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Factorial Models

Factorial Models. Random Effects Gauge R&R studies (Repeatability and Reproducibility) have been an expanding area of application Mixed Effects Models. One-way Random Effects. The one-way random effects model is quite different from the one-way fixed effects model

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Factorial Models

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  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 EffectsE(MSTR)

  7. One-way Random EffectsE(MSTR)

  8. One-way Random EffectsTesting • 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 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

  11. 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

  12. Gauge R&RTwo-way Random Effects Model • P-Part • O-Operator R R

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

  14. Gauge R&R • Repeatability 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)

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

  17. 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

  18. Two-way Random Effects Model • Consider results from our expected mean squares. • 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. • 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 • We construct an approximate F test,

  23. Approximate F test • We require E(MS’)=E(MS”) under Ho • 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 • Returning to our example, how do we test

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

  26. DF for Approximate F tests • The following argument builds approximate c2distributions 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

  27. 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

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

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

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

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

  32. Two-way Mixed Effects Model

  33. 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 • 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.

  34. Two-way Mixed Effects Model

  35. 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)

  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 • SAS and Minitab analysis

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