Edward J. Stanek III Department of Public Health University of Massachusetts Amherst, MA

1. Finite Population Inference for Latent Values Measured with Error that Partially Account for Identifable Subjects froma Bayesian Perspective Edward J. Stanek III Department of Public Health University of Massachusetts Amherst, MA

2. Collaborators Parimal Mukhopadhyay, Indian Statistics Institute, Kolkata, India Viviana Lencina, Facultad de Ciencias Economicas, Universidad Nacional de Tucumán, CONICET, Argentina Luz Mery Gonzalez, Departamentao de Estadística, Universidad Nacional de Colombia, Bogotá, Colombia Julio Singer, Departamento de Estatística, Universidade de São Paulo, Brazil Wenjun Li, Department of Behavioral Medicine, UMASS Medical School, Worcester, MA Rongheng Li, Shuli Yu, Guoshu Yuan, Ruitao Zhang, Faculty and Students in the Biostatistics Program, UMASS, Amherst

3. Review of Finite Population Bayesian Models • Populations, Prior, and Posterior • Notation • Exchangeable distribution • Sample Space and Data • Posterior Distribution, given data Outline

4. Review Bayesian ModelGeneral Idea Posterior Prior Data Populations Populations # Posterior Populations: # Prior Populations: Posterior Probabilities Prior Probabilities

5. Review Bayesian ModelPopulation and Data Notation Populations Data Vector Vector Labels Response Parameter Expected response Measurement Variance Measurement Error Variance Latent Value Label Measurement Error Variance

6. Review Exchangeable Prior PopulationsGeneral Idea When N=3 Joint Probability Density Each Permutation p of subjects in L (i.e. each different listing) Assigns (usually) equal probability to each permutation of subjects in the population. Must be identical Exchangeable Random Variables General Notation The common distribution

7. Review Exchangeable Prior Populations N=3Potential Response for Each Listing of subjects Latent Values for permutations of listing Latent Values for Listing Listings

8. Review Listing p=1 Exchangeable Prior PopulationPermutations Lily Rose Daisy

9. Review Listing p=1 Exchangeable Prior Populations N=3Permutations Lily Rose Daisy

10. Review Listing p=5 Listing p=3 Listing p=6 Listing p=4 Listing p=2 Same Point in Listing Exchangeable Prior Populations N=3Permutations of Listings Listing p=1

11. Prior Sets Vectors Prior Measurement Error ModelPrior Random Variables Population h, # Prior Populations: Assume Random Variables representing a population are exchangeable When p=1, define

12. Data Prior Prior Random Variables and Datawith Measurement Error Prior Random Variables that will correspond to Latent values for subjects In the data Prior Random Variables Remaining Prior Random Variables

13. Review Prior Bayesian Model Exchangeable Prior Populations N=3for h when Sample Space n=2 Listing p=1 10 5 2 2 5 10

14. Review Listing p=5 Listing p=2 Listing p=3 Listing p=1 Listing p=6 Listing p=4 10 10 10 10 10 10 5 5 5 5 5 5 2 2 2 2 2 2 2 5 10 2 2 2 2 2 5 5 5 5 5 10 10 10 10 10 Bayesian Model Exchangeable Prior Populations N=3: Sample Point n=2

15. Review Listing p=1 Exchangeable Prior Populations N=3Permutations Lily Rose Daisy

16. Review Prior Bayesian Model Exchangeable Prior Populations N=3for h when Listing p=1 Listing p=1 Sample Space n=2 when 10 5 2 2 5 10

17. Review Listing p=5 Listing p=2 Listing p=3 Listing p=1 Listing p=6 Listing p=4 10 10 10 10 10 10 5 5 5 5 5 5 2 2 2 2 2 2 2 5 10 2 2 2 2 2 5 5 5 5 5 10 10 10 10 10 Positive Prob. Exchangeable Prior Populations N=3: Sample Points n=2

18. Review Axis Data n=2 Axis 10 5 2 2 5 10

19. Review Axis Data n=2 Axis 10 Daisy 5 2 Rose 2 5 10

20. Review Axis Data n=2 Axis 10 Rose 5 2 2 5 10 Daisy

21. Axis Data n=2Adding Measurement Errorto Rose Axis 10 Daisy 5 2 Rose 2 5 10

22. Review Exchangeable Prior Populations N=3 Sample Points with Positive Probability n=2 Listing p=5 Listing p=3 Listing p=1 10 10 10 5 5 5 2 2 2 2 5 10 2 5 10 2 5 10 Listing p=4 Listing p=2 Listing p=6 10 10 10 5 5 5 2 2 2 2 5 10 2 5 10 2 5 10

23. Review Exchangeable Prior Populations N=3Posterior Random Variables when Data Prior If permutations of subjects in listing p are equally likely: The Expected Value of random variables for the data is the mean for the data. Random variables representing the data are independent of the remaining random variables.

