Controlling Non-Homogeneous Multistream Binomial Processes with a Chi-Squared Control Chart Peter Wludyka Associate Professor of Statistics Director of the Center for Research and Consulting in Statistics University of North Florida Jacksonville, Florida

An 4  8 Multistream Process

A general binomial model with structural components

Here only the column structure is included in the model. This model is non-homogeneous since the 8-streams DO NOT have the same rates at which nonconforming units arise.

A column model corresponds to an 8-Stream Process This is a homogeneous process (model) since the rates at which nonconforming units occur is the same for all eight streams

y 1 ~B(n,p 1 ) y 6 ~B(n,p 6 ) y 5 ~B(n,p 5 ) J-Stream Binomial process Periodically samples of size n are taken, and the number of nonconforming units y j is recorded for each stream — with the goal of controlling the process.

J-Stream Non-homogeneous Processes Control Schemes: J stream p-charts Overall p-chart Chi-Squared Chart Note: initially assume that the p j are known (or have been estimated).

J Stream p-Charts On each chart plot the stream sample proportion nonconforming on the chart with control limits where k j should be larger than 3 to ensure that the false alarm rate is not too high.

Recall: the average run length (ARL) for a control scheme is the average number of samples (epochs) to observe an out-of-control signal. Using J stream charts the

Overall p-Chart Plot the overall sample proportion nonconforming on a chart with control limits

Chi-Squared Chart Sample proportion nonconforming for each stream At each epoch calculate: Quantity plotted on the chart at each epoch

The in-control target ARL is used to determine the control limit (UCL) for the chart. One wants the smallest UCL such that: Asymptotic Distributions:

Determining the UCL (given the target in-control ARL) The quantiles of W, which are used to determine the UCL, can be found using the Chi-Squared distribution (equation (5)) Monte Carlo estimates of the empirical distribution of W.

Example A 4-stream process from which samples of size n = 100 are to be taken. The rates at which nonconforming units are produced are known to be. stream1234 pjpj Suppose the user wishes a target in-control ARL of 370. The “theoretical” UCL = since

The Empirical Distribution of W Based on 1,000,000 Monte Carlo sample values (SAS®) UCL = Note Granularity since W is discrete —» cannot achieve predetermined in-control ARL; that is, only a finite set of available in-control ARLs

Spreadsheet Summary for 10 Epochs of Surveillance

Three signals: samples 3, 6, and 9

Searching for an assignable cause 1.Process shift (across the board) 2.Shift in one stream 3.Shift in several streams Most likely under most circumstances & rational subgroups Useful is determining whether one has a:

Signal 1 Strong evidence that only stream 2 has shifted (not process): counts are close to expected counts in other streams Zsum is not very large max stream is 112% of total z Only one stream with big shift

Signal 2 Strong evidence that process has shifted (upward): several counts far from expected counts (above) Zsum is large and positive (—» upward shift) max stream only 30% of total Z Three streams with large shifts

Signal 3 Strong evidence that process has shifted (downward): several counts far from expected counts (below) Zsum is large and negative (—» downward shift) max stream only 32% of total Z Three streams with large shifts

Design and Implementation Issues Choosing the sample size Estimating stream proportions Homogeneous versus Non-homogeneous Models? Establishing initial Control

References: General Multistream Processes Variables processes: Mortell, R.M. and Runger, G.C., “Statistical process Control for Multiple Stream Processes,” JQT, Nelson, L.S., “Control Chart for Multiple Stream Processes,” JQT, Nelson, P.R. and Stephenson, P.L., “Runs Tests for Group Control Charts,” Communications in Statistics, Binomial, but only for comparing the streams Duncan, A.J., “A Chi-Squared Chart for Controlling a Set of Percentages,” Industrial Quality Control, 1950.

References: For homogeneous processes: control with 1.runs rules 2.group p-charts 3.overall p-charts Wludyka, P and Jacobs, S., “Runs Rules and P-Charts for Multistream Binomial Processes,” Communications in Statistics — Simulation and Computation, 2002, pp Wludyka, P and Jacobs, S., “Using SAS® to Control Multistream Binomial Processes,” Proceedings of the Joint Conference of the South East and South Central SAS® Users Groups, New Orleans, LA, 2001, pp

References For homogeneous processes: control with 1.Chi-squared chart 2.Chi-squared chart with runs rule Wludyka, P and Jacobs, S., “Controlling Homogeneous Multistream Binomial Processes with a Chi-Squared Chart,” Proceedings of Decision Sciences Institute, November 2002, San Diego California (submitted). Wludyka, P., Cavey, D., and Friedlin, B., “Using SAS® to Control Multistream Binomial Processes with a Chi-Squared Control Chart,” Proceedings of the the South East SAS® Users Groups, September 2002, Savannah, GA. (to appear).

Appendix A Smidgen of Monte Carlo Results Provided by two UNF graduate students: Dan Cavey Brett Friedlin Three streams (p =.1,.2,and.3) In control ARL is approximately 370 for three control streams: Overall p-chart J (independent) p-charts Chi-Squared chart

ARL Charts for n=100

ARL Charts for n=200