Experimental Design and the Analysis of Variance

Transcription

1 Expermenl Desgn nd he nlyss of Vrnce

2 Comprng > Groups - Numerc Responses Exenson of Mehods used o Compre Groups Independen Smples nd Pred D Desgns Norml nd non-norml d dsruons D Desgn Independen Smples (CRD) Pred D (RBD) Norml -Tes 1-Wy NOV -Tes -Wy NOV Nonnorml Kruskl- Wlls Tes redmn s Tes

3 Compleely Rndomzed Desgn (CRD) Conrolled Expermens - Sujecs ssgned rndom o one of he remens o e compred Oservonl Sudes - Sujecs re smpled from exsng groups Sscl model y j s mesuremen from he j h sujec from group : yj j j where s he overll men, s he effec of remen, j s rndom error, nd s he populon men for group

4 1-Wy NOV for Norml D (CRD) or ech group on he men, sndrd devon, nd smple sze: 1 ) (.. j j j j n y y s n y y On he overll men nd smple sze N y N y n y n y n n N j j

5 nlyss of Vrnce - Sums of Squres Tol Vron n ( ) 1 1 j1 j.. Tol TSS y y df N Beween Group (Smple) Vron SST n ( y j y n y y dft ) ( 1...) 1 Whn Group (Smple) Vron SSE n ( y j j y n 1 1.) ( 1) 1 s df E N TSS SST SSE df Tol df T df E

6 nlyss of Vrnce Tle nd -Tes Source of Vron Sum of Squres Degrres of reedom Men Squre Tremens SST -1 MST=SST/(-1) =MST/MSE Error SSE N- MSE=SSE/(N-) Tol TSS N-1 ssumpon: ll dsruons norml wh common vrnce H 0 : No dfferences mong Group Mens ( 1 =0) H : Group mens re no ll equl (No ll re 0) T. S.: R. R.: os os MST MSE P vl : P(, 1, N os ) ( Tle 9)

7 Expeced Men Squres Model: y j = + + j wh j ~ N(0,s ), Sn = 0: 1 ) ( ) ( rue), s ( oherwse 1 ) ( ) ( rue, s 0 : When 1) ( 1 1 ) ( ) ( 1 ) ( ) ( MSE E MST E H MSE E MST E H n n MSE E MST E n MST E MSE E s s s s s

8 Expeced Men Squres 3 cors effec mgnude of -ssc (for fxed ) True group effecs ( 1,, ) Group smple szes (n 1,,n ) Whn group vrnce (s ) os = MST/MSE When H 0 s rue ( 1 = = =0), E(MST)/E(MSE)=1 Mrgnl Effecs of ech fcor (ll oher fcors fxed) s spred n ( 1,, ) E(MST)/E(MSE) s (n 1,,n ) E(MST)/E(MSE) (when H 0 flse) s s E(MST)/E(MSE) (when H 0 flse)

9 ) =100, 1 =-0, =0, 3 =0, s = 0 E( MST ) E( MSE) B) =100, 1 =-0, =0, 3 =0, s = n B C D C) =100, 1 =-5, =0, 3 =5, s = 0 D) =100, 1 =-5, =0, 3 =5, s = 5

10 Exmple - Sesonl De Perns n Rvens Tremens - = 4 sesons of yer (3 replces ech) Wner: Novemer, Decemer, Jnury Sprng: erury, Mrch, prl Summer: My, June, July ll: ugus, Sepemer, Ocoer Response (Y) - Vegeon (percen of ol pelle wegh) Trnsformon (or pproxme normly): Y ' rcsn Y 100 Source: K.. Engel nd L.S. Young (1989). Spl nd Temporl Perns n he De of Common Rvens n Souhwesern Idho, The Condor, 91:37-378

11 Sesonl De Perns n Rvens - D/Mens y y y y y Y Wner(=1) ll(=) Summer(=3) ll (=4) j= j= j= Y' Wner(=1) ll(=) Summer(=3) ll (=4) j= j= j=

12 Sesonl De Perns n Rvens - D/Mens Plo of Trnsformed D y Seson Trnsformed % Vegeon Seson

13 Sesonl De Perns n Rvens - NOV Tol Vron TSS Beween Group SST :(df Tol ( ) Vron 3 ( ) Whn Group Vron : ) :(df (df SSE ( ) E... ( ) T 4-1 3)... ( ) 1-4 8)... ( ) NOV Source of Vron SS df MS P-vlue cr Beween Groups Whn Groups Tol Do no conclude h sesons dffer wh respec o vegeon nke

14 Sesonl De Perns n Rvens - Spredshee Monh Seson Y' Seson MenOverll Men TSS SST SSE NOV DEC JN EB MR PR MY JUN JUL UG SEP OCT Sum Tol SS Beween Seson SS Whn Seson SS (Y -Overll Men) (Group Men-Overll Men) (Y -Group Men)

15 CRD wh Non-Norml D Kruskl-Wlls Tes Exenson of Wlcoxon Rnk-Sum Tes o > Groups Procedure: Rnk he oservons cross groups from smlles (1) o lrges ( N = n n ), djusng for es Compue he rnk sums for ech group: T 1,...,T. Noe h T T = N(N+1)/

16 Kruskl-Wlls Tes H 0 : The populon dsruons re dencl (M 1 =...=M ) H : No ll dsruons re dencl (No ll M re equl) 1 T N( N 1) 1 n T. S.: H 3( N 1) R. R.: H, 1 P vl : P( H ) n djusmen o H s suggesed when here re mny es n he d. ormul s gven on pge 344 of O&L.

17 Exmple - Sesonl De Perns n Rvens Monh Seson Y' Rnk NOV DEC JN EB MR PR MY JUN JUL UG SEP OCT T 1 = = 6 T = = 4.5 T 3 = = 8 T 4 = = 19.5 H 0 : Nosesonl dfference H :Sesonl Dfference s T. S.: H 1 (6) 1(1 1) 3 R. R.( 0.05) : H.05,41 (4.5) (8) 3 (19.5) 3 3(1 1) P vlue : P( H 5.1).163

18 s Trnsformons for Consn Vrnce k y y or y y Used for Posson Dsruon k 1 T T s k y ln y or y ln y 1 T ^ 1 s k 1 y sn y Used for Bnoml Dsruon k, y T T 1 n Box-Cox Trnsformon: Power Trnsformon used o on normly nd ofen consn vrnce y T y 0 ln y 0

