Effectiveness of Split-Plot Design over Randomized Complete Block Design in Some Experiments

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1 Journ of Boogy, Agruture nd Hethre ISSN (Pper) ISSN 5-093X (Onne) Vo.4, No.19, Effetveness of Spt-Pot Desgn over Rndomzed Compete Bo Desgn n Some Experments 1 Dvd, I. J. nd Adeh, M. U. 1 Deprtment of Mthemts, Ahmdu Beo Unversty, Zr-Nger Deprtment of Mthemts, Nssrw Stte Unversty, Nsrw-Nger Correspondene: Dvd, I. J. Em: wuohe8@gm.om ABSTRACT One of the mn fetures tht dstngush spt-pot experments from other experments s tht they nvove two types of experment errors; the whoe pot (WP) error nd the spt-pot (SP) error. Ths reserh pper ompred the effetveness of spt-pot desgn (SPD) over rndomzed ompete o desgn (RCBD). The dt used for omprson s 1 x 5 spt-pot experment wth three reptes. It s een rred out to evute the threshng effeny of n mproved sorghum thresher; the three ftors onsdered n the experment re the feed t two dfferent rtes, mosture ontent t fve dfferent eves nd speed t fve dfferent rtes. The dt s nyzed s spt-pot nd s rndomzed ompete o desgn nd ther ANOVA resuts nd retve effeny (RE) sttst vues were ompred. The resut reves the effetveness of spt-pot desgn over rndomzed ompete o desgn. Key words: SPD, SP error, WP error, RCBD, Retve Effeny 1. INTRODUCTION Experments performed y nvestgtors n vrtuy fed of nqury, re usuy, to dsover somethng out prtur proess or system. Ltery, n experment s test (Montgomery, 005). The hoe of experment unt hs een one of the sgnfyng proems for vrous types of experments. In fed experments the sze nd shpe of pot nd the sze nd shpe of o re of gret nfuene nd the hoe on these mtters re of two types: sttst nd others. Under sttst onsdertons we nude tops suh s effet of sze nd shpe of pot on error vrne nd ury of estmton whe the non-sttst onsdertons nude suh mtters s the festy of prtur szes nd shpes of pot from the pont of vew of expermentton. Most fed (gronom) experments re rge nd the hoe of sute experment unt my hve some sope of ngenuty. The expermenter hve to me the dvson of the experment mter nto os n suh wy tht the pots wthn os re s homogeneous s posse, tht s, o shoud remove s muh trends n the mter s posse. Kempthorne (195) stted tht f n npproprte desgn for n experment s used t fes onsdere defets nd t my not e esy to fnd resony homogeneous re of dmensons of the experment. In ddton, oumn effets t one end of the experment my e very dfferent from oumn effets t the other end nd suh n effet woud resut n ower effeny. Cox (1958) stted tht n rndomzed ompete o desgn (RCBD) the effets of ertn soures of vrton redued y groupng the experment unts or y the use of dustments sed on onomtnt vre, the remnng vrton onvert nto effetve rndom vrton y rndomzton. He further dsussed, n grutur fed trs n the form of ftor experments rrnged n rndomzed ompete o desgn the tretments must not e too rge f these desgn s to e effetve: often the numer of tretments needs to e ess thn 0 for rndomzed ompete o desgn, nd sometmes the mt s ower thn ths. When the numers of tretments exeed ths mt, the os or rows nd oumns tend to eome too heterogeneous resutng n hgh resdu stndrd devton. He onuded tht the groupng of the mter nto os emntes the effet of onstnt dfferenes etween os nd the rndomzton ows us to tret the remnng vrton etween unts s rndom vrton, so fr s ssessng tretment omprsons s onerned. The suess of the rndomzed ompete o desgn depends on good groupng of the unts nto os, nd the gener de of groupng nto os s of fundment mportne nd s not ony frequenty used n smpe experments ut so forms the ss for most of the more ompted desgns. Cox (1958) gn