Split-plot Exprimnts nd Hirrchicl Dt Introduction Alx Stlzlni invstigtd th ffcts of fding rgim for bf nimls nd muscl typ on th tndrnss of mt. H ssignd ight nimls to ch of thr trtmnts. Th trtmnts wr th numbrs of dys on th fding rgim (DOF): 0 dys, 4 dys, or 84 dys. From ch niml h xtrctd stk from ch of 9 muscl typs (MT). H msurd th shr-forc using stndrd tchniqu clld WBS. Following is portion of th dt st. Dys on Fd Anim Muscl Typ WBS logwbs 1 1 glm 4. 0.6531 1 1 inf 3.04 0.48874 1 1 lt 7.335 0.8654 1 1 lod 1.7 1.104487 1 1 lon 3.435 0.53597 1 1 pso 3.4 0.53406 1 1 rf 3.95 0.59384 1 1 tfl 6.495 0.81579 1 1 vls 6.65 0.79691 1 glm 6.615 0.8053 1 inf 3.59 0.555094 1 lt 4.9 0.63457 1 lod 14.05 1.147676 1 lon 3.705 0.568788 1 pso 3.515 0.54595 1 rf 3.535 0.548389 1 tfl.855 0.455606 1 vls 6.105 0.785686............... 3 8 glm 10.54 1.0841 3 8 inf 3.47 0.54039 3 8 lt 4.855 0.686189 3 8 lod 10.6 1.0615 3 8 lon 6.685 0.85101 3 8 pso 4.36 0.639486 3 8 rf 7.3 0.86333 3 8 tfl 4.965 0.695919 3 8 vls 6.675 0.84451 First of ll, this is fctoril study bcus th scintist ws concrnd with two fctors, dys on fd (DOF) nd muscl typ (MT). Scond, th dt st is hirrchicl in th sns tht thr r two sizs of units; ntir nimls, nd stks cut from n niml. If you compr vlus of two diffrnt lvls of DOF, thn you r ncssrily compring vlus from two diffrnt nimls. But you cn compr two muscl typs within th sm niml. Ths r fturs of split-plot 1
xprimnt. Figur 1 illustrts th rltionship btwn th two fctors nd th sizs of units. DOF 1 DOF DOF 3 = niml = stk Figur 1. Rltionship of Fctors to Exprimntl Units Split-Plot Exprimnts. A Split-Plot xprimnt is fctoril study in which thr r diffrnt sizs of xprimntl units. On st of units r clld whol-units nd th othrs r substs of th whol-units, nd thy r clld sub-units. Th lvls of on fctor r ssignd to th whol-units. This fctor is clld th whol-unit fctor. Th lvls of th othr fctor r ssignd to th subunits, nd this fctor is clld th sub-unit fctor. Th lvls of th whol-unit fctor might b ssignd to th whol-units in vrity of dsigns, such s compltly rndomizd or rndomizd blocks. Figur illustrts split-plot xprimnt in compltly rndomizd dsign. L H L L H H Mthods of Irrigtion = L,H Cultivrs = Figur. A Typicl Split-plot Dsign with Whol-lot Trtmnts in CRD Th trm split-plot stms from th gronomic hritg of xprimntl dsigns in which th xprimntl units usully wr plots of lnd. But th concpt of split-plot xprimnts xtnds to virtully vry disciplin of rsrch. Th nxt svrl figurs illustrt th stps in which split-plot xprimnt is implmntd with th whol-plot trtmnts ssignd in compltly rndomizd dsign. Th hypothticl stting is n gronomic study on ffcts of irrigtion rt nd cultivr. Th lvls of irrigtion r low nd high (L=low nd H=high). Th lvls of
cultivr r gry, grn nd yllow. Irrigtion is th whol-unit fctor nd cultivr is th sub-unit fctor. In this study irrigtion is usd s th whol-unit fctor bcus it is inconvnint to chng lvls of irrigtion btwn smll units du to th difficulty of instlling wtr lins. But diffrnt cultivrs cn b plntd on smll units without difficulty. First of ll, th whol-units r idntifid. Figur 3 illustrts six plot of lnd tht will function s th whol-units. Exprimntl Units Figur 3. Split-plot Dsign: Th Whol-units. Nxt, th lvls of th whol-unit fctor r ssignd to th whol-units, s illustrtd in Figur 4. Assign Trtmnts to WP Units Mthods of Irrigtion: L= H= Figur 4. Trtmnts Assignd to Whol-units. 3
