Using PROC MCMC in a Process Control Setting

Using PROC MCMC in a Process Control Setting: An Illustrative Example with PROC IML Austin Brown, M. S. , M. M. P. University of Northern Colorado

Table of Contents • Introduction • PROC MCMC • Process Control Example • Conclusion

Introduction • Bayesian Estimation Basic Idea • In practical settings, we usually have an idea about the behavior of whatever mean or variance we’re trying to estimate • For example, if a highly selective university was interested in estimating mean SAT scores for incoming freshman, they know that the scores are not going to fall below a certain threshold • So why not use that knowledge in the estimation procedure instead of ignoring it?

Introduction • Bayesian Estimation Basic Idea • Based on the classical Bayes’ Rule from probability, Bayesian Estimation treats the parameter being estimated as a random variable, with its own distribution separate from the likelihood • While classical frequentist estimation also treats parameters as random variables (like in Maximum Likelihood Estimation), where it differs is that Bayesian Estimation conditions the parameter upon the likelihood of whatever random variable we’re observing (SAT scores in the prior example)

Introduction

Introduction • Bayesian Estimation Basic Idea • While nice, clean, closed forms of the posteriors are good, there are many, many cases in practice where closed forms are impossible or at least extremely difficult to find • In such cases, computer simulation techniques are required in order to obtain approximate posterior distributions and estimates

Introduction • SAS provides several procedures with Bayesian estimation built in as part of another analysis • GENMOD, LIFEREG, PHREG, FMM • However, sometimes we may only be interested in obtaining an estimate and an approximate posterior, such as in a process control setting • In such cases, PROC MCMC offers great flexibility and functionality to obtain estimates, posterior estimates (via the coda), as well as a variety of graphs and tests to evaluate the validity of the output

PROC MCMC: An Example

PROC MCMC: An Example

PROC MCMC: An Example

PROC MCMC: An Example • Okay, we’ve got the program to run… How do I know if this is any good? ?

PROC MCMC: An Example • PROC MCMC provides a variety of graphical and test-based assessments to help users determine if the posterior estimates can be reasonably relied upon • Since the purpose of this presentation is illustrative rather than a deep technical explanation, we’ll stick with looking at graphs for now • The SAS documentation for PROC MCMC is very rich with explanation for all tests and graphics provided in the output

PROC MCMC: An Example

Process Control Example • In a process/quality control setting, prior knowledge of and experience with whatever process or phenomenon being observed is commonplace • Additionally, small sample sizes are also commonplace, and consequently, asymptotic results stemming from frequentist estimation may not hold • Bayesian Estimation is a natural solution • Let’s look at an example of how to build and test a process control chart using IML functionality…

Process Control Example

Process Control Example

Process Control Example

Process Control Example

Questions? ?

Contact Information Name: Austin Brown Institution: University of Northern Colorado City/State: Greeley, CO Phone: 970 -380 -7935 Email: austin. brown@unco. edu