1 Finite Population Inference for Latent Values Measured with Error that Partially Account for Identifable Subjects from a 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 Outline Review of Finite Population Bayesian Models 1.Populations, Prior, and Posterior 2.Notation 3.Exchangeable distribution 4.Sample Space and Data 5.Posterior Distribution, given data

4 Bayesian Model General Idea Populations # Posterior Populations: Data Prior Posterior # Prior Populations: Prior Probabilities Posterior Probabilities Review

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

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

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

8 Exchangeable Prior Population Permutations Rose Daisy Lily Listing p=1 Review

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

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

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

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

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

14 Bayesian Model Exchangeable Prior Populations N=3: Sample Point n= Listing p= Listing p= Listing p=1 Listing p= Listing p= Listing p=6 Review

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

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

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

18 Data n= Review Axis

19 Data n= Review Axis Rose Daisy

20 Data n= Review Axis Rose Daisy

21 Data n=2 Adding Measurement Error to Rose Axis Rose Daisy

22 Exchangeable Prior Populations N=3 Sample Points with Positive Probability n= Listing p= Listing p= Listing p=1 Listing p= Listing p= Listing p=6 Review

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

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

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

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

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

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

29 Data with and without Measurement Error No Measurement Error Latent Value Data With Measurement Error Vectors Sets Data Response at tPotential Response

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

31 Measurement Error Model The Data Vectors Define Latent ValuesPotential response with error Data

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

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

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

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

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

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

38 Posterior Random Variables with Measurement Error Data Prior Points in the Prior that match the data Finite Population Mixed Model for the subjects in the Data: where random subject effect 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)

39 Capturing Partial Label Information in the Posterior Distribution Usual Posterior Prior Data for listing p=1 for points where where 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

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

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

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

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

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

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

47 An Example Posterior Random Variables with Measurement Error where random subject effect 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) Prior Population Data

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