## Monday, October 25, 2010

### Analysis of Means: A Graphical Decision Tool

JMP version 9 has been out for about two weeks now, and I hope you had a chance to play with it. If you are not ready to buy it you can give it a try by downloading a 30-day trial copy.

Today I want to share with you a new feature in JMP version 9: the Analysis of Means (ANOM). An analysis of means is a graphical decision tool for comparing a set of averages with respect to their overall average. You can think of it as a control chart but with decision limits instead of control limits, or as an alternative to an analysis of variance (ANOVA). In an ANOVA a significant F test just indicates that the means are different, but it does not reveal where the differences are coming from. By contrast, in an ANOM chart if an average falls outside the decision limits it is an indication that this average is statistically different, with a given risk α, from the overall average.

Prof. Ellis Ott introduced the analysis of means in 1967 as a logical extension of the Shewhart control chart. Let's look at an example. The plot below shows measurements of an electrical assembly as a function of six different types of ceramic sheets used in their construction. The data appears in Table 13.1 of the first edition of Prof. Ott's book Process Quality Control.

One can see some differences in the average performance of the six ceramic sheets. A Shewhart Xbar and R chart shows that the ranges are in control, indicating consistency within a ceramic sheet, but that the average of the ceramic sheet #6 is outside the lower control limit. Based on this we can say that there is probably an assignable cause responsible for this low average, but we can not claim any statistical significance.

The question of interest, quoting from Prof. Ott's book, is: "Is there evidence from the sample data that some of the ceramic sheets are significantly different from their own group average?". We can perform an analysis of variance to test the hypothesis that the averages are different. The F test is significant at the 5% level, indicating that the average electrical performance of the six ceramic sheets differ from each other. The F test, being an 'omnibus' type test, does not, however, tells which, or which ones, are different. For this we need to perform multiple comparisons tests, or an analysis of means.

A ANOM chart can be easily generated by selecting Analysis of Means Methods > ANOM from within the Analyze>Fit Y by X>Oneway Analysis window

The ANOM chart clearly reveals that the assemblies built using the ceramic sheet #6 have an average that is (statistically) lower than the overall average of 15.952. The other five averages are within the 5% risk decision limits, indicating that their electrical performance can be assumed to be similar.

The ANOM chart with decision limits 15.15 and 16.76, provide a graphical test for simultaneously comparing the performance of these six averages. What a great way to perform the test and communicate its results. Next time you need to decide which average, or averages, are (statistically) different from the overall average, give the ANOM chart a try.