Overview of Analysis of Means
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Analysis of Means (ANOM) is a graphical analog to ANOVA, and tests the equality of population means. ANOM [25] was developed to test main effects from a designed experiment in which all factors are fixed. This procedure is used for one-factor designs. Minitab uses an extension of ANOM or ANalysis Of Mean treatment Effects (ANOME) [33] to test the significance of mean treatment effects for two-factor designs.

An ANOM chart can be described in two ways: by its appearance and by its function. In appearance, it resembles a Shewhart control chart. In function, it is similar to ANOVA for detecting differences in population means [21]. The null hypotheses for ANOM and ANOVA are the same: both methods test for a lack of homogeneity among means. However, the alternative hypotheses are different [25]. The alternative hypothesis for ANOM is that one of the population means is different from the other means, which are equal. The alternative hypothesis for ANOVA is that the variability among population means is greater than zero.

For most cases, ANOVA and ANOM will likely give similar results. However, there are some scenarios where the two methods might be expected to differ:

·    If one group of means is above the grand mean and another group of means is below the grand mean, the
F-test for ANOVA might indicate evidence for differences where ANOM might not.

·    If the mean of one group is separated from the other means, the ANOVA F-test might not indicate evidence for differences whereas ANOM might flag this group as being different from the grand mean.

Refer to [30], [31], [32], and [33] for an introduction to the analysis of means.

ANOM can be used if you assume that the response follows a normal distribution, similar to ANOVA, and the design is one-way or two-way. You can also use ANOM when the response follows either a binomial distribution or a Poisson distribution.