Team Stattc and Art: Samplng, Repone Error, Mxed Model, Mng Data, and nference Ed Stanek Unverty of Maachuett- Amhert, USA 9/5/8 9/5/8 Outlne. Example: Doe-repone Model n Toxcology. ow to Predct Realzed Random Effect. Fnte Populaton Mxed Model 4. Reult on Fnte Populaton Predctor of Realzed Subject True Value 5. llutraton and Dlemma 6. Godambe Lnear Etmator 7. Extence of BLUE for pecal parameter pace 8. Example of SRS w/o Rep. When n, 9. Obervaton/Drecton for the Future 9/5/8. Example: Doe-repone Model Threhold v ormetc Model eat data- 89 chemcal, yeat tran, 5 doe x replcaton- Focu on doe below BMD Repone (% control 8 6 4 ormetc Doe-Repone 5 5 Doe Threhold Doe-Repone 9/5/8 4 Repone (% control 8 6 4 5 5 Doe What orme? Goal: Evaluate orme for each Chemcal ormetc Doe-Repone ormetc Regon Repone (% control 8 6 4 5 5 Doe Repone (above % 8 6 4 4 5 6 7 8 Doe The Problem: Data generaton: a) of chemcal b) For elected chemcal, ample doe c) Meaure repone (n replcate) orme Average Repone (over % Control) n ormetc Regon chemcal J doe ( μ ) B E jk jk k replcaton Fxed Random Aume Repone Error heterogeneou (Bologc Aumpton) ow hould we evaluate orme?
Bet Lnear Unbaed Predctor from Mxed Model μ B Latent Repone of th Selected Chemcal Predctor ( μ ) P ˆ ˆ μ k ˆ or Shrnkage Contant k e / m Per cent of SC 9 8 7 6 5 4 ( μ ) Pˆ ˆ μ k ˆ k e / m - - 4 5 6 7 8 9 95%Pr ed ct on nt er val 9/5/8 7 hme7p4. a 9/ / 7 by EJS 9/5/8 8 Per cent of SC Whch predctor hould be ued? 9 8 7 6 f mxed model developed drectly ung amplng from a fnte populaton: Ue Average of Repone Error Varance- Even f RE heterogenou 5 4 f mxed model developed from a heterogenou varance uperpopulaton model: Ue Chemcal Specfc Repone Error Var. - 7-6 - 5-4 - - - 4 5 6 7 8 9 95%Pr ed ct on nt er val 9/5/8 hme7p4. a 9/ / 7 by EJS 9 Bac Reearch Queton Can Fnte Populaton Mxed Model be developed that ncorporate Realzed Random Effect (chemcal) Specfc nformaton n the Predctor? nvolve dentfablty of chemcal n model (abent n mxed model) Very cloely connected to predctng realzed random effect n mxed model. Relate to fnte populaton amplng model where label are dentfable. Fnte Populaton Mxed Model Populaton, ubject, true repone Subject Label:,..., True Repone: Populaton Parameter Mean: μ μ Subject Devaton: β μ μ Varance: μ μ μ on Stochatc Model: μ μ β
Data for Subject : Repone Error Model: (, k ) μ β E k k Repone Error k,..., r E E R ( k) ( E ) var R k Samplng Select n of ubject wthout Replacement Aume all ample are equally lkely. # of Set : # of Mean:! n ( n)! n!! n! Sequence: ote dfference wth: k n " ample" n k n 9/5/8 9/5/8 4 a part of a Sequence (part of a Permutaton) Populaton Repreent Poton n a Permutaton:,...,! Aume all Permutaton Equally Lkely: P ( Permutaton " p" ) Defne: poton,..., n n k n Mean: 9/5/8 5 Julo Ed Wenjun 9/5/8 6 Poton n Permutaton Poton n Permutaton 9/5/8 7 9/5/8 8
Poton n Permutaton Poton n Permutaton 9/5/8 9 9/5/8 Poton n Permutaton 9/5/8 Remander Remander 9/5/8 Poton n Permutaton Populaton ze () mot lkely > We only ee n ubject n the ample For example: Suppoe n, and 7 We may ee 9/5/8 5 4 9/5/8 4 Poton n Permutaton Remander 4 4
Poton n Permutaton 4 7 9/5/8 5 Remander Mean: T Tradtonal Samplng Approach (ung a ample et) " ample" y n " ample" orvtz-thompon Etmator: y π 9/5/8 6 Mng Data y y y y y π π π π Tradtonal Model Approach Wth Repone Error: Mean: (ung a ample equence) μ β E k k n k n,..., n poton equence: k U k 9/5/8 7 k U k Poton n Permutaton k k 9/5/8 8 Poton n Sequence k U U U k U U U Remander k k k U U U k 9/5/8 9 Bac Random Varable U U U U U U U U U k k k k k k k k k 9/5/8 Remander ( ) k k k Populaton k k k Degn Baed Model Baed 5
Repone Error Model Smple Repone Error Model k β Wk μ β W k k k β W k Fxed Random Fnte Populaton Mxed Model k B W k U UW k μ B W k k B W k 9/5/8 Fxed Random (ung a ample equence) Mxed Model k B W k U k μ B W k k B W k k U k Xα ZB W 9/5/8 B U β Properte of Bac Random Varable () U k U k U k U k U k U k U k U k U k Sum ( k k k) Expected Value Sum k k k y y y Average Expected Value μ μ μ Average 9/5/8 μ Sum Random Varable (n) U U U U U U k k k k k k n n n U U U k k k k k Expected n n n Value y y y k T " ample" π 9/5/8 4 Sum Expected Value μ μ Predcton of Mean n a Smple Cae: o Repone Error (, n) B U μ B B Remander ote: μ ( ) n Crtera: Lnear Functon of ample Unbaed Smallet Mean Squared Error ( n ) 9/5/8 5 Predcton of Mean o Repone Error (, n) μ Β Target P L μ n n y Data Realzed y y Bet Lnear Unbaed ˆ n P y n Predctor: n 9/5/8 n 6 6
