Lectue 5 Sngle fcto desgn nd nlss
Completel ndomzed desgn (CRD
Completel ndomzed desgn In the desgn of expements, completel ndomzed desgns e fo studng the effects of one pm fcto wthout the need to tke othe nusnce vbles nto ccount The expement compes the vlues of esponse vble bsed on the dffeent levels of tht pm fcto. Fo completel ndomzed desgns, the levels of the pm fcto e ndoml ssgned to the expementl unts.
Completel ndomzed desgn stud desgn wth onl one ndependent fcto (e.g. ctego of tetment n whch the fcto s mnpulted t multple levels. Often used n expementl desgn to detemne the effect of cetn tetment o nteventon. M be contsted wth fctol desgn, whch evlutes the effects of two o moe fctos smultneousl.
Completel ndomzed desgn In the expement, onl one fcto, nd levels,,,. In ech level, thee e eplctons, =,, 3,, If = = =, the desgn s blnced. Othewse, t s n unblnced desgn. s the esult of th level nd th eplcton.
Exmple Fo n unblnced desgn, hs 7 smples, hs 5 smples, 3 hs 6 smples, nd 4 hs 6 smples. In totl, thee e 4 smples. Levels of fcto Numbe of expement unts,, 3, 4, 5, 6, 7 (7 8, 9, 0,, (5 3 3, 4, 5, 6, 7, 8 (6 4 9, 0,,, 3, 4 (6
Exmple Cn we nge the 4 expement unts n the ode of the fou levels? No. The ttenton nd skll pofcenc of mnpultos m chnge dung the expements. nd the lght ntenst m lso be dffeent. The obsevtons m not be ndependent! So we should use ndom desgn to solve ths poblem. RndomzRndNum Plot 0.0694 Plot 3 0.040356 Plot3 4 0.0434 Plot4 3 0.06343 Plot5 0.557 Plot6 3 0.66 Plot7 4 0.0830 Plot8 4 0.04 Plot9 0.55363 Plot0 0.334577 Plot 4 0.366835 Plot 0.483 Plot3 4 0.44407 Plot4 0.635086 Plot5 3 0.76573 Plot6 0.799705 Plot7 3 0.80349 Plot8 0.89444 Plot9 4 0.83865 Plot0 3 0.84884 Plot 0.853357 Plot 0.860988 Plot3 0.964504 Plot4 0.975
Exmple 4 expements fo Folc cd content n geen te Levels of fcto Obseved dt (mg Smple men 7.9, 6., 6.6, 8.6, 8.9, 0., 9.6 8.7 5.7, 7.5, 9.8, 6., 8.4 7.50 3 6.4, 7., 7.9, 4.5, 5.0, 4.0 5.8 4 6.8, 7.5, 5.0, 5.3, 6., 7.4 6.35
Dot-plot 0 8 6 4 obs obs obs3 obs4 obs5 obs6 obs7 men 0 0 3 4 5 e the dffeences cused b chnce o not? We use nlss of vnce (NOV fo futhe nlss.
Levels of fcto Dt Sum Men T T T T T Genel Dt T
Bsc ssumptons. Nomlt: smples,,, unde level hve the Noml dstbuton N(,,,,,. Homogenet of Vnce: the vnces of the Noml dstbutons e the sme,.e. 3. Rndomness. ll dt e ndependent.
Tgets. e the mens of the levels,,, the sme? (usng One-w NOV If the mens e not the sme, whch dffeence between mens s sgnfcnt? (usng multple compson
The model s The lne model,,,, ;,,, s the expementl eo of the th level nd th expement. ~ N(0,,..d Theo : e sum of constnt ndom eo Theo : E( E( 0, V(, V(, so nd
The lne model N N(, Theo 3: ~ (0,, so ~ Theo 4: The ndom eos e ndependent, so ll e lso ndependent.
Lest sque estmton Mnmze The lest sque estmto of s In pevous exmple, μ μ μ μ ( ( ( (,,,, ˆ 6.35 ˆ 5.8, ˆ 7.50, ˆ 8.7, ˆ 4 3
One-w NOV
Hpothess n one-w NOV The one-w nlss of vnce s used to test the clm tht moe thn populton mens e equl Ths s n extenson of the two ndependent smples t-test H 0 : H :,,, e not equl. If we eect H 0 unde the sgnfcnce level α, then fcto s sgnfcnt unde the level α. Othewse, fcto s not sgnfcnt.
