Peter R. Nelson

Additional Uses for the Analysis of Means and Extended Tables of Critical Values

The analysis of means (ANOM) is a technique for comparing a group of treatment means to see if any of them are significantly different from the overall mean. As such, 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 the advantages that it identifies any treatments that are different and provides a graphical display that aids in assessing practical significance. When the exact critical values are used, no price is paid for these advantages in terms of decreased power. The first purpose of this article is to show that these exact critical values are appropriate not only for balanced complete designs, but also for Latin squares, Graeco-Latin squares, balanced incomplete block (BIB) designs, Youden squares, and axial mixture designs. Exact critical values for the ANOM in the equal-sample-size case when the degrees of freedom for error are at least as large as the number of treatments being compared have been previously p...

Exact critical points for the analysis of means

Tables of the exact critical points for the analysis of means are given for k means with degrees of freedom m and level of significance a for k = 3(1)20, 24, 30, 40, 60 m = k(l)20, 24, 30, 40, 60, 120, ∞ α = 0.10, 0.05, 0.01.

An analysis-of-means-type test for variances from normal populations

After a brief review of the literature, a test for homogeneity of variances is presented. The test, which is the variances analog of the analysis of means (ANOM), can be presented in a graphical form. This graphical presentation makes it easy for practitioners to assess both practical and statistical significance. Tables of critical values are presented for balanced designs. A large-sample version of the test, which uses ANOM critical values, is also presented, as well as a version of the test for unbalanced designs for which approximate critical values can be calculated. Monte Carlo results indicate that this test has power comparable to well-known normal-based tests for homogeneity of variances.

Power Curves for the Analysis of Means

A brief description of the analysis of means is given with the method used to compute power curves for detecting differences among K treatments at level of significance α when two of the treatment means differ by at least a specified amount δ (measured in units of the process standard deviation). Power curves are given for α = .l, .05, .O1; K = 3(1)10; and 1 ≤ δ ≤ 3. Two examples are presented.

A Comparison of Sample Sizes for the Analysis of Means and the Analysis of Variance

Tables of sample sizes for the analysis of means necessary for detecting differences among k treatments when two of the treatment means differ by at least a specified amount delta (measured in units of the process standard deviation) for fixed level for..

Stability prediction using the Arrhenius model

Abstract The numerical and statistical aspects of fitting the Arrhenius model to a set of data are discussed, and a FORTRAN computer program written to perform this task is described.

Multiple Comparisons of Means Using Simultaneous Confidence Intervals

The state of the art of multiple comparisons of means using simultaneous confidence intervals is discussed. Several industrial examples are used to illustrate the procedures. Attention is limited to those procedures that are recommended for use in pra..

Exact Analysis of Means With Unequal Variances

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.

Testing for interactions using the analysis of means

An analysis of means (ANOM) procedure is suggested for testing the significance of two-factor interactions in fixed-effects balanced designs. When at least one factor is at only two levels, the suggested procedure uses the exact critical points for the ANOM. New critical points are given for the case in which both factors are at more than two but fewer than six levels. The procedure can be extended to higher-order interactions, but the quantities being plotted become so complex that they are of little use in assessing practical significance.

Counting a Large Number of Items Using the Weight of a Small Sample

A procedure is described for counting out a large number of items to within a specified percent of the desired number using the weight of a small sample of items. The sample size necessary to ensure that one is within the specified range with confidence 1 – α is derived. This procedure has application in the pharmaceutical industry where one must count out batches of printed materials that are later to be combined with packages of drugs, and one must be able to account within narrow limits for all the printed material. By using this technique, it was found in one instance that instead of counting out 20,000 items it was possible to accurately determine that amount using the weight of only six items. This technique depends on having at least an estimate of the ratio of the mean to the standard deviation of item weights, and a control chart procedure is suggested for monitoring that ratio.