Compliance Testing for Random Effects Models With Joint Acceptance Criteria

doi:10.1080/00401706.2012.680394

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Compliance Testing for Random Effects Models With Joint Acceptance Criteria

Journal Article•DOI•

Compliance Testing for Random Effects Models With Joint Acceptance Criteria

Crystal Linkletter¹, Pritam Ranjan², C. Devon Lin³, Derek Bingham⁴, William A. Brenneman⁵, Richard A. Lockhart⁴, Thomas M. Loughin⁴ - Show less +3 more•Institutions (5)

MathWorks¹, Acadia University², Queen's University³, Simon Fraser University⁴, Procter & Gamble⁵

03 Apr 2012-Technometrics (Taylor & Francis Group)-Vol. 54, Iss: 3, pp 243-255

TL;DR: This article considers the current United States National Institute of Standards and Technology joint acceptance criteria and provides an approximation for the probability of sample acceptance that is applicable for processes with one or more known sources of variation via a random effects model.

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Abstract: For consumer protection, many governments perform random inspections on goods sold by weight or volume to ensure consistency between actual and labeled net contents. To pass inspection, random samples must jointly comply with restrictions placed on the individual sampled items and on the sample average. In this article, we consider the current United States National Institute of Standards and Technology joint acceptance criteria. Motivated by a problem from a real manufacturing process, we provide an approximation for the probability of sample acceptance that is applicable for processes with one or more known sources of variation via a random effects model. This approach also allows the assessment of the sampling scheme of the items. We use examples and simulations to assess the quality and accuracy of the approximation and illustrate how the methodology can be used to fine-tune process parameters for a prespecified probability of sample acceptance. Simulations are also used for estimating variance compon...

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Journal Article•DOI•

Setting Appropriate Fill Weight Targets—A Statistical Engineering Case Study

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William A. Brenneman¹, Michael D. Joner¹•Institutions (1)

Procter & Gamble¹

26 Mar 2012-Quality Engineering

TL;DR: In this paper, a high-level business need was addressed via the development of a solution for setting appropriate targets for product filling processes, and the solution was used to solve the problem of product filling.

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Abstract: A high-level business need was addressed via the development of a solution for setting appropriate targets for product filling processes.

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6 citations

Cites methods or result from "Compliance Testing for Random Effec..."

...In this case, after the short-term approximate solution was implemented, the accompanying theory was also developed that provided validation of the approximation (see Linkletter et al. 2012)....
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...Because new methodology was required, a new solution was developed through simulation to meet the immediate business need and then followed-up with theoretical research (see Linkletter et al. 2012)....
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...…cases, the probabilities of passing a government inspection obtained by simulation are in close agreement to the those obtained computationally in Linkletter et al. (2012); . for several cases, the recommended fill targets obtained by simulation are in close agreement to those obtained using…...
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...for several cases, the probabilities of passing a government inspection obtained by simulation are in close agreement to the those obtained computationally in Linkletter et al. (2012); ....
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...See Linkletter et al. (2012) for a more general formulation....
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Journal Article•DOI•

Comment: Simulation Used to Solve Tough Practical Problems

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William A. Brenneman¹•Institutions (1)

Procter & Gamble¹

20 Feb 2014-Technometrics

TL;DR: The commentator discusses in general the importance of simulation studies and how they can solve very difficult problems quickly.

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Abstract: The commentator discusses in general the importance of simulation studies and how they can solve very difficult problems quickly.

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5 citations

Cites background from "Compliance Testing for Random Effec..."

...However, a solution under more realistic assumptions can be obtained fairly quickly through simulation (see Brenneman and Joner 2012), while the theoretical work can take several years (see Linkletter et al. 2012)....
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Journal Article•DOI•

Confidence Regions and Intervals for Meta-Analysis Model Parameters

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Andrew L. Rukhin¹•Institutions (1)

National Institute of Standards and Technology¹

18 Nov 2015-Technometrics

TL;DR: Confidence regions are obtained based on canonical representations of the restricted and profile likelihood functions in terms of independent normal random variables and χ2 random variables that provide conservative confidence intervals for the common mean and heterogeneity variance in the heteroscedastic, one-way random effects model.

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Abstract: This article obtains confidence regions for the heteroscedastic, one-way random effects model’s parameters the heteroscedastic, one-way random effects model. The confidence regions are based on canonical representations of the restricted and profile likelihood functions in terms of independent normal random variables and χ2 random variables. These regions provide conservative confidence intervals for the common mean and heterogeneity variance. Mathematical details and the R code are available online as supplementary material.

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4 citations

Cites background from "Compliance Testing for Random Effec..."

...Kulinskaya and Koricheva (2010) showed how to use quality control charts in cumulative meta-analysis while Linkletter et al. (2012) illustrated applications of the homoscedastic one-way random effects model in acceptance testing....
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Journal Article•DOI•

A combined attributes-variables plan

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Kondaswamy Govindaraju¹, R. Kissling²•Institutions (2)

Massey University¹, Fonterra²

01 Sep 2015-Applied Stochastic Models in Business and Industry

TL;DR: A new plan, called the combined attributes-variables plan, is proposed incorporating an acceptance number to the regular variables plan for consumer protection and for food manufacturing applications in which the sample size cannot be predetermined because of short production lengths and other analytical testing issues.

