Control treatments in designs with split units generated by Latin squares

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1 DOI:.2478/bile-24-9 iometicl Lettes Vol. 5 (24), o. 2, Contol tetments in designs with split units geneted by Ltin sues Shinji Kuiki, Iwon Mejz 2, Kzuhio Ozw 3, Stnisłw Mejz 2 Deptment of Mthemticl Sciences, Gdute School of Engineeing,Osk Pefectue Univesity, - Gkuen-cho, k-ku, Ski, Osk , Jpn 2 Deptment of Mthemticl nd Sttisticl Methods, Poznn Univesity of Life Sciences, Wojsk Polskiego 28, PL Poznń, Polnd 3 Deptment of using, Gifu College of using, Hshim, Gifu , Jpn SUMMRY This ppe dels with two-fcto expeiments with split units. The whole plot tetments occu in epeted Ltin sue, modified Ltin sue o Youden sue, while subplot tetments occu in block design within the whole plots. The sttisticl popeties of the consideed designs e exmined. Specil ttention is pid to the cse whee one of the tetments is n individul contol o n individul stndd tetment. In ddition, we give bief oveview of wok on the design of expeiments using the consideed designs, s well s possible ngements of contols in the expeiments. Key wods: Ltin sue, Youden sue, split units, efficiency fctos, meging tetment method.. Intoduction In gicultul field expeiments, ow-column design o epeted ow-column design is uite often used to eliminte el o potentil othogonlly disposed heteogeneity of the expeimentl mteil. Fom sttisticl point of view, Ltin sue is then the ppopite design. This design is especilly efficient in gicultul field expeiments. It possesses mny desible nd optiml sttisticl popeties. In the Ltin sue evey tetment occus once in ech ow nd once in ech column. This mens tht the component designs, both with espect to the ows nd with espect to the columns, e ndomized complete blocks. nothe ow-column design with desible sttisticl

2 26 S. Kuiki, I. Mejz, K. Ozw, S. Mejz popeties is the Youden sue. In this design the tetments occu in ndomized complete blocks with espect to ows (o columns), while with espect to columns (o ows) they occu in symmeticl blnced incomplete block design. In the pesent wok, whole plots occu in one of the bove designs o in some modifiction of them. Then the whole plots e divided into subplots s in the usul split-plot design. Hence the finl design is clled split-plot design geneted by Ltin sue. This mens tht the ow-column stuctue (Ltin sue, modified Ltin sue o Youden sue) is epeted ove the supeblocks. The ngement of tetments my be the sme o diffeent in ech supeblock. Let us conside two-fcto expeiments in which the fist fcto (o set of fctos) occus t levels fctos) occus t t levels, 2,,, nd the second fcto (o set of, 2, t., We conside hee sitution whee the levels of fcto e nged on plots (whole plots) in (modified) Ltin sue, while the levels of the second fcto e nged within the whole plots on the subplots. The levels of fcto will then be clled whole plot tetments, while the levels of fcto will be clled subplot tetments. The im of this ppe is to popose new design, s well s giving suvey of existing methods of constucting new design in which the whole plot tetments occu in Ltin sue o in cetin modifictions theeof, while subplots occu on subplots within whole plots. Specil ttention is pid to the use of contol tetments in such designs. The subplot tetments will occu in ny pope design. Let us ssume tht the expeimentl units e divided into R supeblocks, nd ech supeblock is divided into p whole plot units which fom lttice with ows nd p columns. dditionlly let ech of the whole plots be divided into k subplots. In the ow-column design we cn conside two component block designs, one of them with espect to ows, nd the othe with espect to columns.

