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

Abstract

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 published in several places. There are several types of designs, such as Graeco-Latin squares and BIB designs, however, where for certain parameter combinations the degrees of freedom for error are less than the number of levels for some (or possibly all) of the factors. Therefore, the second purpose of this article is to provide critical values for this situation. Computation of the exact critical values is discussed, and a new table of critical values is presented for α = .25 to aid in assessing the descriptive level of significance associated with a group of treatment means. Two examples are given.

KEY WORDS:

• Axial mixture designs
• Balanced incomplete block designs
• Graeco-Latin squares
• Latin squares
• Multivariate t distribution
• Youden squares