Chapter 13 Prime Charts for Count Data with Very Large Subgroups
With count data involving very large subgroup sizes, SPC charts often produce very tight control limits with many data points falling outside the limits. Figure 13.1 demonstrates this phenomenon, known as overdispersion, using data from from Mohammed et al. (2013) included in qicharts2. The data are tabulated at the end of this chapter.
Figure 13.1: P chart of percent emergency attendances seen within 4 hours
At first sight, this may suggest that the process is highly unstable, even when such a conclusion may not be justified. Overdispersion occurs when the natural, common cause variation in the data exceeds what is expected under the assumed theoretical distribution – binomial for proportions and Poisson for rates.
One way to diagnose this problem is to plot the same data on an I chart, which bases its control limits on the observed variation between successive subgroups rather than on theoretical distributional assumptions. Figure 13.2 is an I chart of the same data as Figure 13.1 and suggests only common cause variation.
Figure 13.2: I chart of percent emergency attendances seen within 4 hours
As a general rule of thumb, when the control limits from an I chart differ markedly from those of the corresponding P or U chart, this suggests that the observed variation is not consistent with the binomial or Poisson assumptions underlying the P and U charts.
Several solutions to the problem of overdispersion and overly tight control limits have been proposed. One pragmatic solution is simply to use I charts for count data as well as for individual measurements. However, I charts have straight control limits and do not take varying subgroup sizes into account. If subgroup sizes vary only a little, this may not matter. But sometimes – particularly in healthcare – variation in subgroup size does matter.
Laney proposed a solution in which the assumed within-subgroup variation in count data is adjusted by a factor based on the moving ranges of successive standardised data points, \(\sigma_z\) (Laney 2002).
13.1 Laney’s prime chart
As shown earlier in Table 6.1, the control limits for a traditional P chart are calculated as:
\[\bar{p}\pm3\sigma_{pi}\]
where \(\bar{p}\) is the average proportion and \(\sigma_{pi}\) is the within-subgroup standard deviation for the i-th subgroup:
\[\sigma_{pi}=\sqrt{\bar{p}(1-\bar{p})/{n_{i}}}\]
where \(n_i\) is the sample size of the i-th subgroup.
The P′ chart, pronounced P prime chart, takes both within and between subgroup variation into account. its control limits are given by:
\[\bar{p}\pm3\sigma_{pi}\sigma_z\]
where \(\sigma_z\) represents the between-subgroup variation. This is calculated from the moving ranges of the standardised proportions, defined as:
\[z_i=(p_i - \bar{p})/\sigma_{pi}\]
The moving ranges are then:
\[MR_i=|z_i-z_{i-1}|\]
The estimat of between-subgroup variation, \(\sigma_z\), is the average moving range devided by the constant 1.128:
\[\sigma_z=\overline{MR_z}/1.128\] Thus, the control limits for the P′ chart become:
\[\bar{p}\pm3\sigma_{pi}\sigma_z \]
Figure 13.3: P’ chart of percent emergency attendances seen within 4 hours
The control limits in Figure 13.3 are very close to those of the I chart, though they vary slightly because subgroup sizes vary.
Similarly, control limits for the U’ chart are:
\[\bar{u}\pm3\sigma_{ui}\sigma_z\]
where
\[\sigma_{ui}=\sqrt{\bar{u}/n_i}\]
and
\[\sigma_{z}=\overline{MR_z}/1.128\]
and the moving ranges are computed from the standardised rates defined as
\[z_i=(u_i-\bar{u})/\sigma_{ui}\]
In qicharts2, the moving ranges of the standardised values are, by default, screened for extreme values, following the same approach used for I charts, as described in Chapter 11.
13.2 When to use prime charts
Laney’s prime charts were developed to handle overdispersion, and also underdispersion, that is, situations in which the data show more or less variation than expected under a binomial or Poisson distribution. This is often seen when subgroup sizes are very large. But how do we know when to suspect overdispersion?
A warning sign is when a traditional P or U chart shows unusually tight control limits that seem unnatural. If plotting the same data on an I chart produces substantially wider limits, this is a strong indication that the assumptions behind the P or U chart may not hold. In such cases, Laney’s P′ or U′ chart will often be the better choice.
When the adjustment factor (\(\sigma_z\)) is close to one, indicating little or no overdispersion, the P′ or U′ chart closely resembles the traditional P or U chart. For this reason, Laney recommends prime charts as a generally safe and robust default.
In the next chapter, we introduce a modified I chart – the I′, or I prime, chart – which produces control limits for count data that closely match those of Laney’s prime charts while also accommodating measurement data with varying subgroup sizes.
Example data for P prime charts
| Week (i) | Attendances seen within 4 hours (r) | Number of attendances (n) |
|---|---|---|
| 1 | 266,501 | 280,443 |
| 2 | 264,225 | 276,823 |
| 3 | 276,532 | 291,681 |
| 4 | 281,461 | 296,155 |
| 5 | 269,071 | 282,343 |
| 6 | 261,215 | 275,888 |
| 7 | 270,409 | 283,867 |
| 8 | 279,778 | 295,251 |
| 9 | 270,483 | 284,468 |
| 10 | 270,320 | 282,529 |
| 11 | 267,923 | 279,618 |
| 12 | 271,478 | 283,932 |
| 13 | 255,353 | 266,629 |
| 14 | 256,820 | 268,091 |
| 15 | 261,835 | 276,803 |
| 16 | 259,144 | 271,578 |
| 17 | 255,910 | 266,005 |
| 18 | 260,863 | 273,520 |
| 19 | 264,465 | 278,574 |
| 20 | 260,989 | 273,772 |