Seiichi Yasui

Generalization of the Run Rules for the Shewhart Control Charts

It is well-known that the Shewhart control charts are useful to detect large shifts of a process mean, but it is insensitive for small shifts and/or other types of variation. We extend the Shewhart’s three sigma rule and propose two new rules based on successive observations. One is that a signal occurs when m successive observations exceed k 1 sigma control limit. The other is that a signal occurs when m − 1 of m successive observations exceed k 2 sigma control limit. The original Shewhart control chart is included in the first generalized rule as m = 1. The performance of the proposed rules is evaluated under several out-of-control situations by both the average run length and the standard deviation of the run length. These rules are more powerful than Shewhart’s three sigma rule at detecting moderate step shifts.

Evaluating Adaptive Paired Comparison Experiments

Paired comparison experiments are effective tools when the characteristics of the objects cannot be measured directly. In paired comparison experiments the characteristics of the objects are estimated from the result of the comparisons. The concept of paired comparison experiments was introduced by Thurstone (1927). The method by Scheffe (1952) is widely used for complete paired comparison experiments and the method by Bradley and Terry (1952) is popularly used in incomplete paired comparison experiments.

An Alternative Expression of the Fractional Factorial Designs for Two-level and Three-level Factors

Two-level and three-level fractional factorial designs are widely and effectively applied in many practical fields for finding active factors. We introduce an operation for obtaining interaction effects in the design matrices of the experiments in this paper. The relationship between two factors and their interaction can be found by applying group theory with the operation, which is called the Hadamard product for two-level factors, and the generalized Hadamard product for three-level factors, based on the columns of the design matrix. The group structure can be applied to solve many related problems including the fold-over technique. The operation, i.e. the Hadamard product and the expression of the design matrices for three-level factorial experiment, are established by introducing the complex cubic root of 1 and some other required generalization.

A Practical Variable Selection for Linear Models

In the analysis of experiments, there are many variable selection algorithms for linear models. Most of these approaches select the best model based on some criteria such as AIC. These criteria do not allow for any relationship between predictors. However, in practice, the analysis is driven by following three principles: Effect Hierarchy, Effect Sparsity, and Effect Heredity Principle. The approach depending solely on those criteria ignore these principles, so it would often select a hard to interpretable models, for instance, which are consisted with only interaction terms. In this article, we extend the LASSO method to identify significant interaction terms mainly focusing on the heredity principle. And we compare the proposed method with ordinary LASSO and traditional variable selection approach. In the example, we analyze the data obtained from designed experiments such as Placket-Burman design and supersaturated design.

D-Optimal Three-Stage Unbalanced Nested Designs for the Determination of Measurement Precision

The precision of measurement results can be quantified by variance components of random effect models. The variance components are estimated from measurement results that are obtained by performing a collaborative assessment experiment. The measurement results are statistically modeled by a nested design. Although balanced nested designs are widely used, staggered nested designs, which are one type of unbalanced nested designs, have the statistical advantage that the degrees of freedom in all stages except for the top stage are equal. Thus, balanced nested designs do not necessarily have a better performance from the statistical point of view. In this study, D-optimal designs are identified in general nested designs that include both balanced and unbalanced designs and consider the practical feasibility of collaborative assessment experiments as well.

Bayesian Lasso with Effect Heredity Principle

The Bayesian Lasso is a variable selection method that can be applied in situations where there are more variables than observations; thus, both main effects and interaction effects can be considered in screening experiments. To apply the Bayesian framework to experiments involving the effect heredity principle, which governs the relationships between interactions and their corresponding main effects, several initial tunings of the Bayesian framework are required. However, it is rather unnatural to specify these tuning values before running an experiment. In this paper, we propose models that do not require the initial tuning values to be specified in advance. The proposed methods are demonstrated with screening examples such as Plackett–Burman and mixed-level design.

Proposal of Advanced Taguchi’s Linear Graphs for Split-Plot Experiments

Taguchi’s orthogonal arrays and linear graphs are convenient tools for the design of fractional factorial experiments, especially for practitioners. Taguchi also proposed how to use them in split-plot designs and prepared linear graphs for split-plot designs. For the orthogonal array of order 16, Taguchi proposed one which is called L16 orthogonal array. Taguchi presented 18 linear graphs when a L16 orthogonal array is used in split-plot designs. Those linear graphs are capable of showing main effects of whole plots, subplots, sub-subplots, and so on, but they are not capable of showing interaction effects of plots of different levels. Also, those linear graphs do not cover all the possible designs, and there exist a lot of other linear graphs that can be applied when using L16 orthogonal arrays. The primary objective of this paper is to propose an improved version of linear graphs. Another purpose of this paper is to investigate how to list all the possible linear graphs that can be applied when using L16 orthogonal arrays. A proposal is made and many new linear graphs are presented.

A Robust Detection Procedure for Multiple Change Points of Linear Trends

A flexible manufacturing system (FMS) enables the production of multiple-items with short production run. By using an automatic measurement system, it is possible to observe a large amount of items in a short time. The observations from the FMS include some variation patterns and outliers, thereby making it difficult to implement a conventional statistical process control. In this study, a retrospective analysis of such a process dataset is proposed. Our procedure detects multiple change points for a dataset with outliers and variation patterns such as shifts and trends. The locally weighted scatter plot smoothing and the jump/roof/valley detection procedure based on a local polynomial kernel smoothing are useful to develop our procedure. We modify these procedures and propose a robust procedure for detecting multiple change points.

On Identifying Dispersion Effects in Unreplicated Fractional Factorial Experiments

The analysis of dispersion effects is as important as the location effect analysis in the quality improvement. Unreplicated fractional factorial experiments are useful to analyse not only location effects but also dispersion effects. The statistics introduced by Box and Meyer (1986) to identify dispersion effects are based on residuals subtracting the estimates for large location effects from observations. The statistic is a simple form, however, the property is not completely discovered. In this article, the distribution of the statistic under the null hypothesis is derived in unreplicated fractional factorial experiments using an orthogonal array. The distribution under the null hypothesis cannot be expressed uniquely. The statistic has different null distributions depending on the combination of columns allocating factors. We concluded that the distributions can be classified into three types, i.e. the F distribution, unknown distributions close to the F distribution and the constant (not stochastic variable) which is one. Finally, the power of the test for detection of a single active dispersion effect is evaluated.

Approximated Interval Estimation in the Staggered Nested Designs for Precision Experiments

Staggered nested experimental designs are the most popular class of unbalanced nested designs in practical fields. Reproducibility is one of the important precision measures. In our study, interval estimation of reproducibility is proposed and evaluated by precision experiments, which we call the staggered nested experimental design. In this design, the reproducibility estimator is expressed as linear combination of the variance components from ANOVA (analysis of variance). By using a gamma approximation, the shape parameter that is needed for the approximation is introduced in our study. Additionally, general formulae for the shape parameter are proposed. Applying the formulae, we constructed the confidence interval for the reproducibility of three-factor and four-factor staggered nested designs. The performance of the proposed gamma approximations is evaluated with the goodness of fit and compared with each other. The interval estimation of reproducibility is evaluated with the coverage probability through a Monte-Carlo simulation experiment. We also compared it to the method ignoring the covariance terms. As a result, the proposed approximations were better than the method without the covariance terms. The performance of the proposed interval estimation was also superior to that without covariance terms. Additionally, some practical recommendations were obtained for designing precision experiments, including the number of participating laboratories.