19 Welch s Tes Unequl Vrnces 1 1 * pprox * 1, ~ 1 W W W W W W W W n w w w s wy w y w w C m C n w C m

20 C m * W Exmple Sesonl De Perns n Rvens W W Seson n_ s_ yr_ s_^ w_ w_y_ w_y_^c_w Sum wy w y 1 1 w w n 1 w 4 1 C W * C W W mw 0.760(1.3390) qf (.95,3, 4.3) n R 0.05,3,4.3

21 Lner Conrss Lner funcons of he remen mens (populon nd smple) such h he coeffcens sum o 0. Used o compre groups or prs of remen mens, sed on reserch queson(s) of neres Populon Conrs: l... where 0 ^ 1 1 Esmed Conrs: l y... y y s s... s V l V l MSE SE l MSE 1 ^ ^ ^ ^ ^ 1 n1 n 1 n 1 n 1 n ^ ^ MSE ^ ^ MSE n... n n V l SE l n 1 n 1

22 Orhogonl Conrss & Sums of Squres Two Conrss: l l l1, l re orhogonl f: 0 for lnced d, 0 n 1 1 Conrs Sum of Squres: SSC ^ y l 1 n n 1 1 for lnced d, SSC ^ nl 1 mong remens, we cn on 1 prwse orhogonl Conrss: l,..., l 1 1 Then, we cn decompose he Beween Tremen Sum of Squres no he Conrss: SST SSC... SSC where ech of he Conrs Sums of Squres hs 1 degree of freedom 1 1 or ny conrs: Tesng H : l 0 H : l 0 0k k k k SSCk l -Tes: TS : k RR : k,1, N -Tes: k ^ MSE SE ^ k ^ l k RR : k /; N

23 Smulneous Tess of Mulple Conrss Usng m conrss for comprsons mong remens Ech conrs o e esed sgnfcnce level, whch we lel s I for ndvdul comprson Type I error re Proly of mkng les one flse rejecon of one of he m null hypoheses s he expermenwse Type I error re, whch we lel s E Tess re no ndependen unless he error (Whn Group) degrees re nfne, however Bonferron nequly mples h E m I Choose I = E / m

24 Scheffe s Mehod for ll Conrss Cn e used for ny numer of conrss, even hose suggesed y d. Conservve (Wde CI s, Low Power) l s ^ ^ ^ l y SE l MSE 1 1 n H : l 0 H : l ^ ^ ^ Rejec H f l SE l ( 1) df 1, df df N 0, df, df 1 E 1 Error Smulneous 1 ^ ^ ^ 100% Confdence Inervls: lse l ( 1) E, df, df 1

25 Pos-hoc Comprsons of Tremens If dfferences n group mens re deermned from he - es, reserchers wn o compre prs of groups. Three populr mehods nclude: sher s LSD - Upon rejecng he null hypohess of no dfferences n group mens, LSD mehod s equvlen o dong prwse comprsons mong ll prs of groups s n Chper 6. Tukey s Mehod - Specfclly compres ll (-1)/ prs of groups. Ulzes specl le (Tle 11, p. 701). Bonferron s Mehod - djuss ndvdul comprson error res so h ll conclusons wll e correc desred confdence/sgnfcnce level. ny numer of comprsons cn e mde. Very generl pproch cn e ppled o ny nferenl prolem

26 sher s Les Sgnfcn Dfference Procedure Proeced Verson s o only pply mehod fer sgnfcn resul n overll -es or ech pr of groups, compue he les sgnfcn dfference (LSD) h he smple mens need o dffer y o conclude he populon mens re no equl 1 1 LSDj / MSE wh df N n n j Conclude f y y LSD j. j. j

27 Tukey s W Procedure More conservve hn sher s LSD (mnmum sgnfcn dfference nd confdence nervl wdh re hgher). Derved so h he proly h les one flse dfference s deeced s (expermenwse error re) MSE Wj q (, ) q gven n Tle 11, p. 701 wh N - n Conclude f y y W j. j. j Tukey's Confdence Inervl:. j. y y W j When he smple szes re unequl, use q (, ) 1 1 Wj MSE n n j

28 Bonferron s Mehod (Mos Generl) Wsh o mke C comprsons of prs of groups wh smulneous confdence nervls or -sded ess When ll pr of remens re o e compred, C = (-1)/ Wn he overll confdence level for ll nervls o e correc o e 95% or he overll ype I error re for ll ess o e 0.05 or confdence nervls, consruc (1-(0.05/C))100% CIs for he dfference n ech pr of group mens (wder hn 95% CIs) Conduc ech es =0.05/C sgnfcnce level (rejecon regon cu-offs more exreme hn when =0.05) Crcl -vlues re gven n le on clss wese, we wll use noon: /,C, where C=#Comprsons, = df

29 Bonferron s Mehod (Mos Generl) B j /, C, v ( gven on clss wese wh Conclude MSE j f 1 n y. 1 n j y j. v N-) B j Bonferron 's Confdence Inervl : y y Bj. j.

30 Exmple - Sesonl De Perns n Rvens Noe: No dfferences were found, hese clculons re only for demonsron purposes MSE n 3.05,8.306 q.05, 4, df , C6, df 8 LSD W B j j j ( ) ( ) 1 3 ( ) E E Comprson( vs j) Group Men Group j MenDfference 1 vs vs vs vs vs vs

31 Rndomzed Block Desgn (RBD) > Tremens (groups) o e compred Blocks of homogeneous uns re smpled. Blocks cn e ndvdul sujecs. Blocks re mde up of suuns Suuns whn lock receve one remen. When sujecs re locks, receve remens n rndom order. Oucome when Tremen s ssgned o Block j s leled Y j Effec of Tr s leled Effec of Block j s leled j Rndom error erm s leled j Effcency gn from removng lock-o-lock vrly from expermenl error

32 Y 1 Rndomzed Complee Block Desgns j j j j j 0 E( ) 0 V( ) s j j Noe: 1 1 Y 1 Y11... Y Y Y Y Tes for dfferences mong remen effecs: H 0 : ( 1... ) H : No ll = 0 (No ll re equl)

33 RBD - NOV -Tes (Norml D) D Srucure: ( Tremens, Blocks) Men for Tremen : Men for Sujec (Block) j: Overll Men: y.. y. Overll smple sze: N = y. j NOV:Tremen, Block, nd Error Sums of Squres 1 j1 1 j1 1 j1 TSS y y df j.. SST y y df 1 SSB y y df 1 s EMST j.... j.. 1 Tol T B SSE y y y y TSS SST SSB df ( 1)( 1) E MSE j s 1 E

34 RBD - NOV -Tes (Norml D) NOV Tle: Source SS df MS Tremens SST -1 MST = SST/(-1) = MST/MSE Blocks SSB -1 MSB = SSB/(-1) Error SSE (-1)(-1) MSE = SSE/[(-1)(-1)] Tol TSS -1 H 0 : ( 1... ) H : No ll = 0 (No ll re equl) T. S.: R. R.: P vl os os : P( MST MSE, 1,( 1)( 1) os )

35 Prwse Comprson of Tremen Mens Tukey s Mehod- q n Sudenzed Rnge Tle wh = (-1)(-1) W j q (, v) MSE Conclude j f y. y j. W j Tukey' s Confdence Inervl : y y Wj. j. Bonferron s Mehod - -vlues from le on clss wese wh = (-1)(-1) nd C=(-1)/ B j /, C, v Conclude MSE j f y. y j. B j Bonferron 's Confdence Inervl : y y Bj. j.