Sum of sques Defnton Becuse Thee e onl - ndependent devtons n Q, we cll the numbe of ndependent devtons n sum of sques s degee of feedom fo sum of sques whch s often denoted s f. 0 ( ( Q ( ( ( (
Dstbuton of sum of sque (Q Theoem: ssume,,, s smple fom noml dstbuton N(μ, σ. Then. Smple men. Rto of sum of sques to σ s 3. Q ~ N(, nd Q e ndependent ~ (
Decomposng the sum of sques Men of ll dt s Defne n n feedom degee of, ( T n f T ε Defne T ( ( ( (
Sum of Sques s the sum between goups, wth degee of feedom -; ε s the sum wthn goups (.e. sum sques of eos, wth degee of feedom n-
Sum of Sques Then, ( T n f n T (,f n n f T,
Exmple The dt nd sum of sques Level Dt Rep Men 7.9, 6., 6.6, 8.6, 8.9, 0., 9.6 =7 8.7 5.7, 7.5, 9.8, 6., 8.4 =5 7.5 3 6.4, 7., 7.9, 4.5, 5.0, 4.0 4 6.8, 7.5, 5.0, 5.3, 6., 7.4 3 =6 5.8 4 =6 6.35 n=4 7.0
Exmple So 3.50 7.0 4 6.35 6 5.8 6 7.5 5 8.7 7 65.7 7.0 4 7.4 6. 6. (7.9 T 4.77 3.50.7 65
Men sque Men sque s sum of sques dvded b ts degee of feedom MS Theo: Unde the bsc ssumpton of sngle fcto desgn, we hve: E Whee n MS ( ( n n ( E( ( E(
Dstbutons unde H 0 It s poved, unde H 0, ( ~ ( n ~ nd ε e ndependent. Then ( ( n MS MS ~ F(-,n-.e. F MS MS ~ F(-,n-
The nlss of vnce tble fo the sngle fcto Souce Fcto Eo Totl T Degees of feedom f =- f ε =n- f T =n- Sum of sques ( ( T ( Gven sgnfcnce level the F (, n, then Men sque MS MS n, fnd If F F (, n, eect H 0 F to MS F MS If F F (, n, ccept H 0
Exmple (contnued We hve clculted the sum of sques. The tble of NOV s Souce Degees of feedom (DF Sum of sques ( Men sque (MS F vlue Fcto 3 3.50 7.83 3.75* Eo 0 4.77.09 Totl T 3 65.7 0.05, F 0. 95(3, 0 3.0, F 3.0, eect H 0,.e. the fou clsses hve sgnfcnt dffeence.
Exmple (contnued Menwhle, we cn get the unbsed estmton of σ : ˆ.09 Estmton of mens e ˆ 8.7, ˆ 7.50, ˆ 3 5.8, ˆ 4 The men unde s the lgest. 0.05, t So t ( n ( n t ˆ / 0.975 (0 8.7 The ntevl estmton of μ s [7.3, 9.4] 3.5.0860,.0860.45/ 7, ˆ.45 7 8.7.4
Blnced expement If the expement hs the sme numbe of eplctons n eve level, the desgn s blnced expement. dvntges: Exclude the mpct of dffeent eplctons The equtons fo clculton e smple.
NOV of blnced expement T (, f T (, f T, f ( Unde H 0, ~ ( ~ ((
NOV tble Souce Degees of feedom Sum of sques Men sque Expected MS F to Fcto f =- ( MS F MS MS Eo f ε =(- ( MS ( Totl T f T =- T (
Let s wok on the pevous exmple Levels of fcto togethe on ou computes Obseved dt (mg 7.9, 6., 6.6, 8.6, 8.9, 0., 9.6 5.7, 7.5, 9.8, 6., 8.4 3 6.4, 7., 7.9, 4.5, 5.0, 4.0 4 6.8, 7.5, 5.0, 5.3, 6., 7.4 Do the ndomzton of the 4 plots Buld the NOV tble fo the obseved dt
Rndoml complete block desgn (RCBD Blockng to ncese pecson b goupng the expementl unts nto homogeneous blocks to compe tetments wthn moe unfom envonment
Complete block desgn (CBD If eve tetment s used nd eplcted the sme numbe of tmes n eve block, the desgn s complete block desgn (CBD. If ech tetment s used once n eve block, t s ndoml complete block desgn (RCBD. Hee we consde expement wth tetments nd b blocks (eplctons.
