Robert W. Mee

Better Foldover Fractions for Resolution III 2k-p Designs

This article describes a family of Resolution IIIdesigns for which the usual advice regarding foldover—reversing all factors—is ill advised. The smallest such designs have 16 runs. For designs in this family, alternative foldover fractions not only increase the resolution to IV, but also separate some of the aliased two-factor interactions. Although Resolution IV designs obtained by reversing all factors provide fewer than half the degrees of freedom for estimating two-factor interactions, the recommended foldovers for designs in this class permit estimation of many more two-factor interactions. This result implies that designs in this class make attractive initial designs, even if they are not minimum aberration.

Semifolding 2 k–P Designs

This article addresses the varied possibilities for following a two-level fractional factorial with another fractional factorial half the size of the original experiment. Although follow-up fractions of the same size as an original experiment are common practice, in many situations a smaller followup experiment will sufIice. Peter John coined the term “semifolding” to describe using half of a foldover design. Existing literature does include brief mention and examples of semifolding but no thorough development of this follow-up strategy. After a quick examination of the estimation details for semlfoldmg the 24 – 1 IV design, we focus on following 16-run fractions with a semifold design of eight runs. Two such examples are considered—one in which the initial fraction ia resolution IV, the other resolution III. A general result is proven for semifolding 2 k – p IV designs. Conducting full foldover designs in two blocks is also recommended.

Regression toward the Mean and the Paired Sample t Test

Abstract This article shows how one should properly conduct a paired-sample comparison of means in a test–retest situation where only a subset of the population retakes the exam. For example, suppose all who fail an exam are given the opportunity to retake a similar exam. If the usual paired-sample t test is conducted from data in this setting, the “regression effect” may lead to the incorrect conclusion that some intervention (such as remedial tutoring) has been effective in raising the scores when this is not necessarily true. A correct test when the mean score for the first exam is known is a test involving the intercept for a simple regression model.

Confidence Intervals for Probabilities and Tolerance Regions Based on a Generalization of the Mann-Whitney Statistic

Abstract Inference procedures are proposed for any specified function of two random variables f(X, Y), assuming that independent random samples from the X and Y populations are available. A generalization of the Mann-Whitney statistic is used to obtain point and interval estimates for the probability that f(X, Y) falls in a given interval. One proposed confidence interval is a modification and improvement to the approach of Halperin, Gilbert, and Lachin (1987) for estimating Pr(X < Y). A thorough review is made of the literature on nonparametric confidence bounds for Pr(X < Y), with emphasis on methods based on the central limit theorem. In addition to the improvement of Halperin et al.s method, bounds analogous to the Clopper-Pearson confidence interval for a binomial parameter are proposed. Simulation results show the performance of the alternative procedures. Approximate nonparametric tolerance limits for f(X, Y) are also proposed. The article closes by constructing a two-sided tolerance interval for ...

Calibration and simultaneous tolerance intervals for regression

Simultaneous calibration (or discrimination) intervals in regression were proposed by Lieberman, Miller, and Hamilton (1967) and by Scheffe (1973). Those procedures enable one to construct confidence intervals for the unobserved values of the independent variable corresponding to an unlimited sequence of observations of the dependent variable in a regression model. These calibration intervals are conservative in that they are obtained from simultaneous tolerance intervals for which the actual confidence level exceeds the nominal level. Furthermore, all other existing simultaneous tolerance intervals in regression are likewise conservative. In this article, we propose simultaneous tolerance intervals that are narrower than previous intervals. Given the tables of factors included in this article, they are also simple to construct and use in straightline calibration applications.

Using factorial experiments to study the toxicity of metal mixtures

Two-level factorial experiments were employed in this study for understanding and predicting the toxicity of binary and ternary metal mixtures. Toxicity of metal mixtures with concentrations between the respective EC10 and EC80 values was experimentally measured. Models were fit to the experimental data and the resultant models were of high quality as reflected by R2 (coefficient of determination). Interactions between mixture components were indicated by the existence of statistically significant interaction terms in the models. Toxicity predictions based on the models were compared with observed toxicity for binary and ternary metal mixtures. The models developed did not assume additivity between metals, were simple and interpretable, and gave satisfactory predictions of the toxicity of metal mixtures in aqueous solutions without requiring knowledge on synergism or antagonism.

Computing factors for exact two-sided tolerance limits for a normal distribution

A self-contained FORTRAN subroutine is provided which computes factors for Wald-Wolfowitz type tolerance limits allowing arbitrary combinations of sample size n and degrees of freedom ν. The exact calculations from our program reveal inadequacies of two existing approximations, especially when ν ≫ n. Numerous applications where ν ≠ n − 1 are cited; two of these are discussed and illustrated.

Split-lot designs: experiments for multistage batch processes

The fabrication of integrated circuits (ICs) is accomplished through a vast sequence of processing steps. Moreover, the silicon wafers on which the ICs are produced move through the process in lots of size 24 or more. Although some processing steps are applied to individual wafers, for other steps several wafers (or even several lots) are processed simultaneously as a group. To facilitate experimentation with such a multistage batch process, “split-lot” experimental designs are attractive because they allow the experimental wafers to be split into sublots for processing. The designs are obtained by using different sets of factorial effects to define the composition of the sublots at each step. Specific examples are given with up to nine processing steps. A split-lot design balances the way in which the wafers are repartitioned at each stage in the experiment. Taguchi refers to such experiments as multiway split-unit designs. Two-way split-unit experiments arise naturally in agriculture, where some facto...

Efficient Two-Level Designs for Estimating All Main Effects and Two-Factor Interactions

Regular designs for a large number of factors require many more treatment combinations than there are lower order effects to be estimated. For example, 29−2 = 128 treatment combinations are required to estimate only nine main effects and their 36 two-factor interactions. Nonregular orthogonal designs as well as nonorthogonal designs provide attractive alternatives in such cases. This article reviews the literature regarding alternatives to the usual orthogonal resolution V designs and suggests questions for future research. The topic is illustrated with a ballistic missile simulation application requiring estimation of main effects and two-factor interactions for 47 factors. For such large examples, run-size efficiency is more important than variance efficiency.

Prediction limits for the Weibull distribution utilizing simulation

Abstract A simulation-based procedure is suggested for constructing prediction limits for Weibull populations. This procedure is based on maximum likelihood (ML) estimation. Although computation of the ML estimates and determination of a needed percentile via simulation require a computer, we assert that the proliferation of personal computers makes these procedures more convenient than alternative procedures in the literature which require several specialized tables. This is particularly true for censored samples, since then the needed tables are generally incomplete and interpolation is required. Simulation may be used in all situations, i.e., any sample size, any level of Type II (failure) censoring, and any order statistic from a future sample. We illustrate our methods with three data sets from the literature.