rgued tht the use of spt-pot desgn s neessry not when the whoe pot tretments re ey to e ompred wth ess preson thn the supot tretments euse the whoe pot tretments need not e restrted to snge ftor ut my onsst of omntons of the eves of sever ftors. He stted tht the spt-pot prnpe s pped to experments on proesses n whh there re sever stges nd t my then e onvenent to wor wth rge thes of mter t the frst stge, dvdng nto smer thes for the ppton of the tretment t the seond stge. Cohrn nd Cox (1956) stted tht the hef prt dvntge of the spt-pot rrngement s, t enes ftors tht requre retvey rge mount of mters nd ftors tht requre ony sm mount of mters to e used. In ddton, ftors of seond type n often e nuded t very tte extr ost, nd some ddton nformton otned very hepy. They summrzed tht, the spt-pot desgn s dvntgeous f the 75

2 Journ of Boogy, Agruture nd Hethre ISSN (Pper) ISSN 5-093X (Onne) Vo.4, No.19, su-pot (B) tretment nd the nterton of the whoe pot (A) nd su-pot (B) tretment effets re of greter nterest thn the whoe pot tretment (A) effet, or f the whoe pot (A) effet nnot e tested on sm mount of mters. They gve two dsdvntges mentoned y expermenters, whh re; sometmes the whoe-unt error s muh rger thn the su-unt error. It my our t tmes tht the effet of whoe-pot (A), though rge nd extng, s not sgnfnt, wheres tht of su-pot (B), whh s too sm to e of prt nterest, s sttsty sgnfnt. The expermenter tends to e unomforte n reportng resuts of ths type. Seondy, the ft tht dfferent tretment omprsons hve dfferent s error vrnes mes the nyss more ompex thn wth rndomzed ompete o desgn, espey f some unusu type of omprson s eng mde. They ommended tht the spt-pot Ltn squre emntes error vrton, whh rse from two types of groupng, nd s prefere to rndomzed ompete o desgn (RCBD). They ted n exmpe presented y tes (1935) where he summrzed fed experments n Ltn squres where the pots were spt nto hves, he found sustnt net nrese n preson over rndomzed ompete o desgn, nd the superorty ws so pronouned tht even the whoe-pot omprsons woud hve een ess presey determned n rndomzed ompete o desgn. Montgomery (005) stted tht the norret testng of effets for runnng n experment n ompetey rndomzed fshon nsted of the spt-pot desgn s euse the error ssoted wth the hrd-tohnge ftor nftes the vrne of the regresson oeffent for the esy hnged ftor. Jones nd Nhthem (009) stted tht when n ndustr experment, s presred wrongy t ed to norret nyss, whh nftes the Type I, error rte for whoe pot su ftors, s does the Type II error rte for spt pot su ftors nd whoe pot y spt-pot ntertons. Therefore, one wy to vod these mstes s to pn the experment s spt-pot desgn n the frst pe. Not ony does ths vod mstes, t so everges the eonom nd sttst effenes. Besdes the ess expensveness of runnng the spt-pot desgn, t s often more sttsty effent desgn ompred to other experment desgns euse t hndes rge dt wth ess error. Bsgrd et. (1996) dsussed ths ssue nd stted tht f spt-pot experment s wrongy nyzed s ompetey rndomzed experment, some ftors my e dered sgnfnt when they re not nd ve vers. Kows nd Potner (003) stted tht when desgned experment uses os suh s dys or thes, the nyss of the experment nudes term for these os. When desgned experment s performed y fxng ftor nd then runnng the omntons of the other ftors, usng dfferent szed experment unts or usng dfferent rndomzton for the ftors ( spt-pot desgn), the nyss shoud norporte these fetures. They nyzed 3 ftor tretments desgn wth two reptes usng the spt-pot pproh. The responses were frst nyzed norrety s f they me from ompetey rndomzed desgn nd then rn orrety s spt-pot desgn. It ws oserved tht some tretment effets