Sttstcl model of RCBD =,,, fo the tetments; =,,,b fo the b eplctons.. (... (... : the genel men α : the tetment effect β : the block effect [ (... (.....] ε : the expementl eo
Feld lout of 8 mutnts nd 3 blocks (.e. 3 eplctons Block RndNBlock RndNBlock 3 RndN B 0.3 D 0.3 G 0.07 0.33 E 0.37 F 0. E 0.38 H 0.43 B 0.39 F 0.4 G 0.45 0.56 C 0.45 0.64 C 0.78 D 0.68 B 0.68 H 0.79 H 0.73 C 0.87 D 0.8 G 0.96 F 0.99 E 0.94
Mutnts Exmple: Obsevtons of 8 mutnts nd 3 blocks (.e. 3 eplctons Obsevtons Rep I Rep II Rep III Men coss eplctons 0.9 9.. 0.7-0.9 B 0.8.3 4.0.4 0.8 C..5 0.5.4-0. D 9. 0.7 0. 0.0 -.6 E.8 3.9 6.8 4..6 F 0. 0.6.8 0.8-0.8 G 0.0.5 4..9 0.3 H 9.3 0.4 4.4.4-0. Men coss mutnts. 0.4.4 3.0.6 ( Block effects β -. -0..4. Mutnt effects α
NOV of RCBD 84.6 4 (,3,,8;,,3,,8;, T 34.08 3 4 3,8, 8. 7.56 8 4 8 3 3. B
NOV of RCBD T ; ( ; ].. (. (. [( n ;.. (. (. ( B.97 B T
Souce of vton Tble of NOV Degee of feedom (df Sum of sques ( Totl b- = 3 84.6 Men sques (MS F-test P > F Mutnts - = 7 34.08 4.87.97* 0.0395 Blocks b- = 7.56 3.78 8.40** 0.004 Eo (- (b- = 4.97.64 sum of sques fo blocks s pttoned out of the sum of sques of expementl eo. The blocked desgn wll mkedl mpove the pecson on the estmtes of tetment mens f the educton n ε wth blockng s substntl.
Confdence ntevl of tetment men Stndd eo of tetment men s MS b.64 3. 0.74 The 95% confdence ntevl (CI CI.. t0.975 ( 4 s...4 s...58 Mutnt : (9.5,.3; B: (0.79, 3.95; C: (9.79,.95; D: (8.39,.55; E: (.59, 5.75; F: (9.5,.4; G: (9.79,.95
Test of hpothess of tetment men F sttstc to test the null hpothess of no eld dffeence mong the eght mutnts F MS MS Ctcl vlue F ( 7,4 4.87.64.97 Obseved sgnfcnce level 0.05.76 P F F(.76,7,4 0.0395
Estmton of vnce component Souce DF MS Expected MS Totl b- = 3 Mutnts - = 7 4.87 Blocks b- = 3.78 b G Eo (- (b- = 4.64 Eo vnce Genotpc vnce Repetblt (H.64 G ( MS MS H G /( G / b.08 39.7%
Repotng the expement nlss of eld dt ndctes sgnfcnt dffeences n eld mong the eght whet mutnts Mutnt E poduces the hghest eld Mutnt D s clel nfeo to the othes E B G C H F D 4. ±.58.4 ±.58.9 ±.58.4 ±.58.4 ±.58 0.8 ±.58 0.7 ±.58 0.0 ±.58
Comped to One-w NOV Souce Degees of feedom Sum of sques Men sque 0.05, F 0. 95(7, 6.66, F.66, we cn t eect H 0,.e. the eght mutnts doesn t hve sgnfcnt dffeence. Hee eo vnce=3.6>.64 (eo usng the lst nlss method F to Mutnts 7 34.08 4.87.54 Eo 6 50.53 3.6 Totl T 3 84.6
Let s wok on the pevous exmple togethe on ou computes Mutnts Rep I Rep II Rep III 0.9 9.. B 0.8.3 4.0 C..5 0.5 D 9. 0.7 0. E.8 3.9 6.8 F 0. 0.6.8 G 0.0.5 4. H 9.3 0.4 4.4 Do the ndomzton of the thee blocks Buld the NOV tble fo the obseved dt