36 Expeced Men Squres / Relve Effcency Expeced Men Squres: s wh CRD, he Expeced Men Squres for Tremen nd Error re funcons of he smple szes (, he numer of locks), he rue remen effecs ( 1,, ) nd he vrnce of he rndom error erms (s ) By ssgnng ll remens o uns whn locks, error vrnce s (much) smller for RBD hn CRD (whch comnes lock vron&rndom error no error erm) Relve Effcency of RBD o CRD (how mny mes s mny replces would e needed for CRD o hve s precse of esmes of remen mens s RBD does): RE ( RCB, CR) MSE MSE CR RCB ( 1) MSB ( 1) MSE ( 1) MSE

37 Exmple - Cffene nd Endurnce Tremens: =4 Doses of Cffene: 0, 5, 9, 13 mg Blocks: =9 Well-condoned cyclss Response: y j =Mnues o exhuson for cycls dose D: Dose \ Sujec

38 Plo of Y versus Sujec y Dose Tme o Exhuson mg 5 mg 9mg 13 mg Cycls

39 Exmple - Cffene nd Endurnce Sujec\Dose 0mg 5mg 9mg 13mg Suj MenSuj Dev Sqr Dev Dose Men Dose Dev Squred Dev TSS TSS ( ) SST TSS 9 ( ) SSB 4 ( ) SST SSB ( ) ( ) ( ) SSE ( ) df Tol 9(103.68) ( ) (9) 1 35 ( ) df E df T df (4 1)(9 1) 4 B

40 Exmple - Cffene nd Endurnce Source df SS MS Dose Cycls Error Tol H H : No Cffene : Dfference R. R.( P 0 T. S.: os vlue Conclude MST MSE 0.05) : : P( s h rue Dose Exs os mens Effec mong.05,3,4 5.9).0036 ( 1 Doses re no ll (rom EXCEL) equl 4 0)

41 Exmple - Cffene nd Endurnce Tukey' sw : q.05,4, W Bonferron 's B :.05/,6,4.875 B Doses Hgh Men Low Men Dfference Conclude 5mg vs 0mg mg vs 0mg mg vs 0mg mg vs 5mg NSD 13mg vs 5mg NSD 13mg vs 9mg NSD

42 Exmple - Cffene nd Endurnce Relve Effcency of Rndomzed Block o 4 9 MSB MSE 5.57 ( 1) MSB ( 1) MSE RE ( RCB, CR) ( 1) MSE Compleely Rndomzed 8(694.75) 9(3)(5.57) (9(4) 1)(5.57) Desgn : Would hve needed 3.79 mes s mny cyclss per dose o hve he sme precson on he esmes of men endurnce me. 9(3.79) 35 cyclss per dose 4(35) = 140 ol cyclss

43 RBD -- Non-Norml D redmn s Tes When d re non-norml, es s sed on rnks Procedure o on es ssc: Rnk he remens whn ech lock (1=smlles, =lrges) djusng for es Compue rnk sums for remens (T ) cross locks H 0 : The populons re dencl (M 1 =...=M ) H : Dfferences exs mong he group medns 1 T S T ( 1)..: r 3 ( 1) 1 R. R.: r, 1 P vl P : ( r )

44 Exmple - Cffene nd Endurnce H H Sujec\Dose 0mg 5mg 9mg 13mg Rnks 0mg 5mg 9mg 13mg Tol T S : No Dose Dfferences : Dose Dfferences Exs (4)(4 1) 180..: r (10) (8) 3(9)(4 1) R. R.( 0.05) : P r.05,41 -vlue: P( 14.).006 (rom EXCEL) Conclude Medns re no ll equl

45 Ln Squre Desgn Desgn used o compre remens when here re wo sources of exrneous vron (ypes of locks), ech oserved levels Bes sued for nlyses when 10 Clssc Exmple: Cr Tre Comprson Tremens: 4 Brnds of res (,B,C,D) Exrneous Source 1: Cr (1,,3,4) Exrneous Source : Poson (Drver ron, Pssenger ron, Drver Rer, Pssenger Rer) Cr\Poson D P DR PR 1 B C D B C D 3 C D B 4 D B C

46 Ln Squre Desgn - Model Model ( remens, rows, columns, N= ) : y jk k j jk Overll Men Effec of Tremen k y y k k.. k... j jk Effec due o row y y ^ ^ ^... ^..... Effec due o Column j y y Rndom Error Term y j. j....

47 Ln Squre Desgn - NOV & -Tes Tol Sum of Squres : TSS Tremen Sum of Squres SST Row Sum of Squres SSR 1 j1 1 Column Sum of Squres SSC y.. j1 y k 1 jk y. j. y y.. k... y... y... y... df R df 1 df df C T Error Sum of Squres SSE TSS SST SSR SSC df E ( 1) 3( 1) ( 1)( ) H 0 : 1 = = = 0 H : No ll k = 0 TS: os = MST/MSE = (SST/(-1))/(SSE/((-1)(-))) RR: os, -1, (-1)(-)

48 Prwse Comprson of Tremen Mens Tukey s Mehod- q n Sudenzed Rnge Tle wh = (-1)(-) W j q (, v) MSE Conclude j f y. y j. W j Tukey' s Confdence Inervl : y y Wj. j. Bonferron s Mehod - -vlues from le on clss wese wh = (-1)(-) nd C=(-1)/ B j /, C, v Conclude MSE j f y. y j. B j Bonferron 's Confdence Inervl : y y Bj. j.

