ON THE RELATIVE EFFICIENCY OF SPLIT SPLIT PLOT DESIGN TO SPLIT PLOT SPLIT BLOCK DESIGN

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1 Colloquium Biometricum ON THE RELATIVE EFFICIENCY OF SPLIT SPLIT PLOT DESIGN TO SPLIT PLOT SPLIT BLOCK DESIGN Katarzyna Ambroży-Deręgowska Iwona Mejza Stanisław Mejza Department o Matematical and Statistical Metods Poznań University o Lie Sciences Wojska Polskiego Poznań Poland s: ambrozy@up.poznan.pl imejza@up.poznan.pl smejza@up.poznan.pl Summary In te paper we consider two te most popular in practice designs or tree-actorial experiments i.e. split-split-plot () design and split plot split block () design. Discussed ere models o observations are called randomized-derived models and are strictly connected wit randomization perormed or nested and crossed structures o experimental units. Statistical properties result rom two dierent scemes o randomization applied in te experiments. Te aim o te paper is comparing bot models wit respect to te relative eiciency in order to decide wic a design is te best or estimation and testing ypoteses or actors and interactions between tem. Te considerations are illustrated wit an example involving a winter weat trial. Keywords and prases: relative eiciency split-split-plot design split-plot split-block design Classiication AMS 00: 6K0 6K5. Introduction In design o tree-actorial experiments two experimental designs are te most popular i.e. split-split-plot () design and split plot split block

2 70 KATARZYNA AMBROŻY-DERĘGOWSKA IWONA MEJZA STANISŁAW MEJZA () design. In practice tey are mostly used to ortogonal (complete) experiments (ere by wic we mean tat all treatment combinations occur once in eac block). Design o suc experiments modelling and statistical analysis or bot te complete designs and incomplete designs were discussed in many papers (e.g. Ambroży i Mejza a 008b 009 0a 0b Mejza I. 997a 997b). Models o observations wic are presented in tis paper are called randomized-derived models and are strictly connected wit nested and crossed structures o experimental units. Statistical properties result rom te types o randomization scemes applied in te experiments. Te aim o te paper is comparing bot experiments wit respect to te relative eiciency in order to estimation and testing ypoteses or actors and interactions between tem. It sould be noted tat selecting experimental design we ollow mainly possibility (or ease) to apply it in practice. Additionally an order o actors plays an important role in te experiment. However economic considerations and practical reasons cannot obscure statistical purposes o te designs. Tis subject was raised also in te papers o Ambroży and Mejza ( a 008b 0b). Te considerations are illustrated wit an example involving a winter weat trial.. Structure o an experimental material Consider an s t w experiment designed to test te eects o two (s = ) levels o nitrogen ertilization (kg/a) A 90 A 50 and two (w = ) cemical growt regulators (kg/a) C 0 C on te grain yields o ive (t = 5) weat varieties B Grana B Dana B Eka Nowa B Kaukaz B 5 Mironowskaja 808. So we ave v = 0 treatment combinations. Te original experiment was perormed in Słupia Wielka (Poland) on a complete split-plot split-block (say ) design in tree blocks (replications); see Muca (975). Te structure o te experimental material (in tis case: o te ield) is described in.. Ten or comparison te same data were analyzed under a mixed linear model as data rom a complete split-split-plot (say ) design (see.). All analyzes were perormed wit te elp o te STATISTICA package (Ambroży and Mejza 006 0b)... Split-plot split-block () design In te design every block (b = ) o te experimental material orms a row-column design wit s (= ) rows and t (= 5) columns o te irst order called I-columns or sort. Ten eac I-column as to be split into w (= ) columns o te second order (called II-columns). In tis case te rows also

3 ON THE RELATIVE EFFICIENCY OF SPLIT SPLIT PLOT DESIGN 7 correspond to te levels o actor A termed row treatments te I-columns correspond to te levels o actor B termed I-column treatments and te II-columns are to accommodate te levels o actor C termed II-column treatments. Here te tird actor is in a split-plot design in a relation to te I-column treatments (wic in turn are in a split-block design wit te row treatments)... Split-split-plot () design In te design it is assumed tat te experimental material can be divided into b (= ) blocks. Every block can be divided into s (= ) wole plots. Ten eac wole plot is divided into t (= 5) subplots and ten eac subplot can be divided into w (= ) sub-subplots. Here te wole plots correspond to te levels o actor A (wole plot treatments) te subplots correspond to te levels o actor B (subplot treatments) and te sub-subplots are to accommodate te levels o actor C (sub-subplot treatments). Hence te tird actor is in a splitplot design in relation to te wole plot and subplot treatment combinations (i.e. combinations o te levels o actor A and actor B wic are also in a split-plot design). In bot and designs every block consists o stw = v = 0 plots and te number o observations is equal to n = bstw = 60.. Mixed linear models We consider a randomized-derived models o observations o wic te orms and properties are strictly connected wit perormed randomization processes in te experiment. Te randomization scemes used ere consist o our randomization steps perormed independently. Dierent scemes o randomization in te considered designs lead to te ollowing models y m D e (.) were m = 6 or te design (c. Ambroży and Mejza 00 00) and m = or te design (Mejza 997a 997b). Te considered models o te orm (.) ave te ollowing properties: m E( y) τ Cov( y) D V D σe In (.)

