Cumulative correspondence analysis using orthogonal polynomials

Taguchi's statistic has long been known to be a more appropriate measure of association of the dependence for ordinal variables compared to the Pearson chi-squared statistic. Therefore, there is some advantage in using Taguchi's statistic in the correspondence analysis context when a two-way contingency table consists at least of an ordinal categorical variable. The aim of this paper, considering the contingency table with two ordinal categorical variables, is to show a decomposition of Taguchi's index into linear, quadratic and higher-order components. This decomposition has been developed using Emerson's orthogonal polynomials. Moreover, two case studies to explain the methodology have been analyzed.     

Cumulative correspondence analysis of ordered categorical data from industrial experiments

Most studies of quality improvement deal with ordered categorical data from industrial experiments. Accounting for the ordering of such data plays an important role in effectively determining the optimal factor level of combination. This paper utilizes the correspondence analysis to develop a procedure to improve the ordered categorical response in a multifactor state system based on Taguchi's statistic. Users may find the proposed procedure in this paper to be attractive because we suggest a simple and also popular statistical tool for graphically identifying the really important factors and determining the levels to improve process quality. A case study for optimizing the polysilicon deposition process in a very large-scale integrated circuit is provided to demonstrate the effectiveness of the proposed procedure.     

The alpha error of the Taguchi method with L16 array for the LTB response variable using simulation

Taguchi's quality engineering concepts are of great importance in designing and improving product quality and process. However, most of the controversy and mystique have been centred on Taguchi's tactics. This research proposes an extension to the on-going research by investigating the probability of identifying insignificant factors as significant, or the so-called alpha error, with L₁₆ orthogonal array for the larger-the-better type response variable using simulation. The response variables in the L₁₆ array are generated from a normal distribution with the same mean and standard deviation. Consequently, the null hypothesis that all factors in the L₁₆ array will be identified as insignificant is true. Simulation results, however, reveal that some insignificant factors are wrongly identified as significant with a very high probability, which may provide a risky parameter design. Therefore, efficient and more valid statistical tactics should be developed put out Taguchi's important quality engineering concepts into practice.     

Decomposition of the main effects and interaction term by using orthogonal polynomials in multiple non symmetrical correspondence analysis

The multiple non symmetric correspondence analysis (MNSCA) is a useful technique for analyzing a two-way contingency table. In more complex cases, the predictor variables are more than one. In this paper, the MNSCA, along with the decomposition of the Gray–Williams Tau index, in main effects and interaction term, is used to analyze a contingency table with two predictor categorical variables and an ordinal response variable. The Multiple-Tau index is a measure of association that contains both main effects and interaction term. The main effects represent the change in the response variables due to the change in the level/categories of the predictor variables, considering the effects of their addition, while the interaction effect represents the combined effect of predictor categorical variables on the ordinal response variable. Moreover, for ordinal scale variables, we propose a further decomposition in order to check the existence of power components by using Emerson's orthogonal polynomials.     

A note on a parametric extension of rank tests to the analysis of multifactor experiments within blocks

A general rank test procedure based on an underlying multinomial distribution is suggested for randomized block experiments with multifactor treatment combinations within each block. The Wald statistic for the multinomial is used to test hypotheses about the within–block rankings. This statistic is shown to be related to the one–sample Hotellingt's T² statistic, suggesting a method for computing the test statistic using the standard statistical computer packages.     

Iso‐chi‐squared testing of 2 × k: ordered tables

The Pearson chi‐squared statistic for testing the equality of two multinomial populations when the categories are nominal is much less appropriate for ordinal categories. Test statistics typically used in this context are based on scorings of the ordinal levels, but the results of these tests are highly dependent on the choice of scores. The authors propose a test which naturally modifies the Pearson chi‐squared statistic to incorporate the ordinal information. The proposed test statistic does not depend on the scores and under the null hypothesis of equality of populations, it is asymptotically equivalent to the likelihood ratio test against the alternative of two‐sided likelihood ratio ordering.     

