The Mxed Models Controversy Revsted Vvana eatrz Lencna Departamento de Metodología e Investgacón, FM Unversdad Naconal de Tucumán Julo da Motta Snger Departamento de Estatístca, IME Unversdade de São Paulo Edward J. Stanek III Department of ostatstcs and Epdemology, SPH Unversty of Massachusetts at Amherst www.unx.ot.umass.edu/~cluster
ANOVA WITH TWO FIXED FACTORS Example I: Evaluate changes n the average bond strength values of metal parts as a functon of manufacturng technque and orgn of raw materal Populaton averages Manufact urng technque Orgn of raw materal 3 4 A 3 4 A 3 4 A 3 3 3 33 34 : average bond strength values of metal parts manufactured accordng to technque A wth raw materal, =,, 3, =,, 3, 4. : average bond strength values of metal parts manufactured accordng to technque A, =,, 3;. 4 = 4 = : smlar defnton for orgn of raw materal. : average bond strength values of metal parts ; 4 3 3 4 = = = 4 = 3 = = =
3 Defnton of factor effects Man effect of level of factor A α = α = α Man effect of level of factor β = β = β Interacton between level of factor A and level of factor γ + γ = γ or equvalently, γ = { + ) + ( )} = ( Interacton patterns
4 Model Yk = + ek =,,3; =,,3,4; k =,..., r constants { }.. d. N ( 0, ) e k or equvalently, Y = + α + β + γ + e =,,3; =,...,4; k =,..., r k k, α, β e γ constants subect to α β = γ = γ { }.. d. N ( 0, ) e k = No nteracton between factors A and, γ γ When there s no nteracton or when the nteracton s not essental, we may consder the hypotheses of no man effects No factor A man effects, α α No factor man effects, β 0 β =
5 ANOVA Table (=,..., I; =,..., J e k=,..., r) Source Sum of Squares d.f.. EMS Factor A SSA = Jr ( y y ) I- + Jrφ ( α) Factor SS = Ir ( y y ) J- + Irφ ( β ) SS = r A* A Error SS y ( y y y + y ) (I-)(J-) + rφ ( γ ) E = k y ) k ( IJ(r-) φ ( α) = α, φ ( β ) = β, φ ( γ ) = γ I J ( J )( I )
6 ANOVA WITH ONE FIXED AND ONE RANDOM FACTOR Example II: Evaluate changes n the average bond strength values of metal parts as a functon of manufacturng technque and machne Averages Manufact urng technque A Machne w w w 3 w 4 m ( w ) m( w ) m3( w 3) m4( w 4) A 3 m ( w ) m( w ) m3( w 3) m4( w 4) A 3 m ( w ) m3( w ) m33( w 3) m34( w 4) OS: The machnes consttute a smple random sample of a populaton, Ω. Then s a random vector. ( ( ), ( ), ( )) m w m w m w 3 Mxed model Fxed factor: A Random factor: Two models, defned n analogy to the fxed factor model have been proposed n the lterature.
7 UNCONSTRAINED PARAMETER MODEL (UP) k ( ) E, Y = + α + + α + ( 0, ), ( α) ~ N( 0, α ) e E ~ N ( 0, ) ~ N ndependent, =,..., a; =,..., b e k =,..., r. k k, CONSTRAINED PARAMETER MODEL (CP) D k ( D) E, Y = η + τ + D + τ + ( a ) ( 0, ), ( τd) ~ N 0, τ e E ~ N ( 0, ) ~ N D D k a ( ) ( ) τ, τd, ; Cov ( τd),( τd) = τ D / a otherwse, ndependent; =,..., a; =,..., b e k =,..., r. k, ANOVA Table (=,..., a; =,..., b e k=,..., r) Source Sum of Squares EMS (UP) EMS (CP) Factor A SS + rα + brφ ( α ) + r + φ τd br ( τ ) A Factor SS α + + r ar + ar D A* SS A + r α + r τd Error SS E
8 Tests for no effects Effect UP CP Hypothess Test statstc Hypothess Test statstc Factor A φ ( α ) MSS A MSS φ () τ A MSS A MSS E Factor MSS MSSA D MSS MSSE Interacton α MSS A MSS E τd MSS A MSS E Controversy: test for no factor man effects? Equvalence between parameters One may show that = D D = a + a Dfferent models apparently generate dfferent hypotheses. Problem Whch one should we test? How should we defne man effects and nteracton? Some authors suggest that the choce must be made by the researcher Suggeston: nterpret the parameters under both models τ D α