4 7 KATARZYNA AMBROŻY-DERĘGOWSKA IWONA MEJZA STANISŁAW MEJZA were is a known design matrix or v treatment combinations and (v) is te vector o ixed treatment combination parameters. According to te ortogonal block structure o te considered designs te covariance matrix Cov(y) can be expressed by Cov ( y) 0P0 P P mp m (.) were γ 0 and P } is a amily o known pairwise ortogonal matrices { summing to te identity matrix (c. Houtman and Speed 98). Te range space o P or = 0.. m is termed te -t stratum o te model and } are unknown strata variances. Tey are some unctions o unknown variance components resulting rom te suitable or te design sceme o randomization. In te tey are: { 0 e stw e tw e sw e s (.) e w 5 5 e 6 6 e were ( =... 6) denote variances components o eects o blocks rows columns I columns II wole plots and subplots respectively and means a variance component o a tecnical error. In te design tey are: e 0 e stw e tw e w e (.5) e were ( = ) denote variance components o eects o blocks wole plots subplots sub-subplots respectively and component o a tecnical error. e means a variance Models (.) can be analyzed using te metods developed or multistratum experiments (see e.g. Ambroży and Mejza 006).

5 ON THE RELATIVE EFFICIENCY OF SPLIT SPLIT PLOT DESIGN 7. Relative eiciency and empirical relative eiciency Deinition.. Let and denote any experimental designs ten te relative eiciency o and is deined as (see Yates 95) RE( / ) Eiciency Eiciency (.) were and denote te variances o te same contrast in te respective designs. Te relative eiciency as deined in (.) depends on te true stratum variances } o te designs wic are usually unknown. Moreover te { stratum variances are unctions o variance components deined during te randomization processes. Relations among tem allow comparison o te eiciencies o te considered designs. Usually te same relations occur among estimates o te stratum variances (except or sampling errors). Comparing eiciency o te design in relation to eiciency o te design or an estimation o contrasts in some cases we can use te measure deined in (.) but in oters we sould take into account te estimation o te RE called empirical relative eiciency wic we sall denote by ERE (see (.)). Following Yates (95) we consider uniormity trials tat is trials wit dummy treatments wit b blocks and experimental units in eac block (see also Hinkelmann and Kemptorne 008). Te ANOVA tables or data wit structures corresponding to design and design are given in Table and Table respectively. Deinition.. Empirical relative eiciency (ERE) is deined as ollows: (c. Sie and Jan 00; Wang and Hering 005) ERE ( / ) [( c ) [( c ) ] ˆ ˆ ] (.) were ˆ - estimates o variance components in te and designs (see Tables ) K K K K K K K K A B C AB AC BC ABC A B K AC KBC K AB were K { :... v } and K A KB KC K C

6 7 KATARZYNA AMBROŻY-DERĘGOWSKA IWONA MEJZA STANISŁAW MEJZA denote sets o numbers o ortogonal contrasts connected wit main eects and dierent types o interaction eects o te actors A B C and... m... m'. Table. ANOVA or dummy experiment in te design Source D.. Mean Square () Blocks b ˆ = MSE () Rows b(s ) ˆ = MSE () I-columns b(t ) ˆ = MSE () II-columns bt(w ) ˆ = MSE (5) Wole plots b(s )(t ) ˆ 5 = MSE 5 (6) Subplots bt(s )(w ) ˆ 6 = MSE 6 Total n Table. ANOVA or dummy experiment in te design Source D.. Mean Square () Blocks b () Wole plots b(s ) () Subplots bs(t ) () Sub-subplots bst(w ) Total n ˆ = MSE ˆ = MSE ˆ = MSE ˆ = MSE Following Ambroży and Mejza ( b) and rom (.) - (.5) it can be assumed (except or sampling errors) tat te estimates o variance components ˆ in te designs satisy te ollowing inequalities: or design: ˆ ˆ ˆ ˆ 5 6 and ˆ ˆ ˆ ˆ ˆ 5 6 or design: ˆ ˆ ˆ ˆ. (.) Additionally it can be expected tat te appropriate stratum variances and ence also teir estimates in te designs will be identical (to some extent) i.e.

