XCHART Statement: ANOM Procedure

Constructing ANOM Charts for Means

The following notation is used in this section:

$X_{ij}$

jth response in the ith group

k

number of groups

$n_{i}$

sample size of ith group

N

total sample size $= n_1 + \cdots + n_ k$

$\mu _ i$

expected value of a response in the ith group

$\sigma $

standard deviation of a response

$\bar{X}_{i}$

average response in the ith group

$\overline{\overline{X}}$

weighted average of k group means

$s_{i}^{2}$

sample variance of the responses in the ith group

$ \widehat{\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 $(n_ i \equiv n)$

$h(\alpha ; k, n_1,\ldots ,n_ k, \nu )$

critical value for analysis of means when the sample sizes $n_ i$ are not equal

Plotted Points

Each point on an ANOM chart indicates the value of a group mean ($\bar{X}_{i}$).

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

\[  \overline{\overline{X}} = \frac{n_{1}\bar{X}_{1} + \cdots + n_{k}\bar{X}_{k}}{n_{1} + \cdots + n_{k}}  \]

You can specify a value for $\overline{\overline{X}}$ with the MEAN= option in the XCHART 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 ith group are at least approximately normally distributed with a constant variance:

\[  X_{ij} \sim N( \mu _{i}, \sigma ^2 ), \; \; \; \; \;  j = 1, \ldots , n_ i  \]

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:

\begin{eqnarray*}  \mbox{lower decision limit (LDL)} &  = &  \overline{\overline{X}} - h(\alpha ; k, n, \nu ) \sqrt {\mbox{MSE}} \sqrt { \frac{k-1}{N}} \\ \mbox{upper decision limit (UDL)} &  = &  \overline{\overline{X}} + h(\alpha ; k, n, \nu ) \sqrt {\mbox{MSE}} \sqrt {\frac{k-1}{N}} \end{eqnarray*}

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

\[  \mbox{MSE} = \widehat{\sigma ^2} = \frac{1}{k} \sum _{j=1}^{k} s_{j}^2  \]

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

When the group sizes are not constant (the unbalanced case), $\nu = N-k$ and the decision limits for the ith group are computed as follows:

\begin{eqnarray*}  \mbox{lower decision limit (LDL)} &  = &  \overline{\overline{X}} - h(\alpha ; k, n_1,\ldots ,n_ k, \nu ) \sqrt {\mbox{MSE}} \sqrt { \frac{N-n_ i}{Nn_ i}} \\ \mbox{upper decision limit (UDL)} &  = &  \overline{\overline{X}} + h(\alpha ; k, n_1,\ldots ,n_ k, \nu ) \sqrt {\mbox{MSE}} \sqrt {\frac{N - n_ i}{Nn_ i}} \end{eqnarray*}

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

\[  \mbox{MSE} = \widehat{\sigma ^2} = \frac{(n_{1} - 1)s_1^{2} + \cdots + (n_{k} - 1)s_{k}^{2}}{n_{1} + \cdots + n_{k} - k}  \]

This requires that $\nu $ be positive. A chart is not produced if $\nu >0$ but MSE is equal to zero (unless you specify the ZEROSTD option). For details concerning the function $h(\alpha ; k, n_1,\ldots ,n_ k, \nu )$, see Nelson (1991).

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, n is the observed sample size in the balanced case.

  • Specify k with the LIMITK= option or with the variable _LIMITK_ in a LIMITS= data set. By default, k 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 $\widehat{\sigma ^2}$ with the MSE= option or with the variable _MSE_ in a LIMITS= data set. By default, $\widehat{\sigma ^2}$ 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.