BOXCHART Statement: ANOM Procedure

Constructing ANOM Boxcharts

The following notation is used in this section:

$\text{[math]}$	$\text{[math]}$ th response in the $\text{[math]}$ th group
$\text{[math]}$	number of groups
$\text{[math]}$	sample size of $\text{[math]}$ th group
$\text{[math]}$	total sample size $\text{[math]}$
$\text{[math]}$	expected value of a response in the $\text{[math]}$ th group
$\text{[math]}$	standard deviation of response
$\text{[math]}$	average response in $\text{[math]}$ th group
$\text{[math]}$	weighted average of $\text{[math]}$ group means
$\text{[math]}$	sample variance of the responses in the $\text{[math]}$ th group
$\text{[math]}$	mean square error (MSE)
$\text{[math]}$	degrees of freedom associated with the mean square error
$\text{[math]}$	significance level
$\text{[math]}$	critical value for analysis of means when the sample sizes $\text{[math]}$ are equal $\text{[math]}$
$\text{[math]}$	critical value for analysis of means when the sample sizes $\text{[math]}$ are not equal

Elements of Box-and-Whisker Plots

A box-and-whisker plot is displayed for the measurements in each group on the ANOM boxchart. Figure 4.9 illustrates the elements of each plot.

Figure 4.9 Box-and-Whisker Plot

The skeletal style of the box-and-whisker plot shown in Figure 4.9 is the default. You can specify alternative styles with the BOXSTYLE= option; see the entry for the BOXSTYLE= option in Dictionary of Options: SHEWHART Procedure.

Central Line

By default, the central line on an ANOM chart for means represents the weighted average of the group means, which is computed as

$\text{[math]}$

You can specify a value for $\text{[math]}$ with the MEAN= option in the BOXCHART statement or with the variable _MEAN_ in a LIMITS= data set.

Decision Limits

In the analysis of means for continuous data, it is assumed that the responses in the $\text{[math]}$ th group are at least approximately normally distributed with a constant variance:

$\text{[math]}$

When the group sizes are constant ( $\text{[math]}$ ), then $\text{[math]}$ and the decision limits are computed as follows:

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

Here the mean square error (MSE) is computed as follows:

$\text{[math]}$

For details concerning the function $\text{[math]}$ , see Nelson (1981, 1982a, 1993).

When the group sizes $\text{[math]}$ are not constant (the unbalanced case), $\text{[math]}$ and the decision limits for the $\text{[math]}$ th group are computed as follows:

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

Here the mean square error (MSE) is computed as follows:

$\text{[math]}$

This requires that $\text{[math]}$ be positive. A chart is not produced if $\text{[math]}$ but MSE is equal to zero (unless you specify the ZEROSTD option). For details concerning the function $\text{[math]}$ , see Fritzsch and Hsu (1997), Nelson (1982b, 1991), and Soong and Hsu (1997).

You can specify parameters for the limits as follows:

Specify $\text{[math]}$ with the ALPHA= option or with the variable _ALPHA_ in a LIMITS= data set. By default, $\text{[math]}$ = 0.05.
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. By default, $\text{[math]}$ is the observed sample size in the balanced case.
Specify $\text{[math]}$ with the LIMITK= option or with the variable _LIMITK_ in a LIMITS= data set. By default, $\text{[math]}$ is the number of groups.
Specify $\text{[math]}$ with the MEAN= option or with the variable _MEAN_ in a LIMITS= data set. By default, $\text{[math]}$ is the weighted average of the responses.
Specify $\text{[math]}$ with the MSE= option or with the variable _MSE_ in a LIMITS= data set. By default, $\text{[math]}$ is computed as indicated above.
Specify $\text{[math]}$ with the DFE= option or with the variable _DFE_ in a LIMITS= data set. By default, $\text{[math]}$ is determined as indicated above.