49 Expeced Men Squres / Relve Effcency Expeced Men Squres: s wh CRD, he Expeced Men Squres for Tremen nd Error re funcons of he smple szes (, he numer of locks), he rue remen effecs ( 1,, ) nd he vrnce of he rndom error erms (s ) By ssgnng ll remens o uns whn locks, error vrnce s (much) smller for LS hn CRD (whch comnes lock vron&rndom error no error erm) Relve Effcency of LS o CRD (how mny mes s mny replces would e needed for CRD o hve s precse of esmes of remen mens s LS does): RE ( LS, CR) MSE MSE CR LS MSR MSC ( 1) MSE ( 1) MSE

50 Power pproch o Smple Sze Choce R Code When he mens re no ll equl, he -ssc s non-cenrl : * 1 1 ~ 1,, where where N n 1 1 When ll smple szes re equl: where s The power of he es, when conduced he sgnfcnce level of : n * * Pr 1 ; 1, N ~ 1, N, In R: 1 ; 1, N qf (1, 1, N ) n Power = 1 1 pf qf (1, 1, N ), 1, N, s N

51 -Wy NOV nomnl or ordnl fcors re eleved o e reled o qunve response ddve Effecs: The effecs of he levels of ech fcor do no depend on he levels of he oher fcor. Inercon: The effecs of levels of ech fcor depend on he levels of he oher fcor Noon: j s he men response when fcor s level nd cor B j

52 j -Wy NOV - Model y 1,..., j 1,..., k 1,..., n jk j j jk h y Mesuremen on k un recevng cors level, B level j jk Overll Men jk Effec of Effec of j h h level of fcor level of fcor B h h j Inercon effec when level of nd j level of B re comned Rndom Error Terms Model depends on wheher ll levels of neres for fcor re ncluded n expermen: xed Effecs: ll levels of fcors nd B ncluded Rndom Effecs: Suse of levels ncluded for fcors nd B Mxed Effecs: One fcor hs ll levels, oher fcor suse

53 xed Effecs Model cor : Effecs re fxed consns nd sum o 0 cor B: Effecs re fxed consns nd sum o 0 Inercon: Effecs re fxed consns nd sum o 0 over ll levels of fcor B, for ech level of fcor, nd vce vers Error Terms: Rndom Vrles h re ssumed o e ndependen nd normlly dsrued wh men 0, vrnce s 1 0, j 0 j 0 j j 0 jk ~ N 0, s j1 1 j1

54 Exmple - Thldomde for IDS Response: 8-dy wegh gn n IDS pens cor : Drug: Thldomde/Plceo cor B: TB Sus of Pen: TB + /TB - Sujecs: 3 pens (16 TB + nd 16 TB - ). Rndom ssgnmen of 8 from ech group o ech drug). D: Thldomde/TB + : 9,6,4.5,,.5,3,1,1.5 Thldomde/TB - :.5,3.5,4,1,0.5,4,1.5, Plceo/TB + : 0,1,-1,-,-3,-3,0.5,-.5 Plceo/TB - : -0.5,0,.5,0.5,-1.5,0,1,3.5

55 NOV pproch Tol Vron (TSS) s proned no 4 componens: cor : Vron n mens mong levels of cor B: Vron n mens mong levels of B Inercon: Vron n mens mong comnons of levels of nd B h re no due o or B lone Error: Vron mong sujecs whn he sme comnons of levels of nd B (Whn SS)

56 nlyss of Vrnce Tol Vron: TSS y y df n 1 1 j1 k 1 cor Sum of Squres: SS n y y df 1 1 j cor B Sum of Squres: SSB n y y df 1 Inercon Sum of Squres: n jk. j.... Tol B SSB n y y y y df ( 1)( 1) n 1 j1 k 1 1 j1 j.... j.... Error Sum of Squres: SSE y y df ( n 1) TSS = SS + SSB + SSB + SSE df Tol = df + df B + df B + df E jk j. E B

57 NOV pproch - xed Effecs Source df SS MS cor -1 SS MS=SS/(-1) =MS/MSE cor B -1 SSB MSB=SSB/(-1) B=MSB/MSE Inercon (-1)(-1) SSB MSB=SSB/[(-1)(-1)] B=MSB/MSE Error (n-1) SSE MSE=SSE/[(n-1)] Tol n-1 TSS Procedure: rs es for nercon effecs If nercon es no sgnfcn, es for cor nd B effecs Tes for Inercon: Tes for cor Tes for cor B H :... 0 H :... 0 H : H : No ll 0 H : No ll 0 H : No ll 0 j MSB MS MSB TS : B TS : TS : B MSE MSE MSE RR : RR : RR : B,( 1)( 1), ( n1),( 1), ( n1) B ( 1), ( n1) j

58 Exmple - Thldomde for IDS 7.5 Indvdul Pens Negve Posve Group Mens wgn menwg Plceo T hldomde Plceo T hldomde drug drug Repor WTGIN GROUP TB+/Thldomde TB-/Thldomde TB+/Plceo TB-/Plceo Tol Men N Sd. Devon

59 Exmple - Thldomde for IDS Dependen Vrle: WTGIN Source Correced Model Inercep DRUG TB DRUG * TB Error Tol Correced Tol Tess of Beween-Sujecs Effecs Type III Sum of Squres df Men Squre Sg R Squred =.5 (djused R Squred =.471) There s sgnfcn Drug*TB nercon ( DT =5.897, P=.0) The Drug effec depends on TB sus (nd vce vers)

60 NOV ddve Model If he Inercon s no sgnfcn, he Inercon erm cn e removed, nd n ddve model cn e f. n j Y SSE Y Y Y Y df n 1 jk j jk jk E 1 j1 k 1 Source df SS MS cor -1 SS MS=SS/(-1) =MS/MSE cor B -1 SSB MSB=SSB/(-1) B =MSB/MSE Error n--+1 SSE MSE =SSE /[n--+1] Tol n-1 TSS

61 Comprng Mn Effecs (No Inercon) Tukey s Mehod- q n Sudenzed Rnge Tle wh = n--+1 MSE B W q (, v) W q (, v) j j n Conclude: f y y W f y y W j.. j.. j... j... j..... j. Tukey's CI: ( ) : y y W ( ) : y y W j j j j MSE n Bonferron s Mehod - -vlues n Bonferron le wh =n--+1 MSE B MSE B j /, ( 1)/, v B j /, ( 1)/, v n n Conclude: f y y B f y y B j.. j.. j... j... j..... j. Bonferron's CI:( α - α ): y y B ( - ): y y B j j j j B j j B j B B j

62 Comprng Mn Effecs (Inercon) Tukey s Mehod- q n Sudenzed Rnge Tle wh = (n-1) W q (, v) j MSE n h Whn k level of cor B, Conclude: f y y W j k. jk. j j k. jk. Tukey's CI: ( ) : y y W Smlr for cor B n Bonferron s Mehod - -vlues n Bonferron le wh = (n-1) B j /, ( 1)/, v MSE n h Whn k level of B, Conclude: Bonferron's CI:( α - α ): y y B j k. jk. j f y y B j k. jk. j j