PARAMETER INTERPRETATION AND DEFINITION OF EFFECTS IN MIXED MODELS 9 Fnte number of levels of factor Ω : # Ω = b fnte Populaton of levels of factor, ( ) * Fxed effects model for the populaton of levels of factor sk s sk sk { }..d ( 0, ) Y = + e e N * =,..., a, s=,..., b e k=,..., r No factor man effects s, s Mxed model after samplng levels of factor ) Y M E a b b k r * k = + k, =,...,, =,..., ( < ) e =,..., where b * * M = U, E = U e U s s k s sk s= s= s b f th level of factor s level s 0 otherwse = OS: Randomness s clearly nduced by samplng
0 One may show that ( M ) 0 s s Ω Var M = Ths mples that no factor man effects s equvalent to ( M M ) Var For fnte number of levels of factor the test s complcated Infnte number of levels of factor Populaton of levels of factor : Ωwth #( Ω ) = Model: Yk = + α + M + Ek, =,, a, =,, b, k =,, r M N(0, p+ q), Cov( M, M ) = q ' Ek N(0, ), ndependent and ndependent of M Var( M,, M ) = pi+ qj Scheff'e (959) a UP model: M = + α N N (0, ), α (0, α), ndep p=, q= α CP model: M = D + τ D D N D N a a (0, D), τ {0,( /( )) τ D}, ndep Cov D D a ( τ, τ ' ) = τd/ q= / a, p= / a D τd τd
Random varables: M : Ω Θ R ( ωθ, ) M ( ω) E k : Ω Θ R ( ωθ, ) E ( ωθ, ) k Factor A man effects: ( ) ( ) E Y (,) E Y (,) = ( + α ) = α Ω Θ Ω Θ No factor A man effects α Factor man effects: No factor ( ω ) ( ) EΘ Y (,) E E Y (,) Ω Θ = b( ω) man effects b ω ω Var( b) ( ), Ω Var( M )
Expresson n terms of UP and CP models V V ( M ) UP ( M ) = + a α = a CP D Concluson: The hypothess of nterest s D under the CP model or equvalently, + a α under the UP model COPMPUTATIONAL ASPECTS Many authors suggest the UP model: Analyss s easly performed n terms of lkelhoods Analyss s easly mplemented for unbalanced data Statstcal software avalable (SAS proc MIXED) Ths s not so! Lkelhood rato tests under non-standard condtons (parameters le on the border of parametrc space) Exact tests (based on ANOVA technques) are avalable but not ncluded n most statstcal software packages
3 REFERENCES Chrstensen, R. (996) Exact tests for varance components. ometrcs, 5, 309-34. Hnkelmann, K. (000). Resolvng the Mxed Models Controversy, Comments. The Amercan Statstcan, 54, 8. Hockng, R.R. (973). A Dscusson of the Two-Way Mxed Model. The Amercan Statstcan, 7, 48-5. McLean, R.A., Sanders, W.L. and Stroup, W.W. (99). A Unfed Approach to Mxed Lnear Models. The Amercan Statstcan, 45, 54-59. Neter, J., Wasserman, W., Kutner, M.H. and Nachtshem, C.J. (996). Appled Lnear Statstcal Models (4 rd ed.). Homewood, Ill: Irwn. Öfversten, N. (993). Exact tests for varance components n unbalanced mxed lnear models. ometrcs, 49, 45-57. Samuels, M.L., Casella, G. and McCabe, G.P. (99). Interpretng locks and Random Factors. Journal of the Amercan Statstcal Assocaton, 86, 798-8. Scheffé, H. (959). The Analyss of Varance. New York: Wley. Searle, S.R. (97). Lnear Models. New York: Wley. Stram, D.O. and Lee, J.W. (994). Varance components tetstng n the longtudnal mxed effects model. ometrcs, 50, 7-77. Voss, D.T. (999). Resolvng the Mxed Models Controversy. The Amercan Statstcan, 5, 35-356. Wolfnger, R. and Stroup, W.W. (000). Resolvng the Mxed Models Controversy, Comments. The Amercan Statstcan, 54, 8.