Th nxt stp is to idntify th sub-units, s illustrtd in Figur 5. L H L L H H Mthods of Irrigtion = L,H Sub-plot Unit Figur 5. Sub-units Idntifid. Finlly, ssign lvls of cultivr to th sub-units within th whol units. Assign Trtmnts to SP Units Mthods of Irrigtion = L,H Cultivrs = Figur 6. Trtmnts Assignd to Sub-units. Notic th hirrchicl structur of th units: sub-units r nstd within whol units. An nlysis of vrinc for split-plot xprimnt in compltly rndomizd dsign hs th form s shown hr. Th whol-unit fctor is dnotd A nd th sub-unit fctor is dnotd B. Th fctor A hs nd th fctor B hs b. Ech lvl of A is ssignd to r units. Sourc of Vrition DF A -1 Error() = Rps(A) (r-1) B b -1 A*B (-1)(b-1) Error(b) = Rps*B(A) (b-1)(r-1) Totl rb-1 4
A split-plot xprimnt with whol-units is rndomizd blocks dsign is illustrtd in Figur 7. Blocks 1 3 4 5 W-P S-P Figur 7. Split-plot Exprimnt with Whol-units in Rndomizd Blocks Dsign. An nlysis of vrinc for split-plot with whol-units in rndomizd blocks is shown blow. Thr r r blocks, lvls of th whol-unit fctor, nd b lvls of th sub-unit fctor: Sourc of Vrition DF Blocks r-1 A -1 Error() = Blk*A (r-1)(-1) B (b-1) A*B (-1)(b-1) Error(b) = Blk*B+Blk*A*B (r-1)b-1) Totl rb-1 Th Mt Tndrnss Study s Split-Plot Exprimnt Tchniclly, th mt tndrnss is not tru split-plot xprimnt. Thr r two fctors, DOF nd MT. Also, thr r two sizd of units, nimls nd stks. Th lvls of DOF r indd rndomly ssignd to nimls. This prt constituts tru compltly rndomizd dsign. But th lvls of MT wr not rndomly ssignd to stks. Instd, th stks r cut from typ of muscl, mking thm of tht prticulr typ. But th study hs mny of th sm ssntil fturs s split-plot xprimnt in tht thr is hirrchicl structur of units (nimls nd stks), nd lvls of on fctor chng btwn units of on siz nd lvls of th othr fctor chng btwn sub-units within whol-units. Thrfor, on should considr nlyzing th dt following th mthods for split-plot xprimnts. 5
Modl for Dt Th distributions of dt for ch muscl nd indict htrognous vrincs btwn th muscl typs. Th logrithmic trnsformtion stbilizs th vrincs. A sttisticl modl is y = µ + β + + γ + δ +, ijk i ij k ik ijk whr: y ijk is th log(wbs) for muscl k in niml j in trtmnt (DOF) i, µ ik = µ + βi + γk + δik is th popultion mn for trtmnt i muscl j, µ is th ovrll mn, β i is th fixd ffct of DOF i, γ is th fixd ffct of MT k, k δ ik is th fixd ffct of intrction btwn DOF i nd MT k, ij is th rndom ffct of niml j in trtmnt i, ijk is th rndom ffct of th stk from muscl k in niml j in DOF i. Th prmtrs in µ ik = µ + βi + γk + δik dfin th mn structur nd r clld fixd ffcts. Th rndom ffcts r rndom vribls nd hv probbility distributions. W ssum ij is distributd normlly with mn 0 nd vrinc σ nd ijk is distributd normlly with mn 0 nd vrincσ. Sinc th modl contins both fixd nd rndom ffcts, it is clld mixd ffcts modl, or simply mixd modl. An ANOVA for th dt, showing dgrs of frdom nd xpctd mn squrs, is: Sourc of Vrition DF Exp Mn Squrs DOF Animl(DOF) 1 MT 8 DOF*MT 16 Error(b) 168 σ + 9σ + φ DOF σ + 9σ σ + φmt σ + φb* Mt σ Totl 15 6
