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

2. One-way Random Effects
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
unbalanced designs
Smith-Satterthwaite approximations
Restricted ML estimates
We will defer the last two topics to general random and mixed effects models

4. One-way Random Effects

5. One-way Random Effects

6. One-way Random Effects E(MSTR)

7. One-way Random Effects E(MSTR)

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

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

10. One-way Testing

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

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

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

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

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

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

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

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

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

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

21. Approximate F test
Source EMS

22. Approximate F test
Source EMS

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

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

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

26. DF for Approximate F tests
Restating the result:

27. DF for Approximate F tests
The following argument builds approximate c2 distributions for the numerator and denominator mean squares (and assumes they are independent)
We will review the argument for the numerator
The argument computes the variance of the mean square two different ways

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

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

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

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

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

33. Two-way Mixed Effects Model

34. Two-way Mixed Effects Model
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.
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.

35. Two-way Mixed Effects Model

36. Two-way Mixed Effects Model
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)

37. 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
SAS and Minitab analysis