SPLIT PLOT AND STRIP PLOT DESIGNS

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1 1. Split Plot Design SPLIT PLOT AND STRIP PLOT DESIGNS D.K. SEHGAL Indin Agriculturl Sttistics Reserch Institute Lirry Avenue, New Delhi Introduction In conducting experiments, sometimes some fctors hve to e pplied in lrger experimentl units while some other fctors cn e pplied in comprtively smller experimentl units. Further some experimentl mterils my e rre while the other experimentl mterils my e ville in lrge quntity or when the levels of one (or more) tretment fctors re esy to chnge, while the ltertion of levels of other tretment fctors re costly, or time-consuming. One more point my e tht lthough two or more different fctors re to e tested in the experiment, one fctor my require to e tested with higher precision thn the others. In ll such situtions, design clled the split plot design is dopted. A split plot design is design with t lest one locking fctor where the experimentl units within ech lock re ssigned to the tretment fctor levels s usul, nd in ddition, the locks re ssigned t rndom to the levels of further tretment fctor. The designs hve nested locking structure. In lock design, the experimentl units re nested within the locks, nd seprte rndom ssignment of units to tretments is mde within ech lock. In split plot design, the experimentl units re clled split-plots (or su-plots), nd re nested within whole plots (or min plots). In split plot design, plot size nd precision of mesurement of effects re not the sme for oth fctors, the ssignment of prticulr fctor to either the min plot or the su-plot is extremely importnt. To mke such choice, the following guidelines re suggested: Degree of Precision- For greter degree of precision for fctor B thn for fctor A, ssign fctor B to the su-plot nd fctor A to the min plot e.g. plnt reeder who plns to evlute ten promising rice vrieties with three levels of fertiliztion, would proly wish to hve greter precision for vrietl comprison thn for fertilizer response. Thus, he would designte vriety s the su-plot fctor nd fertilizer s the min plot fctor. Or, n gronomist would ssign vriety to min plot nd fertilizer to su-plot if he wnts greter precision for fertilizer response thn vriety effect. Reltive Size of the Min effects- If the min effect of one fctor (A) is expected to e much lrger nd esier to detect thn tht of the other fctor (B), fctor A cn e ssigned to the min plot nd fctor B to the su-plot. This increses the chnce of detecting the difference mong levels of fctor B which hs smller effect. Mngement Prctices- The common type of sitution when the split plot design is utomticlly suggestive is the difficulties in the execution of other designs, i.e. prcticl execution of plns. The culturl prctices required y fctor my dictte the use of lrge plots. For prcticl expediency, such fctor my e ssigned to the min plot e.g. in n

2 experiment to evlute wter mngement nd vriety, it my e desirle to ssign wter mngement to the min plot to minimize wter movement etween djcent plots, fcilitte the simultion of the wter level required, nd reduce order effects. Or, if ploughing is one of the fctors of interest, then one cnnot hve different depths of ploughing in different plots scttered rndomly prt. 1. Rndomiztion nd Lyout There re two seprte rndomiztion processes in split plot design one for the min plot nd nother for the su-plot. In ech repliction, min plot tretments re first rndomly ssigned to the min plots followed y rndom ssignment of the su-plot tretments within ech min plot. This procedure is followed for ll replictions. A possile lyout of split plot experiment with four min plot tretments (=4), three suplot tretments (=), nd four replictions (r=4) is given elow: Rep. I Rep. II Rep. III Rep. IV The ove lyout hs the following importnt fetures: the size of the min plot is times the size of the su-plot ech min plot tretment is tested r times wheres ech su-plot tretment is tested r times, thus the numer of times su-plot tretment is tested will lwys e lrger thn tht for the min plot nd is the primry reson for more precision for the suplot tretments reltive to the min plot tretments This concept of splitting ech plot my e extended further to ccommodte the ppliction of dditionl fctors. An extension of this design is clled the split-split plot design where the su-plot is further divided to include third fctor in the experiment. The design llows for different levels of precision ssocited with the fctors. Tht is, the degree of precision ssocited with the min fctor is lowest, while the degree of precision ssocited with the su-su plot is the highest. 1. Model The model for simple split plot design is Y ijk = µ + ρ i + τ j + δ ij + β k + (τβ) jk + ε ijk for i = 1,,,r, j = 1,,,, k = 1,,, where, Y ijk : oservtion corresponding to k th level of su-plot fctor(b), j th level of min plot fctor(a) nd the i th repliction. µ : generl men ρ i : i th lock effect : j th min plot tretment effect τ j