whr: φ = (8*9 / ) ( µ µ ) DOF i... i φ = (8*3/ 8) ( µ µ ) MT. k.. k φ = (8/16) ( µ µ µ + µ ) DOF* MT ik i.. k.. ik Expctd mns squrs cn b usd to dtrmin th pproprit wys to comput F sttistics for tsting th vrious ffcts in th ANOVA tbl. Considr first DOF*MT typ intrction. Th null hypothsis is H0 : φ DOF* MT = 0 nd th ltrntiv is H : φ DOF* MT > 0. If H 0 is tru, thn th EMS for DOF*Muscl typ is qul to th EMS for Error(b). But if H is tru, thn th EMS for DOF*Muscl typ is lrgr thn th EMS for Error(b). Thus, if H0 is tru, w xpct MS(DOF*MT) to b bout th sm mgnitud s MS(Error(b)). But if H is tru, w xpct MS(DOF*MT) to b lrgr thn MS(Error(b)). This motivts compring th mgnituds of MS(DOF*MT) nd MS(Error(b)) s tst critrion, so w us th rtio F = MS(DOF*MT)/ MS(Error(b)) s th tst sttistic. This hs th F distribution with 16 nd 168 dgrs of frdom. Likwis, w cn motivt tst for muscl typ, i.. rtio F = MS(MT)/ MS(Error(b)) H φ = vrsus H : φ > 0, using th : 0 0 Mt Mt for th tst sttistic. Now considr tsting th ffct of DOF; : 0 H0 φ DOF = vrsus H : φ B > 0. Th EMS for DOF is qul to th EMS for Error() whn H : 0 0 φ B = is tru nd EMS for DOF is lrgr thn th EMS for Error() whn H : φ B > 0 is tru. MS(DOF) cn b comprd with MS(Animl(DOF)) s critrion. This suggsts using F = MS(DOF)/ MS(Animl(DOF)) s tst sttistic. In othr words, us MS(Animl(DOF)) s n rror trm whn tsting DOF ffcts. For this rson, th sourc of vrition Animl(DOF) is clld Error(). 7
Sourc of Vrition DF Mn Squrs F rtios DOF MS(DOF) MS(Br)/ MS(Error()) Error() 1 MS(Animl(DOF)) MT 8 MS(MT) MS(Mt)/ MS(Error(b) DOF*MT 16 MS(DOF*MT) MS(DOF*Mt)/ MS(Error(b) Error(b) 168 MS(Error(b) Totl 15 Th ANOVA tbl contining computtions is Sourc of Vrition DF SS MS F rtios p-vlu DOF 0.047.036.036/0.047 = 0.71.5038 Error() 1 0.699.0333 MT 8 3.159.3948.3948/0.017 = 3.91 <.0001 DOF*MT 16 0.748.0468.0468/0.017 =.83.0004 Error(b) 168.774 0.017 Totl 15 Th sttisticl significnc of DOF*MT intrction is notbl. It implis tht diffrncs btwn th DOFs r not th sm for ll muscl typs. Hr r th mns of th log wbs vlus for combintions of DOF nd muscl typ. It might sms pculir tht min ffcts of DOF r not significnt (p=0.5083) whn DOF*MT is highly significnt. Th phnomnon occurs bcus in som MTs on DOF hs th lrgst mn nd in othr MTs nothr DOF hs th lrgst mn, so tht in th vrg cross MTs th DOF mns r similr. GLM INF LAT LOD LON PSO REF TFL VLS Mrgin DOF 1 0.805 0.518 0.791 1.05 0.681 0.501 0.619 0.640 0.818 0.71 DOF 0.851 0.499 0.770 0.800 0.813 0.473 0.685 0.757 0.849 0.7 DOF 3 0.645 0.50 0.746 0.711 0.783 0.507 0.716 0.705 0.850 0.684 Mrgin 0.767 0.51 0.769 0.846 0.759 0.494 0.673 0.701 0.839 Thr r rltivly lrg diffrncs btwn DOFs for som muscl typs (.g. LOD) nd quit smll diffrncs for othr muscl typs (.g. VLS). In ddition, DOF 1 hs th lrgst mn nd DOF 3 hs th smllst mn for LOD, but DOF 3 hs th lrgst mn for REF. S Figur 8. 8