Target Data Predcton of a Subject Mean n Poton wth o Rep. Error (, n) Bet Lnear Unbaed Predctor: P L μ Β μ Β Realzed y y y 9/5/8 7 Predcton of a Subject Mean n Poton wth Repone Error k k Target P L μ Β L L Realzed Bet Lnear ˆ Pˆ k P y f n ( y ) f n Unbaed Predctor: f > n k 9/5/8 8 f > n e μ Β W y y w UW Xμ ZB W Data Predcton of Realzed Random Effect Other Example SRS Subject Pˆ k( y ) Rep. Error SRS Poton P ˆ k Rep. Error ( y ) Cluter Samplng: Balanced Cluter Samplng: Un-Balanced k k e e n n P ˆ k k e Smlar form, more complcated 9/5/8 9 Per cent of SC 9 8 7 6 5 4 - - 4 5 6 7 8 9 95%Pr ed ct on nt er val hme7p4. a 9/ / 7 by EJS 9/5/8 4 Per cent of SC Delmma 9 8 7 6 5 4 Pooled Subject Repone Error Varance hould be ued for K (Ung theoretcal Reult) Emprcal example llutrate maller MSE reult wth K dependng on realzed Subject -- but no theory! - 7-6 - 5-4 - - - 4 5 6 7 8 9 95%Pr ed ct on nt er val 9/5/8 hme7p4. a 9/ / 7 by EJS 4 Can fnte populaton model nclude nformaton on realzed ubject? 9/5/8 4 7
Godambe Lnear Etmator of a Total: et: h,..., ndcator RV for ample et h : Etmate for ample et h : Godambe Lnear Etmator : h n e β y h hj hj j e h h h E T y Clam Etmator general, wth pecal cae correpondng to degn baed etmator and model baed etmator. Coeffcent n the etmator depend on the ample and the ubject (label) t uffcent to ue ample et of ubject. The equence doe not contrbute addtonal nformaton. There no unque bet lnear unbaed etmator of the populaton total. (on-extance reult) 9/5/8 4 9/5/8 44 Current Reearch dea Godambe etmator (f baed on equence) nclude poton and label. We are tudyng the etmator n a mple cae (SRS wth or wthout replacement, n, ). ntal Obervaton Godambe etmator doe not make ene f y Godambe etmator doe not make ene f y y n ettng where all y and are dtnct, the model over-parameterzed. Removng the overpararmeterzaton, a unque oluton ext wth zero MSE. 9/5/8 45 Example: SRS wthout Replacement: n, otaton otaton (et) h,..., {, } uh uh uh uh uh uh u h β β β h h h e h yu hβh β hj β vec β β β ( ) S h h uhj h {, } h {, } u u u j,..., n h {,} poton n et E S yu h β h y ( S ) u β h 9/5/8 h 46 Example: SRS w/o Rep- Contrant: n, Etmator E y ( S ) u hβ T y h Snce Ep( h) p E S p Unbaed Contrant: E E T y puβ [ ] p p Addtonal Contrant: p u β p β 9/5/8 47 u u u... u Example: SRS w/o Rep.- Etmatng Equaton: n, Contrant Summary: Varance pl β u L ( ) E E T p T p( ) β uh( yy ) u hβ h h φ pβ uh( yy ) u hβ T λ pl β h h Etmatng Equaton: n ˆ uh yy u h L h h β ˆ p L λ 9/5/8 48 8
Example: SRS w/o Rep.- General Soluton: n, Ung QR Decompoton of L We olve the etmatng equaton: where XX L S L S T ( ) Q Q XXQ Q R L Q ˆ n XX L β ˆ L λ p SQR -Q QXXQ QXXQR ˆ β S 9/5/8 49 Example: SRS w/o Rep.- Specfc Soluton: n, y y Ung QR Decompoton of L y y ˆ β () y y ˆ β y y Q L R 4 y y ˆ β () y y ˆ β ˆ β y y ˆ β y y 9/5/8 5 Obervaton: The oluton unque, but the coeffcent depend on all populaton value. Emprcal oluton (formed by replacng unknown value by an etmate equal to the ample mean) wll reult n an etmator equal to the ample mean. Dfferent oluton wll reult when the populaton ha zero value or nclude te. on-homogeneou etmator eem to be poble (ubtractng a contant). A oluton can be explctly developed for mple random amplng wth replacement where n and. That oluton requre the ame addtonal over-parameterzaton contrant. 9/5/8 5 Addtonal Obervaton: t appear poble to generalze the oluton to n and when amplng wthout replacement. Coeffcent n Godambe lnear etmator can be factored nto coeffcent that depend on poton, coeffcent that depend on ubject, common factor, and cro term. Repone error can be added. Un-equal probablty amplng can be ncorporated. Specal ettng where repone or wll gve re to pecal oluton. Auxlary varable attached to ubject may be ncorporated. 9/5/8 5 All thee area and more need addtonal work! Thank 9/5/8 5 9