3 Contol tetments in designs with split units geneted by Ltin sues 27 The ngement of the whole plot nd subplot tetments on the expeimentl mteil is bsed on pope scheme of ndomiztions (cf. Kchlick nd Mejz, 996). This scheme includes ndomiztions of the supeblocks, the ows (columns), the columns (ows) nd the subplots. s esult of such ndomiztions nd some dditionl ssumptions, we cn expess the obsevtions by line mixed model with ndom supeblock, ow nd column effects nd fixed tetment combintion effects. The pplied scheme of ndomiztions nd the stuctue of the expeimentl mteil led to line model of obsevtions, possessing n othogonl block stuctue. Then the ovell nlysis cn be split into so-clled stt, s in multisttum expeiments. In ou cse we hve five stt, i.e. the inte-supeblock sttum (I), the inte-ow sttum (II), the inte-column sttum (III), the inte-whole plot sttum (IV) nd finlly the inte-subplot sttum (V). In Kchlick nd Mejz (996) method ws given fo nlyzing twofcto expeiment cied out in design in which whole plot tetments occu in epeted ow-column designs while the subplot tetments occu in complete o incomplete block designs whee whole plots e teted s blocks. Howeve, we continue to obseve lck of constuction methods fo the consideed clss of designs. The pesent wok constitutes n impotnt supplement to the foementioned ppe. In pticul the ppes by Kchlick nd Mejz (23), Kchlick et l. (24), Kuiki et l. (29) nd Mejz et l. (29) exmine the sttisticl popeties of design in which whole plot tetments occu in Youden sue nd subplot tetments occu in blnced incomplete block design (ID) o in goup-divisible ptilly blnced incomplete block designs with two efficiency clsses (GDPID(2)s). The ID nd GDPID(2)s e vey useful in biologicl nd gicultul expeiments, nd hence they e often used to genete new moe complex designs with split units (cf. Mejz nd Mejz, 996, Heing nd Mejz, 22, Kchlick nd Mejz, 26, Mejz nd Kuiki, 23). The tetment combintions e consideed s tetments with the ntul lexicogphicl ode of combintions sh (s =, 2,, ; h =, 2,, t) nd

4 28 S. Kuiki, I. Mejz, K. Ozw, S. Mejz the usul expession of the tetment effect s the sum of the fcto effects nd the intection effects. Let v = t denote the numbe of tetments. The sttisticl popeties of the design e connected with the lgebic popeties of the so-clled infomtion mtices the design consideed hee hve the foms: Rpk R, pk p, 22 pk p p 2 3, f,2,3,4, 5 which in pk, () k 33, whee,, 2, 3 e the incidence mtices of tetments vs. supeblocks, tetments vs. ows, tetments vs. columns nd tetments vs. whole plots espectively, denotes the vecto of tetment eplictes, nd stnds fo the digonl mtix with digonl elements eul to the numbes of tetment eplictes. desible sttisticl popety which we shll be exmining is clled the genel blnce (cf. iley, 995). design is genelly blnced iff the infomtion mtices stisfy the conditions (cf. Mejz, 992): f f f, f f, f, f =, 2, 3, 4, 5. f Let us ssume tht the bove commuttivities of the infomtion mtices hold, nd hence the consideed designs e genelly blnced. Let p j define contsts of the fom p j τ, j =, 2,..., v-, whee the vectos p e connected with eigenvectos of infomtion mtices, nd τ denotes the j vecto of effects of tetment combintions. Then the p j τ e clled the bsic contsts (cf. Pece et l., 974). ext, the eigenvlues of the infomtion mtices cn be identified s sttum efficiency fctos of the design with espect to the j-th bsic contst in the f-th sttum, f =, 2, 3, 4, 5; j =, 2,...,v- (cf. Houtmn nd Speed, 983). f,

5 Contol tetments in designs with split units geneted by Ltin sues 29 The sttum efficiency fctos mesue the mount of infomtion tht is included in the stt fo estimting the tetment effect contst. sttum efficiency fcto eul to mens tht the pticul contst is estimble with full efficiency only in tht sttum. sttum efficiency fcto eul to mens tht the contst is not estimble in tht sttum (it is confounded). The sum of sttum efficiency fctos concening ech tetment contst is eul to. We shll ssume thoughout tht the contents of whole plots within ech supeblock e ll the sme with espect to subplot tetments. This mens tht the ngements of whole plot nd subplot tetments cn be expessed by the Konecke poduct of pope incidence mtices. 2. Whole plot design 2.. Ltin sue Let J denote the tetments vs. ows nd tetments vs. columns incidence mtix in Ltin sue, while denotes the incidence mtix fo the subplot tetments. The vectos nd denote the ppopite vectos of whole plot nd subplot tetment eplictes, whee denotes the vecto of ones nd J. Let the whole plot tetments be nged in Ltin sue, nd the subplot tetments in completely ndomized design. Fom () it is esy to check tht the efficiency fctos of the finl design fo the estimtion of tetment contst effects e s in Tble. Tble. Sttum efficiency fctos of Ltin sue with split units Type of contsts umbe of contsts I II Stt III IV V t ( )( t ) Fom Tble we cn infe tht ll contsts mong the whole plot tetment effects e estimted with full efficiency in the fouth sttum. The contsts

