BOXCHART Statement: Constructing ANOM Boxcharts

Product	Release
SAS/QC	9.2

BOXCHART Statement

Constructing ANOM Boxcharts

The following notation is used in this section:

$x_{ij}$	th response in the th group
	number of groups
$n_{i}$	sample size of th group
	total sample size
$\mu_i$	expected value of a response in the th group
$\sigma$	standard deviation of response
$\bar{x}_{i}$	average response in th group
$\overline{\overline{x}}$	weighted average of group means
$s_{i}^2$	sample variance of the responses in the th group
$\hat{\sigma^2}$	mean square error (MSE)
$\nu$	degrees of freedom associated with the mean square error
$\alpha$	significance level
$h(\alpha; k, n, \nu )$	critical value for analysis of means when the sample sizes $n_{i}$ are equal $(n_i \equiv n)$
$h(\alpha; k, n_1, ... , n_k, \nu )$	critical value for analysis of means when the sample sizes $n_{i}$ 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 5.8 illustrates the elements of each plot.

Figure 5.8: Box-and-Whisker Plot

The skeletal style of the box-and-whisker plot shown in Figure 5.8 is the default. You can specify alternative styles with the BOXSTYLE= option; see the entry for the BOXSTYLE= option in Chapter 56, "Dictionary of Options."

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

$\overline{\overline{x}} = \frac{n_{1}\bar{x_{1}} + ... + n_{k}\bar{x_{k}}} {n_{1} + ... + n_{k}}$

You can specify a value for $\overline{\overline{x}}$ 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 th group are at least approximately normally distributed with a constant variance:

$x_{ij} \sim n( \mu_{i}, \sigma^2 ), \;\;\;\;\; j = 1, ... , n_i$

When the group sizes are constant ( $n_i \equiv n$ ), then $\nu = n-k=k(n-1)$ and the decision limits are computed as follows:

${lower decision limit (ldl)} & = & \overline{\overline{x}} - h(\alpha; k, n, \... ...rline{\overline{x}} + h(\alpha; k, n, \nu ) \sqrt{mse} \sqrt{\frac{k-1}n}$

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

${mse} = \hat{\sigma^2} = \frac{1}k \sum_{j=1}^k s_{j}^2$

For details concerning the function $h(\alpha; k, n, \nu )$ , see Nelson (1981, 1982a, 1993).

When the group sizes $n_{i}$ are not constant (the unbalanced case), $\nu = n-k$ and the decision limits for the th group are computed as follows:

${lower decision limit (ldl)} & = & \overline{\overline{x}} - h(\alpha; k, n_1,... ... h(\alpha; k, n_1, ... ,n_k, \nu ) \sqrt{mse} \sqrt{\frac{n - n_i}{nn_i}}$

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

${mse} = \hat{\sigma^2} = \frac{(n_{1} - 1)s_1^2 + ... + (n_{k} - 1)s_{k}^2} {n_{1} + ... + n_{k} - k}$

This requires that $\nu$ be positive. A chart is not produced if $\nu\gt$ but MSE is equal to zero (unless you specify the ZEROSTD option). For details concerning the function $h(\alpha; k, n_1, ... , n_k, \nu )$ , see Fritsch and Hsu (1997), Nelson (1982b, 1991), and Soong and Hsu (1997).

You can specify parameters for the limits as follows:

Specify $\alpha$ with the ALPHA= option or with the variable _ALPHA_ in a LIMITS= data set. By default, $\alpha$ = 0.05.
Specify a constant nominal sample size $n_i \equiv n$ for the decision limits in the balanced case with the LIMITN= option or with the variable _LIMITN_ in a LIMITS= data set. By default, is the observed sample size in the balanced case.
Specify with the LIMITK= option or with the variable _LIMITK_ in a LIMITS= data set. By default, is the number of groups.
Specify $\overline{\overline{x}}$ with the MEAN= option or with the variable _MEAN_ in a LIMITS= data set. By default, $\overline{\overline{x}}$ is the weighted average of the responses.
Specify $\hat{\sigma^2}$ with the MSE= option or with the variable _MSE_ in a LIMITS= data set. By default, $\hat{\sigma^2}$ is computed as indicated above.
Specify $\nu$ with the DFE= option or with the variable _DFE_ in a LIMITS= data set. By default, $\nu$ is determined as indicated above.

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SAS/QC(R) 9.2 User's Guide

Constructing ANOM Boxcharts

Elements of Box-and-Whisker Plots

Central Line

Decision Limits