BLOCKS REPLICATION EXPERIMENTAL UNITS RANDOM VERSUS FIXED EFFECTS

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Nicholas Watkins
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1 DESIGN CONCEPTS: BLOCKS REPLICATION EXPERIMENTAL UNITS RANDOM VERSUS FIXED EFFECTS TREATMENT DESIGNS (PLANS) VS. EXPERIMENTAL DESIGNS Outln: Blockd dsgns Random Block Effcts REML analyss Incomplt Blocks Scrnng Dsgns Fractonal Factorals Alasng Confoundng Rspons Surfacs Quadratc Functons Cntral Compost Dsgns Sarchng for Optma

2 Extra topcs f tm allows Nstd Dsgns Splt Plots & Rpatd Masurs Blocks- collctons of xprmntal unts that ar homognous. Exampls: rgons, customr clustrs, ndvdual customrs. Dsgn (Randomzd Complt Block or RCB) Blocks hav as many unts n thm as w hav tratmnts and all tratmnts ar randomly assgnd to th unts wthn ach block. Blocks ar usually random ffcts (dfnton blow) Exampl.: A grocry chan wants to look at sals as a functon of promoton typ. Thy want to tst thr partcular promoton typs. Thy thnk thr may b rgonal ffcts but can t afford to tst ths out n all of thr rgons so thy pck 5 rgons at random, stors pr rgon, and assgn ach of th thr promoton typs randomly to th stors. Th rspons s sals n th promoton prod. Modl: j = µ α Bj, j µ: a basln valu applyng to all stors.

3 α : th ffct of promoton, =,,. ths s a fxd ffct. B j : th ffct of rgon j, j =,,,,5. ths s a random ffct.,j : th rror trm for th stor n rgon j that got assgnd promoton Random vrsus fxd ffcts. Fxd: Infrnc s only for th lvls w usd ( promoton typs) Random: Infrnc s for all lvls, whthr chosn for th xprmnt or not (rgons n our cas) Fxd: α ar fxd but unknown constants. Random: B j ~N(,σ B ) for all j (rgons), not just th ons w slctd. Fxd: Would b th sam f w ran th xprmnt agan. Random: Mght b dffrnt n anothr run (randomly pckng rgons, w d lkly not gt ths sam ons agan).

4 Exampl : W gv customrs dffrnt offrs (5 possblts) from tm to tm wth ach customr gttng xposd to all 5 offrs. Rspons s spndng as a rsult of th offr. Ar offrs random? No. W do NOT clam that our rsults tll us about othr typs of offrs bsds ths 5. Ths wr not randomly slctd from an nfnt (or vry larg) populaton of offrs but rathr, wr purposfully slctd. Ar customrs random? s. W would not want to say that our nfrnc s rstrctd to only thos customrs that wr slctd from our customr databas for th xprmnt, and w should hav slctd thm at random! Dos t mattr? It all dpnds No, for th balancd cas (ach customr got all 5 promotons) hr. s, f somhow som customrs dd not gt all 5 offrs durng th tst prod. Exampl : In clncal trals th blocks mght b coopratng doctors and ach doctor assgns th

5 thr drugs undr nvstgaton randomly to thr patnts. Doctors ar th blocks, drugs th tratmnts, and patnts ar th xprmntal unts. Exampl : W hav dffrnt machns for prformng an outdoor task. W block on days bcaus wathr, traffc, polluton, tc. can vary from day to day. On ach tral day (w mght want to hav days n dffrnt sasons), w assgn workrs to th thr machns and masur som rspons lk tm ndd to complt th task or qualty of th rsult (or both). For smplcty w ll assum dffrnt sts of workrs for ach tst day. Exampl 5: W slct dlvry popl from our dlvry popl natonwd, and typs of trucks on loan. W gv of thm typ A trucks, typ B, and typ C. W lt thm drv ths for months and masur mls pr gallon. W hop to pck on of th typs and gt a dal on a volum ordr of trucks.