3 β k : k th su-plot tretment effect (τβ) jk : interction etween j th level of min-plot tretment nd the k th level of suplot tretment The error components δ ij nd ε ijk re independently nd normlly distriuted with mens zero nd respective vrinces σ δ nd σ ε. 1.4 Anlysis Whole-Plot nlysis: This prt of the nlysis is sed on comprisons of whole-plot totls: The levels of A re ssigned to the whole plots within locks ccording to rndomized complete lock design, nd so the sum of squres for A needs no lock djustment. There re 1 degrees of freedom for A, so the sum of squres is given y ssa y. j. / r y.../ r = j [The dot nottion mens dd over ll vlues of the suscript replced with dot ] There re r 1 degrees of freedom for locks, giving lock sum of squres of ssr yi.. / y... / r = i There re whole plots nested within ech of the r locks, so there re, in totl, r( - 1) whole-plot degrees of freedom. Of these, 1 re used to mesure the effects of A leving (r 1)( 1) degrees of freedom for whole-plot error. Equivlently, this cn e otined y the sutrction of the lock nd A degrees of freedom from the whole-plot totl degrees of freedom, i.e., (r 1) (r 1) ( 1) = (r 1)( 1). So, the whole plot error sum of squres, is otined s sse1 = yi j. / y... / r - ssr - ssa i j The whole plot error men squre mse 1 = sse 1 / (r 1)( 1), is used s the error estimte to test the significnce of whole plot fctor(a). Su-plot nlysis: This prt of the nlysis is sed on the oservtions rising from the split-plots within whole plots: There re r 1 totl degrees of freedom, nd the totl sum of squres is sstot = i j k j k y i y... / r Due to the fct tht ll levels of B re oserved in every whole plot s in rndomized complete lock design, the sum of squres for B needs no djustment for whole plots, nd is given y - ssb= y.. k / r y.../ r, corresponding to 1 degrees of freedom. k The interction etween the fctors A nd B is lso clculted s prt of the split-plot nlysis. Agin, due to the complete lock structure of oth the whole-plot design

4 nd the split-plot design, the interction sum of squres needs no djustment for locks. The numer of interction degrees of freedom is ( 1)( 1), nd the sum of squres is ss(ab) = y. j k j k / r y... / r - ssa - ssb Since there re split plots nested within the r whole plots, there re, in totl, r( 1) split-plot degrees of freedom. Of these, 1 re used to mesure the min effect of B, nd ( 1)( 1) re used to mesure the AB interction, leving r( 1) ( 1) ( 1)( 1) = (r 1)( 1) degrees of freedom for error. Equivlently, this cn e otined y sutrction of the whole plot, B, nd AB degrees of freedom from the totl, i.e., (r 1) (r 1) ( 1) ( 1)( 1) = (r 1)( 1). The split-plot error sum of squres cn e clculted y sutrction: sse = sstot ssr ssa sse 1 ssb ss(ab). The split-plot error men squre mse = sse / (r 1)( 1) is used s the error estimte in testing the significnce of split-plot fctor(b) nd interction(ab). The nlysis of vrince tle is outlined s follows: ANOVA Source of Vrition DF SSs MS F Whole plot nlysis Repliction r-1 SSR - - Min plot tretment(a) -1 SSA MSA msa/mse 1 Min plot error(e 1 ) (r-1)(-1) SSE 1 MSE 1 =E Su-plot nlysis Su-plot tretment(b) -1 SSB MSB msb/mse Interction (-1)(-1) SS(AB) MS(AB) ms(ab)/mse (A B) Su-plot error(e ) (r-1)(-1) SSE MSE =E Totl r-1 SST 1.5 Stndrd Errors nd Criticl Differences: Estimte of S.E. of difference etween two min plot tretment mens = E r E Estimte of S.E. of difference etween two su-plot tretment mens = r Estimte of S.E. of difference etween two su-plot tretment mens E t the sme level of min plot tretment = r Estimte of S.E. of difference etween two min plot tretment ( -1)E mens t the sme or different levels of su-plot tretment = r [ ] Criticl difference is otined y multiplying the S.E d y t 5% tle vlue for respective error d.f. for (i), (ii) & (iii). For (iv), s the stndrd error of men difference involves two error terms, we use the following eqution to compute the weighted t vlues: 4