24. Review Populations Data # Prior Populations: Prior If permutations of subjects in listing p are equally likely: Posterior Random Variables no Measurement Error

25. Review Populations Data # Prior Populations: Prior If permutations of subjects in listing p are equally likely: Posterior Random Variables no Measurement Error nxn random permutation matrix Data

26. Review Populations Data # Prior Populations: Prior If permutations of subjects in listing p are equally likely: Posterior Random Variables no Measurement Error

27. Review Populations Data # Prior Populations: Prior If permutations of subjects in listing p are equally likely: Posterior Random Variables no Measurement Error

28. Let to be an Data (set of vectors) permutation matrix, k=1,…,n! and Data (set) Data without Measurement Error Vectors Latent Value

29. No Measurement Error With Measurement Error Data Data with and without Measurement Error Data Sets Response at t Potential Response Latent Value Data Data Vectors Potential Response

30. the realization of The realization of on occasion t the latent value Sets Data Data with Measurement Error Assume: Measurement errors are independent when repeatedly measured on a subject

31. Data Vectors Measurement Error ModelThe Data Latent Values Potential response with error Define

32. Prior Populations Population h Measurement Error ModelPrior Random Variables Vector Labels # Prior Populations: Parameter Assume Random Variables representing a population are exchangeable Defines the axes for a cloud of points in the prior When p=1, define

33. Daisy Rose Lily Exchangeable Prior Populations N=3 No Measurement Error Single Point

34. Daisy Rose Lily Exchangeable Prior Populations N=3 with Measurement Error Cloud of Points

35. Prior Prior Sets Vectors Measurement Error ModelPrior Random Variables Population h, Population # Prior Populations: Vectors of Labels Latent Values Potential Response Assume Random Variables representing a population are exchangeable Defines the axes for a cloud of points in the prior When p=1, define , and

36. Response for subject Prior Population Data Prior Random Variables and Datawith Measurement Error Since If permutations of subjects in listing p are equally likely: Assume Random Variables representing a population are exchangeable in each population or

37. Data Points in the Prior that match the data Prior If permutations of subjects in listing p are equally likely: Posterior Random Variables with Measurement Error where

38. Data Points in the Prior that match the data Prior Posterior Random Variables with Measurement Error Finite Population Mixed Model for the subjects in the Data: Use this model to obtain the best linear unbiased predictor of the latent value for a subject in the data (which we call the BLUP for a realized subject) where random subject effect

39. Prior for listing p=1 Capturing Partial Label Information in the Posterior Distribution Data for points where Usual Posterior We want to use the label information for the response error, but not use it for the mean. In the posterior, we want to replace by where

40. Prior Capturing Partial Label Information in the Posterior Distribution In the posterior, we can list the subjects that are in the data in an order, say Data Usual Posterior =realized order for posterior Now We want to define for all k=1,…,n! such that Measurement Error variance will be block diagonal, in the realized order for the posterior.

41. ??? Prior Partially Labeled Posterior Distribution Data Latent values for response in the posterior are in random order Partially Labeled Posterior =realized order for posterior for realized response Measurement error variance is heterogenous and matches the order of the subjects in the posterior.

42. How do we define a prior distribution that will result in such a posterior distribution? ??? Prior Partially Labeled Posterior Distribution Data Latent values for response in the posterior are in random order Partially Labeled Posterior =realized order for posterior for realized response Measurement error variance is heterogenous and matches the order of the subjects in the posterior.

43. Capturing Partial Label Information in the Posterior Distribution Prior Data Usual Posterior for points where To form a partially labeled posterior, we need to be equal to for all k=1,…,n!

44. Capturing Partial Label Information in the Posterior Distribution How do we define a prior distribution that will result in such a posterior distribution? Prior ??? Data Partially Labeled Posterior for points where since

45. Partially Labeled Prior Capturing Partial Label Information in the Posterior Distribution Data Define =realized order for posterior for realized response Partially Labeled Posterior for points where

46. =realized order for posterior for realized response Capturing Partial Label Information in the Posterior Distribution Partially Labeled Prior Data Partially Labeled Posterior Measurement error variance is heterogenous and matches the order of the subjects in the posterior.

47. Population Data Prior An ExamplePosterior Random Variables with Measurement Error Use this model to obtain the best linear unbiased predictor of the latent value for a subject in the data (which we call the BLUP for a realized subject) where random subject effect

48. An Exchangeable Prior N=3No Measurement Error Permutation p*=1 p*=2 p*=3 p*=4 p*=5 p*=6 L L R D D Rose Lily Daisy R R D D R L D R L L R L D R R L L D D Lily Rose Daisy L D R D L R D L D R R L L L R R D D Daisy Lily Rose D R L R D L R D R L L D Prior For Population h p=1 Listings p=2 p=6=N!