Why is Six Sigma so successful?

There can be little doubt that Motorola, General Electric, Black and Decker, Allied Signal (now Honeywell), ABB and Bombardier, have achieved impressive business performance in recent years. Their annual reports document this success. Furthermore, in several cases, the Annual Report clearly attributes this success to having followed a Six Sigma strategy. Not surprisingly, many other companies wish to learn what Six Sigma can do for them, and their first question is 'What exactly is Six Sigma?'. Unfortunately it is rather difficult, if not impossible, to define Six Sigma in one or two sentences. This paper identifies the essential elements of Six Sigma. Some are obvious, such as the extensive use of statistical techniques by employees known as Blackbelts. However, other more subtle, but very important, features of Six Sigma are concealed within the business culture of these successful companies. It is clear to those who have participated in this success, that any company embarking on Six Sigma will not succeed if it focuses on statistics whilst failing to develop a supporting culture.     

Why is Six Sigma so successful?

There can be little doubt that Motorola, General Electric, Black and Decker, Allied Signal (now Honeywell), ABB and Bombardier, have achieved impressive business performance in recent years. Their annual reports document this success. Furthermore, in several cases, the Annual Report clearly attributes this success to having followed a Six Sigma strategy. Not surprisingly, many other companies wish to learn what Six Sigma can do for them, and their first question is 'What exactly is Six Sigma?'. Unfortunately it is rather difficult, if not impossible, to define Six Sigma in one or two sentences. This paper identifies the essential elements of Six Sigma. Some are obvious, such as the extensive use of statistical techniques by employees known as Blackbelts. However, other more subtle, but very important, features of Six Sigma are concealed within the business culture of these successful companies. It is clear to those who have participated in this success, that any company embarking on Six Sigma will not succeed if it focuses on statistics whilst failing to develop a supporting culture.     

Methods to test for equality of two normal distributions

Statistical tests for two independent samples under the assumption of normality are applied routinely by most practitioners of statistics. Likewise, presumably each introductory course in statistics treats some statistical procedures for two independent normal samples. Often, the classical two-sample model with equal variances is introduced, emphasizing that a test for equality of the expected values is a test for equality of both distributions as well, which is the actual goal. In a second step, usually the assumption of equal variances is discarded. The two-sample t test with Welch correction and the F test for equality of variances are introduced. The first test is solely treated as a test for the equality of central location, as well as the second as a test for the equality of scatter. Typically, there is no discussion if and to which extent testing for equality of the underlying normal distributions is possible, which is quite unsatisfactorily regarding the motivation and treatment of the situation with equal variances. It is the aim of this article to investigate the problem of testing for equality of two normal distributions, and to do so using knowledge and methods adequate to statistical practitioners as well as to students in an introductory statistics course. The power of the different tests discussed in the article is examined empirically. Finally, we apply the tests to several real data sets to illustrate their performance. In particular, we consider several data sets arising from intelligence tests since there is a large body of research supporting the existence of sex differences in mean scores or in variability in specific cognitive abilities.     

On power and sample size calculations for Wald tests in generalized linear models

A Wald test-based approach for power and sample size calculations has been presented recently for logistic and Poisson regression models using the asymptotic normal distribution of the maximum likelihood estimator, which is applicable to tests of a single parameter. Unlike the previous procedures involving the use of score and likelihood ratio statistics, there is no simple and direct extension of this approach for tests of more than a single parameter. In this article, we present a method for computing sample size and statistical power employing the discrepancy between the noncentral and central chi-square approximations to the distribution of the Wald statistic with unrestricted and restricted parameter estimates, respectively. The distinguishing features of the proposed approach are the accommodation of tests about multiple parameters, the flexibility of covariate configurations and the generality of overall response levels within the framework of generalized linear models. The general procedure is illustrated with some special situations that have motivated this research. Monte Carlo simulation studies are conducted to assess and compare its accuracy with existing approaches under several model specifications and covariate distributions.     