6 3 S. Kuiki, I. Mejz, K. Ozw, S. Mejz mong the subplot tetment effects nd the intection effects e estimted with full efficiency in the fifth sttum. The Ltin sue will be used s stting point design fo futhe constuctions. We cn constuct new designs using tetment meging method (cf. Cenk nd Mejz, 987). Moeove, by this method it is possible to constuct desible designs fo subplot tetments lso. Meging of two tetments leds to gete pecision in the est of the design (cf. Cliński nd Kgeym, 23). Fo exmple, let h g s denote the numbe of tetments. We pefom the tetment meging method by multiplying the incidence mtix by ( g ) h meging mtix of the fom: I g P. s The numbe s descibes how mny tetments we would like to mege into one new tetment (cf. Cliński nd Kgeym, 23). Then the numbe of tetments is eul to g. Hee will denote the incidence mtix fo the subplot tetments. Let us mege the lst s whole plot tetments into new tetment. This mens tht togethe we hve g ( g s) whole plot tetments in the finl design. The new incidence mtix hs the fom M P. 2 Then we hve: D 22 D 3 3 D whee,, J g s I g g D 2, D s s, s s,, 2 D nd denotes the Konecke poduct of mtices. The infomtion mtices e s follows: ), k D ( R 2 3, (2)

7 Contol tetments in designs with split units geneted by Ltin sues 3 ], 4 k [( D D) ). 5 D ( k Fom (2) we cn infe tht ll contsts mong the whole plot tetments e estimted with full efficiency in the fouth sttum. The contsts mong the subplot tetments e estimted in the fist nd the fifth sttum, while the contsts mong the intection effects e estimted with efficiencies depending on the efficiencies of the subplot design. Finlly, let us note tht the tetments being meged cn be egded s the contol tetment o stndd tetment, which is eplicted moe times thn othe tetments. This is lso one of the possible wys to incopote contol tetment into design Youden sue Le us ssume tht ech supeblock hs Youden sue stuctue with ows nd columns. Moeove, the subdesign of the Youden sue with espect to columns is symmeticl blnced incomplete block design (ID) with pmetes s follows: ID( v, b,, k, ). Then the following eltionships hold: k, b v nd ( ) Let be the whole plot tetments vs. columns incidence mtix in the Youden sue. Then the so-clled C-mtix fo the Youden sue subdesign with espect to columns is eul to C I k. The so-clled efficiency fcto connected with this mtix is eul to /, while is its multiplicity (cf. Cliński nd Kgeym, 2). Let D (t, b, k, ) denote ny pope block design in which t subplot tetments occu in b blocks of size k ccoding to the incidence mtix, nd let be the vecto of tetment eplictes ( n ), whee n stnds fo the vecto of ones, nd n bk.

8 S. Kuiki, I. Mejz, K. Ozw, S. Mejz 32 Let k C be the C-mtix of the block design D (t, b, k, ) nd let h be n eigenvlue of the mtix C coesponding to n eigenvecto h c with espect to, i.e. let h h h c c C, h =, 2,..., t. The eigenvectos cn be chosen to be - othonoml in pis, i.e., i i c c nd h i c c, fo i h; i, h =, 2,..., t. Since C t, the lst eigenvecto my be chosen s t c t n 2. The sttisticl popeties of the design e connected with the lgebic popeties of the infomtion mtices f, 5,2,3,4, f, which in the design consideed hee hve the foms: n b R Rk J, 2 k, k k J I, (3), ) ( k k J I k b k I I, whee J 2, I, 2 J, J, 2 2, 3 3 I, J I J I ) (. The mtices,, 2, 3 e the incidence mtices of tetments vs. supeblocks, tetments vs. ows, tetments vs. columns nd tetments vs. whole plots espectively, denotes the vecto of tetment eplictes, nd

9 Contol tetments in designs with split units geneted by Ltin sues 33 stnds fo the digonl mtix with digonl elements eul to the numbes of tetment eplictes. Fom (3) we cn infe tht the contsts mong the whole plot tetment effects e estimted in the thid nd fouth stt with efficiencies nd espectively. The contsts mong the subplot tetment effects e estimted in the fist nd fifth stt, with efficiencies depending on the design used fo subplot tetments. Similly, contsts mong the intection effects cn be estimted in the thid, fouth nd/o fifth stt. ll of these emks e summized in Tble 2 (cf. Kuiki et l., 29). The sttum efficiency fctos e functions of nd h, h =, 2,..., t-. Tble 2. Sttum efficiency fctos of the design unde considetion Type of contsts umbe of contsts h h, 2,, t ( ) h h, 2,, t Stt I II III IV V h h ( )( h) ( h) h 3. Subplot designs 3.. Subplot tetments in ID In this section we will conside two pticul cses of optiml designs fo subplot tetments. Let the whole plot tetment be nged in Youden sue s in Section 2.2, while subplot tetments e nged in blnced incomplete block design (ID) with the pmetes v t, b,, k,, R b k k.,