6 Ths dsgn s NOT blockd. It s calld a compltly randomzd dsgn. Data for xampl 5: ( = ovrall man = 5 ) T D j ( ) ( ) ( j ) ( j ) ( j ) ( j ) MPG truck man A A A A B B B B C C C C == == ==== ===== ==== ===== Total Sum of Squars = Tratmnt Sum of Squars = ( ) = ( ) = 8, = = j= = df = = df ( j ) 7.88, = j=

7 Error Sum of Squars = Total SS Tratmnt SS = = = df from ach group ( j ) 6.88, 9 ( ) = j= Man Squar = (Sum of Squars)/df F = (Tratmnt Man Squar)/(Error Man Squar) Analyss of Varanc (ANOVA) Sourc df SSq MnSq F Truck 8 /6.987 =7 Error Total 7.88 Modl: MPG = man truck ffct rror j =mt j () Matrx Formulaton th modrn approach

8 =, X= µ. = Assumng j ~N(,σ ) ths s just rgrsson! = Xb ˆ X X = ( X X ) =? How to solv ( X X ) β = ( X ' )? Many solutons! Problm can t comput nvrs. Back to squar! Rformulat th quaton to gv

9 Omt last column of X: = µ = Xb Chcks: lns,6,9, = µ OK lns,,7, = = ) ( µ µ OK lns,5,8, = = ) ( µ µ OK Can comput ths on! = X X = ) ( X X (can you chck ths?) = = = = = = = j j j j j j X

10 Sampl mans truck typ A 6, truck typ B, truck typ C 5. /\ µ ˆ β = = ( X X ) X ( ) = 8 6 = C man = = Aman C man B man C man *** Run program DOE.sas *** () Dscusson: Suppos = 7 µ, µ = 6, µ = Evn knowng ths, w cannot unquly fnd µ,,,, lt alon whn ths ar stmatd by th sampl mans. For xampl µ =, =, =, = 9 would work as would µ = 5, =, =, = ( ffcts or dvatons soluton) or th rfrnc cll or GLM soluton µ =, = 5, =, = Usng obsrvd sampl mans (all w rally hav) w stmat mt as 6, mt as, and mt as5 gvng GLM (rfrnc cll) solutons 5,, -. W can stmat mt, mt, and mt and any lnar combnaton of thm, lk t (t t )/ ( mpg)

11 Dmo shows no truck ffct, lot of drvr ffcts (capturd n rror trm). Nw problm, sam data (for llustraton). Now suppos ach of drvrs drov all typs of truck. Truck comparson lk t (t t )/ wll b fr of drvr ffcts. If drvr has ffct - for all trucks, w s that (t -) (t -t -)/ = t (t t )/ : drvr ffct cancls out. Nw Modl: MPG = man truck ffctsdrvr ffcts rror j = m t D j j Assumptons: D j ~N(, σ D ), j ~N(, σ ) Implcatons: s = s D D s = s s Expctd valu: E{ Varanc (varanc of s s = s} s / ) σ n σ / ns *********Usng th thr ruls ***********

12 ********* Th thr ruls *********** () Varanc of a man s σ / n () Varanc of a sum or dffrnc of ndpndnt varabls s sum of varancs. () Varanc of C (constant) tms a varabl s C tms varanc of orgnal varabl. Fxd block (drvr) ffcts (unusual): = µ D D D D D D D = Xb Sampl mans: Expctd valu = µ D E{ } = µ D, varanc s /

13 Random block (drvr) ffcts (usual): = µ D D D D = XbZg Assumptons: g~n(,g) ~N(,R) Our xampl: G= D D D D σ σ σ σ = σ D = I D σ R=Is Sampl mans: = D µ Expctd valu: E } = µ {, (not D µ ) Varanc (of D ) s / / σ σ D (not / σ ) *** Run program DOE.sas *** *** What s standard rror for A man (LSMEANS)?? *** *** How dos ths rlat to formulas abov??***

