Estimating Realized Random Effects in Mixed Models

Transcription

1 Etimating Realized Random Effect in Mixed Model (Can parameter for realized random effect be etimated in mixed model?) Edward J. Stanek III Dept of Biotatitic and Epidemiology, UMASS, Amhert, MA USA Julio Singer Department of Statitic, U. of São Paulo, Brazil Preented at the EAR Biometric Meeting March 6, 00 Charlotte, C. c0ed.doc 3//00 7:0 AM

2 Outline Example- The Mixed Model Repreenting Random Effect The Population: Recovering Random Effect -Stage Random Permutation Model Etimation: Model-baed Survey Sampling Interpretation Further Reearch/ Summary c0ed.doc 3//00 7:0 AM

3 Example Clutered Randomized Trial (Diet Intervention)! Phyician practice (cluter)! How effective are MD Two Stage Cluter Sample of State School! School (cluter)! Math tet core c0ed.doc 3//00 7:0 AM 3

4 Mixed Model Sample n of cluter; m of j M ubject j = µ +Β + E jk j jk B j = the effect of the ( µ + B ) j = expected value of the th j elected cluter th j elected cluter c0ed.doc 3//00 7:0 AM 4

5 Randomly Selected Cluter Mean: ( ) µ + B = U µ + U µ +! + U µ j j j j = U = j µ where =,,!, cluter in the Population, U = if the j elected cluter i cluter " " j = th 0 otherwie. c0ed.doc 3//00 7:0 AM 5

6 Expanded Random Effect U µ j j U µ j j + = = j " " U µ ( µ B ) j j c0ed.doc 3//00 7:0 AM 6

7 Taking the Expected Value: ( ξ over cluter; ξ over ubject in a cluter) E ξξ µ j µ j = " " µ j c0ed.doc 3//00 7:0 AM 7

8 ( µ µ ) U + E µ j jk jk µ U ( µ µ ) + E jk= X + j jk j " " " jk µ U ( µ µ ) + E j jk ( ) or ( ) * = µ + * j j j X E where X = j c0ed.doc 3//00 7:0 AM 8

9 Expanded Model: (( µ )) X E E ( *) = X ( µ ) ξξ * = + * j j j Uual Model: ( ) j j = µ +Β + E E ( ) = µ ξξ jk j jk jk Can Realized Random Effect be Etimated in the Uual Model? c0ed.doc 3//00 7:0 AM 9

10 c0ed.doc 3//00 7:0 AM 0 The Population: Recovering Random effect Full Model for Population : M µ = + + X ZB E where M M M M = " and j j j M jm = ".

11 c0ed.doc 3//00 7:0 AM A Linear Combination a permuation M M U U M U µ µ µ µ µ µ = = = +Β +Β = = +Β I " "

12 Concluion: A linear combination of Random Variable in the Population i a Permutation of the Cluter Parameter When we realize part of the permutation, we know which cluter correpond to the realized portion. A realized random effect i the mean for the realized cluter. c0ed.doc 3//00 7:0 AM

13 -Stage Random Permutation Model Ue RV to repreent the POPULATIO: Randomly permute PSU (Cluter) Randomly permute SSU in PSU (Subject) c0ed.doc 3//00 7:0 AM 3

14 Ue eyman/cochran Indicator RV for Permutation: M M ( ) U U µ = j kt t = t= ( ) = M = U I U ( ) where U = ( U ) j ( ), and (( µ )) t (( )) kt ( ) ( ) U = U. M M c0ed.doc 3//00 7:0 AM 4

15 Summary: = Xµ + ZB+ E M ( ) = X + ZB and ( ) E ξ ξ µ E = X µ ξξ ( ) var ξξ = σ I J J M + σ I e I M J M M where σ e = σ =. c0ed.doc 3//00 7:0 AM 5

16 Etimation: Model-baed Survey Sampling Re-arrange Pop. into Sample and Remainder Specify Function (linear) of RV of Interet Oberved Sampled part; Predict remainder Set Criteria for prediction Combine Sampled + Predicted = Etimator. c0ed.doc 3//00 7:0 AM 6

17 Re-arrange Pop. into Sample and Remainder K K = = K r r Alo: X r var ξξ X and V V = r r V V r r where K = I 0 I 0 and n m n ( n) m ( M m) K r In 0 0 IM m n ( n) ( M m) m = 0 I n IM ( n) n. c0ed.doc 3//00 7:0 AM 7

18 Specify Function (linear) of RV of Interet = g + g P r r g 0 m = " M 0 n ; 0 M m gr = 0 " M 0 n ( n) M (firt elected cluter mean) c0ed.doc 3//00 7:0 AM 8

19 Set Criteria for prediction ( ) Linear in the ample data: ˆ P = g + ˆ =0 ξξ ( ) Zero Expected Bia: E P P Minimum MSE where: ( ) P P ˆ ξξ a var = ava av g + g Vg r r r r r c0ed.doc 3//00 7:0 AM 9

20 Combine Sampled + Predicted = Etimator ˆ = g g X ˆ V V X ˆ r r r + + ( ) P α α ( ) where αˆ = X V X V X Look like a BLUP! c0ed.doc 3//00 7:0 AM 0

21 g m M Sampled Part: = Remaining Part: g ˆ ( ) Shrinkage Contant: r M m = + k r M k ( mσ f σ ) m e σ e mσ + Mσ = f where m m =. M c0ed.doc 3//00 7:0 AM

22 Interpretation Target parameter i clearly identified. Terminology for Prediction ha meaning Finite number of SSU in PSU i directly accounted for Eaier to explain why etimator doe not equal the PSU ample mean. o aumption needed apart from two-tage ampling (random permutation) c0ed.doc 3//00 7:0 AM

23 Account for repone error Allow for unequal M Allow for unequal probability ampling Evaluate how to etimate variance. How to contruct interval etimate? Further Reearch Summary Realized random effect are mean for realized cluter. Sampling model i unified with modeling implication for urvey ampling Finite ize of Cluter matter. c0ed.doc 3//00 7:0 AM 3