10 34 S. Kuiki, I. Mejz, K. Ozw, S. Mejz Let C be the C-mtix of the ID fo subplot tetments, nd let t( k )/ k( t ) denote the efficiency fctos, with multiciplities t. The finl design is genelly blnced, nd the efficiency fctos e pesented in Tble 3 (cf. Kchlick nd Mejz, 24). Tble 3. Sttum efficiency fctos of the design unde considetion Type of umbe of Stt contsts contsts I II III IV V t ( )( t ) )( ) ( 3.2. Subplot tetments in GDPID(2) Let the whole plot tetments be nged in Youden sue, while the subplot tetments e nged in goup-divisible (GD) ptilly blnced incomplete block design with two ssocition clsses (GDPID(2)) (cf. Cltwothy, 973). The pmetes of the GDPID(2) fo subplot tetments e s follows: v t mn, b,, k ( k ),, 2. The pmetes, 2 especttively denote the numbes of occuing pis of tetments fom the sme goup nd diffeent goups in the blocks. The concuence mtix thee eigenvlues Let nd let i with multiplicities i, whee, k,, 2 k v2, m( n ), m. 2 hs C denote the C -mtix of the GDPID(2) fo subplot tetments, / k denote the efficiency fctos, with multiplicities i i, i,, 2, whee. The ovell sttisticl popeties i 2 i i v (estimbility of tetment effect contsts nd thei efficiencies) of the finl design (Tble 4) e connected with these efficiency fctos (cf. Mejz et l.,

11 Contol tetments in designs with split units geneted by Ltin sues 35 29). The nks of the infomtion mtices, i, 2, 3, 4, 5, depend on the type of GDPID(2) pplied. Tble 4. Sttum efficiency fctos of the design unde considetion Type of umbe of Stt contsts contsts I II III IV V () (2) () ( ) ( )( ) (2) ( ) 2 ( )( 2) 2 2 i 4. Contol tetments In the ntul sciences, expeiments e pefomed to compe known tetments (lso clled test tetments) with set of contol tetments o stndds. Hee we conside minly sitution in which we would like to compe whole plot tetments with some contols (whole plot contols). One of the possible wys to nge contol tetments in Ltin sue is to use meging method, s descibed in Section 2.. nothe wy is to supplement the Youden sue by dditionl contol tetments. This cn be expessed by the following incidence mtix:. J sb Moe exctly, test tetments of the fcto e ssigned in the ( ) Youden sue, nd dditionlly s contol tetments e dded. Using the pevious chcteiztion of the Youden sue with the whole plot test tetments vs. columns incidence mtix we hve k, p b v, ( ),,

12 36 S. Kuiki, I. Mejz, K. Ozw, S. Mejz with multiplicity. The finl design with espect to the whole plot tetments hs the following pmetes: v s, k s, b b, [ s ] k., s,, k s Similly, using the chcteiztion of subplot tetments occuing in the blnced incomplete block design (ID) with the pmetes v t, b,, k, nd incidence mtix, we hve the following pmetes of the design with espect to subplot tetments: ( k ) t k R b, s, k k,,, t k t with multiplicity t. Finlly, the sttum efficiency fctos of tht design e pesented in Tble 5 (cf. Mejz nd Kuiki, 23). In Tble 5 the bbevition effects of whole plot test tetments only; () epesents the set of contsts mong (2) mong effects of whole plot contol tetments only; nd epesents the set of contsts (3) epesents the contst mong the effects of the whole plot test nd contol tetments. Tble 5. Sttum efficiency fctos of the design unde considetion Type of contsts o of contsts () (2) (3) Stt I II III IV V s t ( ( )( t ) ) 2) ( )( ) ( ( )( t ) 3) ( s t

13 Contol tetments in designs with split units geneted by Ltin sues ppliction Let us conside two-fcto expeiment in which we hve fou whole plot tetments ( = 4) nd thee subplot tetments (t = 3). The expeiment ws set up in fou supeblocks (R = 4), ech divided into fou ows ( = 4) nd fou columns ( = 4). dditionlly, ech whole plot is divided into thee subplots (k = 3) to ccommodte the subplot tetments. possible ngement of whole plot tetments in Ltin sue could be the following: The incidence mtix is s follows: Let us mege the lst two whole plot tetments. This we cn do by enumbeing 4 into 3 o multiplying the incidence mtix P. Then we hve P by the mtix P