63 Mscellneous Topcs -cor NOV cn e conduced n Rndomzed Block Desgn, where ech lock s mde up of expermenl uns. nlyss s drec exenson of RBD wh 1-fcor NOV corl Expermens cn e conduced wh ny numer of fcors. Hgher order nercons cn e formed (for nsnce, he B nercon effecs my dffer for vrous levels of fcor C). When expermens re no lnced, clculons re mmensely messer nd you mus use sscl sofwre pckges for clculons

64 Unequl Smple Szes When smple szes re unequl, clculons nd prmeer nerpreons (especlly mrgnl ones) ecome messer Oservonl sudes ofen hve unequl smple szes due o vlly of smplng uns for cern comnons of fcor levels (vllgers of cern ypes n rurl sudy for nsnce) Expermenl sudes, even when plnned wh equl smple szes cn end up unlnced hrough echncl prolems or drop ous Some condons my e cheper o mesure hn ohers, nd wll hve lrger smple szes Some suons hve prculr conrss of hgher mpornce

65 Regresson pproch - I Smple Szes: # of Cses when cor s level, j: n nj j j j j T j j jk j j1 1 1 j1 k 1 nj n n n n n n Y Y Y j Y N Model: Y ~ 0, s (ndependen) jk j j jk jk Resrcons on Effecs: j j j 1 j1 1 j , 1 j 1 j j 1 j

66 Regresson Model: Regresson pproch - II Y X... X X... X jk 1 jk1 1 jk, 1 1 jk 1 jk,... X X X X 11 jk1 jk 1, 1 jk, 1 jk, jk 1 f cse from level 1 of fcor where: X jk1 1 f cse from level of fcor 0 oherwse 1 f cse from level -1 of fcor X jk, 1 1 f cse from level of fcor 0 oherwse 1 f cse from level 1 of fcor B where: X jk 1 f cse from level of fcor B 0 oherwse 1 f cse from level -1 of fcor B X jk, 1 f cse from level of fcor B 0 oherwse

67 Regresson pproch Exmple I Wrer Type (B) Syle (cor ) Poes (B1) Yer Pek Novelss (B) Yer Pek Concepulss (1) Elo 3 zgerld 9 (nders) Cummngs 6 Hemngwy 30 Plh 30 Melvlle 3 Pound 30 Lwrence 35 Wlur 34 Joyce 40 n & Men n11= n1= Expermenlss () Bshop 9 Jmes 38 (Seekers) Moore 3 ulkner 39 Wllms 40 Dckens 41 Lowell 41 Woolf 45 Sevens 4 Conrd 47 ros 48 Twn 50 Hrdy 51 n & Men n1= n= Y X X X X jk 1 jk1 1 jk 11 jk1 jk jk X_jk1 X_jk X1X

68 Tesng Sreges Models Model 1: ll cor, cor B, nd Inercon B Effecs Model :ll cor, cor B Effecs (Remove Inercon) Model 3: ll cor B,Inercon B Effecs (Remove ) Model 4:ll cor,inercon B Effecs (Remove B) To es for Inercon Effecs, Model 1 s ull Model, Model s Reduced df Numeror =(-1)(-1) df den =n T - Tesng for cor Effecs, ull=model 1, Reduced=Model 3 df Numeror =(-1) df den =n T - Tesng for cor B Effecs, ull=model 1, Reduced=Model 4 df Numeror =(-1) df den =n T -

69 ^ Regresson pproch Exmple - Connued Y X X X X jk 1 jk1 1 jk 11 jk1 jk jk Model 1: jk 1 jk1 1 jk 11 jk1 jk Y X.59X 0.9X X 1 1 Model : E Y X X Y X.63X Model 3: E Y X X X X E Y jk 1 jk1 1 jk 1 jk ^ X X X Y X 0.47X X ^ 1 jk 11 jk1 jk 1 Model 4: E Y X X X Y X 0.6X X jk 1 jk1 11 jk1 jk 1 1 NOV Model1 Model Model3 Model4 df SS df SS df SS df SS Regresson Resdul Tol ^

70 H Regresson pproch Exmple - Connued : 0 H : Inercon Exss SSE R df R 0 SSE df 19 E SSE R SSE df * E R dfe : B SSE dfe * RR B TS :.95,1, E NOV Model1 Model Model3 Model4 df SS df SS df SS df SS Regresson Resdul Tol

71 H Regresson pproch Exmple - Connued E * * * B : 0 H : cor Effecs Exs: SSE R df R 1.80 RR :.95,1, H : 0 H : cor B Effecs Exs: 0 1 E SSE R df R * 5.16 RR :.95,1, NOV Model1 Model Model3 Model4 df SS df SS df SS df SS Regresson Resdul Tol B

72 Esmng Tremen nd cor Level Mens/Conrss Tremen Mens: ^ jk k 1 Prmeer: j Esmor: j = Yj Esmed Sndrd Error: nj cor Mens: j ^ j1 j1 n j Y ^ s Prmeer: = Esmor: Esmed Sndrd Error: cor B Mens: Y Prmeer: = Esmor: Esmed Sndrd Error: j Y j j ^ MSE n j j ^ ^ 1 1 MSE 1 j s j 1 nj Conrs or Lner uncon of cor Mens: s j MSE MSE Prmeer: L c Esmor: L c Esmed Sndrd Error: s L c ^ ^ ^ j1 nj Conrs or Lner uncon of cor B Mens: ^ ^ ^ MSE Prmeer: L c Esmor: LB c j j Esmed Sndrd Error: s B L c B j j j1 Conrs or Lner uncon of Tremen Mens: j j1 j1 1 nj ^ ^ B jj B j j d Sndrd Error: sl B 1 j1 1 j1 Prmeer: L c Esmor: L c Y Esme j1 1 n j MSE 1 1 c n j 1 j1 j

73 Sndrd Error Mulplers Sngle Comprsons: / ; N Generl Mulple Comprsons of Tremen (Cell) Mens : Scheffe: S 1 ; 1, N Bonferron: B g, N g # of comprsons Tukey (ll prs of remen mens): T 1 q ;, N Generl Mulple Comprsons of cor Level Mens : B B g N g Scheffe: cor : S 1 ; 1, N cor B: S 1 1 ; 1, N Bonferron: cor or cor B :, # of comprsons 1 1 Tukey: cor : T q ;, N cor B: TB q ;, N