logwbs 1. 1 1. 0 0. 9 0. 8 0. 7 0. 6 0. 5 0. 4 gl m i nf l t l od l on pso r f t f l vl s MT 3 DOF 1 Figur 8. DOF Mns Plottd vrsus MT showing DOF*MT intrction. Significnt intrction btwn DOF nd Muscl typ implis tht DOFs should b comprd sprtly for ch muscl typ. In ordr to ssss th sttisticl significnc of diffrnc btwn DOF mns t ch muscl typ, w nd th stndrd rror of th diffrncs. Th tbl blow shows stndrd rror xprssions nd thir stimts for four typs of diffrncs btwn mns: 1) Min ffct of MT, ) Min ffct of DOF, 3) Simpl ffct of MT, nd 4) Simpl ffct of DOF. Comprison Stndrd rror s.. stimt df 1/ 1) Diffrnc btwn two ( σ / 4) (MS(b)/4) 1/ 168 MTs vrgd cross DOFs ) Diffrnc btwn two DOFs vrgd cross MTs 3) Diffrnc btwn two MTs in th sm DOF 4) Diffrnc btwn two DOFs in th sm MT (( σ / 8 + σ / 7)) (MS()/4) 1/ 1 1/ ( σ /8) (MS(b)/8) 1/ 168 1/ (( σ + σ )/8) ((MS()+8MS(b))/7) 1/ 175 1/ Notic tht thr r diffrnt stndrd rror xprssions for ch typ of comprison. In ddition, thr r diffrnt dgrs for frdom for th stimts of th vrious stndrd rrors. Th stndrd rror from th first nd third typs of comprison cn b stimtd bsd on MS(Error(b)) bcus th stndrd rror involvs only th rsidul rror vrinc, σ. Stndrd rror stimts for ths two comprisons thrfor hv df=168, th df for Error(b). Th 9
stndrd rror for th scond typ of comprison cn b stimtd from MS(Error()), nd it hs 1 df. But th fourth typ of comprison cnnot b stimtd bsd on ithr MS(Error()) or MS(Error(b)) lon. Its stimt is combintion of MS(Error()) nd MS(Error(b)) nd th dgrs of frdom r bsd on th so-clld Sttrthwit pproximtion. Following r rprsnttivs of ll four typs of comprison, showing th stimt of th diffrnc (stimt), th stimt of th stndrd rror of th diffrnc (s..), th dgrs of frdom for stimting th stndrd rror, nd finlly 95% confidnc intrvl for th popultion diffrnc: Diffrnc stimt s.. df 95% CI 1) glm-inf.55.037 168.55 ± (.037) ) DOF1-DOF3.04.030 1.04 ±.1(.030) 3) glm-lt in DOF1.87.064 168.064 ± (.064) 4) DOF1-DOF3 in MT1.160.068 175.160 ± (.068) Th tbl blow shows stndrd rror xprssions nd thir stimts for thr typs of mns: 1) Mrginl MT, ) Mrginl DOF, nd 3) Combintion of MT nd DOF. Mn Stndrd rror s.. stimt df 1/ 1) Mrginl MT (( σ + σ ) / 4) ((MS()+8MS(b))/) 1/ 175 (vrgd cross DOFs) 1/ ) Mrginl DOF (( σ + 9 σ ) / 7) (MS()/7) 1/ 1 (vrgd cross MTs) 3) Combintion of MT nd DOF (( σ + σ ) / 8) ((MS()+8MS(b))/7) 1/ 168 1/ Following r rprsnttivs of ths thr typs of mns, showing th stimt of th mn (stimt), th stimt of th stndrd rror of th mn (s..), th dgrs of frdom for stimting th stndrd rror, nd finlly 95% confidnc intrvl for th popultion mn: Mn stimt s.. df 95% CI 1) glm.767.08 175.767 ± (.08) ) DOF1.711.0 1.711 ±.1(.0) 3) glm in DOF1.805.048 175.805 ± (.048) 10
A Littl Bit of Thory In this sction w prsnt drivtion of th vrincs of on of th mns nd on of th diffrncs btwn mns prviously usd in th stndrd rror xprssions. Thy utiliz th modl y = µ + β + + γ + δ + ijk i ij k ik ijk whr: y ijk is th log(wbs) for muscl k in niml j in trtmnt (DOF) i, µ = µ + β + γ + ( βγ) is th popultion mn for trtmnt i muscl j, ik i k ik ij is th rndom ffct of niml j in trtmnt i, ijk is th rndom ffct of th stk from muscl k in niml j in DOF i. First, w driv th vrinc of th mrginl of MT=glm, vrgd cross DOF. Th mn hs th xprssion: y = (1/ 3 8) ( µ + β + + γ + δ + ).. glm i ij k ik ijk ij = (1/ 4)[4µ + 8β + + γ + 8 δ + ]... k. k.. glm = [ µ + β + + γ + δ + ]... k. k.. glm Th vrinc is V( y ) = V{[ µ + β + + γ + δ + ]}.. glm... k. k.. glm = V( ) + V( ).... glm = σ /4 + σ /4 = + ( σ σ )/4 Now w driv th vrinc of th min-ffct of th diffrnc btwn MT=glm nd MT=lt. Th diffrnc is: y y = ( γ + δ + ) ( γ + δ + ).. glm.. lt glm. glm.. glm lt. lt.. lt Notic tht th diffrnc dos not contin th rndom ffct of nimls. 11
