PCHART Statement: ANOM Procedure

Constructing ANOM Charts for Proportions

The following notation is used in this section:

$X_{i}$

response number (count) in the ith group

k

number of groups

$n_{i}$

sample size of the ith group

N

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

$p_{i}$

proportion in the ith group, where $p_ i = X_ i / n_ i$

$\bar{p}$

weighted average of proportions across groups:

\[ \bar{p} = \frac{n_1p_1 + \cdots + n_ kp_ k}{N} = \frac{X_1 + \cdots + X_ k}{N} \]

$\alpha $

significance level

$h(\alpha ;k,n,\infty )$

critical value for ANOM for normal data in the balanced case $(n_ i \equiv n)$

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

critical value for ANOM for normal data in the unbalanced case

Plotted Points

Each point on an ANOM p chart represents the response proportion ($p_ i=X_ i/n_ i$) for a group.

Central Line

By default, the central line on an ANOM p chart is computed as $\bar{p}$, the weighted average of the group proportions. You can specify $\bar{p}$ with the P= option or with the variable _P_ in a LIMITS= data set.

Decision Limits

For the ith group, the response counts are assumed to have the binomial distribution $B( n_ i, p_ i)$. The ANOM method for proportions tests the null hypothesis that $p_1 = p_2 = \cdots = p_ k$, that is, that the proportions are the same, against the alternative that at least one of the $p_ i$’s is different from the average of the k proportions.

The decision limits are computed using the normal approximation to the binomial distribution, which is appropriate when the sample sizes for the groups are large; refer to Ramig (1983). A commonly recommended check for this assumption is that $n_ ip_ i > 5$ and $n_ i(1-p_ i) > 5$ for all the groups. The critical values in the ANOM method for normally distributed data are adapted to the binomial case by using infinite degrees of freedom for the variance.

When the sample sizes are constant across groups ($n_ i \equiv n$), the decision limits are computed as follows:

\begin{eqnarray*} \mbox{lower decision limit (LDL)} & = & \mbox{max} \left(\bar{p} - h(\alpha ;k, n, \infty ) \sqrt {\bar{p}(1-\bar{p})}\sqrt {\frac{k-1}{N}}\; , 0 \right) \\ \mbox{upper decision limit (UDL)} & = & \mbox{min}\left(\bar{p} + h(\alpha ;k, n, \infty ) \sqrt {\bar{p}(1-\bar{p})}\sqrt {\frac{k-1}{N}}\; , 1 \right) \end{eqnarray*}

For the theoretical derivation of the decision limits, refer to Nelson (1982a).

When the sample sizes ($n_ i$) are different across groups (the unbalanced case), the decision limits are computed as follows:

\begin{eqnarray*} \mbox{lower decision limit (LDL)} & = & \mbox{max} \left(\bar{p} - h(\alpha ;k, n_1,\ldots ,n_ k, \infty ) \sqrt {\bar{p}(1-\bar{p})}\sqrt {\frac{N - n_ i}{Nn_ i}}\; , 0 \right) \\ \mbox{upper decision limit (UDL)} & = & \mbox{min}\left(\bar{p} + h(\alpha ;k, n_1,\ldots ,n_ k, \infty ) \sqrt {\bar{p}(1-\bar{p})}\sqrt {\frac{N - n_ i}{Nn_ i}}\; , 1 \right) \end{eqnarray*}

Note that the decision limits for the ith group depend on $n_ i$. If the sample sizes are constant across groups ($n_ i \equiv n$), the decision limits in the unbalanced case reduce to the formulas given for the balanced case since $n_ i = n$ and $N = kn$ so

\begin{eqnarray*} \sqrt {\frac{N - n_ i}{Nn_ i}} = \sqrt {\frac{kn - n}{Nn}} = \sqrt {\frac{k - 1}{N}} \end{eqnarray*}

For the derivation of the decision limits for unequal sample sizes, refer to Nelson (1991).

Exact critical values $h(\alpha ;k, n, \infty )$ were first tabulated by L. S. Nelson (1983). Refer to Nelson (1993) for derivation of critical values.

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 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 $\bar{p}$ with the P= option or with the variable _P_ in a LIMITS= data set. By default, $\bar{p}$ is the weighted average of the group proportions.