5 ( -1)E t t t = ( -1)E where t nd t re t-vlues t error d.f. (E ) nd error d.f.(e ) respectively. Exercise: In study crried y gronomists to determine if mjor differences in yield response to N fertiliztion exist mong different vrieties of jowr, the min plot tretments were three vrieties of jowr (V 1 : CO-18, V : CO-19 nd V : C0-), nd the su-plot tretments were N rtes of 0, 0, nd 60 Kg/h. The study ws replicted four times, nd the dt gthered for the experiment re shown in Tle 1. Anlyse the dt nd drw conclusions. Tle 1: Repliction-wise yield dt. N rte, Kg/h Repliction Vriety Yield, kg per plot I V V V II V V V III V V V IV V V V Steps of nlysis: Clculte the repliction totls(r), nd the grnd totl(g) y first constructing tle for the repliction vriety totls shown in Tle 1.1, nd then second tle for the vriety nitrogen totls s shown in Tle 1.. Tle 1.1 Repliction vriety (RA) - tle of yield totls. Vriety Repliction V 1 V V Rep.Totl(R) I II III IV Vriety Totl(A) Grnd Totl(G) 69.0 Tle 1. Vriety Nitrogen (AB) - tle of yield totls. Vriety 5

6 Nitrogen V 1 V V Nitrogen Totl(B) N N N Compute the vrious sums of squres for the min plot nlysis y first computing the correction fctor: G (69) C.F. = r = 4 = Totl S.S. (sstot) = [ (15.5) + (0.5) + + (10.5) ] - C.F.= R Repliction S.S. (ssr) = C.F. (181.) + (168.1) + (16.9) + (15.7) = = A S.S. due to Vriety (ssa) = C.F. r (0.) + (9.6) + (196.) = = (RA) Min plot error S.S. (sse 1 ) = C.F. ssr ssa (5.8) + (6.6) (4.9) = = Compute the vrious sums of squres for su-plot nlysis: B S.S. due to Nitrogen (ssb) = C.F. r (189.4) + (1.1) + (6.5) = = (AB) S.S. due to Interction (A B) = C.F. ssa ssb r (60.) + (65.7) (69.8) = = Su-plot error S.S. (sse ) = Totl S.S. All other sum of squres = ( ) = 81. Clculte the men squre for ech source of vrition y dividing the S.S. y its corresponding degrees of freedom nd compute the F vlue for ech effect tht needs to e tested, y dividing ech men squre y the corresponding error men squre, s shown in Tle 1.. 6

7 Tle 1. ANOVA results. Source of vrition DF SSs MS F Repliction Vriety(A) ns Error() (E ) Nitrogen(B) ** Vriety Nitrogen <1 (A B) Error() (E ) Totl ns not significnt, ** - significnt t 1% level. Compute the coefficient of vrition for the min plot nd su-plot s: E E cv() = 100, nd cv() = 100 respectively. G.M. G.M. Compute stndrd errors nd to mke specific comprisons mong tretment mens compute respective criticl differences only when F-tests show significnce differences nd interpret. Conclusion: There ws no significnt difference mong vriety mens. Yield ws significntly ffected y nitrogen. However, the interction etween N rte nd vriety ws not significnt. All the vrieties gve significnt response to 0 kg N/h s well s to 60 kg N/h. SAS input sttements for the split plot experiment dt split plot; input rep vr nit yield; crds;

8 ; proc print; proc glm; clss rep vr nit; model yield = rep vr rep* vr nit vr*nit; test h = vr e = rep*vr; men vr nit vr*nit; run; SAS Output ANOVA 8

9 . Strip Plot Design.1 Introduction Sometimes sitution rises when two fctors ech requiring lrger experimentl units re to e tested in the sme experiment, e.g., suppose four levels of spcing nd three levels of methods of ploughing re to e tested in the sme experiment. Here oth the fctors require lrge experimentl units. If the comintions of the two fctors t ll possile levels re llotted in R.B.D. in the norml wy, the experimentl plots shll hve to e very lrge therey ringing heterogeneity. So, it will not e pproprite. On the other hnd if one fctor (spcing) is tken in min plots nd other fctor (methods of ploughing) is tken in su-plots within min plots, the su-plots shll hve to e lrge enough. Hence split plot design lso will not e pproprite. In such situtions design clled strip plot design is dopted. The strip plot is -fctor design tht llows for greter precision in the mesurement of the interction effect while scrificing the degree of precision on the min effects. The experimentl re is divided into three plots, nmely the verticl-strip plot, the horizontlstrip plot, nd the intersection plot. We llocte fctors A nd B, respectively, to the verticl nd horizontl-strip plots, nd llow the intersection plot to ccommodte the interction etween these two fctors. As in the split plot design, the verticl nd the horizontl plots re perpendiculr to ech other. However, in the strip plot design the reltionship etween the verticl nd horizontl plot sizes is not s distinct s the min nd su-plots were in the split plot design. The su-plot tretments insted of eing rndomized independently within ech min plot s in the cse of split plot design re rrnged in strips cross ech repliction. The intersection plot, which is one of the chrcteristics of the design, is the smllest in size.. Rndomiztion nd Lyout In this design ech lock is divided into numer of verticl nd horizontl strips depending on the levels of the respective fctors. Let A represent the verticl fctor with levels, B represent the horizontl fctor with levels nd r represent the numer of replictions. To lyout the experiment, the experimentl re is divided into r locks. Ech lock is divided into horizontl strips nd tretments re rndomly ssigned to these strips in ech of the r locks seprtely nd independently. Then ech lock is divided into verticl strips nd tretments re rndomly ssigned to these strips in ech of the r locks seprtely nd independently. A possile lyout of strip plot experiment with =5 ( 1,,, 4, nd 5 ), = ( 1,, nd ) nd four replictions is given elow: Rep. I Rep. II Rep. III Rep. IV