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Abstract: The design of single sampling plans in which the lot acceptance decision is based on both variables and attribute measurement of quality is discussed. A new plan, called the combined attributes-variables plan, is proposed incorporating an acceptance number to the regular variables plan for consumer protection. A design approach for the new plan is also developed for food manufacturing applications in which the sample size cannot be predetermined because of short production lengths and other analytical testing issues. Copyright © 2014 John Wiley & Sons, Ltd.

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4 citations

Journal Article•DOI•

Confidence limits for compliance testing using mixed acceptance criteria

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Pasan M. Edirisinghe¹, Thomas Mathew², T. S. G. Peiris¹•Institutions (2)

University of Moratuwa¹, University of Maryland, Baltimore County²

26 Jan 2020-Quality and Reliability Engineering International

3 citations

Cites background from "Compliance Testing for Random Effec..."

...We note that in the case of r = 0, so that we have the expression (5) for the corresponding parameter θ10, Monte Carlo simulation is not necessary for computing the gpq....
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...Thus, we conclude that in the special case when r = 0, the parameter θ10 = g0(μ, σ) given in (5) is the cdf of a multivariate normal distribution of dimension (n + 1)....
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References

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Book•

Continuous univariate distributions

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Norman L. Johnson, Samuel Kotz, Narayanaswamy Balakrishnan

01 Jan 1994

TL;DR: Continuous Distributions (General) Normal Distributions Lognormal Distributions Inverse Gaussian (Wald) Distributions Cauchy Distribution Gamma Distributions Chi-Square Distributions Including Chi and Rayleigh Exponential Distributions Pareto Distributions Weibull Distributions Abbreviations Indexes

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Abstract: Continuous Distributions (General) Normal Distributions Lognormal Distributions Inverse Gaussian (Wald) Distributions Cauchy Distribution Gamma Distributions Chi-Square Distributions Including Chi and Rayleigh Exponential Distributions Pareto Distributions Weibull Distributions Abbreviations Indexes

...read moreread less

7,270 citations

Book•

Monte Carlo Statistical Methods

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Christian P. Robert¹, George Casella¹•Institutions (1)

University of Florida¹

01 Jan 1999

TL;DR: This new edition contains five completely new chapters covering new developments and has sold 4300 copies worldwide of the first edition (1999).

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Abstract: We have sold 4300 copies worldwide of the first edition (1999). This new edition contains five completely new chapters covering new developments.

...read moreread less

6,884 citations

Book•

Statistical Inference

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George Casella, Roger L. Berger

24 Apr 1990

6,235 citations

Journal Article•DOI•

Sample Criteria for Testing Outlying Observations

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Frank E. Grubbs¹•Institutions (1)

University of Michigan¹

01 Mar 1950-Annals of Mathematical Statistics

TL;DR: In this article, the significance of the largest observation in a sample of size $n$ from a normal population was investigated and the authors proposed a new statistic, S^2_n/S^2, to test whether the two largest observations are too large.

