found to be significantly poorer than some new methods that are proposed.
Abstract: Several methods of estimating error rates in Discriminant Analysis are evaluated by sampling methods. Multivariate normal samples are generated on a computer which have various true probabilities of misclassification for different combinations of sample sizes and different numbers of parameters. The two methods in most common use are found to be significantly poorer than some new methods that are proposed.
1,513 citations
TL;DR: In this paper, Measurement and Analysis of Random Data (MADR) is used for the analysis of random data in the context of measurement and analysis of statistical data sets.
Abstract: (1968). Measurement and Analysis of Random Data. Technometrics: Vol. 10, No. 4, pp. 869-871.
1,041 citations
TL;DR: This review of some of the recent work in the study of errora of measurement focuses on the type of mathematical model used, the extent to which standard techniques of analysis become erroneous and misleading if certain types of errors are present, and the techniques that are available for the numerical study of errors of measurement.
Abstract: In this review of some of the recent work in the study of errora of measurement, attention is centered on the type of mathematical model used to represent errom of measurement, on the extent to which standard techniques of analysis become erroneous and misleading if certain types of errors are present (and the possible remedial procedures), and the techniques that are available for the numerical study of errors of measurement
449 citations
438 citations
TL;DR: In this article, three basic methods of non-linear least squares estimation and the best known modifications to the popular Gauss-Newton procedure are reviewed and an example is given to illustrate its effectiveness.
Abstract: This paper reviews three basic methods of non-linear least squares estimation and the best known modifications to the popular Gauss-Newton procedure. With regard to the Gauss-Newton procedure the paper identifies the problem of poor parameterization and gives simple examples to show how it may arise. A simple example is also given to show that one popular method of handling constrained boundaries can lead to erroneous results. A modification, based on stepwise linear regression techniques, to handle both of these problems is presented and discussed and an example is given to illustrate its effectiveness. The required fundamentals of stepwise linear regression are reviewed.
214 citations
TL;DR: A Monte Carlo study of the distribution of (symmetrically) Winsorized t shows that when the population is normally distributed, t behaves, to a satisfactory approximation, like a multiple of Student's t with the degrees of freedom that would be assigned intuitively as mentioned in this paper.
Abstract: A Monte Carlo study of the distribution of (symmetrically) Winsorized t shows that when the population is normally distributed Winsorized t behaves, to a satisfactory approximation, like a multiple of Student's t with the degrees of freedom that would be assigned intuitively The approximate multiplier of t is (n – l)/(h – l), where h of the original n observations were not touched by the symmetrical Winsorzation, and the degrees of freedom are h – 1 This approximation is precise enough for most routine uses More precise values are tabulated
208 citations
TL;DR: The Advanced Theory of Statistics, Vol. 10, No. 1, pp. 211-212 as mentioned in this paper, is the most cited work in the field of statistician and statistician.
Abstract: (1968). The Advanced Theory of Statistics, Vol. 3. Technometrics: Vol. 10, No. 1, pp. 211-212.
174 citations
TL;DR: In this paper, the Incomplete Beta Function (IBF) is described and a table of the complete Beta function is presented, along with a discussion of its properties and properties.
Abstract: (1968). Tables of the Incomplete Beta Function. Technometrics: Vol. 10, No. 4, pp. 879-880.
167 citations
TL;DR: The noncentral t-distribution has been applied to tolerance limits, to variables sampling plans, to confidence limits on a quantile, to a proportion, to the distribution of the sample coefficient of variation, and to the power of Student's t-test as mentioned in this paper.
Abstract: Applications are outlined of the noncentral t-distribution to tolerance limits, to variables sampling plans, to confidence limits on a quantile, to confidence limits on a proportion, to the distribution of the sample coefficient of variation, and to the power of Student's t-test. The basic assumption is that there is available a random sample from a normal distribution with mean and variance unknown. Some of the mathematical properties of the noncentral t-distribution are also outlined. An extensive bibliography has been prepared and cross-referenced to several review journals and books. The bibliography covers references to tolerance limits and sampling plans based on the normal distribution whether or not the noncentral t-distribution is directly involved. Some selected references to Student's t-distribution are also included.
