This section provides the computational details for constructing an ANOM chart for the lth factor in an experiment involving two factors (l = 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:
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kth response at the ith level of factor 1 and the jth level of factor 2, where |
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number of groups (levels) for the lth factor, |
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number of replicates in cell |
N |
total sample size |
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variance of a response |
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average response in cell |
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average response for ith level of factor 1 |
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average response for jth level of factor 2 |
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|
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sample variance of the responses for the ith level of factor 1 and the jth level of factor 2 |
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mean square error (MSE) in the two-way analysis of variance |
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degrees of freedom associated with the mean square error in the two-way analysis of variance |
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significance level |
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critical value for analysis of means in a one-way layout for groups (treatment levels) when the sample sizes for each level are constant and is the degrees of freedom associated with the mean square error; see the section Constructing ANOM Charts for Means. |
The points on the ANOM chart for factor 1 represent , and the points on the ANOM chart for factor 2 represent , .
The central line on the ANOM chart for the lth factor is the overall weighted average . Some authors use the notation for this average.
It is assumed that
where the quantities are independent and at least approximately normally distributed with
The correct decision limits for a given factor in a two-way layout are not computed by default when the lth 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, and 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 (), the appropriate decision limits are:
where the mean square error (MSE) is computed as in the ANOVA or GLM procedure:
and the degrees of freedom for error is . For details concerning the function , see Nelson (1982a, 1993).
You can provide the appropriate values of MSE
and by
specifying with the MSE= option or with the variable _MSE_
in a LIMITS= data set
specifying with the DFE= option or with the variable _DFE_
in a LIMITS= data set
In addition you can:
Specify with the ALPHA= option or with the variable _ALPHA_
in a LIMITS= data set. By default, .
Specify a constant nominal sample size for the decision limits in the balanced case with the LIMITN= option or with the variable _LIMITN_
in a LIMITS= data set.
Specify with the LIMITK= option or with the variable _LIMITK_
in a LIMITS= data set.
Specify with the MEAN= option or with the variable _MEAN_
in a LIMITS= data set.