were dered sgnfnt for the ompetey rndomzed desgn nd not for the spt-pot desgn nd ve vers. They stted tht the use s, n the ompetey rndomzed desgn, ftors effet use the men squre error s the estmte of experment error. In spt-pot experment, however, there re two dfferent experment error strutures: one for the WP ftor nd one for the SP ftor. They onuded tht t s euse of the two seprte rndomztons tht our when runnng the experment.. MATERIALS AND METHODS A 1 x 5 spt-pot experment wth three reptes ws rred out to evute the threshng effeny of n mproved sorghum thresher. The three ftors onsdered n the experment re the feed (whoe pot ftor) t two dfferent rtes, mosture ontent (frst su-pot ftor) t fve dfferent eves nd speed (seond su-pot ftor) t fve dfferent rtes. The dt w e nyzed s spt-pot desgn nd s rndomzed ompete o desgn, ths s to he the effetveness of the spt-pot desgn over rndomzed ompete o desgn. Ther respetve modes re s foows;.1 Spt-Pot Desgn Mode The mode s ner ddtve mode, t s gven s; = µ + α + γ + + β + η + ( αβ ) + ( αη ) + ( βη) + (1) where: s the response; µ s onstnt; α s the WP ftor; ~ (0, σ ); β s the frst SP ftor; η s the seond SP ftor; ftors; s the SP error, NID ~ (0, σ γ s the o effet; (αβ ), (αη ), ) s the WP error, NID (βη re the nterton ); = 1,, 3,, (eves of WP ftor), = 1,, 3,, (eves of SP ftor), = 1,, 3,, r (numer of reptes or os). Note, the mode dd not nude the three-ftor nterton. 76

3 Journ of Boogy, Agruture nd Hethre ISSN (Pper) ISSN 5-093X (Onne) Vo.4, No.19, Te I: Seth of the ANOVA Te for Spt-Pot Desgn Mode Soure Df Sum of Squre Men Squre F Bo r 1 SS BLOCK BLOCK Bo / MPE A 1 SS A A A / MPE Mn pot error (r 1)( 1) SS MPE MPE B 1 SS B B B / SPE C 1 SS C C C / SPE AB ( 1)( 1) SS AB AB AB / SPE AC ( 1)( 1) SS AC AC AC / SPE BC ( 1)( 1) SS BC BC BC / SPE Spt-pot error (r-1)( 1)( 1 ) SS SPE SPE TOTAL r-1 SS TOTAL Soure: Jones nd Nhthem (009) where, SS TOTAL = r r, SS BLOCK = r... r r r SS MPE = r SS AB = r r, SS A =......, r r... +, SS B = r, SS C = r r... r r r SS BC =... + r..... r r r r, SS AC = r r hene, the spt-pot error w e, SS SPE = SS TOTAL SS BLOCK SS A SS MPE SS B SS C SS AB SS AC SS BC. Rndomzed Compete Bo Desgn Mode (RCBD) Three modes n e dentfed from rndomzed ompete o desgn, they re; 1. RCBD mode wthout repton wthn os nd one oservton per e. r r = µ + α + β + () where; s the response yed; µ s the over men; α s the effet of the th row (tretment); β s the effet of the th oumn (o) nd s the experment error whh s NID ~ (0, σ ); = 1,, 3,, p = 1,, 3,,.. RCBD mode wth repton wthn os ut no nterton etween tretments. = µ + α + β + δ ( ) + ( ) (3) where; s the response from the th experment unt n o, the th rndomzton nd gven the th tretment; µ s the over men; α s the th o effet; β s the th tretment effet; the th o (.e. Df for δ () s zero) nd s the experment error, NID~ (0, σ ). 3. RCBD mode wth repton wthn os nd nterton etween tretments. where; nd δ () s the th restrton error wthn = µ + α + δ + β + η + ( αβ ) + ( αη ) + ( βη ) +.. (4), s the th response from the th nd th effets; µ, s the over men onstnt; tretment A; δ, s the o effet; β, s the th effet of tretment B; α, s the th effet of η, s the th effet of tretment C; (αβ ), (αη ), (βη) re the nterton of the th, th nd th effet of tretment A nd B, A nd C, B nd C; 77