14 (optonal: mathmatcally mor dffcult) REML stmaton: Data (, 8,, ) man 6 SSq=695=5 Smpl modl ~N(m,s ) Log lklhood: St drvatv w.r.t. m to n ( µ ) = σ = ( µ ) n = ln(πσ ) σ n = n ( µ ) = µ MLE = maxmum lklhood stmat or MLE. Lklhood bcoms n ( ) n n = n n 5 ln(π ) ln( σ ) = ln(π ) ln( σ ) σ σ St drvatv w.r.t s to : n ( ) n 5 = = nσ 5 = σ 5 / MLE = = σ ( σ ) n n

15 Ths ( ) n MLE n σ = = s basd downward bcaus µ TRUE, xpctd valu s σ n n (/6), ~ σ µ N whr w usd (/6) σ σ σ σ σ σ = = Ths shows that ths orthogonal contrasts satsfy, ~ σ N Thr ndpndnt random varabls wth known man (). Maxmz thr lklhood.

16 8 = 7 sum of squars 9=5 (sam as bfor! always). Now comput Maxmum Lklhood Estmator (mans known to b, obsrvatons) 5/ = unbasd. n = ( n ) whch s Gnral: = XbZg Multply through by I X ( X X ) X ( I X X X ) X ( ) = ( I X ( X X ) X )Xb ( I X ( X X ) X )Zg( I X ( X X ) X ) ( I X X X ) X ( ) =( I X ( X X ) X )Zg( I X ( X X ) X ) From ths pck n-(rank of X) lnarly ndpndnt lnar combnatons wth man and maxmz lklhood. Ths s calld REML stmaton. Ths s how PROC MIXED works. REML gvs lss basd (not always unbasd) varanc componnt stmats than dos unrstrctd maxmum lklhood. ****** nd of optonal mathmatcal matral ***** Excutv summary: Us PROC MIXED whn you hav random ffcts othr than th rror trm!

17 Gnral mxd modls wthout balanc drvr s sck on last day whn h was supposd to drv truck typ B and drvr skps out on hr truck typ A run. *** run DOE.sas *** GLM rsults fxd ffct stmats Th GLM Procdur Last Squars Mans LSMEAN truck MPG LSMEAN Numbr PROC MIXED - s blow A (6.9) B.786 (.69) C 5. (5.) Last Squars Mans for ffct truck Pr > t for H: LSMan()=LSMan(j) Dpndnt Varabl: MPG /j NOTE: To nsur ovrall protcton lvl, only probablts assocatd wth pr-plannd comparsons should b usd. PROC MIXED rsults Last Squars Mans Standard Effct truck Estmat Error DF t Valu Pr > t truck A truck B truck C Dffrncs of Last Squars Mans Standard Effct truck _truck Estmat Error DF t Valu Pr > t GLM - s abov truck A B (.9) truck A C (.5) truck B C (.9)

18 What ar LSMEANS? Tratng drvrs as fxd, LSMEANS ar stmats of µ D GLM: 7.886( )/.7 = 6.7 Intrcpt B <. truck A.7857 B truck B B truck C. B... drvr B drvr -. B drvr B drvr. B... Tratng drvrs as random, LSMEANS ar stmats of µ Why? Random ffct modl assums E{D}=, no nd to adjust by sampl avrag of ths partcular drvrs. MIXED: 5..9 = 6.9 Soluton for Fxd Effcts Standard Effct truck Estmat Error DF t Valu Pr > t Intrcpt truck A truck B truck C.... Why LSMEANS ar bttr than MEANS () For balancd data thy ar th sam () Salary data: Excutvs Workrs Man Mals 5, 8, 9, 5 8 Fmals 7, 9, 8 8, Concluson: Fmals mak $ K mor than mals.

