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Analysis of Variance
Published in Robert M. Bethea, R. Russell Rhinehart, Applied Engineering Statistics, 2019
Robert M. Bethea, R. Russell Rhinehart
The best (and only) estimates we have for the expected mean squares are the mean squares calculated from sample data. The mean square for any variance component is nothing more than the sum of squares (of deviations from the appropriate mean) divided by the corresponding degrees of freedom. The ratio of the appropriate mean squares provides the calculated value of F for testing the null hypothesis (no treatment effect). If the calculated F is “significantly different” from one, the null hypothesis is rejected and you must assume that the treatments (or levels of a single treatment type) probably affect the outcome. The AOV table is, therefore, a summary of all necessary calculations for evaluating the null hypothesis. This format is followed throughout this chapter and everywhere AOV is used in this text.
Dorfman–Berbaum–Metz–Hillis (DBMH) analysis
Published in Dev P. Chakraborty, Observer Performance Methods for Diagnostic Imaging, 2017
The expected mean squares in Table 9.1 are variance-like quantities; specifically, they are weighted linear combinations of the variances appearing in Equation 9.7. For single factors, the column headed degrees of freedom (df) is one less than the number of levels of the corresponding factor. Estimating a variance requires first estimating the mean, which imposes a constraint, thereby decreasing df by one. For interaction terms, df is the product of the degrees of freedom for the individual factors. As an example, the term (τRC)ijk contains three individual factors, and therefore df = (I − 1)(J − 1)(K − 1). The number of degrees of freedom can be thought of as the amount of information available in estimating a mean square. As a special case, with no replications, the ε term has zero df as N − 1 = 0, With only one observation Yn(ijk there is no information to estimate the variance corresponding to the ε term. To estimate this term, one needs to replicate the study several times—each time the same readers interpret the same cases in all treatments—a very boring task for the reader and totally unnecessary from the researcher's point of view.
The use of D-optimal design in an expanded gauge R&R study
Published in Quality Engineering, 2021
To estimate the variance components in Eq. [2], several approaches are used:The average and range method using sample ranges as estimators of standard deviations is still referred to as an option (see AIAG 2010 and other manuals), and it is also implemented in various software products, such as Minitab or Statgraphics, although it is “a relic of classical quality control” according to John (1994).The expected mean square method (EMS) is based on ANOVA; the mean squares in an ANOVA table are equated to their expected values. The necessary formulas can be found in many sources (see Burdick, Borror, and Montgomery 2003; Montgomery and Runger 1993b; Montgomery 2013a, 2013b). The shortcoming of the EMS method is that, occasionally, it produces a negative estimate of some variance component. Then there are two courses of action—either the negative estimate is replaced by zero or the corresponding effect is omitted and the model reduced, as Montgomery and Runger (1993a) suggest.The method of restrictive maximum likelihood (REML) (see Searle, Casella, and McCulloch 2006) is based on a restricted likelihood function. If the likelihood equation solutions are nonnegative, the variance component estimates are directly equal to these solutions and also to the ANOVA estimates. The REML method is not widely implemented in software products; therefore, it may not be available.