4 Journ of Boogy, Agruture nd Hethre ISSN (Pper) ISSN 5-093X (Onne) Vo.4, No.19, s the rndom error used y the th response from the th effet of B nd th effet of C n the th effet of A, NID ~ (0, σ ); = 1,,, = 1,,, = 1,,, = 1,, n. The thrd mode s the dopted mode for ths reserh n omprson wth the spt-pot desgn mode so; the three-ftor nterton ws not nuded n the mode ust to otn dequte degree of freedom for error to estmte the nterton effet dequtey. Te II: Seth of the ANOVA Te for the Thrd RCBD Mode Soure Df Sum of Squre Men Squre F Bo r-1 SS BLOCK BLOCK Bo /E A -1 SS A A A / E B -1 SS B B B / E C -1 AB (-1)(-1) SS AB AB AB / E AC (-1)(-1) BC (-1)(-1) Error (-1)(r-1) SSE E TOTAL r-1 SS TOTAL Soure: Montgomery (005). where, SS TOTAL = r r, SS BLOCK = r... r... =, SS C = r r... r r SS AC = r r r, SS AB =... + r hene, the sum of squre error w e; SS Error = SS TOTAL SS BLOCK SS A SS B SS C SS AB SS AC SS BC, SS A =..., SS B r r..... r r r r... nd SS BC = Effeny of Spt-Pot Desgn Retve to RCBD Aordng to Hnemnn nd Kempthorne (008), expermenters utze the spt-pot desgn n mny nstnes nd rumstnes for tehn resons. Under most rumstnes, the spt-pot desgn s een used for purey tehn nd prt resons, s the eves of some ftor n e pped ony to rge experment unts, whh n then e spt nto smer experment unts for ppton of the eves of the other ftor. Ths nudes so the dstnton etween hrd-to-hnge nd esy-to-hnge ftors n ndustr expermentton. It s, however, of nterest to evute the effeny of the spt-pot desgn retve to the RCBD wth r os. The queston then s gven tht we hve rred out spt-pot experment, wht woud hve een E for the RCBD? Ths, of ourse, determnes how muh nformton woud hve een ve for tretment omprsons. Usng the pooed tretment sums of squres wth pproprte error sum of squres, tht r ( 1) E = r ( 1) WPE + r( 1) SPE dvde through y r we hve E = ( 1) WPE + ( 1) 1 SPE r r, r 5 where E = E = s weghted verge of men squre WP error nd men squre SP error; WPE, s the men squre WP error; SPE, s the men squre spt-pot error;, s the numer of eves of WP ftor;, s the numer of eves of SP ftor. From equton (1) nd equton (4) the nformton on tretment omprsons woud then hve een proporton to 1/ E. The nformton on WP tretments from the spt-pot experment retve to rndomzed ompete o desgn s then E / MPE tht s ess thn 1. For SP tretments nd nterton effets from the spt-pot experment retve to the rndomzed ompete o, desgn s E / SPE tht s greter thn 1. These resuts express the ovous: tht the rrngement of spt-pot tretments together wthn whoe pot resuts n ower ury on whoe pot tretment omprsons nd n nresed ury on other tretment omprsons, the formus ene quntttve evuton of these effets. Reourse shoud e ten to spt-pot desgn r 78

5 Journ of Boogy, Agruture nd Hethre ISSN (Pper) ISSN 5-093X (Onne) Vo.4, No.19, when experment ondtons neessttes the spe rrngement, or when the expermenter s more nterested n one ftor, whh he/she rrnges wthn whoe pots, thn n the other (Hnemnn nd Kempthorne, 008). Equton (5) ove s presented y Hnemnn nd Kempthorne (008) s ndequte for our desgned modes sne ts ppty s for two ftors ony. Hene, for three-ftor desgn the retve effeny of the spt-pot desgn over RCBD w e otned from Te I nd II, usng the pooed tretment sums of squres wth pproprte error sums of squres, tht r( 1)E = r( 1) WPE + r( 1)( 1) SPE dvde through y r we hve E = ( 1) WPE + ( 1)( 1) 1 SPE. 6 The nterprettons for the two ftors s gven y Hnemnn nd Kempthorne (008) nd ther respetve onusons hod for the three ftors too. 3. RESULT Te III: ANOVA Te for the Spt-Pot Mode Soure Df SS F CAL P- Vues Bos <.0001 FR Mp error GM <.0001 BFS <.0001 FR*GM <.0001 FR*BFS GM*BFS SP Error Tot Soure: Author s omputton Te IV: ANOVA Te for the RCBD Mode Soure Df SS F CAL P- Vues Bos <.0001 FR <.0001 GM <.0001 BFS <.0001 FR*GM <.0001 FR*BFS GM*BFS Error Tot Soure: Author s omputton 4. DISCUSSION From equton (5) the weghted men squre error, E s estmted s, E = Hene, the nformton on WP tretment from the spt-pot experment retve to rndomzed ompete o desgn s then, (E / MPE ) = /3.4 = < 1 Whe the nformton on SP tretments from the spt-pot experment retve to rndomzed ompete o desgn s then (E / SPE ) = /