Design of acceptance sampling plan using a modified EWMA statistic

In this paper, a modified exponentially weighted moving average (EWMA) statistic is proposed. The approximate distribution of the proposed modified EWMA statistic is derived. A variable acceptance sampling plan is designed using the proposed EWMA statistic. The plan parameters of the proposed sampling plan are determined such that the given producer's risk and consumer's risk are satisfied. The efficiency of the proposed plan based on the new EWMA statistic is compared with the existing EWMA plan in terms of the sample size required. The application of the proposed plan is given with the help of an example.     

Asymptotic Comparison of Two Discriminants Used in Normal Covariate Classification

In this paper, we examine the performance of Anderson's classification statistic with covariate adjustment in comparison with the usual Anderson's classification statistic without covariate adjustment in a two-population normal covariate classification problem. The same problem has been investigated using different methods of comparison by some authors. See the bibliography. The aim of this paper is to give a direct comparison based upon the asymptotic probabilities of misclassification. It is shown that for large equal sample size of a training sample from each population, Anderson's classification statistic with covariate adjustment and cut-off point equal to zero, has better performance.     

The disagreeable behaviour of the kappa statistic

It is often of interest to measure the agreement between a number of raters when an outcome is nominal or ordinal. The kappa statistic is used as a measure of agreement. The statistic is highly sensitive to the distribution of the marginal totals and can produce unreliable results. Other statistics such as the proportion of concordance, maximum attainable kappa and prevalence and bias adjusted kappa should be considered to indicate how well the kappa statistic represents agreement in the data. Each kappa should be considered and interpreted based on the context of the data being analysed. Copyright © 2014 JohnWiley & Sons, Ltd.     

Detecting Outliers in Gamma Distribution

Zerbet and Nikulin presented the new statistic Z _k for detecting outliers in exponential distribution. They also compared this statistic with Dixon's statistic D _k . In this article, we extend this approach to gamma distribution and compare the result with Dixon's statistic. The results show that the test based on statistic Z _k is more powerful than the test based on the Dixon's statistic.     

Pena's statistic for the Liu regression

In fitting regression model, one or more observations may have substantial effects on estimators. These unusual observations are precisely detected by a new diagnostic measure, Pena's statistic. In this article, we introduce a type of Pena's statistic for each point in Liu regression. Using the forecast change property, we simplify the Pena's statistic in a numerical sense. It is found that the simplified Pena's statistic behaves quite well as far as detection of influential observations is concerned. We express Pena's statistic in terms of the Liu leverages and residuals. The normality of this statistic is also discussed and it is demonstrated that it can identify a subset of high Liu leverage outliers. For numerical evaluation, simulated studies are given and a real data set has been analysed for illustration.     

Relations between statistics for testing exponentiality and uniformity

It is shown that Greenwood's statistic for uniformity and the Hahn-Shapiro and Stephens statistics for exponentiality with known origin are equivalent. It is also shown that the distribution of the Shapiro-Wilk statistic for testing the hypothesis of exponentiality with unknown origin is obtainable from the distribution of Greenwood's statistic.     

Exact and resampling probability values for the Piccarreta nominal–ordinal index of association

Exact, resampling, and Pearson type III permutation methods are provided to compute probability values for Piccarreta's nominal–ordinal index of association. The resampling permutation method provides good approximate probability values based on the proportion of resampled test statistic values equal to or greater than the observed test statistic value.     

An application of taguchi's methods reconsidered

Two aspects of Taguchi's methods for analyzing parameter design experiments that can be improved upon are considered. It is shown how using interaction graphs instead of marginal graphs, and how using the sample variance instead of a signal-to-noise ratio, can lead to product designs that are more robust to variation. The advantages of the alternative analysis will be illustrated by reanalyzing a case study considered by Barker (1986).     