Monitoring Multivariate Simple Linear Profiles in the Presence of between Profile Autocorrelation

In recent years, statistical profile monitoring has emerged as a relatively new and potentially useful subarea of statistical process control and has attracted attention of many researchers and practitioners. A profile, waveform, or signature is a function that relates a dependent or a response variable to one or more independent variables. Different statistical methods have been proposed by researchers to monitor profiles where each method requires its own assumptions. One of the common and implicit assumptions in most of the proposed procedures is the assumption of independent residuals. Violation of this assumption can affect the performance of control procedures and ultimately leading to misleading results. In this article, we study phase II analysis of monitoring multivariate simple linear profiles when the independency assumption is violated. Three time series based methods are proposed to eliminate the effect of correlation that exists between multivariate profiles. Performances of the proposed methods are evaluated using average run length (ARL) criterion. Numerical results indicate satisfactory performance for the proposed methods. A simulated example is also used to show the application of the proposed methods.     

A Comparison of Two Group Classification Approaches to Fat-tailed and Skewed Data

The problem of two-group classification has implications in a number of fields, such as medicine, finance, and economics. This study aims to compare the methods of two-group classification. The minimum sum of deviations and linear programming model, linear discriminant analysis, quadratic discriminant analysis and logistic regression, multivariate analysis of variance (MANOVA) test-based classification and the unpooled T-square test-based classification methods, support vector machines and k-nearest neighbor methods, and combined classification method will be compared for data structures having fat-tail and/or skewness. The comparison has been carried out by using a simulation procedure designed for various stable distribution structures and sample sizes.     

Testing equality of variances after a preliminary test of significance for correlation coefficient

On the basis of the outcome of a preliminary test of significance for the population correlation coefficient it is decided as to whether the variance ratio or the sample correlation coefficient between u=(x+y)/2 and v=x?y)/2 is to be used as a test statistic for testing the equality of variances. A method for determining the critical points for the preliminary test and the main test has been suggested. The power of the test procedure is compared with those of standard tests.     

C55. Quick and accurate rule of thumb for determining the critical value of the coefficient of correlation

Life table analysis techniques in epidemiology depend upon the asymptotic properties of the statistical test methods employed. In some instances, the statistical procedures indicate highly significant results which are, in reality, unjustified. The phenomenon may occur when the asymptotic methods are applied in situations where the cases of interest are few in number. This situation is illustrated by the 20 multiple myeloma deaths observed in the RERF Life Span Study cohort. A permutation test is applied to the life table data, although the test requires the false assumption that the censoring distribution is independent of the radiation dose. A simulation test is developed which does not require equal censoring, which has the same asymptotics as the usual test methods, and which is less likely to overestimate significance in small samples. It is found that both of these small-sample tests provide reasonable numerical solutions. In addition, the simulation test is recommended in general for analyzing life table data with unequal censoring. Finally, by using the small-sample tests, the frequency of death from multiple myeloma is shown to be positively associated with radiation dose (P<0.01).     

A comparative study of confidence intervals to assess biosimilarity from analytical data

Assessment of analytical similarity of tier 1 quality attributes is based on a set of hypotheses that tests the mean difference of reference and test products against a margin adjusted for standard deviation of the reference product. Thus, proper assessment of the biosimilarity hypothesis requires statistical tests that account for the uncertainty associated with the estimations of the mean differences and the standard deviation of the reference product. Recently, a linear reformulation of the biosimilarity hypothesis has been proposed, which facilitates development and implementation of statistical tests. These statistical tests account for the uncertainty in the estimation process of all the unknown parameters. In this paper, we survey methods for constructing confidence intervals for testing the linearized reformulation of the biosimilarity hypothesis and also compare the performance of the methods. We discuss test procedures using confidence intervals to make possible comparison among recently developed methods as well as other previously developed methods that have not been applied for demonstrating analytical similarity. A computer simulation study was conducted to compare the performance of the methods based on the ability to maintain the test size and power, as well as computational complexity. We demonstrate the methods using two example applications. At the end, we make recommendations concerning the use of the methods.     

A robust coefficient of determination for regression

To assess the quality of the fit in a multiple linear regression, the coefficient of determination or R² is a very simple tool, yet the most used by practitioners. Indeed, it is reported in most statistical analyzes, and although it is not recommended as a final model selection tool, it provides an indication of the suitability of the chosen explanatory variables in predicting the response. In the classical setting, it is well known that the least-squares fit and coefficient of determination can be arbitrary and/or misleading in the presence of a single outlier. In many applied settings, the assumption of normality of the errors and the absence of outliers are difficult to establish. In these cases, robust procedures for estimation and inference in linear regression are available and provide a suitable alternative.     