74 Creve Lfe Cycles Comprng Tremen Mens Comprng ll 4 Tremen Mens(hough no nercon ws presen): MSE 9.3 MSE 9.3 Y 8.60 n 5 s Y.4 Y 33.0 n 5 s Y n11 n1 MSE sy Y 1 n s Y 1 Y n n T q 0.95, 4, sy j Y ' j ' MSE n j n Y 11 Y 1 s Y 11 Y 1 HSD Y 11 Y s (3.43) Y Y HSD.81(3.8) Y 11 Y s Y 11 Y HSD Y 1 Y 1 s Y 1 Y 1 HSD ' j ' MSE n (3.17) ( 3.8) Y 1 Y s Y 1 Y HSD Y 1 Y s Y 1 Y HSD (3.17) (3.01) Concepulss/Poes Concepulss/Novelss Expermenlss/Poes Expermenlss/Novelss

75 Creve Lfe Cycles Comprng cor Level Mens cor (Syle): 1 j ^ j1 Y11 Y j ^ j1 Y 1 Y ^ ^ 1 Y Y s ^ ^ ( 1) , % CI for (Concepulss - Expermenlss): (.093)(3.3) , cor B (Wrer Type): 1 ^ 1 Y11 Y ^ ^ j1 Y Y Y ^ ^ 1 ^ s Y ( 1) , % CI for (Poes - Novelss): (.093)(3.3) ,1.57

76 1-Wy Rndom Effecs Model Tremen Levels n Expermen re Smple from Populon of Levels Effecs re Rndom Vrles (No xed Consns) Y 1,..., ; j 1,..., n j j NID ~ NID 0, s ~ 0, s, ndependen j j NOV oned s n xed Effecs Model: T 1 SST SST n Y Y df E MST E s ns 1 1 n SSE SSE Yj Y dfe n 1 EMSE E s 1 j1 n 1 ^ ^ MST Esmng Populon Men: Y SE 1 100% CI: n ^ ^ MST MSE Pon Esmes of Vrnce Componens: s MSE s n MST Tesng for Tr Effecs: H : s 0 H : s 0 TS : RR : MSE ^ /; 1 0 ; l, n1 MST n

77 Rndomzed Complee Block Desgns Sujecs s Blocks Y 1,..., ; j 1,..., j j j j j 1 0 E( ) 0 V ( ) s Rndom Blocks: ~ N 0, s j j j Noe: 1 1 Y 1 Y11... Y Y 1 1 Y Y Tes for dfferences mong remen effecs: H 0 : H : No ll = 0 TS: = MST/MSE RR: ;-1,(-1)(-1)

78 ssume: Mxed Effecs Models cor xed (ll levels of neres n sudy) 1 0 cor B Rndom (Smple of levels used n sudy) j ~ N(0,s ) (Independen) B Inercon erms Rndom ) j ~ N(0,s (Independen) nlyss of Vrnce s compued excly s n xed Effecs cse (Sums of Squres, df s, MS s) Error erms for ess chnge (See nex slde).

79 Expeced Men Squres for -Wy NOV Men Squre df xed Model Rndom Model Mxed Model ( xed, B xed) ( Rndom, B Rndom) ( xed, B Rndom) n n 1 1 MS -1 s + s ns ns s ns 1 1 MSB -1 s + n j j1 s ns ns s ns ns 1 n j 1 j1 MSB 1 1 s + s ns s ns 11 MSE n 1 s s s

80 NOV pproch Mxed Effecs Source df SS MS cor -1 SS MS=SS/(-1) =MS/MSB cor B -1 SSB MSB=SSB/(-1) B=MSB/MSB Inercon (-1)(-1) SSB MSB=SSB/[(-1)(-1)] B=MSB/MSE Error (n-1) SSE MSE=SSE/[(n-1)] Tol n-1 TSS Procedure: rs es for nercon effecs If nercon es no sgnfcn, es for cor nd B effecs Tes for Inercon: Tes for cor Tes for cor B H : s 0 H :... 0 H : s 0 H : s 0 H : No ll 0 H : s 0 MSB MS MSB TS : B TS : TS : B MSE MSB MSB RR : RR : RR : B,( 1)( 1), ( n1),( 1),( 1)( 1) B,( 1),( 1)( 1)

81 Comprng Mn Effecs for (No Inercon) Tukey s Mehod- q n Sudenzed Rnge Tle wh = (-1)(-1) W q (, v) j MSB n Conclude: f y y W j.. j.. Tukey's CI: ( ) : y y W j.. j.. Bonferron s Mehod - -vlues n Bonferron le wh = (-1)(-1) B j Conclude: /, ( 1)/, v MSB n f y y B j.. j.. Bonferron's CI:( α - α ): y y B j.. j.. j j j j

82 ssume: Rndom Effecs Models cor Rndom (Smple of levels used n sudy) ~ N(0,s ) (Independen) cor B Rndom (Smple of levels used n sudy) j ~ N(0,s ) (Independen) B Inercon erms Rndom ) j ~ N(0,s (Independen) nlyss of Vrnce s compued excly s n xed Effecs cse (Sums of Squres, df s, MS s) Error erms for ess chnge (See nex slde).

83 NOV pproch Rndom Effecs Source df SS MS cor -1 SS MS=SS/(-1) =MS/MSB cor B -1 SSB MSB=SSB/(-1) B =MSB/MSB Inercon (-1)(-1) SSB MSB=SSB/[(-1)(-1)] B =MSB/MSE Error (n-1) SSE MSE=SSE/[(n-1)] Tol n-1 TSS Procedure: rs es for nercon effecs If nercon es no sgnfcn, es for cor nd B effecs Tes for Inercon: Tes for cor Tes for cor B H : s 0 H : s 0 H : s 0 H : s 0 H : s 0 H : s 0 MSB MS MSB TS : B TS : TS : B MSE MSB MSB RR : RR : RR : B,( 1)( 1), ( n1),( 1),( 1)( 1) B ( 1),( 1)( 1)

84 Nesed Desgns Desgns where levels of one fcor re nesed (s opposed o crossed) wr oher fcor Exmples Include: Clssrooms nesed whn schools Lers nesed whn eed Vrees Hr swches nesed whn shmpoo ypes Swmps of vryng szes (e.g. lrge, medum, smll) Resurns nesed whn nonl chns

85 Nesed Desgn - Model Y 1,... j 1,..., k 1,..., n jk j() jk where: Y jk jk j () h h h Response for k rep of cor level, B j level whn Overll Men h Effec of level of (xed or Rndom) h h Effec of j level of B whn level of (xed or Rndom) h Rndom error erm for k rep when s, B s j()

86 Nesed Desgn - NOV Tol Vron: n TSS Y Y df n 1 cor : jk... Tol 1 j1 k 1 1 SS n Y.. Y... df 1 1 cor B Nesed Whn SSB( ) n Y Y df Error: SSE Y Y j... B( ) 1 j1 1 n jk j. E 1 j1 k1 1 df ( n 1)

