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Heteroscedastic analysis of means (HANOM)

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Abstract

The analysis of means (ANOM) is a method that can compare a group of treatment means to see if any of those means are significantly different from the overall mean. It can be thought of as an alternative to the analysis of variance for analyzing fixed main effects in a designed experiment. The ANOM has advantages: it identifies any treatments that are different from the overall mean and has a graphical display that helps one to assess practical significance. Sample size tables and power curves have previously been developed for using ANOM to detect differences among I treatments when two of them differ by at least a specified amount δσ, where σ is the common standard deviation of the treatments (or processes).

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... For unknown and unequal variances, Nelson and Dudewicz (2002) developed a two-stage sampling procedure which was originally proposed by Stein (1945) to solve ANOM under heteroscedasticity (HANOM). Furthermore, Dudewicz and Nelson (2003) showed that while comparing HANOM with heteroscedastic ANOVA, the differences in powers are small, and HANOM has the graphical advantages as ANOM. Although the two-stage sampling procedure can solve problems caused by unequal variances, it has some drawbacks. ...
... 2. Specify the significance α, power γ , and the difference δ between any two treatment means that will lead to rejection of the all of treatment means are equal hypothesis. From Figure A1-A36 in Dudewicz and Nelson (2003), given k, α, γ , and degree of freedom df = n 0 − 1 combination, find the corresponding value of ω. ...
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The analysis of means (ANOM) is a method that can compare the mean of each treatment to the overall mean. According to the graphical result of a statistical data analysis, we can specify which one is different from another. One of the assumptions of the classical ANOM model is that the variances are equal. However, it is not always true for the practice. To solve unknown and unequal population variances, Nelson and Dudewicz (2002) proposed a two-stage sampling procedure. However, additional samples need to be added in the second stage of the two-stage sampling procedure, so it is not practical all the time due to limited time and insufficient budget. Thus, under heteroscedasticity, we applied Chen and Lam’s (1989) single-stage sampling procedure to solve the drawback of the two-stage sampling procedure. In addition, we also provided an illustrative example and critical values for practical uses. In order to make the procedure user-friendly, we built an interface by using R Shiny.
... Bernard and Wludyka [21] and Wludyka and Sa [22] suggested the robustness of ANOMV with the combination of the Fligner and Killeen test and the Levene test. An extension of the ANOM test under a heteroscedastic model, known as heteroscedastic analysis of means (HANOM), was proposed by Nelson and Dudewicz [23] and Dudewicz and Nelson [24]. A nonparametric version of the ANOM test was introduced by Bakir [25], and a comparison between ANOM and ANOVA tests using parametric bootstrap was conducted by Chang et al. [26] The exact control limits for the balanced design with equal sample sizes were presented by Nelson [27], Nelson [28], while for the unbalanced design with unequal sample sizes were given by Soong and Hsu [29]. ...
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In many experiments, our interest lies in testing the significance of means from the grand mean of the study variable. Sometimes, an additional linearly related uncontrollable factor is also observed along with the main study variable, known as a covariate. For example, in Electrical Discharge Machining (EDM) problem, the effect of pulse current on the surface roughness (study variable) is affected by the machining time (covariate). Hence, covariate plays a vital role in testing means, and if ignored, it may lead to false decisions. Therefore, we have proposed a covariate-based approach to analyze the means in this study. This new approach capitalizes on the covariate effect to refine the traditional structure and rectify misleading decisions, especially when covariates are present. Moreover, we have investigated the impact of assumptions on the new approach, including normality, linearity, and homogeneity, by considering equal or unequal sample sizes. This study uses percentage type Ⅰ error and power as our performance indicators. The findings reveal that our proposal outperforms the traditional one and is more useful in reaching correct decisions. Finally, for practical considerations, we have covered two real applications based on experimental data related to the engineering and health sectors and illustrated the implementation of the study proposal.
... Because of this paper aims at exploring ANOM using control limits of extreme value statistics we have considered only the control chart aspects but not any recently developed ANOM tables or techniques. However, a detailed literature about ANOM is available in Rao (2005) and some related works in this direction are Enrick (1976), Schilling (1979), Ohta (1981), Ramig (1983), Mason et al. (1989), Bakir (1994, Bernard and Wludyka (2001), Montgomery (2001, Nelson and Dudewicz and Nelson (2003), Farnum (2004, Guirguis and Tobias (2004) and references therein. The rest of the paper is organized as follows. ...
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A measurable quality characteristic is assumed to follow Inverse Rayleigh Distribution. Variable control charts based on extreme values of each subgroup are constructed. The technique of analysis of means (ANOM) is adopted to workout the decision lines of Inverse Rayleigh Distribution(IRD). The preferability of the proposed ANOM decision lines over that of Ott(1967) is illustrated by some examples.
... When variances cannot be assumed equal across groups, a heteroscedastic version of ANOM is expedient. The so-called HANOM was introduced in the past decade [11,41]. In the context of MCTs, two distinct approaches dealing with heterogeneous variances are available. ...
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The quality is assessable by its feature is implicit with a probability model which follow Gumbel distribution. From each subgroups extreme values are used to construct extreme value charts and variable control charts. Probability model of the extreme order statistics and the size of each sub group are used in control chart constants. To find the decision lines of Gumbel distribution we implement a method of analysis of means (ANOM). The proposed ANOM decision lines are constructed for given number of subgroup within means category and between means category given by Ott (1967). These decision lines are illustrated by giving few examples.
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The analysis of means (ANOM) is a technique for comparing a group of treatment means to see if any one of them differs significantly from the overall mean. It can be viewed as an alternative to the analysis of variance (ANOVA) for analyzing fixed main effects in a designed experiment. The ANOM has the advantages that it identifies any treatment means that differ from the overall mean (something the ANOVA does not do), and enables a graphical display that aids in assessing practical significance. Sample size tables and power curves have previously been developed for detecting differences among I treatments when two of them differ by at least a specified multiple of the common population standard deviation. Here, we consider the heteroscedastic situation where the different processes or populations from which the samples are drawn do not necessarily have equal standard deviations. In addition, we provide power curves that enable an experimenter to design a study for detecting differences among I treatment means when any two of them differ by at least a specified amount δ, independent of these standard deviations.
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