A revisit to test the equality of variances of several populations

We revisit the problem of testing homoscedasticity (or, equality of variances) of several normal populations which has applications in many statistical analyses, including design of experiments. The standard text books and widely used statistical packages propose a few popular tests including Bartlett's test, Levene's test and a few adjustments of the latter. Apparently, the popularity of these tests have been based on limited simulation study carried out a few decades ago. The traditional tests, including the classical likelihood ratio test (LRT), are asymptotic in nature, and hence do not perform well for small sample sizes. In this paper we propose a simple parametric bootstrap (PB) modification of the LRT, and compare it against the other popular tests as well as their PB versions in terms of size and power. Our comprehensive simulation study bursts some popularly held myths about the commonly used tests and sheds some new light on this important problem. Though most popular statistical software/packages suggest using Bartlette's test, Levene's test, or modified Levene's test among a few others, our extensive simulation study, carried out under both the normal model as well as several non-normal models clearly shows that a PB version of the modified Levene's test (which does not use the F-distribution cut-off point as its critical value), and Loh's exact test are the “best” performers in terms of overall size as well as power.     

George W. Snedecor,Pioneer Statistician, 1881–1974

Users of statistical packages need to be aware of the influence that outlying data points can have on their statistical analyses. Robust procedures provide formal methods to spot these outliers and reduce their influence. Although a few robust procedures are mentioned in this article, one is emphasized; it is motivated by maximum likelihood estimation to make it seem more natural. Use of this procedure in regression problems is considered in some detail, and an approximate error structure is stated for the robust estimates of the regression coefficients. A few examples are given. A suggestion of how these techniques should be implemented in practice is included.     

Inferences on the lognormal mean for complete samples

In many engineering problems it is necessary to draw statistical inferences on the mean of a lognormal distribution based on a complete sample of observations. Statistical demonstration of mean time to repair (MTTR) is one example. Although optimum confidence intervals and hypothesis tests for the lognormal mean have been developed, they are difficult to use, requiring extensive tables and/or a computer. In this paper, simplified conservative methods for calculating confidence intervals or hypothesis tests for the lognormal mean are presented. In this paper, “conservative” refers to confidence intervals (hypothesis tests) whose infimum coverage probability (supremum probability of rejecting the null hypothesis taken over parameter values under the null hypothesis) equals the nominal level. The term “conservative” has obvious implications to confidence intervals (they are “wider” in some sense than their optimum or exact counterparts). Applying the term “conservative” to hypothesis tests should not be confusing if it is remembered that this implies that their equivalent confidence intervals are conservative. No implication of optimality is intended for these conservative procedures. It is emphasized that these are direct statistical inference methods for the lognormal mean, as opposed to the already well-known methods for the parameters of the underlying normal distribution. The method currently employed in MIL-STD-471A for statistical demonstration of MTTR is analyzed and compared to the new method in terms of asymptotic relative efficiency. The new methods are also compared to the optimum methods derived by Land (1971, 1973).     

A density based empirical likelihood approach for testing bivariate normality

Sample entropy based tests, methods of sieves and Grenander estimation type procedures are known to be very efficient tools for assessing normality of underlying data distributions, in one-dimensional nonparametric settings. Recently, it has been shown that the density based empirical likelihood (EL) concept extends and standardizes these methods, presenting a powerful approach for approximating optimal parametric likelihood ratio test statistics, in a distribution-free manner. In this paper, we discuss difficulties related to constructing density based EL ratio techniques for testing bivariate normality and propose a solution regarding this problem. Toward this end, a novel bivariate sample entropy expression is derived and shown to satisfy the known concept related to bivariate histogram density estimations. Monte Carlo results show that the new density based EL ratio tests for bivariate normality behave very well for finite sample sizes. To exemplify the excellent applicability of the proposed approach, we demonstrate a real data example.     