87 Esmors, nlyss of Vrnce, -ess ^ ^ ^ Y Y Y Y Y j () j ^ ^ ^ ^ ed Vlues: Y = Y Y Y Y Y Y jk j () j j ^ n jk jk jk jk j jk j E 1 j1 k 1 Resduls: e Y Y Y Y SSE Y Y df n 1 cor Sum of Squres: SS n Y Y df j1 j B( ) cor B Whn Sum of Squres: SSB( ) n Y Y df 1 xed, B xed xed, B Rndom Rndom, B Rndom n n ( ) ( ) 1 1 E MS s E MS s ns E MS s ns ns 1 1 E MSB( ) n j () 1 j1 EMSB( ) s ns ( ) EMSB( ) s ns ( ) s 1 s EMSE E MSE s s E MSE MS MS MS H0 : TS : H0 : TS : H0 : s 0 TS : MSE MSB( ) MSB( ) MSB( ) MSB( ) MSB( ) H0 : 1(1)... ( ) 0 TS : B ( ) H0 : s ( ) 0 TS : B ( ) H0 : s ( ) 0 TS : B ( ) MSE MSE MSE

88 cors nd B xed 0 0 1,..., ~ N 0, s j() jk 1 j1 Tess for Dfferences mong cor Effecs H :... 0 H : No ll MS Tes Ssc: P-vlue: P MSE Rejecon Regon:, 1,( n1) Tess for Dfferences mong cor B Effecs H : 0, j H : No ll 0 0 j( ) j( ) MSB( ) Tes Ssc: P-vlue: P MSE Rejecon Regon: B( ),,( n1) B( ) B( )

89 xed Effecs Model ( nd B xed) MS cor : H0 : TS : RR : 1 ; 1, n 1 MSE s MSE EY s Y s Y n n Conrss mong Mens of : c 0 1 ^ MSE ^ L c c L c Y s L c n 1 ^ MSE 1 100% CI for L: L 1 / ; n 1 c n ll Possle Comprsons mong prs of mens of : c 1 1 C 1 MSE Tukey: Y Y ' q 1 ;, n 1 Bonferron: Y Y ' 1 ; n 1 n C MSE n MSB( ) cor B(): H0 : 1(1)... ( ) 0 TS : B ( ) RR : B ( ) 1 ; 1, n 1 MSE s MSE EY j j j() s Y j s Y j n n 1 ll Possle Comprsons mong prs of mens of B whn gven level ( ) of : CB( ) MSE MSE Tukey: Y j Y j ' q 1 ;, n 1 Bonferron: Y j Y j ' 1 ; n 1 n C B( ) n

90 Comprng Mn Effecs for Tukey s Mehod- q n Sudenzed Rnge Tle wh = (n-1)s W q (, v) j MSE 1 1 n n Conclude: f y y W j.. j.. Tukey's CI: ( ) : y y W j.. j.. Bonferron s Mehod - -vlues n Bonferron le wh = (n-1)s 1 1 B j /, ( 1)/, v MSE n n j Conclude: j.. j.. j f y y B Bonferron's CI:( α - α ): y y B j.. j.. j j j j

91 Comprng Effecs for cor B Whn Tukey s Mehod- q n Sudenzed Rnge Tle wh = (n-1)s B W q (, v) j ( k ) k Conclude: MSE n f y y W ( k ) j( k ) k. kj. Tukey's CI: ( ) : y y W ( k ) j( k ) k. kj. Bonferron s Mehod - -vlues n Bonferron le wh = (n-1)s B B j ( k ) /, k ( k 1)/, v MSE n Conclude: ( k ) j( k ) k. kj. f y y B ( k ) j( k ) k. kj. j ( k ) B j ( k ) Bonferron's CI:( - ): y y B B B j ( k ) B j ( k )

92 cor xed nd B Rndom r B B B jk j P MSE MSB H j H P MSB MS H H N N 1),(, ) ( ) ( ) ( 0 1,, 1 0 ) ( 1 : Regon Rejecon : P - vlue ) ( : Tes Ssc 0 :, 0 : cor B Effecs mong s for Dfference Tess : Regon Rejecon : P - vlue ) ( : Tes Ssc 0 No ll : 0... : cor Effecs mong s for Dfference Tess 0, ~ 0, ~ 0 s s s s

93 Mxed Effecs Model ( xed nd B Rndom) MS cor : H0 : TS : RR : 1 ; 1, 1 MSB( ) s ns ( ) MSB( ) E Y s Y s Y n n 1 C ll Possle Comprsons mong prs of mens of : c 1 Tukey: ( ) MSB( ) q 1 ;, 1 Bonferron: Y Y 1 ; 1 n C n MSB Y Y' ' MSB( ) cor B(): H0 : s( ) 0 TS : B ( ) RR : B ( ) 1 ; 1, n 1 MSE 1 1 EMSB( ) s ns ( ) EMSE s EMSB( ) EMSE s n n 1 1 s( ) ( ) ( ) pproxme df (Serhwe): ( ) s MSB MSE df n n MSB( ) MSE n n 1 n 1 pproxme 1 100% CI or s : ( ) df s 1 ; df ( ) ( ) ( ) df ( ) s ( ), ; df ( ) ( )

94 Comprng Mn Effecs for (B Rndom) Tukey s Mehod- q n Sudenzed Rnge Tle wh = S - W q (, v) j MSB( ) 1 1 n n Conclude: f y y W j.. j.. Tukey's CI: ( ) : y y W j.. j.. Bonferron s Mehod - -vlues n Bonferron le wh = S - B /, ( 1)/, ( ) j v MSB n n j Conclude: j.. j.. j.. j f y y B Bonferron's CI:( α - α ): y y B j j j j j

95 cors nd B Rndom r B B B jk j P MSE MSB H j H P MSB MS H H N N N 1),(, ) ( ) ( ) ( 0 1,, 0 ) ( : Regon Rejecon : vlue P - ) ( : Tes Ssc 0 :, 0 : cor B Effecs mong s for Dfference Tess : Regon Rejecon : vlue P - ) ( : Tes Ssc 0 : 0 : cor Effecs mong s for Dfference Tess 0, ~ 0, ~ 0, ~ s s s s s s s

96 Rndom Effecs Model ( nd B Rndom) MS MSB( ) cor : H0 : s 0 TS : RR : 1 ; 1, 1 ( ) E MS s ns ns E MSB s ns ( ) ( ) 1 1 EMS) EMSB( ) s n n 1 1 s MS MSB( ) pproxme df (Serhwe): df n n df s df s pproxme 1100% CI or s :, 1 ; df ; df cor B Whn s sme s n Mxed Effecs Model s MS MSB( ) n n 1 1

