1 / 36

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

ilya
Download Presentation

Factorial Models

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related