Analysis of Means Methods

Analysis of means (ANOM) methods compare means and variances and other measures of location and scale across several groups. You might want to use these methods under these circumstances:

•	to test whether any of the group means are statistically different from the overall (sample) mean

•	to test whether any of the group standard deviations are statistically different from the root mean square error (RMSE)

•	to test whether any of the group ranges are statistically different from the overall mean of the ranges

Note: Within the Contingency platform, you can use the Analysis of Means for Proportions when the response has two categories. For details, see the Contingency Analysis section.

For a description of ANOM methods and to see how JMP implements ANOM, see the book by Nelson et al. (2005).

Analysis of Means for Location

You can test whether groups have a common mean or center value using the following options:

•

ANOM

•	ANOM with Transformed Ranks

ANOM

Use ANOM to compare group means to the overall mean. This method assumes that your data are approximately normally distributed. See Example of an Analysis of Means Chart.

ANOM with Transformed Ranks

This is the nonparametric version of the ANOM analysis. Use this method if your data is clearly non-normal and cannot be transformed to normality. ANOM with Transformed Ranks compares each group’s mean transformed rank to the overall mean transformed rank. The ANOM test involves applying the usual ANOM procedure and critical values to the transformed observations.

Transformed Ranks

Suppose that there are n observations. The transformed observations are computed as follows:

•	Rank all observations from smallest to largest, accounting for ties. For tied observations, assign each one the average of the block of ranks that they share.

•	Denote the ranks by R1, R2, ..., Rn.

•	The transformed rank corresponding to the ith observations is:

The ANOM procedure is applied to the values Transformed Ri. Since the ranks have a uniform distribution, the transformed ranks have a folded normal distribution. For details, see Nelson et al. (2005).

Analysis of Means for Scale

You can test for homogeneity of variation within groups using the following options:

•	ANOM for Variances

•	ANOM for Variances with Levene (ADM)

•	ANOM for Ranges

ANOM for Variances

Use this method to compare group standard deviations (or variances) to the root mean square error (or mean square error). This method assumes that your data is approximately normally distributed. To use this method, each group must have at least four observations. For details about the ANOM for Variances test, see Wludyka and Nelson (1997) and Nelson et al. (2005). For an example, see Example of an Analysis of Means for Variances Chart.

ANOM for Variances with Levene (ADM)

This method provides a robust test that compares the group means of the absolute deviations from the median (ADM) to the overall mean ADM. Use ANOM for Variances with Levene (ADM) if you suspect that your data is non-normal and cannot be transformed to normality. ANOM for Variances with Levene (ADM) is a nonparametric analog of the ANOM for Variances analysis. For details about the ANOM for Variances with Levene (ADM) test, see Levene (1960) or Brown and Forsythe (1974).

ANOM for Ranges

Use this test to compare group ranges to the mean of the group ranges. This is a test for scale differences based on the range as the measure of spread. For details, see Wheeler (2003).

Note: ANOM for Ranges is only available for balanced designs and specific group sizes. See Restrictions for ANOM for Ranges Test.

Restrictions for ANOM for Ranges Test

Unlike the other ANOM decision limits, the decision limits for the ANOM for Ranges chart uses only tabled critical values. For this reason, ANOM for Ranges is only available for the following:

•	groups of equal sizes

•	groups specifically of the following sizes: 2–10, 12, 15, and 20

•	number of groups between 2 and 30

•	alpha levels of 0.10, 0.05, and 0.01

Analysis of Means Charts

Each Analysis of Means Methods option adds a chart to the report window that shows the following:

•	an upper decision limit (UDL)

•	a lower decision limit (LDL)

•	a horizontal (center) line that falls between the decision limits and is positioned as follows:

‒	ANOM: the overall mean

‒	ANOM with Transformed Ranks: the overall mean of the transformed ranks

‒	ANOM for Variances: the root mean square error (or MSE when in variance scale)

‒	ANOM for Variances with Levene (ADM): the overall absolute deviation from the mean

‒	ANOM for Ranges: the mean of the group ranges

If a group’s plotted statistic falls outside of the decision limits, then the test indicates that there is a statistical difference between that group’s statistic and the overall average of the statistic for all the groups.

Analysis of Means Options

Each Analysis of Means Methods option adds an Analysis of Means red triangle menu to the report window.

Descriptions of the Analysis of Means Methods Options

Set Alpha Level

Select an option from the most common alpha levels or specify any level with the Other selection. Changing the alpha level modifies the upper and lower decision limits.

Note: For ANOM for Ranges, only the selections 0.10, 0.05, and 0.01 are available.

Show Summary Report

•	For ANOM, creates a report showing group means and decision limits.

•	For ANOM with Transformed Ranks, creates a report showing group mean transformed ranks and decision limits.

•	For ANOM for Variances, creates a report showing group standard deviations (or variances) and decision limits.

•	For ANOM for Variances with Levene (ADM), creates a report showing group mean ADMs and decision limits.

•	For ANOM for Ranges, creates a report showing group ranges and decision limits.

Graph in Variance Scale

(Only for ANOM for Variances) Changes the scale of the y-axis from standard deviations to variances.

Display Options

Display options include the following:

•	Show Decision Limits shows or hides decision limit lines.

•	Show Decision Limit Shading shows or hides decision limit shading.

•	Show Center Line shows or hides the center line statistic.

•	Point Options

‒	Show Needles shows the needles. This is the default option.

‒	Show Connected Points shows a line connecting the means for each group.

‒	Show Only Points shows only the points representing the means for each group.