19 Tabl of mans blow: mans of mans ar LSMEANS for data lk ths. LSMEANS us th stmats from any data, balancd or not, to stmat what mans would hav bn f th data had bn balancd and all contnuous varabls (lk basln blood prssur, ag) wr st at th avrag obsrvd valu. Excutvs Workrs LSMEAN Mals 5 5 Fmals 8 5 LSMEANS show mals mak mor than fmals ***** run DOE.sas ******* Factoral Tratmnt Arrangmnts: Room Prc:,, 6, 8 Locaton: Cty, Rsort, Othr Problm: hghr prc lowr occupancy lowr proft Problm: lowr prc wth no occupancy chang lowr proft = proft Twlv tratmnt combnatons, hotls n ach catgory Prc and locaton ar fxd ffcts. Q: Dos optmal prcng dpnd on locaton? Q: Avragd ovr prc, how do locaton profts compar?

20 Tabl of Proft Mans (of hotls ach): Prc $ $ $6 $8 Locaton Mans Cty Othr Rsort mans SS(tabl) = [( )... ( ) ]= 77,98,98.8 ( df) SS(prc) = 9[(.-8.5)... (9.-8.5) ]=,85,77. ( df) SS(locaton) = [( )... ( ) ]= 7,,558 ( df) SS(PxL ntracton) = SS(tabl)-SS(locaton)-SS(prc)=9,65,698.7 (--=6 df) Splttng prc ffcts nto lnar, quadratc, cubc ffcts () (optonal by hand) Locat tabl of orthogonal polynomal coffcnts c j for lvls (.g. lnar: c =-, c =-, c =, c = mans ffct dn SSq=(ffct) /dn lnar ,75,95 df quadratc ,76,9.9 df cubc ,5.7 df Comput ffct = c and dnomnator: c / n j j j= Sum of squars s ffct /dn j (n obs pr man) j= () Optonal: How dd thy gt thos orthogonal polynomal coffcnts? Rgrss ach column on th ons to ts lft. Rplac ach column by ts rsduals (multpld by a constant to mak th ntrs ntgrs). Nw columns contan orthogonal polynomal coffcnts.

21 Dtals: Rgrss column (call t ) on column (trat t as X). X = X ' X = ( ) = b= X X X= ( ) = ( ' ) ' rsdual = Xb = = Gvs lnar coffcnts Quadratc: Rgrss column on column and column. X = X ' X = = / / 7 5 b= ( X' X) X ' = = = = / 9 / rsdual = Xb = = = Cubc: Rgrss column on othrs. / / 5 8 b= ( X' X) X ' = / = / 8. 7 = / / rsdual = Xb =. = = = **** Run DOE5.sas ****.

22 Th N xprmnt: Svral (N) factors, ach at lvls. Smpl xampl : Gndr: Fmal, Mal cod as XG = -, Dos: 5, 75 cod as XD = -, Tn clncs, patnts ach. Masur sd ffct: chang n blood prssur (aftr drug bfor drug). Mans of : Fmal Mal (Man) Dos Dos (Man) (. ovrall) Dscusson: If w look at th avrag rspons at th two doss, t s frst of all vry small (.) and scond, t s th sam at both doss (w say thr s no man ffct of dos) whch s ntrstng n lght of th farly larg and potntally harmful ffcts wthn th fmals and wthn th mals. Ths w call th smpl ffcts of dos wthn th mals and fmals. Not that t would b dangrous and ncorrct to say that thr s a nglgbl ( pont) chang n blood prssur whn dos s ncrasd from 5 to 75.

23 Soluton: S f thr s ntracton. If so, rport smpl ffcts. W dfn and dscuss ntractons and smpl ffcts but frst - ** Run DOE6.sas ** Mans of : Fmal Mal (Man) Dos Dos (Man) (. ovrall) Smpl ffct of Dos wthn Fmals (-)(-.) ()(-.) ()(.) ()(6.)=- Smpl ffct of Dos wthn Mals: ()(-.) ()(-.) (-)(.) ()(6.)= Smpl ffct of gndr at low dos (-)(-.) ()(-.) ()(.) ()(6.)=6 Smpl ffct of gndr at hgh dos ()(-.) (-)(-.) ()(.) ()(6.)=5!!! Dfnton: Intracton s ½ (dffrnc of smpl ffcts): ½(-(-))= = ½ (5-6) Not that on way to comput th man ffct of gndr s to tak ½ (sum of th smpl ffcts of gndr) ½ (65)=8=5-(-) Wth n obsrvatons pr man and th s, -s, and s calld coffcnts c thr s a on dgr of frdom sum of squars for ach ffct. Dvdng that man squar by th rror man squar gvs an F tst wth numrator dgr of frdom. That F s also th squar of th t tst for th assocatd coffcnt.

