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Evaluating the Effectiveness of BLUPs in Estimating Clustered Latent Values

This study investigates the limitations of Best Linear Unbiased Predictors (BLUPs) in estimating latent values from clustered data, focusing on scenarios such as bullying rates in schools and health metrics in clinics. We explore various models including fixed-effect and mixed models, and compare their predictive accuracy through simulation studies. Results indicate that while BLUPs are theoretically optimal, they may yield higher Mean Squared Error (MSE) under certain conditions, particularly with small sample sizes. Future research is suggested to refine predictive techniques.

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Evaluating the Effectiveness of BLUPs in Estimating Clustered Latent Values

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  1. When are BLUPs Bad Ed Stanek UMass- Amherst Julio Singer USP- Brazil George Reed UMass- Worc. Wenjun Li UMass- Worc.

  2. Outline Description of the Problem Examples Basic Models & Ideas Approaches Results Conclusions/Further Research

  3. Problem: Estimate/Predict Cluster Latent Value Example: 11 Middle Schools 8-16 Classrooms/School Question: How much bullying occurs at school? Example: 166 School Districts 6-18 Schools/District Question: Does the District have a sun-protection policy for students? Example: 25,000 Members in an HMO 30 Days/Member before Cholesterol measure Question: What is a Patient’s Ave Sat fat intake (g)?

  4. Description of the Problem Clustered Finite Population (cluster=group of subjects) Schools, Clinics, Hospitals, Cities, Neighborhoods, Physician Practices, Families, Litters, States, Studies… Data is available only on some clusters Observational studies, multi-stage samples, group-randomized trials For selected clusters, response is observed on a subset of subjects (possibly with response error). How to Best Estimate/Predict the Cluster’s Latent Value?

  5. Basic Models & Ideas Simple Response Error Model for a Subject: School: Student: where

  6. Combining terms, the Response Error Model: becomes where ith school effect jth student effect kth response error

  7. Model Properties Over Schools Over Students The Latent Value of the School in the ith position:

  8. Approaches to Estimate/Predict Latent Value 1. Fixed Models 2. Mixed Models 3. Super-population Models 4. Random Permutation Models

  9. 1. Fixed Effect Models: (Random Sample of Students) Latent Cluster Mean: (i.e. School “s”) OLS Estimate:

  10. 2. Mixed Models To predict Solution: BLUP= where

  11. 3. Super-population Models Super-population Finite Population (Realization) Latent Value Latent Value

  12. Breaking Up the Latent Value Parameter where

  13. Superpopulation Model Prediction Approach (Scott and Smith, 1969) Sample: Remainder: BLUP:

  14. 4. Random Permutation Model Predictors Finite Population: 2-Stage Permutation: To Predict : BLUP where

  15. Approaches 1. Fixed: 2. Mixed: 3. Superpop. 4. RP Model

  16. If variance components are known, then the RP Model Predictor is “best”- (i.e. has small expected MSE) How much better depends on: Sampling fraction: Cluster-intra-class correlation: Unit- Reliability:

  17. Figure 1a. Percent Increase in Expected MSE for Mixed Model (──) and Scott and Smith Model (- - - ) Predictors Relative to the Random Permutation Model Predictors of the Latent Value of a Realized Sample PSU

  18. Figure 1b. Percent Increase in Expected MSE for Mixed Model (──) and Scott and Smith Model (- - - ) Predictors Relative to the Random Permutation Model Predictors of the Latent Value of a Realized Sample PSU

  19. Figure 1c. Percent Increase in Expected MSE for Mixed Model (──) and Scott and Smith Model (- - - ) Predictors Relative to the Random Permutation Model Predictors of the Latent Value of a Realized Sample PSU

  20. What happens when variances must be estimated? Are BLUPs still best? Not always.

  21. Simulation Study: Two-stage finite population sampling Settings: N, M, n, m Variance Components: Distribution of Clusters: normal, uniform, beta, gamma Unit Variance Prop Cluster Mean: yes/no

  22. Simulation Results (No response Error) N=3 N=5 N=10 N=25 N=50 N=100 N=500 N=1000

  23. Simulation Results (with Response Error) N=100, n=10, M=25, m=10 Clusters Units Normal Normal Gamma(2) Normal

  24. Summary and Conclusions eBLUPs can have higher MSE than the Cluster Ave Small #s of sample clusters (n<5) RP predictors (while theoretically best), may be worse than MM empirical predictors, or Super-population predictors. Properties don’t seem to depend on distributions. Simulations are limited- fall short of providing guidelines

  25. Thanks

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