Th vrinc of th diffrnc is V( y y ) = V( ) =.. glm.. lt glm.. lt ( σ ) / 4 Multipl Split-plot Exprimnts. In som xprimnts thr r vn mor thn two sizs of xprimntl units. Such ws th cs with Alx Stlzlni s xprimnt. In rlity, h xtrctd two stks from ch of th 9 muscls. H rndomly ssignd on of th stks from ch muscl to b gd for 10 dys nd th othr for 0 dys. Aftr ging, h cut six cors (smpls) from ch stk nd msurd th shr-forc on ch cor. Thus, thr r ctully thr sizs of xprimntl units nd multipl msurmnt units from th smllst xprimntl unit: Exprimntl units Exprimntl Fctor Animls Dys on Fd Muscls within nimls Muscl Typ Stks within muscls within nimls Dys of Aging Cors within stks within muscls within nimls Figur 9 illustrts th vrious xprimntl units nd msurmnt units: Animls Muscls Stks Cors Figur 9. Digrm showing rltionships of xprimntl units. You cn s tht th units ctully constitut nstd clssifiction. It is only ftr w ssign trtmnts tht w hv split-plot xprimnt. A modl for th study might b yijklm = µ + βi + ij + γk + ( βγ) ik + cijk + τ + ( βτ) + ( γτ) + ( βγτ) + d + l il kl ijkl ijklm 1
whr, in ddition to dscriptions givn rlir, nd c ijk is th rndom ffct of muscl within niml d ijkl is th rndom ffct of stk within muscl within niml ijklm is th rndom ffct of cor within stk within muscl within niml τ l is th fixd ffct of DOA=l ( βτ ) il is th fixd intrction ffct btwn DOF=I nd DOA=l ( γτ ) kl is th fixd intrction ffct btwn MT=k nd DOA=l ( βγτ ) ikl is th fixd intrction btwn DOF=i, MT=k nd DOA=l An nlysis of vrinc for th dt hs sourcs of vrition, dgrs of frdom nd xpctd mn squrs s shown: Sourc of Vrition DF ExpMS DOF σ + 6σ + 1σ + 108σ + φ E()=Anim(DOF) 1 σ + 6σd + 1σc + 108σ MT 8 σ + 6σd + 1σc + φmt DOF*MT 16 σ + 6σd + 1σc + φdoa * MT E(b)= MT*Anim(DOF) [Muscl] 168 σ + 6σd + 1σc DOA 1 σ + 6σd + φdoa DOF*DOA σ + 6σd + φdof* DOA MT*DOA 8 σ + 6σd + φmt* DOA DOF*MT*DOA 16 σ + 6σd + φdof* MT* DOA E(c)=DOA*MT*Anim(DOF) [Stk] 189 σ + 6σ d E(d)=Cor(Stk Muscl Anim DOF) 149 σ d c DOA* MT 13
Th compltd ANOVA tbl is: Sourc of Vrition DF MS F p-vlu DOF 0.056 0.37.6935 E()=Anim(DOF) 1 0.161 3.44.0001 MT 8 4.4763 93.57.0001 DOF*MT 16 0.370 7.76.0001 E(b)= MT*Anim(DOF) [Muscl] 168 0.0478 1.66.0003 DOA 1 0.314 10.65.0013 DOF*DOA 0.0048 0.18.8369 MT*DOA 8 0.037 1.31.417 DOF*MT*DOA 16 0.048 0.89.5770 E(c)=DOA*MT*Anim(DOF) [Stk] 189 0.086.31.0001 E(d)=Cor(Stk Muscl Anim DOF) 149 0.014 Finl tbls of mns for this study r stright-forwrd: No intrctions involving DOA indict tbl of DOA mns. Mn s.. p-vlu DOA=10.697.0086 - DOA=0.675.0086 - Diffrnc.0.00067.0010 Intrction btwn DOF nd MT indicts tbl of DOF*MT mns. GLM INF LAT LOD LON PSO REF TFL VLS Mrgin DOF 1.766.507.694.935.699.49.617.648.817.686 DOF.798.495.759.758 c.776 b.47.639.734 b.86.694 DOF 3.641 b.55.79.694 c.734 b.494.700 b.70 b.876.677 Slic p.0001.685.19.0001.099.10.055.054.16 Stndrd rror of diffrnc btwn two DOF mns in sm MT: 0.03 LSD for compring two DOF mns in sm MT: 0.070 14
Figur 10 shows th mns plottd vrsus MT. l ogwbs 1. 0 0. 9 0. 8 0. 7 0. 6 0. 5 0. 4 gl m i nf l t l od l on pso r f t f l vl s MT DOF 1 3 Figur 10. DOF Mns Plottd vrsus MT. Th pttrn of mns in Figur 10 (for th ntir dt st) is bsiclly th sm s w sw rlir in Figur 8 (for only singl msur pr stk). 15