24 As bfor, th sum of squars s: k ( ffct) / ( c / n) = whr k s th numbr of mans ( n our cas) bng combnd. Notng that man ffcts and ntractons hav th ½ multplr whch th othrs do not, w can organz th varous ffcts and thr sums of squars n a tabl. Not that ths s bng prsntd to llustrat what your statstcal softwar s computng. Furthr, ths formulas wll no longr work whn th data bcom unbalancd. Th matrx approach that SAS uss s gnral! Effct D low G low - D low G hgh D h G lo - D h G h 6 ffct Dnom ( c /) = SSq D -½ -½ ½ ½ ½ (). /.= G -½ ½ -½ ½ ½(56). 7,8 D n - -., G_lo D n -., G_h G n -. D_lo G n - 5.,5 D_h D*G ntrctn ½ -½ -½ ½ ½ (). 8 Exampl: Mor factors Gnralzatons nclud mor factors and factors at mor than lvls. Lt s consdr a marktng xampl. I may hav dsplays at th nd of an asl or n th asl tslf for my product

25 (Factor E, lvls -, ) and thr may b a sgn or not. My product could b dsplayd on a low, mddl, or hgh lvl shlf (factor H, lvls -,, ) and I thnk thr may b dffrnt purchasng habts n dffrnt towns so I wll block my xprmnt on towns, assumng a random slcton of 5 towns wth ach havng at last stors (why?). Altrnatvly I could block on (5) stors and randomly apply ach tratmnt for a month. How long would that xprmnt tak? *** Run DOE8.SAS *** Hr s th analyss of varanc tabl usng th cod proc glm data=nxt; class E H sgn town; modl sals = town E H sgn; Sum of Sourc DF Squars Man Squar F Valu Pr > F Modl <. Error Total Th modl sum of squar contans th block (TOWN) sum of squars 6. plus th tratmnt sum of squars 8.8. Ths sums of squars can b obtand from th town and tratmnt mans. Each sum of squars s obtand by th famlar formula n whch ach man, for xampl th tratmnt mans, s subtractd from th ovrall man and that dffrnc s squard and multpld by th numbr of obsrvatons gong nto th tratmnt man, 5 n our xampl, to gv a contrbuton to th sum of squars. Ths ar totald up across th lvls, for xampl th lvls of tratmnt

26 combnatons, to gv th sum of squars. Hr ar th computatons for town and tratmnts: _TPE_=8 (ths ar town mans) town E H sgn _FREQ_ mn_sals dff contrbuton _TPE_= (ths ar th tratmnt mans) E H sgn _FREQ_ mn_sals dff contrbuton no - no no - ys no no no ys no no no ys ys - no ys - ys ys no ys ys ys no ys ys W now want to brak th tratmnt sum of squars down nto pcs du to man ffcts and ntractons. Th thr way (E by sgn by H) tabl of mans w usd so far contans all th

27 man ffcts, two way ntractons and th thr way ntracton. Usng two-way tabls w can gt th sums of squars for any two man ffcts and thr ntracton. Onc w hav all of ths, a smpl subtracton from th tratmnt sum of squars wll gv th rmanng sum of squars, th on for th thr way ntracton. For xampl, lt s xamn th by tabl for E*H. Frst, th PROC MEANS prntout: _TPE_=6 E H sgn _FREQ_ mn_sals dff contrbuton no no no ys ys ys Mn= Mn=7. Mn=9.85 Mn=98. Mn=6.7 E H sgn _FREQ_ mn_sals dff contrbuton E H sgn _FREQ_ mn_sals dff contrbuton no ys