150 citations
TL;DR: In this paper, conditions are established under which, when the number of experiments is a multiple of the number number of parameters, replication of the best design for p experiments is an optimal design for N experiments.
Abstract: This paper is concerned with the design of experiments to estimate the parameters in a model of known form, which may be nonlinear in the parameters. This problem was discussed in detail by Box and Lucas for the case where N, the number of experiments, is equal to p, the number of parameters. The present work is an extension to cases where N is greater than p. Conditions are established under which, when the number of experiments is a multiple of the number of parameters, replication of the best design for p experiments is an optimal design for N experiments. Several chemical examples are discussed; in each instance, the best design consists of simply repeating points of the original design for p experiments. An example is also mentioned where the best design does not consist of such replication.
132 citations
TL;DR: In this article, the authors present a design criterion that emphasizes model discrimination when there is considerable doubt as to which model is best and gradually shifts the emphasis to parameter estimation as experimentation progresses and discrimination is accomplished.
Abstract: Two objectives of much experimentation in science and engineering are (i) to establish the form of an adequate mathematical model for the system being investigated and (ii) to obtain precise estimates of the model parameters. In the past, statistical design procedures have been proposed for tackling either one of these problems separately. Investigators, however, frequently want to perform experiments which will shed light on both questions simultaneously. In this paper, therefore, we present a design criterion which takes both objectives into account. The basic design strategy is to emphasize model discrimination when there is considerable doubt as to which model is best and gradually shifting the emphasis to parameter estimation as experimentation progresses and discrimination is accomplished. It is assumed that experiments can be performed sequentially. The use of the design criterion is illustrated with an example.
TL;DR: In this article, the cumulative sum chart for controlling the mean of a manufacturing process having normally distributed quality with known variance is studied, and the costs of repairing the process, of operating out-of-control, and of maintaining the chart are assumed known.
Abstract: The cumulative sum chart for controlling the mean of a manufacturing process having normally distributed quality with known variance is studied. The costs of repairing the process, of operating out-of-control, and of maintaining the chart are assumed known. A formula giving approximately the long-run time average cost of operation as a function of the control chart parameters is given. Some rules of thumb for choosing these parameters are also suggested.
TL;DR: In this article, point estimators of parameters of the first asymptotic distribution of smallest (extreme) values, or, the extreme value distribution, are surveyed and compared, including maximum likelihood and moment estimators, inefficient estimators based on only a few ordered observations, and various linear estimation methods.
Abstract: Point estimators of parameters of the first asymptotic distribution of smallest (extreme) values, or, the extreme-value distribution, are surveyed and compared. Those investigated are maximum-likelihood and moment estimators, inefficient estimators based on only a few ordered observations, and various linear estimation methods. A combination of Monte Carlo approximations and exact small-sample and asymptotic results has been used to compare the expected loss (with loss equal to squared error) of these various point estimators. Since the logarithms of variates having the two-parameter Weibull distribution are variates from the extreme-value distribution, the investigation is applicable to the estimation of Weibull parameters. Interval estimation procedures are also discussed.
TL;DR: In this paper, two modes of combination of a collection of W statistics are considered, namely the standardized mean of the normal transforms of W and the sum of the x 2(2) transforms of the significance levels of w; and these are proposed for use in conjunction with the probability plotting of the collection of these transforms.
Abstract: Statistical methods are presented for the joint assessment of the supposed normality of a collection of independent (small) samples, which may derive from populations having differing means and variances. The procedures are based on the use of the W statistic (Shapiro and Wilk, 1965) as a measure of departure from normality. Two modes of combination of a collection of W statistics are considered, namely the standardized mean of the normal transforms of W and the sum of the x 2(2) transforms of the significance levels of w; and these are proposed for use in conjunction with the probability plotting of the collection of these transforms. Tables and formulae for practical implementation are provided. Some summary empirical sampling results are given on the comparative sensitivities of these procedures, along with detailed consideration of several specific examples which illustrate the additional informative value of probability plotting. The proposed techniques appear to have substantial data analysis value ...
TL;DR: In this paper, the authors describe efficient computational procedures for calculating all possible 2 k − 1 regressions of a dependent variable ariable upon subsets of k independent variables, and provide efficiency comparisons among five different procedures.
