Section 7.2 Two-way ANOVA with random effect(s)
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1 Secto 7. Two-wy ANOVA wth rdom effect(s) 1
2 1. Model wth Two Rdom ffects The fctors hgher-wy ANOVAs c g e cosdered fxed or rdom depedg o the cotext of the study. or ech fctor: Are the levels of tht fctor of drect terest? Or do they ust represet some lrger populto of levels tht could hve ee cluded? If the study were to e coducted g would the exct sme levels of tht fctor e used g? Or would other levels e used?
3 xmples xmple 1. Potssum mesuremets cross ls Te commercl lortores cross the UK were chose y dmstrtors of the Ntol Qulty Cotrol Scheme. They re terested dffereces potssum mesuremets from serum smples cross ls. Te serum smples re creted ech of whch cots pre-determed qutty of potssum. ch specme s dvded to te equl portos. Oe porto from ech specme s set to ech l completely rdomzed desg. xmple. Reserch ursg stff eed to e tred to mesure su-scpulr s fold wth clpers for ogog sgle-ceter clcl trl. Wth the ceter ptets re rdomly chose; rdom smple of urses t the ceter mesure ech ptet rdomzed order. Multple mesuremets my e te r tmes. 3
4 4. 0 Vr the respose : The covrce structure of depedet. re ll d d where wth two rdom effects (lced desg) ANOVA Model Cov Cov Cov Cov N N N N d d d d
5 1.. Devto of SSTO SSTR SS Devto Decomposto of the ( ) - th cell me : SSA SSB respose from the grd me ( SSAB SSTR SSA SSB ) - th cell: SSTO SS SSA SSB
6 ANOVA wth two rdom effects Source SS D MS A SSA -1 MSA = SSA / (-1) A = MSA / MSAB B SSB -1 MSB = SSB / (-1) B = MSB / MSAB AB SSAB (-1)(-1) MSAB = SSAB / ((-1)(-1)) rror SS (-1) MS = SS/((-1)) Totl SSTO -1 AB = MSAB / MS xpecttos : MSA MSB MSAB MS Uder uder uder : : : A B AB vs. vs. vs. 1 1 : : 1 : ; ; 6
7 Vrce Compoets stmto stmtors of the vrce compoets : ˆ MS MSAB MS MSB MSAB ˆ ˆ ˆ MSA MSAB XAMPL 1: A Mesuremet Systems Cplty Study Sttstclly desged expermets re frequetly used to vestgte the sources of lty tht ffect system. A commo dustrl pplcto s to use desged expermet to study the compoets of vrlty mesuremet system. These studes re ofte clled guge cplty studes or guge repetlty d reproduclty (R&R) studes ecuse these re the compoets of vrlty tht re of terest (for more dscusso of guge R&R studes. A strumet or guge s used to mesure crtcl dmeso o prt. Twety prts hve ee selected from the producto process d three rdomly selected opertors mesure ech prt twce wth ths guge. The order whch the mesuremets re mde s completely rdomzed so ths s two-fctor fctorl expermet wth desg fctor prts d opertors wth two replctos. Both prts d opertors re rdom fctors. Source D Type I SS Me Squre Vlue Pr > prt <.0001 opertor prt*opertor
8 . Model wth fxed d rdom effects Mxed - effects ANOVA Model (1). (). where fxed fctor A : tercto effect : Cov d Vr Cov Cov Cov N rdom fctor B : 1 0. d 1 1 N0 suect to restrctos 0 1 re prwse depedet d d N 0
9 ANOVA wth mxed effects Source SS D MS A SSA -1 MSA = SSA / (-1) A = MSA / MSAB B SSB -1 MSB = SSB / (-1) B = MSB / MS AB SSAB (-1)(-1) MSAB = SSAB / ((-1)(-1)) rror SS (-1) MS = SS/((-1)) Totl SSTO -1 AB = MSAB / MS xpecttos of MSs : MSA MSB MSAB MS 1 Uder A uder 0 : 0 vs. 1 : 0 1 1; uder 0 0 : 0 : B AB 0 vs. vs. 1 : : t lest oe ; 0. 9
10 stmto of fxed ffects Mxed Model 1. Lest Squre stmtor of -th me d effect of fxed fctor A :. Vrce of 3. ˆ ollow - up Test : Vr ˆ Squred of (1). Prwse comprso ˆ fxed effects : stdrd error : Cofdece Itervl for : t q 1 1 (3). Cofdece Itervl for cotrst L t ˆ MSAB / t 1 1 C.I.for d (). Multple Comprso Tueys smulteous C.I.for c s MSAB 1 1. : MSAB MSAB c : MSAB c MSAB 1...
11 Vrce Compoet stmto Mxed Model Vrce estmtor for rdom effect d tercto effect : ˆ MSB MS ˆ xmple : MSAB MS A compy trg progrm tres to vestgte the effets of methods (fctor A). ve structors (fctor B) re selected t rdom to use ll trg methods dfferet clsses. our clsses re ssged to ech trg - structor comto. Respose vrle s the me mprovemet per studet the clss t the ed of four dfferet trg the trg progrm. Source SS D MS A (trg) 4.1 B (structor) 53.9 AB tercto 46.7 rror 16.4 Totl 69.1 SAS code: xmple 1 (cot). Cplty Study
12
Soo King Lim Figure 1: Figure 2: Figure 3: Figure 4: Figure 5: Figure 6: Figure 7: Figure 8: Figure 9: Figure 10: Figure 11:
Soo Kg Lm 1.0 Nested Fctorl Desg... 1.1 Two-Fctor Nested Desg... 1.1.1 Alss of Vrce... Exmple 1... 5 1.1. Stggered Nested Desg for Equlzg Degree of Freedom... 7 1.1. Three-Fctor Nested Desg... 8 1.1..1
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