Laney P’ and U’ Charts

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Have you ever created a P or U Control chart with large subgroup sizes? Were your limits so narrow that an unexpected number of points were flagged as beyond the control limits? P and U charts are based on assumptions such as the underlying distribution (binomial and Poisson) and the average of the distribution being the same over time. These assumptions do not always hold. Your data may show overdispersion (more variation than expected given the distribution) or underdispersion (less variation than expected given the distribution). 

Example

Consider the following simulated example:

dt=New Table( "Defectives",
	Add Rows( 20 ),
	New Column( "# of Defectives",
		Numeric,
		"Continuous",
		Format( "Best", 12 ),
		Set Values(
		[3023, 3014, 2987, 2956, 2963, 2985, 3001, 2997, 2935, 3056, 3005, 2985,
		 3029, 3027, 3001, 3082, 2954, 2971, 3061, 2999]
		)
	),
	New Column( "Subgroup Size",
		Numeric,
		"Continuous",
		Format( "Best", 12 ),
		Set Selected,
		Set Values(
		[4077, 4072, 5142, 5433, 5053, 5216, 4801, 4579, 4789, 4751, 5045, 4617,
		4760, 5644, 6336, 5144, 5450, 5146, 4929, 5705]
		)
	)
);
obj=dt<<Control Chart Builder(
	Show Two Shewhart Charts( 0 ),
	Class( Shewhart Attribute ),
	Variables( Y( :"# of Defectives"n ), n Trials( :Subgroup Size ) ),
	Chart( Points( Statistic( "Proportion" ) ), Limits( Sigma( "Binomial" ) ) )
);

Notice that 14 out of 20 points fall beyond the control limits.  Also notice that the subgroup sizes are large (4072 -6336). For this example, the within subgroup variation is much less than the between subgroup variation.

This problem can be solved with a Laney P’ chart. Laney P’ and U’ charts were added in JMP 17. Let’s look at the formulas used to create the limits of both the P and the Laney P’ charts.

The only difference in these formulas is the σ_z, which is a measure of overdispersion. Laney described this term as “…the relative amount of process variation not explained by the binomial assumption alone. As n increases, the variation due to sampling diminishes, thus making the batch-to-batch components relatively larger.” σ_z is based on standardized values, which have a mean of 0 and variance of 1, and therefore, σ_z approaches 1. Any discernable difference in σ_z from 1 indicates the Laney (P’) is recommended over the P chart.

You can create this type of chart in Control Chart Builder by altering the Sigma drop down box from Binomial (P, NP) to Laney (P’).

With the Laney P’ chart, we find that no points fall beyond the limits for this example.

Conclusion

Laney P’ and U’ control charts were added to Control Chart Builder in JMP 17. These charts are useful when subgroup sizes are large, which causes narrow limits in the traditional P and U charts. Laney charts do this by adding a term to the limit formulas that account for the amount of variation that is not explained by the model assumptions.