1. Factorial Models Random Effects Random Effects Gauge R&R studies (Repeatability and Reproducibility) have been an expanding area of application Gauge R&R studies (Repeatability and Reproducibility) have been an expanding area of application Mixed Effects Models Mixed Effects Models

2. One-way Random Effects The one-way random effects model is quite different from the one-way fixed effects model The one-way random effects model is quite different from the one-way fixed effects model –Yandell has a real appreciation for this difference –We should be surprised that the analytical approaches to the main hypotheses for these models are so similar

3. One-way Random Effects In Chapter 19, Yandell considers In Chapter 19, Yandell considers –unbalanced designs –Smith-Satterthwaite approximations –Restricted ML estimates We will defer the last two topics to general random and mixed effects models We will defer the last two topics to general random and mixed effects models

4. One-way Random Effects

6. One-way Random Effects E(MSTR)

8. One-way Random Effects Testing By a similar argument, we can show E(MSE)=  2 By a similar argument, we can show E(MSE)=  2 The familiar F-test statistic for testing The familiar F-test statistic for testing

9. One-way Testing Under the true model, Under the true model, So power analysis for balanced one-way random effects can be studied using a central F-distribution So power analysis for balanced one-way random effects can be studied using a central F-distribution

10. One-way Random Effects Method of Moment point estimates for  2 and   2 are available Method of Moment point estimates for  2 and   2 are available Confidence intervals for  2 and   2 /  2 are available Confidence intervals for  2 and   2 /  2 are available A confidence interval for the grand mean  is available A confidence interval for the grand mean  is available

11. Two-way Random Effects Model We will concentrate on a particular application—the Gauge R&R model We will concentrate on a particular application—the Gauge R&R model 20.2 addresses unbalanced models 20.2 addresses unbalanced models –Material is accessible Topics in 20.3 will be addressed later Topics in 20.3 will be addressed later 20.4 and 20.5 can safely be skipped 20.4 and 20.5 can safely be skipped

12. Gauge R&R Two-way Random Effects Model P-Part P-Part O-Operator O-Operator RR

13. Gauge R&R With multiple random components, Gauge R&R studies use variance components methodology With multiple random components, Gauge R&R studies use variance components methodology

14. Gauge R&R Repeatability is measured by Repeatability is measured by Reproducibility is measured by Reproducibility is measured by

15. Gauge R&R Unbiased estimates of the variance components are readily estimated from Expected Mean Squares (a=# parts, b=# operators, n=# reps) Unbiased estimates of the variance components are readily estimated from Expected Mean Squares (a=# parts, b=# operators, n=# reps)

16. Gauge R&R Use Mean Sums of Squares for estimation Use Mean Sums of Squares for estimation

17. Gauge R&R Minitab has a Gauge R&R module Minitab has a Gauge R&R module –Output is specific to industrial methods Consider an example with 3 operators, 5 parts and 2 replications Consider an example with 3 operators, 5 parts and 2 replications

18. Two-way Random Effects Model Consider results from our expected mean squares. Consider results from our expected mean squares. What would be appropriate tests for A, B, and AB? What would be appropriate tests for A, B, and AB?

19. Approximate F tests Statistics packages may do this without your being aware of it. Statistics packages may do this without your being aware of it. Example Example –A, B and C random –Replication

20. Approximate F test SourceEMS

21. Approximate F test SourceEMS

22. Approximate F test No exact test of A, B, or C exists No exact test of A, B, or C exists We construct an approximate F test, We construct an approximate F test,

23. Approximate F test We require E(MS’)=E(MS”) under H o We require E(MS’)=E(MS”) under H o F has an approximate F distribution, with parameters F has an approximate F distribution, with parameters

24. Approximate F test Note that MS’, MS’’ can be linear combinations of the mean squares and not just sums Note that MS’, MS’’ can be linear combinations of the mean squares and not just sums Returning to our example, how do we test Returning to our example, how do we test

25. DF for Approximate F tests Restating the result: Restating the result:

26. DF for Approximate F tests The following argument builds approximate   distributions for the numerator and denominator mean squares (and assumes they are independent) The following argument builds approximate   distributions for the numerator and denominator mean squares (and assumes they are independent) We will review the argument for the numerator We will review the argument for the numerator The argument computes the variance of the mean square two different ways The argument computes the variance of the mean square two different ways

27. DF for Approximate F tests Remember that the numerator for an F random variable has the form: Remember that the numerator for an F random variable has the form: Note that we already have this result for the constituent MS i Note that we already have this result for the constituent MS i

28. DF for Approximate F tests For each term in the sum, we have For each term in the sum, we have

29. DF for Approximate F tests We can derive the variance by another method: We can derive the variance by another method:

30. DF for Approximate F Tests Equating our two expressions for the variance, we obtain: Equating our two expressions for the variance, we obtain:

31. DF for Approximate F Tests Replacing expectations by their observed counterparts completes the derivation. Replacing expectations by their observed counterparts completes the derivation.

32. Two-way Mixed Effects Model

33. Both forms assume random effects and error terms are uncorrelated Both forms assume random effects and error terms are uncorrelated Most researchers favor the restricted model conceptually; Yandell finds it outdated. It is certainly difficult to generalize. Most researchers favor the restricted model conceptually; Yandell finds it outdated. It is certainly difficult to generalize. SAS tests the unrestricted model using the RANDOM statement with the TEST option; the restricted model has to be constructed “by hand”. SAS tests the unrestricted model using the RANDOM statement with the TEST option; the restricted model has to be constructed “by hand”. Minitab tests unrestricted model in GLM, restricted model option in Balanced ANOVA. Minitab tests unrestricted model in GLM, restricted model option in Balanced ANOVA.

34. Two-way Mixed Effects Model

35. The EMS suggests that the fixed effect (A) is tested against the two-way effect (AB) for both forms (F=MSA/MSAB) The EMS suggests that the fixed effect (A) is tested against the two-way effect (AB) for both forms (F=MSA/MSAB) The EMS suggests that the random effect (B) is tested against error (F=MSB/MSE) for the restricted model, but tested against the two-way effect (AB) for the unrestricted model (F=MSB/MSAB) The EMS suggests that the random effect (B) is tested against error (F=MSB/MSE) for the restricted model, but tested against the two-way effect (AB) for the unrestricted model (F=MSB/MSAB)

36. Two-way Mixed Effects Model For the Gage R&R study, assume that Part is still a random effect, but that Operator is a fixed effect For the Gage R&R study, assume that Part is still a random effect, but that Operator is a fixed effect SAS and Minitab analysis SAS and Minitab analysis