PROC ANOM: Constructing ANOM Charts for Two-Way Layouts

The ANOM Procedure

Constructing ANOM Charts for Two-Way Layouts

This section provides the computational details for constructing an ANOM chart for the $\text{[math]}$ th factor in an experiment involving two factors ( $\text{[math]}$ = 1 or 2). It is assumed that there is no interaction effect. See Example 4.5 for an illustration.

The following notation is used in this section:

$\text{[math]}$	$\text{[math]}$ th response at the $\text{[math]}$ th level of factor 1 and the $\text{[math]}$ th level of factor 2, where $\text{[math]}$
$\text{[math]}$	number of groups (levels) for the $\text{[math]}$ th factor, $\text{[math]}$
$\text{[math]}$	number of replicates in cell $\text{[math]}$
$\text{[math]}$	total sample size $\text{[math]}$
$\text{[math]}$	variance of a response
$\text{[math]}$	average response in cell $\text{[math]}$
$\text{[math]}$	average response for $\text{[math]}$ th level of factor 1 $\text{[math]}$
$\text{[math]}$	average response for $\text{[math]}$ th level of factor 2 $\text{[math]}$
$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	sample variance of the responses for the $\text{[math]}$ th level of factor 1 and the $\text{[math]}$ th level of factor 2
$\text{[math]}$	mean square error (MSE) in the two-way analysis of variance
$\text{[math]}$	degrees of freedom associated with the mean square error in the two-way analysis of variance
$\text{[math]}$	significance level
$\text{[math]}$	critical value for analysis of means in a one-way layout for $\text{[math]}$ groups (treatment levels) when the sample sizes for each level are constant $\text{[math]}$ and $\text{[math]}$ is the degrees of freedom associated with the mean square error; see the section Constructing ANOM Charts for Means.

Plotted Points

The points on the ANOM chart for factor 1 represent $\text{[math]}$ , $\text{[math]}$ and the points on the ANOM chart for factor 2 represent $\text{[math]}$ , $\text{[math]}$ .

Central Line

The central line on the ANOM chart for the $\text{[math]}$ th factor is the overall weighted average $\text{[math]}$ . Some authors use the notation $\text{[math]}$ for this average.

Decision Limits

It is assumed that

$\text{[math]}$

where the quantities $\text{[math]}$ are independent and at least approximately normally distributed with

$\text{[math]}$

The correct decision limits for a given factor in a two-way layout are not computed by default when the $\text{[math]}$ th factor is specified as the group-variable in the XCHART statement, since the mean square error and degrees of freedom are not adjusted for the two-way structure of the data. Consequently, $\text{[math]}$ and $\text{[math]}$ must be precomputed and provided to the ANOM procedure, as illustrated in Example 4.5.

In the case of a two-way layout with equal group sizes ( $\text{[math]}$ ), the appropriate decision limits are:

	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

where the mean square error (MSE) is computed as in the ANOVA or GLM procedure:

$\text{[math]}$

and the degrees of freedom for error is $\text{[math]}$ . For details concerning the function $\text{[math]}$ , see Nelson (1982a, 1993).

You can provide the appropriate values of MSE and $\text{[math]}$ by

specifying $\text{[math]}$ with the MSE= option or with the variable _MSE_ in a LIMITS= data set
specifying $\text{[math]}$ with the DFE= option or with the variable _DFE_ in a LIMITS= data set

In addition you can:

Specify $\text{[math]}$ with the ALPHA= option or with the variable _ALPHA_ in a LIMITS= data set. By default, $\text{[math]}$ .
Specify a constant nominal sample size $\text{[math]}$ for the decision limits in the balanced case with the LIMITN= option or with the variable _LIMITN_ in a LIMITS= data set.
Specify $\text{[math]}$ with the LIMITK= option or with the variable _LIMITK_ in a LIMITS= data set.
Specify $\text{[math]}$ with the MEAN= option or with the variable _MEAN_ in a LIMITS= data set.

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