28 Th two way tabl sum of squars contans th E*H ntracton sum of squar plus th H sum of squars 5. and th E sum of squars 856. so th ntracton sum of squars s = 57. Wth 6 ntrs th tabl has 6-=5 df whl H has -= and E has so th ntracton has 5--= dgrs of frdom. Not that ths formulas work only for balancd data whras th gnral matrx approach of SAS works for balancd or unbalancd data. Bcaus th data ar balancd, th Typ I (squntal) and Typ III (partal) sums of squars ar th sam as ach othr. Hr ar th Typ III sums of squars from th cod abov: Sourc DF Typ III SS Man Squar F Valu Pr > F town E <. H <. E*H sgn E*sgn <. H*sgn E*H*sgn Th E*H*sgn s whatvr t taks to mak all but th town sum of squars add up to th tratmnt sum of squars w computd arlr. Notc how th man squars masur avrag varaton do to th lvls of th tratmnt n quston and how th F rato compars that varaton to th varaton du to nos. Hnc larg Fs mply addd varablty du to nonzro tratmnt ffcts.

29 In our cas, th thr way ntracton s nsgnfcant at th 5% lvl. Ths wll not always b th cas so t s of ntrst to undrstand that a thr way ntracton s th falur of th two way ntractons of two of th factors to hav th sam pattrn across th lvls of th thrd factor and n gnral a k-way ntracton s th falur of th pattrns of k- ntractons to b th sam across th lvls of th rmanng factor. Hr for xampl ar H*sgn ntracton plots for two lvls of th E factor: Not nd of asl End of asl Th nd of asl sals (rght sd) ar hghr than th md asl sals (lft) but that s an nd of asl ffct, not an ntracton. It s th fact that th pattrn on th lft (two almost straght lns convrgng togthr) dffrs from that on th rght (two crossng knkd lns that ar almost horzontal at frst) that consttuts th thr way ntracton. Apparntly th rror varaton s nough to xplan th dffrncs and thr s not vdnc at th usual lvl of a ral thr way ntracton. Not that w do not

30 s th amount of varaton n th ndvdual data ponts n ths plots. W could us contrasts to ask qustons such as: Is a hgh dsplay n md asl as advantagous as a mor xpnsv nd of asl dsplay at a low lvl, and dos ths answr dpnd on whthr thr s a sgn for my product? (s th lttl X pattrn btwn th two ntracton plots) Coffcnts for ovrall contrast End \ Ht low mddl hgh no ys - Coffcnts for contrast ntracton wth sgn factor Ht low mddl hgh (nd,sgn) (no, ys) (ys,ys) - (no, no) - (ys, no) On way to do ths, as llustratd n th dmo, s just to crat a tratmnt varabl to dlnat th tratmnts thn b sur you assgn th abov coffcnts to th propr lvls. Notcng that w ar just talkng about a lnar combnaton (wghtd sum) of mans, w can also just us th ruls to formulat a t-tst whos squar would b th F tst for th contrast. Now suppos you want to run an xprmnt wth factors ach at lvls. Thn for vn a sngl obsrvaton at ach

31 tratmnt combnaton you would nd = 96 obsrvatons and you would hav man ffcts plus ()/=66 two way ntractons lavng 5 dgrs of frdom for way and hghr ntractons. Prhaps w can rduc th numbr of obsrvatons ndd by gvng up on stmatng hgh ordr ntractons. To do so, w nxt look at fractonal factorals.

ST 524 NCSU - Fall 2008 One way Analysis of variance Variances not homogeneous

ST 524 NCSU - Fall 2008 One way Analysis of variance Variances not homogeneous ST 54 NCSU - Fall 008 On way Analyss of varanc Varancs not homognous On way Analyss of varanc Exampl (Yandll, 997) A plant scntst masurd th concntraton of a partcular vrus n plant sap usng ELISA (nzym-lnkd