97 Elemens of Spl-Plo Desgns Spl-Plo Expermen: corl desgn wh les fcors, where expermenl uns wr fcors dffer n sze or oservonl pons. Whole plo: Lrges expermenl un Whole Plo cor: cor h hs levels ssgned o whole plos. Cn e exended o or more fcors Suplo: Expermenl uns h he whole plo s spl no (where oservons re mde) Suplo cor: cor h hs levels ssgned o suplos Blocks: ggreges of whole plos h receve ll levels of whole plo fcor

98 Spl Plo Desgn Block 1 Block Block 3 Block 4 =1, B=1 =1, B=1 =1, B=1 =1, B=1 =1, B= =1, B= =1, B= =1, B= =1, B=3 =1, B=3 =1, B=3 =1, B=3 =1, B=4 =1, B=4 =1, B=4 =1, B=4 =, B=1 =, B=1 =, B=1 =, B=1 =, B= =, B= =, B= =, B= =, B=3 =, B=3 =, B=3 =, B=3 =, B=4 =, B=4 =, B=4 =, B=4 =3, B=1 =3, B=1 =3, B=1 =3, B=1 =3, B= =3, B= =3, B= =3, B= =3, B=3 =3, B=3 =3, B=3 =3, B=3 =3, B=4 =3, B=4 =3, B=4 =3, B=4 Noe: Whn ech lock we would ssgn rndom he 3 levels of o he whole plos nd he 4 levels of B o he suplos whn whole plos

99 Exmples grculure: Vrees of crop or gs my need o e grown n lrge res, whle vrees of ferlzer or vryng growh perods my e oserved n suses of he re. Engneerng: My need long heng perods for process nd my e le o compre severl formulons of y-produc whn ech level of he heng fcor. Behvorl Scences: Mny sudes nvolve repeed mesuremens on he sme sujecs nd re nlyzed s spl-plo (See Repeed Mesures lecure)

100 Desgn Srucure Blocks: groups of expermenl uns o e exposed o ll comnons of whole plo nd suplo fcors Whole plos: expermenl uns o whch he whole plo fcor levels wll e ssgned o rndom whn locks Suplos: c suuns whn whole plos o whch he suplo fcor levels wll e ssgned o rndom. ully lnced expermen wll hve n=c oservons

101 D Elemens (xed cors, Rndom Blocks) Y jk : Oservon from wp, lock j, nd sp k : Overll men level : Effec of h level of whole plo fcor (xed) j : Effec of j h lock (Rndom) ( ) j : Rndom error correspondng o whole plo elemens n lock j where wp s ppled k : Effec of k h level of suplo fcor (xed) ) k : Inercon wn wp nd sp k (c ) jk : Inercon wn lock j nd sp k (ofen se o 0) jk : Rndom Error= (c ) jk + (c ) jk Noe h f lock/sp nercon s ssumed o e 0, represens he lock/sp whn wp nercon

102 Model nd Common ssumpons Y jk = + + j + ( ) j + k + ( ) k + jk 1 j ~ 0 NID(0, s ) ( ) j ~ NID(0, s ) c k 1 k 0 1 ( ) k c k 1 ( ) k 0 jk ~ COV NID(0. ( j s,( ) j ) ) COV ( j, jk ) COV (( ) j, jk ) 0

103 Tess for xed Effecs ) ~ ( Ssc: Tes, 0 ) ( : SP Inercon : WP ) ~ ( Ssc: Tes 0 : Suplo Tr Effecs : ) ~ ( Ssc: Tes 0 : Plo Tr Effecs : Whole 1) 1)( ( 1). 1)( ( 0 1) 1)( ( ) 1)( 1.( * 1 0 c c SP WP SP WP ERROR SP WP SP WP k c c SP SP ERROR SP SP c WP WP WP BLOCK WP WP P P MS MS k H P P MS MS H P P MS MS H

104 Comprng cor Levels Whole Plo cor Levels : 95% CI for 95% CI for : Y.. Y '.. Su Plo cor Levels : ( ) ( ) : Y. k Y. k ' MSBLOCK WP ( c 1) MSERROR Y. k Y '. k ( c 1) MSERROR MSBLOCK WP ( c 1) MS MS ( 1)( c 1) BLOCK WP ( 1)( 1) MS c BLOCK WP MSERROR 95% CI for ( k k ') : Y.. k Y.. k ' Su Plo Effecs Whn sme whole plo (Inercon) : Whole Plo Effecs whn sme su plo (Inercon) : ^ k ' k ' ERROR k c k ' (df MS ERROR gven elow)

105 Repeed Mesures Desgns Tremens/Condons o compre N sujecs o e ncluded n sudy (ech sujec wll receve only one remen) n sujecs receve r : n = N me perods of d wll e oned Effecs of r, me nd rxme nercon of prmry neres. Beween Sujec cor: Tremen Whn Sujec cors: Tme, TrxTme

106 Model Y jk j( ) overll men k ( ) k jk effec of r j( ) effec of j h 1 0 sujec n r j( ) ~ NID 0, s k effec of k h me perod k 1 k 0 ( ) jk k nercon eween r rndom error erm jk ~ nd me k NID 0, s 1 ( ) k k 1 ( ) k 0 Noe he rndom error erm s cully he nercon eween sujecs (whn remens) nd me

107 Tess for xed Effecs ) ~ ( Ssc: Tes, 0 ) ( : Tremen/Tme Inercon : ) ~ ( Ssc: Tes 0 : Tme Effecs : ) ~ ( Ssc: Tes 0 : Tremen Effecs : 1) 1)( ( 1), 1)( ( 0 1) 1)( ( 1, 1 0 1) ( 1, ) ( 1 0 n TIME TRT TIME TRT ERROR TIME TRT TIME TRT k n TIME TIME ERROR TIME TIME n TRTS TRTS TRTS SUBJECTS TRTS TRTS P P MS MS k H P P MS MS H P P MS MS H

108 Y Comprng cor Levels Comprng Tremen Levels : 95% CI for : Comprng Tme Levels : wh pproxme df : ( 1) MSERROR MSSUBJECT ( TRT ) ( 1) MS MSSUBJECT ( TRT ) ( n 1)( 1) Y MSERROR 95% CI for k k ' : Y.. k Y.. k ' n Comprng Tremen Levels Whn Tme Levels : Y ^. k '. k ' MS ERROR Y.. '.. SUBJECTS ( TRTS ) n ( ( n 1) MS 1) MS SUBJECTS ( TRTS ) n ERROR