We will use a Laney P' chart to correct the overdispersion in our data. The P chart in the previous example calculates the control limits on the binomial distribution. If the process being studied has a natural variation in the proportion that has a variation larger than the binomial distribution, then a P chart can show many more out of control points than it should. The P chart uses the variation between groups to estimate the position of the control limits.
Minitab has a P chart diagnostic tool to look for overdispersion or underdispersion in the data.
Initially, we will run the P chart diagnostic to check our data and then run the P' chart.
The following steps will check for overdispersion in the data and then generate a P' chart:
- Use Open Worksheet… from the File menu to open the
Calls Lost.mtwworksheet. - Navigate to Stat | Control Charts | Attributes Charts and select P Chart Diagnostic.
- Enter
Hung upin the Variables: field. - In Subgroup sizes: field, enter
Calls. - Click on OK.
- Check the results of the P chart diagnostic to see if there is evidence of overdispersion.
- Navigate to Stat | Control Charts | Attribute Charts and click on Laney P', as suggested from the diagnostic tool.
- Enter
Hung upin the Variables: field. - Enter
Callsin the Subgroups sizes field. - Click on P' Chart Options.
- Navigate to the tab for Tests and select Perform all tests for special causes.
- Click on OK in each dialog box.
The traditional P chart assumes that the variation exhibited in the process is all part of the within variation or rather the variation from a binomial distribution. In most cases, the variation in P over time and the variation between groups is usually smaller than the variation that is observed due to the within variation. If the subgroup sizes are very large, then the between variation can be a significant component of the variation in the study.
This results in overdispersion—a P chart where the control limits are too narrow. As the control limits are too narrow, they cause an elevated false alarm rate. The Laney P' charts account for the variation between groups and use it to plot the control limits.
The diagnostic charts for P and U check to see if the variation observed in the proportion or per-unit values is higher or lower than expected from a binomial or Poisson distribution. The diagnostic tool will then tell us if we need to use the P' chart instead of the traditional P chart.
The results here can be checked for comparison against the P chart.
The traditional approach to overdispersion is to use I-MR charts. The problem with individual charts is that while they plot the variation between groups, they do not see the subgroup size for the proportion. The control limits remain flat and do not vary with the subgroup size. The P' chart allows for variable control limits with subgroup size.
When subgroup sizes are constant, the P' chart will be the same as the I-MR chart. Also, P' charts will be the same as P charts when the data follows a binomial distribution.
For more on Laney control charts see the paper Improved Control Charts for Attributes, David B. Laney.