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Sampling, WLS, and Mixed Models

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  1. Sampling, WLS, and Mixed Models II ESAMP Meetings Nov 6, 2009 Natal, Brazil Ed Stanek and Julio Singer U of Mass, Amherst, and U of Sao Paulo, Brazil SPH&HS, UMASS Amherst

  2. Finite Population Mixed Models Research Group Luz Mery Gonzalez, Columbia; Viviana Lencina, Argentina; Julio Singer, Brazil; Silvina San Martino, Argentina; Wenjun Li, US; and Ed Stanek US

  3. Background • Motivation: • 2-stage cluster sample of hospitals • n Hospitals • m Appendectomy operations per hospital • What is the average cost of an operation at a selected hospital (latent value)? • Choices: • Use average cost of m operations for selected hospital • Use ‘shrunk’ cost- regressing to the mean for other sample hospitals. • Which should we use?

  4. Consider/Account for: • Study Design • Sampling • Response Error • Model Assumptions How do we make up models to get better insight from limited information? An Example What is a subject’s saturated fat intake? SPH&HS, UMASS Amherst

  5. Seasons Study UMASS Worc SPH&HS, UMASS Amherst

  6. Seasons Study UMASS Worc- focus on 3 subjects SPH&HS, UMASS Amherst

  7. The Problem-Simplified • Observe: • 1 Measure of SFat on each Subject • Assume: • Response Error (RE) Variance known • Question: • How do we estimate Subject’s True Sat Fat intake? • Begin with a Response Error Model … which leads to…. • Mixed Model • Finite Population Mixed Model Daisy Lily Rose SPH&HS, UMASS Amherst

  8. Population SPH&HS, UMASS Amherst

  9. Set Population SPH&HS, UMASS Amherst

  10. Response 4 11 SPH&HS, UMASS Amherst

  11. Response 0 11 SPH&HS, UMASS Amherst

  12. Response 4 9 SPH&HS, UMASS Amherst

  13. Response 4 11 SPH&HS, UMASS Amherst

  14. Response 4 9 SPH&HS, UMASS Amherst

  15. Response 0 11 SPH&HS, UMASS Amherst

  16. Response…….. 0 11 4 SPH&HS, UMASS Amherst

  17. Response Error Model for Set SPH&HS, UMASS Amherst

  18. Summary Response Error Model Latent Value SPH&HS, UMASS Amherst

  19. Re-parameterized RE Model Mean Latent Value-of what?: or the Set the Population SPH&HS, UMASS Amherst 19

  20. Generating Response in the RE Model SPH&HS, UMASS Amherst

  21. Generating Response in the RE Model SPH&HS, UMASS Amherst

  22. Generating an Observed Response in the RE Model -1 1 -2 2 -2 -1 SPH&HS, UMASS Amherst

  23. 9 11 9 11 4 4 0 0 Sample Space • Response Error Model SPH&HS, UMASS Amherst

  24. 9 11 9 11 4 4 0 0 Response Error Model SPH&HS, UMASS Amherst

  25. Mixed Model (MM) Random Effect SPH&HS, UMASS Amherst

  26. Mixed Model (MM) Latent Value SPH&HS, UMASS Amherst

  27. Mixed Model (MM) in action SPH&HS, UMASS Amherst

  28. Mixed Model (MM) …. SPH&HS, UMASS Amherst

  29. Mixed Model (MM) ….?? Who Are They? ? ? SPH&HS, UMASS Amherst

  30. Mixed Model (MM) ?? ?? What Does it Mean? ? ? SPH&HS, UMASS Amherst

  31. 8 1 8 3 9 9 11 11 4 4 0 0 12 12 3 1 Sample Space (MM) Artificial Real SPH&HS, UMASS Amherst

  32. MM-Latent Values? Samples SPH&HS, UMASS Amherst

  33. What are they (for Daisy)? Samples SPH&HS, UMASS Amherst

  34. What are they (for Rose)? Samples SPH&HS, UMASS Amherst

  35. BLUPs of the MM-Latent Value SPH&HS, UMASS Amherst

  36. MSE of BLUPs for MM-Latent Values Samples Ave=0.986 Ave=3.768 SPH&HS, UMASS Amherst

  37. MSE of BLUPs |P=y Samples MSE=Ave=0.986 MSE=Ave=3.768 SPH&HS, UMASS Amherst

  38. Finite Population Mixed Model (FPMM) Population SPH&HS, UMASS Amherst

  39. Response Error Model Latent Value SPH&HS, UMASS Amherst

  40. Accounting for Sampling Indicator random variable, 1 if ith Selected sample subject is subject “s”

  41. Finite Population Mixed Model (FPMM) SPH&HS, UMASS Amherst

  42. Finite Population Mixed Model (FPMM) SPH&HS, UMASS Amherst

  43. Finite Population Mixed Model (FPMM) SPH&HS, UMASS Amherst

  44. Finite Population Mixed Model (FPMM) SPH&HS, UMASS Amherst

  45. 4 11 0 11 4 9 0 9 FPMM- Sample Space …

  46. FPMM- Sample Space … 11 9 9 11 4 0 0 4 SPH&HS, UMASS Amherst

  47. FPMM- Sample Space … 4 4 -7 13 0 0 13 -7 SPH&HS, UMASS Amherst

  48. FPMM- Sample Space … 13 13 0 4 -7 -7 4 0 SPH&HS, UMASS Amherst

  49. FPMM- Sample Space … 11 11 -7 13 9 9 13 -7 SPH&HS, UMASS Amherst

  50. FPMM- Sample Space … 13 13 9 11 -7 -7 11 9 SPH&HS, UMASS Amherst