Abstract: When the number of contemplated independent variables in a regression analysis is reasonably small, an alternative to the use of step-wise procedures for selecting variables is to base the selection on the calculation of all possible regressions. This paper describes efficient computational procedures for calculating all possible 2 k − 1 regressions of a dependentv ariable upon subsets of k independent variables, and provides efficiency comparisons among five different procedures.
TL;DR: In this paper, the authors developed an approximation to the null distribution of W, a statistic suggested for testing normality by Shapiro and Wilk (1965), based on fitting and smoothing empirical sampling results.
Abstract: The present note deals with the development of an approximation to the null distribution of W, a statistic suggested for testing normality by Shapiro and Wilk (1965). This approximation is based on fitting and smoothing empirical sampling results. The KT statistic is defined as the ratio of the square of a linear combination of the ordered sample to the usual sum of squares of deviations about the mean. For a sample from a normal distribution, the ratio is statistically independent of its denominator and so the moments of the ratio are equal to the ratio of the moments. This enables the simple computation of the 3 and 1 moments of W. Higher moments of W are not available and hence the Cornish-Fisher expansion could not be used as an approximation method. Good approximation was attained, after preliminary investigations, using Johnson’s (1949) S, distribution, which is defined as that of the random variable u, where 2 = y + 6 In u - e X+e-u is distributed as standard normal, and where E and X + e are the minimum and maximum attainable values of U, respectively. For IV, X + E = 1 for all n, while e is a known function of sample size (see Shapiro and Wilk (1965)). Values of e are given in Table 1 of the present note for n = 3(1)50. To obtain suitable values of y and 6, in the case when the bounds are known, Johnson (1949) recommends matching chosen percentage points. An alternative method might be to match two moments, but would require heavy computation. Also, while the matching of two moments could be done solely using theoretical values, in principle, it would not necessarily provide for weighting the fit so as to be good in the tails of the distribution-which is what is wanted. The procedure actually used here was to do, for each 72, the simple least squares regression of the empirical sampling value of
TL;DR: In this paper, the authors describe from first principles the direct calculation of the operating characteristic function, O.C., the probability of accepting the hypothesis θ = θ 0, and the average sample size, A.S.N., required to terminate the test, for any truncated sequential test once the acceptance, rejection, and continuation regions are specified at each stage.
Abstract: This paper describes from first principles the direct calculation of the operating characteristic function, O.C., the probability of accepting the hypothesis θ = θ0, and the average sample size, A.S.N., required to terminate the test, for any truncated sequential test once the acceptance, rejection, and the continuation regions are specified at each stage. What is needed is to regard a sequential test as a step by step random walk, which is a Markov chain. The method is contrasted with Wald's and two examples are included.
TL;DR: In this paper, a study was conducted on eight tests for differences in means under a variety of simulated experimental situations, and the power of the tests and measures of the extent to which they gave similar results were made.
Abstract: A study was conducted on eight tests for differences in means under a variety of simulated experimental situations. Estimates were made of the power of the tests and measures of the extent to which they gave similar results. In particular the performance of a new quick test developed by Neave was studied and was found to be satisfactory: in fact it was by far the best of the quick tests considered. However some of the classical and more general nonparametric tests, such as the runs and the Kolmogorov-Smirnov tests, were found to be less useful when testing for differences in means. Over the range of situations investigated, the Normal Scores test gave the most satisfactory results, followed closely by the Wilcoxon rank-sum test. Even when the populations were normally distributed, these tests were only very slightly inferior to the t-test, and naturally were much superior in the cases of non-normal populations.
TL;DR: In this paper, it was shown that the smallest resolution 4 designs for n factors at two levels must contain at least 2n runs, and that "foldover" designs are available with 2 n runs.
Abstract: Designs of even resolution have the property that a set of parameters, although themselves not estimable, do not appear as aliases of those which are estimable. The most important are designs of resolution 4, which are such that the main effects are estimable with no two-factor interactions as aliases. In this paper it is shown that the smallest resolution 4 designs for n factors at two levels must contain at least 2n runs, and that “foldover” designs are available with 2n runs. It is conjectured that the only minimal resolution 4 designs are foldover designs. The case of resolution 6 designs is also discussed.
