Chapter 14 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 outside the limits.
Figure 14.1 demonstrates this phenomenon, known as overdispersion, using data from from Mohammed A. Mohammed et al. (2013) (example data are tabulated at the end of this chapter).
Figure 14.1: P chart of percent emergency attendances seen within 4 hours
This can suggest that the process is highly unstable, even when such a conclusion may not be warranted. 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.
To diagnose this issue, we can plot the data on an I chart, which bases its control limits on the natural variation between successive subgroups rather than on theoretical distributional assumptions. Figure 14.2 is an I chart of the data from Figure 14.1 suggesting only common cause variation.
Figure 14.2: 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 significantly from those of the P or U chart, this suggests that the observed variation is inconsistent with the assumed binomial or Poisson models underlying the P and U charts, respectively.
Several solution to the problem with overdispersion and tight control limits have been proposed. The obvious pragmatic solution is to simply use I charts for count data as well as for (individual) measurements. However, I charts have straight control limits that do not account for varying subgroup sizes. If the subgroup sizes only vary little, this may not be a problem. But sometimes – not the least in healthcare – varying subgroup sizes matter.
Laney proposed a solution that adjust the within subgroup variation in count data (\(\sigma_{pi}\)) by a factor based on the moving ranges of successive standardised data points (\(\sigma_z\)) (Laney 2002).
14.1 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) accounts for both within and between subgroup variation. The control limits are given by:
\[\bar{p}\pm3\sigma_{pi}\sigma_z\]
where \(\sigma_z\) represents the between-subgroup variation, calculated using the moving ranges of the standardised proportions defined as:
\[z_i=(p_i - \bar{p})/\sigma_{pi}\]
The moving ranges are:
\[MR_i=|z_i-z_{i-1}|\]
Then, the estimated 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 are:
\[\bar{p}\pm3\sigma_{pi}\sigma_z \]
Figure 14.3: P’ chart of percent emergency attendances seen within 4 hours
The control limits in Figure 14.3 are very close to the I chart limits from Figure 14.3, but vary slightly due to varying subgroup sizes.
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 \(z_i=(u_i-\bar{u})/\sigma_{ui}\).
Note that in qicharts2, the moving ranges of the standardized values are, by default, screened for extreme values – following the same approach used for I charts, as described in Chapter 12.
14.2 When to use prime charts
Laney’s prime charts were developed to handle overdispersion (and underdispersion), that is, when data show more (or less) variation than expected under a binomial or Poisson distribution, which is often seen with very large subgroup sizes. But how do we know when to suspect overdispersion?
A red flag 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 significantly wider control limits, it is a strong indication that the assumptions behind the P or U chart may not hold. In such cases, switching to Laney’s P’ or U’ charts is typically the better choice.
Note that when the adjustment factor (\(\sigma_z\)) is close to one – indicating little to no overdispersion – the Laney P′ or U′ chart closely resembles the traditional P or U chart. For this reason, Laney recommends using prime charts as a generally safe and robust default.
In the next chapter, we will explore a modified I chart – the I’ (I prime) chart – which generates control limits for count data that closely align with 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 |