10 The strip plot design scrifices precision on the min effects of oth the fctors in order to provide higher precision on the interction which will generlly e more ccurtely determined thn in either rndomized locks or simple split plot design. Consequently this design is not recommended unless prcticl considertions necessitte its use or unless the interction is the principle oject of study.. Model The model for strip plot design is Y ijk = µ + ρ i + α j + (ρα) ij + β k + (ρβ) ik + (αβ) jk + ε ijk for i = 1,,,r, j = 1,,,, k = 1,,, where, Y ijk : oservtion corresponding to k th level of fctor A, j th level of fctor B nd i th repliction µ : generl men ρ i : i th lock effect α j : effect of j th level of fctor A β k : effect of k th level of fctor B (αβ) jk : interction etween j th level of fctor A nd the k th level of fctor B The error components (ρα) ij, (ρβ) ik nd ε ijk re independently nd normlly distriuted with mens zero nd respective vrinces σ, σ, nd σ ε..4 Anlysis In sttisticl nlysis seprte estimtes of error re otined for min effects of the fctor, A nd B nd for their interction AB. Thus there will e three error men squres pplicle for testing the significnce of min effects of the fctors nd their interction seprtely. Suppose 4 levels of spcings (A) nd levels of methods (B) of ploughing re to e tested in the sme experiment. Ech repliction is divided into 4 strips verticlly nd into strips horizontlly. In the verticl strips the four different levels of spcings re llotted rndomly nd in the horizontl strips three methods of ploughing re llotted rndomly. Let there e 4 replictions(r). The nlysis of vrince is crried out in three prts viz. verticl strip nlysis, horizontl strip nlysis nd interction nlysis s follows: Form spcing repliction (A R) tle of yield totls nd from this tle compute the S.S. due to repliction, S.S. due to spcings nd S.S. due to interction - Repliction Spcing, i.e., error (). Form method repliction (B R) tle of yield totls nd from this tle compute the S.S. due to methods nd S.S. due to interction - Repliction Method, i.e., error (). Form spcing method (A B) tle of yield totls nd from this tle compute the S.S. due to interction - Spcing Method. Totl S.S. will e otined s usul y considering ll the oservtions of the experiment nd the error S.S., i.e., error (c) will e otined y sutrcting from totl S.S. ll the S.S. for vrious sources. Now, clculte the men squre for ech source of vrition y dividing ech sum of squres y its respective degrees of freedom. 10

11 Compute the F-vlue for ech source of vrition y dividing ech men squre y the corresponding error term. The nlysis of vrince tle is outlined s follows: ANOVA Source of DF SS MS F Vrition Repliction(R) (r-1)= SSR - - Spcing(A) (-1)= SSA msa msa/mse 1 Error() (r-1)(-1)= 9 SSE 1 mse 1 =E Method(B) (-1)= SSB msb msb/mse Error() (r-1)(-1)= 6 SSE mse =E Spcing Method (-1)(-1)= 6 SS(AB) ms(ab) ms(ab)/mse (A B) Error(c) (r-1)(-1)(-1)=18 SSE mse =E c Totl (r-1)= 47 SST.5 Stndrd Errors nd Criticl Differences: Estimte of S.E. of difference etween two A level mens = E r E Estimte of S.E. of difference etween two B level mens = r Estimte of S.E. of difference etween two A level mens t the sme level of B mens [( -1)E E ] = c+ r Estimte of S.E. of difference etween two B level mens t the sme level of A mens = [( -1)E E ] c+ r Criticl difference is otined y multiplying the S.E d y t 5% tle vlue for respective error d.f. for (i) & (ii). For (iii) & (iv), s the stndrd error of men difference involves two error terms, we use the following eqution to compute the weighted t vlues: ( -1)E ct c t ( -1)Ect c t t =, nd t = respectively, ( -1)E c ( -1)E c where t, t, nd t c re t-vlues t error d.f. (E ), error d.f.(e ) nd error d.f.(e c ) respectively. 11