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Abstract: The problem of testing outlying observations, although an old one, is of considerable importance in applied statistics. Many and various types of significance tests have been proposed by statisticians interested in this field of application. In this connection, we bring out in the Histrical Comments notable advances toward a clear formulation of the problem and important points which should be considered in attempting a complete solution. In Section 4 we state some of the situations the experimental statistician will very likely encounter in practice, these considerations being based on experience. For testing the significance of the largest observation in a sample of size $n$ from a normal population, we propose the statistic $\frac{S^2_n}{S^2} = \frac{\sum^{n-1}_{i=1} (x_i - \bar x_n)^2}{\sum^n_{i=1} (x_i - \bar x)^2}$ where $x_1 \leq x_2 \leq \cdots \leq x_n, \bar x_n = \frac{1}{n - 1} \sum^{n-1}_{i=1} x_i$ and $\bar x = \frac{1}{n}\sum^{n}_{i=1} x_i.$ A similar statistic, $S^2_1/S^2$, can be used for testing whether the smallest observation is too low. It turns out that $\frac{S^2_n}{S^2} = 1 - \frac{1}{n - 1} \big(\frac{x_n - \bar x}{s}\big)^2 = 1 - \frac{1}{n - 1} T^2_n,$ where $s^2 = \frac{1}{n}\sigma(x_i - \bar x)^2,$ and $T_n$ is the studentized extreme deviation already suggested by E. Pearson and C. Chandra Sekar [1] for testing the significance of the largest observation. Based on previous work by W. R. Thompson [12], Pearson and Chandra Sekar were able to obtain certain percentage points of $T_n$ without deriving the exact distribution of $T_n$. The exact distribution of $S^2_n/S^2$ (or $T_n$) is apparently derived for the first time by the present author. For testing whether the two largest observations are too large we propose the statistic $\frac{S^2_{n-1,n}}{S^2} = \frac{\sum^{n-2}_{i=1} (x_i - \bar x_{n-1,n})^2}{\sum^n_{i=1} (x_i - \bar x)^2},\quad\bar x_{n-1,n} = \frac{1}{n - 2} \sum^{n-2}_{i=1} x_i$ and a similar statistic, $S^2_{1,2}/S^2$, can be used to test the significance of the two smallest observations. The probability distributions of the above sample statistics $S^2 = \sum^n_{i=1} (x_i - \bar x)^2 \text{where} \bar x = \frac{1}{n} \sum^n_{i=1} x_i$ $S^2_n = \sum^{n-1}_{i=1} (x_i - \bar x_n)^2 \text{where} \bar x_n = \frac{1}{n-1} \sum^{n-1}_{i=1} x_i$ $S^2_1 = \sum^n_{i=2} (x_i - \bar x_1)^2 \text{where} \bar x_1 = \frac{1}{n-1} \sum^n_{i=2} x_i$ are derived for a normal parent and tables of appropriate percentage points are given in this paper (Table I and Table V). Although the efficiencies of the above tests have not been completely investigated under various models for outlying observations, it is apparent that the proposed sample criteria have considerable intuitive appeal. In deriving the distributions of the sample statistics for testing the largest (or smallest) or the two largest (or two smallest) observations, it was first necessary to derive the distribution of the difference between the extreme observation and the sample mean in terms of the population $\sigma$. This probability$X_1 \leq x_2 \leq x_3 \cdots \leq x_n$ $s^2 = \frac{1}{n} \sum^n_{i=1} (x_i - \bar x)^2 \quad \bar x = \frac{1}{n} \sum^n_{i=1} x_i$ distribution was apparently derived first by A. T. McKay [11] who employed the method of characteristic functions. The author was not aware of the work of McKay when the simplified derivation for the distribution of $\frac{x_n - \bar x}{\sigma}$ outlined in Section 5 below was worked out by him in the spring of 1945, McKay's result being called to his attention by C. C. Craig. It has been noted also that K. R. Nair [20] worked out independently and published the same derivation of the distribution of the extreme minus the mean arrived at by the present author--see Biometrika, Vol. 35, May, 1948. We nevertheless include part of this derivation in Section 5 below as it was basic to the work in connection with the derivations given in Sections 8 and 9. Our table is considerably more extensive than Nair's table of the probability integral of the extreme deviation from the sample mean in normal samples, since Nair's table runs from $n = 2$ to $n = 9,$ whereas our Table II is for $n = 2$ to $n = 25$. The present work is concluded with some examples.

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1,401 citations

"Compliance Testing for Random Effec..." refers background in this paper

...With this distributional assumption, Schilling and Dodge (1969) gave tables of exact acceptance probabilities under the old U.S. criterion for small sample sizes using tabulated integral values for the distribution of an extreme deviate from the sample mean (developed by Nair 1948; Grubbs 1950)....
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Journal Article•DOI•

Two Generalizations of the Binomial Distribution

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P. M. E. Altham¹•Institutions (1)

University of Cambridge¹

01 Jun 1978-Journal of The Royal Statistical Society Series C-applied Statistics

TL;DR: In this article, the sum of k independent and identically distributed (0, 1) variables has a binomial distribution and two distinct generalizations are obtained, depending on whether the "multiplicative" or "additive" definition of interaction for discrete variables is used.

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Abstract: SUMMARY The sum of k independent and identically distributed (0, 1) variables has a binomial distribution. If the variables are identically distributed but not independent, this may be generalized to a two-parameter distribution where the k variables are assumed to have a symmetric joint distribution with no second- or higher-order "interactions". Two distinct generalizations are obtained, depending on whether the "multiplicative" or "additive" definition of "interaction" for discrete variables is used. The multiplicative generalization gives rise to a two-parameter exponential family, which naturally includes the binomial as a special case. Whereas with a beta-binomially distributed variable the variance always exceeds the corresponding binomial variance, the "additive" or "multiplicative" generalizations allow the variance to be greater or less than the corresponding binomial quantity. The properties of these two distributions are discussed, and both distributions are fitted, successfully, to data given by Skellam (1948) on the secondary association of chromosomes in Brassica.

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227 citations

"Compliance Testing for Random Effec..." refers background or methods in this paper

...dependent indicators have been discussed extensively (Bahadur 1960; Altham 1978; Soon 1996). After exploring several of these options for the distribution of W | (X̄n + bS ≥ V, aL), we chose the approximation from Soon (1996), W | (X̄n + bS ≥ V, aL) D ≈ Bin(n, pW ), for obtaining...
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...dependent indicators have been discussed extensively (Bahadur 1960; Altham 1978; Soon 1996)....
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...…not independent the indicators are identically distributed— WLj | (X̄n + bS ≥ V, aL) ∼ Bin(1, pW ) for every j. Approximations for the distribution of the sum of TECHNOMETRICS, AUGUST 2012, VOL. 54, NO. 3 dependent indicators have been discussed extensively (Bahadur 1960; Altham 1978; Soon 1996)....
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