6 Journ of Boogy, Agruture nd Hethre ISSN (Pper) ISSN 5-093X (Onne) Vo.4, No.19, = > 1 Te III nd IV ove shows the ANOVA resut for the spt-pot desgn nd the rndomzed ompete o desgn respetvey. From oservton on te III, t α = 5% sgnfne eve, the mn effets FR, GM nd BFS re sgnfnt sne, ther p-vues re ess thn α = 5% sgnfne eve. Whe for the nterton effets FR*GM, FR*BFS nd GM*BFS t s oserved tht GM*BFS s not sgnfnt sne ts p-vue of s greter thn α = 5% sgnfne eve. Lewse, from te IV mn effets re sgnfnt whe for the nterton effets t s oserved tht FR*BFS nd GM*BFS re not sgnfnt sne ther p-vues of nd respetvey re greter thn α = 5% sgnfnt eve. Ths s n evdene of the spt-pot desgn superorty over the rndomzed ompete o desgn. Hene, for ompete proof of the ANOVA resuts the retve effeny of the spt-pot desgn over the RCBD ws een omputed. It ws er when the effeny of the WP tretment retve to the RCBD omputed usng equton (6) the vue (0.0483) otned s ess thn one nd tht of the su-pot tretments retve to the RCBD show tht the vue (1.131) otned s greter thn one. Hene, we n gree on the superorty of the SPD over the RCBD. Ths resut otned s not fr from tht otned y Kows nd Potner (003) though they oserved the effeny of SPD retve to ompete rndomzed desgn (CRD) usng ANOVA resuts ony. The dfferene etween the RCBD nd CRD s the o ftor esdes; the ssue of ong s to mprove the desgn nd to remove every eement of heterogenety s muh s posse. Hene, the SPD s not n exepton, ts form of desgn s to deveop n dequte nd prese desgn tht n redue ost, remove heterogenety nd study ftors of ess nd mportnt nterest together. 5. CONCLUSION Ths reserh s n ttempt to ompre the effetveness of SPD over RCBD. The resut otned ery put the SPD hed of the RCBD sne from the ANOVA te resut for the nterton of the feed rte y the mhne owng fn speed (FR*BFS) s sgnfnt from the SPD ANOVA mode ut not sgnfnt from the RCBD ANOVA mode t 5% sgnfne eve. Lewse, the retve effeny sttst omputed ws to ompre the two desgns nd the vues otned so reve tht the SPD mode s more effent retve to the RCBD mode. Before emrng on the pn nd desgn of experment, expermenters shoud dequtey study the type of ftors for ther experments to now how mportnt they re for hevng ther experment gos. Beuse the resuts from ths reserh shows tht the SPD w e more effent thn the RCBD espey when one or some of the ftors to e studed re of ess mportne or s hrd-to-hnge s n the ndustr experments. Insted of vodng the SPD due to omputton ompextes s vewed y some expermenters nd shors, professon shoud e ontted t w go ong wy n redung sed resuts n estmtng ftors sgnfnt ontruton to the experment response. REFERENCES 1. Bsgrd, S., Fuer, H. nd Brros. E Two-eve Ftors run s Spt pot Experments. Quty engneerng 8: Cohrn, W.G. nd Cox, G. M Experment Desgns, John Wey nd Sons New or, pp , Cox, D. R Pnnng of Experments, John Wey nd Sons New or, pp. 6 30, 34 nd Hnemn, K. nd Kempthorne, O. (008). Desgn nd Anyss of Experments, vo. I: Introduton to Experment Desgn, nd ed. Wey-Intersene, New or. 5. Jones, B. nd Nhtshem, C. J Spt-pot Desgns: Wht, why, nd How. Journ of Quty Tehnoogy, 41(4): Kempthorne, O Desgn nd nyss of Experments, John Wey nd Sons New or, pp , nd Kows, S. M. nd Potner, K. J How to Anyze Spt-Pot Experment, Quty Progress, 37(1): Montgomery, D. C Desgn nd Anyss of Experments, 5 th ed. John Wey nd Sons New or, PP nd tes, F Compex Experments. Journ of Roy Sttsts Soety Suppementry:

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