TL;DR: In this article, the effect of first order time trend on the main effects of fractional factorial and eight-run factorial designs was examined from two points of view: (1) the number of levels which must be changed in performing the design, and (2) the effect that a few run orders are desirable from these viewpoints.
Abstract: Eight-run two level factorial and fractional factorial designs are examined from two points of view: (1) the number of levels which must be changed in performing the design, (2) the effect of a first order time trend on the main effects. It is found that only a few run orders are desirable from these viewpoints and thus randomization of the runs is likely, in general, to lead to an unfavorable sequence.
TL;DR: In this article, the p-mean significance levels for the mass-significance method developed by Eklund were derived by formulating and solving an urn problem, in the case of independent tests.
Abstract: This note concerns the derivation of the p-mean significance levels, in the case of independent tests, for a mass-significance method developed by Eklund [l]. The solution is reached by formulating and solving an urn problem. Some comparisons are made with the p-mean significance levels of Duncan's multiple range test.
TL;DR: In this article, a sample of n independent observations from a normal population is used to find a region that has a given probability of containing the next K observations, for the four possible cases of known and unknown mean and variance.
Abstract: Based on a sample of n independent observations from a normal population, formulas are given for a K-dimensional region that has a given probability of containing the next K observations, for the four possible cases of known and unknown mean and variance.
TL;DR: The statistical analysis of multi-dimensional contingency tables is discussed from the point of view of the associated underlying model and different formulations of hypotheses of ‘no interaction’ are considered.
Abstract: The statistical analysis of multi-dimensional contingency tables is discussed from the point of view of the associated underlying model. Different formulations of hypotheses of ‘no interaction’ are considered. The corresponding test statistics are based on a general and computationally simple criterion originally due to Wald [1943]. The suggested methods are illustrated with several numerical examples.
TL;DR: In this paper, an alternative procedure based on the same generd principles is developed and applied to a variety of models, and the estimators obtained are unbiased and consistent, and they are also reasonably easy to compute.
Abstract: The estimates of Koch [1967a] have the undesirable property that they may change in value if the same constant is added to each of the observations. In this paper, an alternative procedure based on the same generd principles is developed and applied to a variety of models. As before, the estimators obtained are unbiased and consistent. They are also reasonably easy to compute. Finally, in the case of balanced experiments, they coincide with those obtained from the analysis of variance. On the other hand, their structure is more complex than that of the estimators considered in the previous paper. In particular, the derivation of their covariance matrix is much more complicated, and hence no attempt has been made here to study its properties.
TL;DR: The Combinatorial Methods in the Theory of Stochastic Processes (CMLP) as mentioned in this paper is a generalization of the theory of stochastic processes, which is used in this paper.
Abstract: (1968). Combinatorial Methods in the Theory of Stochastic Processes. Technometrics: Vol. 10, No. 3, pp. 630-631.
TL;DR: In this article, a general method of constructing response surface designs from familiar response surface design in k − 1 independent variables and the appropriate analysis for a general polynomial is given, with special attention given to the first and second order polynomials.
Abstract: In estimating a response surface where the k variables represent proportions in a mixture, the experimenter is often interested in a reasonably well-defined region of interest which may, for example, center about current operating levels. Previously developed designs are difficult to use except in exploring the entire factor space, and even then there are several disadvantages to these designs. A general method of constructing designs from familiar response surface designs in k − 1 independent variables and the appropriate analysis for a general polynomial is given. Special attention is given to the first and second order polynomials.
TL;DR: In this article, the problem of determining sample size to take for a tolerance limit L(X) is investigated, where X is a function of a random sample X 1, X n from a distribution with density f(x : θ), and is investigated.
Abstract: The problem of determining sample size to take for a tolerance limit L(X), where L(X) is a function of a random sample X 1, …, Xn from a distribution with density f(x : θ), and is investigated. A criteriorl of “goodness” of tolerance limits is developed and a method given, using this criterion, for solving the sample size problem. Examples are given using the uniform, exponential, and normal distributions as underlying models.