7 ON THE RELATIVE EFFICIENCY OF SPLIT SPLIT PLOT DESIGN 75 ˆ = ˆ ˆ = ˆ (.) It sould be pointed out tat ˆ = ˆ i te designs are complete (as or instance in tis paper). Ten it can be sown tat te subplot error and te sub-subplot error in te design is a weigted average between te two errors in te design i.e. ˆ 5 < ˆ < ˆ ˆ 6 < ˆ < ˆ. (.5) 5. Analysis using te STATISTICA sotware package In Tables and given below we present STATISTICA output or a mixed model analyses o te grain yields o weat. Te eects o blocks are deined as random eects and te remaining eects are ixed (except o te errors). Source Blocks A Error () B Error () C B x C Error () A x B Error (5) A x C A x B x C Error (6) Table. ANOVA or te complete design Eect SS DF MS F p Random Fixed Random Fixed Random Fixed Fixed Random Fixed Random Fixed Fixed

8 76 KATARZYNA AMBROŻY-DERĘGOWSKA IWONA MEJZA STANISŁAW MEJZA Source Blocks A Error () B A x B Error () C A x C B x C A x B x C Error () Table. ANOVA or te complete design Eect SS DF MS F p Random Fixed Random Fixed Fixed Random Fixed Fixed Fixed Fixed Results and conclusions Te relations (.) (.5) were applied to examine te empirical relative eiciency o te complete and designs or an estimation o te ortogonal contrasts associated wit te main and interaction eects o te actors. ) For estimation o te contrasts associated wit te main eects o actor A (Nitrogen ertilization) bot considered designs are equally eective ERE A ( / ) [( c ˆ ) ] ˆ [( c ) ] were K A. ) Te design is more eective tan te design or estimation o te contrasts associated wit te main eects o actor B ERE B ( / ) [( c ˆ ) ] ˆ [( c ) ] were K B. ) Te design is less eective tan design or estimation o te interaction contrasts o type A B ERE AB ( / ) [( c ) ] ˆ ˆ [( c ) ] were K A B = 5.

9 ON THE RELATIVE EFFICIENCY OF SPLIT SPLIT PLOT DESIGN 77 ) Te design is more eective tan te design in estimation o te interaction contrasts o types A C and A B C [( c ) ] ˆ τ 6 6 ERE (/) were K A C K A B C. ˆ [( c τ) ] 5) Te design is more eective tan te design or estimation o te contrasts associated wit te main eects o actor C and te interaction eects o type B C ERE (/) [( c ˆ τ) ] ˆ [( c τ) ] were K C K B C. A summary o te conclusions given above is presented in Table 5. Te symbols a b are used to denote eiciency levels in descending order. Table 5. Comparing o te eiciency o te design and design in te estimation o some groups o contrasts Sources A a a B a b A B b a C a b A C b a B C a b A B C b a

10 78 KATARZYNA AMBROŻY-DERĘGOWSKA IWONA MEJZA STANISŁAW MEJZA Reerences Ambroży K. Mejza I. (00). Some split-plot split-block designs. Colloquium Biometryczne 896. Ambroży K. Mejza I. (00). Split-plot split-block type tree actors designs. Proc. o te 9t International Worksop on Statistical Modelling Florence Ambroży K. Mejza I. (006). Doświadczenia trójczynnikowe z krzyżową i zagnieżdżoną strukturą poziomów czynników. Wyd. PTB i PRODRUK Poznań. Ambroży K. Mejza I. (008a). Mixed designs in agriculture. XVIIIt Summer Scool o Biometrics Pavel Fl ak (Ed.)VUZV Nitra Slovakia 7 5. Ambroży K. Mejza I. (008b). Te relative eiciency o split plot split block designs and split block plot designs. Biometrical Letters 5 No. 9. Ambroży K. Mejza I. (009). Analiza danyc z krzyżową i zagnieżdżoną strukturą poziomów na przykładzie doświadczenia z łubinem. Biul. IHAR Nr Ambroży K. Mejza I. (0a). On te eiciency o some non-ortogonal split-plot split-block designs wit control treatments. Journal o Statistical Planning and Inerence. Vol. Issue Ambroży K. Mejza I. (0b). Modelowanie danyc z doświadczeń trójczynnikowyc zakładanyc w układac zależnyc o różnyc strukturac blokowyc. Biul. IHAR 6. Hinkelmann K. Kemptorne O. (008). Design and Analysis o Experiments. Wiley New Jersey. Houtman A.M. Speed T.P. (98). Balance in designed experiments wit ortogonal block structure. Ann. Statist Mejza I. (997a). Doświadczenia trójczynnikowe w niekompletnyc układac o poletkac podwójnie rozszczepionyc. Colloquium Biometryczne Mejza I. (997b). Incomplete split-split-plot designs. 5st Session o te International Statistical Institute ISI 97 Istanbul. Contributed papers book. Muca S. (975). Reakcja odmian pszenicy jarej i ozimej na antywylęgacz. Wiadomości odmianoznawcze rok II zeszyt / COBORU Słupia Wielka. Sie G. Jan S.L. (00). Te eectiveness o randomized complete block design. Statistica Neerlandica 58 No.. Wang M. Hering F. (005). Eiciency o split-block designs versus split-plot designs or ypotesis testing. Journal o Statistical Planning and Inerence 6 8. Yates F. (95). Complex experiments (wit discussion). J. Roy. Statist. Soc. Suppl StatSot Inc. (0). STATISTICA (data analysis sotware system) version 0.