UPPER BOUNDS ON THE MINIMUM COVERAGE PROBABILITY OF CONFIDENCE INTERVALS IN REGRESSION AFTER MODEL SELECTION

We consider a linear regression model, with the parameter of interest a specified linear combination of the components of the regression parameter vector. We suppose that, as a first step, a data-based model selection (e.g. by preliminary hypothesis tests or minimizing the Akaike information criterion – AIC) is used to select a model. It is common statistical practice to then construct a confidence interval for the parameter of interest, based on the assumption that the selected model had been given to us  a priori . This assumption is false, and it can lead to a confidence interval with poor coverage properties. We provide an easily computed finite-sample upper bound (calculated by repeated numerical evaluation of a double integral) to the minimum coverage probability of this confidence interval. This bound applies for model selection by any of the following methods: minimum AIC, minimum Bayesian information criterion (BIC), maximum adjusted  R ², minimum Mallows'   C _P   and  t -tests. The importance of this upper bound is that it delineates general categories of design matrices and model selection procedures for which this confidence interval has poor coverage properties. This upper bound is shown to be a finite-sample analogue of an earlier large-sample upper bound due to Kabaila and Leeb.     

Stringency-based ranking of normality tests

Many parametric statistical inferential procedures in finite samples depend crucially on the underlying normal distribution assumption. Dozens of normality tests are available in the literature to test the hypothesis of normality. Availability of such a large number of normality tests has generated a large number of simulation studies to find a best test but no one arrived at a definite answer as all depends critically on the alternative distributions which cannot be specified. A new framework, based on stringency concept, is devised to evaluate the performance of the existing normality tests. Mixture of t-distributions is used to generate the alternative space. The LR-tests, based on Neyman–Pearson Lemma, have been computed to construct a power envelope for calculating the stringencies of the selected normality tests. While evaluating the stringencies, Anderson–Darling (AD) statistic turns out to be the best normality test.     

Selection of the best with a preliminary test for two-parameter exponential distributions

This paper deals with the problem of selecting the best population from among k(≥ 2) two-parameter exponential populations. New selection procedures are proposed for selecting the unique best. The procedures include preliminary tests which allow the xperimenter to have an option to not select if the statistical evidence is not significant. Two probabilities, the probability to make a selection and the probability of a correct selection, are controlled by these selection procedures. Comparisons between the proposed selection procedures and certain earlier existing procedures are also made. The results show the superiority of the proposed selection procedures in terms of the required sample size.     

A pragmatic approach to experimental design in industry

The importance of statistically designed experiments in industry has been well recognized. However, the use of 'design of experiments' is still not pervasive, owing in part to the inefficient learning process experienced by many non-statisticians. In this paper, the nature of design of experiments, in contrast to the usual statistical process control techniques, is discussed. It is then pointed out that for design of experiments to be appreciated and applied, appropriate approaches should be taken in training, learning and application. Perspectives based on the concepts of objective setting and design under constraints can be used to facilitate the experimenters' formulation of plans for collection, analysis and interpretation of empirical information. A review is made of the expanding role of design of experiments in the past several decades, with comparisons made of the various formats and contexts of experimental design applications, such as Taguchi methods and Six Sigma. The trend of development shows that, from the realm of scientific research to business improvement, the competitive advantage offered by design of experiments is being increasingly felt.     

Experimental comparison of functional and multivariate spectral-based supervised classification methods in hyperspectral image

The aim of this article is to assess and compare several statistical methods for hyperspectral image supervised classification only using the spectral dimension. Since hyperspectral profiles may be viewed either as a random vector or a random curve, we propose to confront various multivariate discriminating procedures with functional alternatives. Eight methods representing three important statistical communities (mixture models, machine learning and functional data analysis) have been applied on three hyperspectral datasets following three protocols studying the influence of size and composition of the learning sample, with or without noised labels. Besides this comparative study, this work proposes a functional extension of multinomial logit model as well as a fast computing adaptation of the nonparametric functional discrimination. As a by-product, this work provides a useful comprehensive bibliography and also supplemental material especially oriented towards practitioners.     

The family of t-designs—part I

The family of t-designs is, without any doubt, the most important family of statistical designs. Their importance is due to their statistical optimalities, desirable symmetries for analyses and interpretations, and uses for constructing other important designs and structures such as Youden designs, generalized Youden designs, optimal fractional factorial designs, error defecting and correcting binary codes, balanced arrays, combinatorial filing systems, Hadamard matrices, finite projective and affine planes, strongly regular graphs, and so on. Research in the area of t-designs has been steadily and rapidly growing, especially during the last three decades. The number of publications in this area is in the several hundreds. Since papers on t-designs are published in a variety of journals, and because of the extensive role of these designs in design of experiments and other areas we believe it is imperative to gather these results and present them in varied form to suit diverse interests. This paper is an instance of such an attempt.