On the asymptotic distributions of high-order spacings statistics

Goodness-of-fit tests for the uniform distribution based on sums of smooth functions of m-spacings are studied. A limiting sum-of-weighted-chi-squareds approximation is shown to be accurate uniformly in m for the special cases of analogues of Greenwoo?s statistic and Moran's statistic. Asymptotic critical points are provided; theory and Monte Carlo studies show they are accurate for all m provided n is moderately large.     

Considering Taguchi loss function on statistically constrained economic sum of squares exponentially weighted moving average charts

In this paper, Duncan's cost model combined Taguchi's quadratic loss function is applied to develop the economic-statistical design of the sum of squares exponentially weighted moving average (SS-EWMA) chart. The genetic algorithm is applied to search for the optimal decision variables of SS-EWMA chart such that the expected cost is minimized. Sensitivity analysis reveals that the optimal sample size and sampling interval decrease; optimal smoothing constant and control limit increase as the mean and/or variance increases. Moreover, the combination of optimal parameter levels in orthogonal array experiment plays an important guideline for monitoring the process mean and/or variance.     

Application of ordinal correspondence analysis for submerged aquatic vegetation monitoring

The European Water Framework states that macrophyte communities (seaweeds and seagrass) are key indicators of the ecological health of lagoons. Furthermore, the restoration of these communities, especially the Zostera meadows, is one of the main objectives of the Berre lagoon restoration plan. Consequently, a monitoring programme of the main macrophyte species still present in the lagoon was initiated in 1996. This monitoring resulted in a sequence of 11 spatially structured annual tables consisting of the observed density of these species. These tables are processed in this study. First, we specify the principles of Beh's ordinal correspondence analysis (OCA), designed for ordered row/column categories, and compare this method to classical correspondence analysis (CA). Then, we show that OCA is straightforwardly adaptable for processing a sequence of ordered contingency tables like ours. Both OCA and CA are afterwards used to reveal and test the main patterns of spatio-temporal changes of two macrophyte species in the Berre lagoon: Ulva and Zostera. The results we obtained are compared and discussed.     

Decomposition of cumulative chi-squared statistics,with some new tools for their interpretation

It is well known that the Pearson statistic \(\chi ^{2}\) can perform poorly in studying the association between ordinal categorical variables. Taguchi’s and Hirotsu’s statistics have been introduced in the literature as simple alternatives to Pearson’s chi-squared test for contingency tables with ordered categorical variables. The aim of this paper is to shed new light on these statistics, stressing their interpretations and characteristics, providing in this way new and different interpretations of these statistics. Moreover, a theoretical scheme is developed showing the links between the different proposals and classes of cumulative chi-squared statistical tests, starting from a unifying index of heterogeneity, unalikeability and variability measures. Users of statistics may find it attractive to understand well the different proposals. Some decompositions of both statistics are also highlighted. This paper presents a case study of optimizing the polysilicon deposition process in a very large-scale integrated circuit, to identify the optimal combination of factor levels. It is obtained by means of the information coming from a correspondence analysis based on Taguchi’s statistic and regression models for binary dependent variables. A new optimal combination of factor levels is obtained, different from many others proposed in the literature for this data.     

A modification of watson's statistic for goodness-of-fit

A modification of Watson's statistic is suggested for dealing with stochastically ordered alternatives in the goodness-of-fit setting.     

A continuously adaptive nonparametric two–sample test

A two–sample test statistic for detecting shifts in location is developed for a broad range of underlying distributions using adaptive techniques. The test statistic is a linear rank statistics which uses a simple modification of the Wilcoxon test; the scores are Winsorized ranks where the upper and lower Winsorinzing proportions are estimated in the first stage of the adaptive procedure using sample the first stage of the adaptive procedure using sample measures of the distribution's skewness and tailweight. An empirical relationship between the Winsorizing proportions and the sample skewness and tailweight allows for a ‘continuous’ adaptation of the test statistic to the data. The test has good asymptotic properties, and the small sample results are compared with other populatr parametric, nonparametric, and two–stage tests using Monte Carlo methods. Based on these results, this proposed test procedure is recommended for moderate and larger sample sizes.