14 38 S. Kuiki, I. Mejz, K. Ozw, S. Mejz The sme meging mtix cn be used, fo exmple, to constuct subdesign fo the subplot tetments. Let the stting point design be the symmetic I design fo fou tetments with incidence mtix. fte meging the lst two tetments in the subplot tetments occu in n efficiency blnced block design with uneul numbes of eplictes (cf. Cliński nd Kgeym, 23). The ngement of the subplot tetments in supeblocks cn be epesented schemticlly s follows: In ech supeblock the subplot tetments e nged on the subplots within the whole plots ccoding to the incidence mtix P. 2 2 Hence we hve: v b 3, R b 4, k 3, [3, 3, 6], 8/ 9, h =, 2, 3. In Figue the ngement of the subplot nd the whole plot tetments in the supeblocks is given. The sttum efficiency fctos of the design e given in Tble 6. Hee, () nd (2) denote espectively the bsic contst between the min effects of whole plot test tetments nd the bsic contst between the min effects of whole plot test nd contol tetments. () nd h (2) denote the sme

15 Contol tetments in designs with split units geneted by Ltin sues 39 Supeblock I Supeblock II Supeblock III Supeblock IV Figue. ngement of the whole plot nd the subplot tetments in the supeblocks

16 4 S. Kuiki, I. Mejz, K. Ozw, S. Mejz Tble 6. Sttum efficiency fctos in the exmple umbe Stt Type of contsts of contsts I II III IV V () (2) () / 9 8 / 9 (2) / 9 8 / 9 () () / 9 8 / 9 () (2) / 9 8 / 9 (2) () / 9 8 / 9 (2) (2) / 9 8 / 9 bsic contsts s i, j, () nd (2) fo the subplot tetments. ( i) ( j), 2, denote the bsic contsts mong the intection effects of the coesponding whole plot nd subplot tetments; fo exmple, () denotes the bsic contst mong the intection effects of the whole plot nd subplot tetments. () 6. Finl emks This ppe hs pesented suvey of the design of expeiments with split units in which whole plot tetments e nged in Ltin sue, modified Ltin sue o Youden sue. The subplot tetments cn be nged in ny pope block design. In pticul we cn find design fo the subplot tetments stisfying ll of the expeimente s suggestions concening sttisticl popeties, especilly with egd to efficiency. The tetment meging method cn be useful in seching fo pope optiml designs. Moeove, the poblem of intoducing whole plot contol tetments o subplot contol tetments is exmined. The poblems exmined in the ppe demonstte the consideble potentil of the designs consideed s egds thei optiml use in pctice.

17 Contol tetments in designs with split units geneted by Ltin sues 4 cknowledgements The uthos would like to thnk JSPS nd P (Polish cdemy of Sciences) fo giving them the oppotunity to cy out this joint esech in Jpn nd in Polnd. The esech ws ptilly suppoted by JSPS, Gnt-in-id fo Scientific Resech (C) nd Gnt-in-id fo Young Scientists () REFERECES iley R. (995): Genel blnce: tificil theoy o pcticl elevnce. Poc. of the Intentionl Confeence on Line Sttisticl Infeence, LISTT 93, Mthemtics nd pplictions. Vol. 36, Kluwe cdemic Publishes, Dodecht: Cliński T., Kgeym S. (2): lock Designs: Rndomiztiom ppoch, Volume I: nlysis, Lectue otes in Sttistics, 5, Spinge-Velg, ew Yok. Cliński T., Kgeym S. (23): lock Designs: Rndomiztiom ppoch, Volume II: Design, Lectue otes in Sttistics, 7, Spinge-Velg, ew Yok. Cenk., Mejz S. (987): Meging of tetments in cetin ptilly efficiency blnced block designs with two diffeent numbes of eplictions. Clcutt Sttist. ssoc. ull. 36: Cltwothy W.H. (973): Tbles of Two-ssocite-Clss Ptilly lnced Designs. S pplied Mthemtics Seies 63. Wshington, D.C, US. Heing F., Mejz S. (22): n incomplete split-block design geneted by GDPID(2)s. Jounl of Sttisticl Plnning nd Infeence 6: Houtmn.M., Speed T.P. (983): lnce in designed expeiments with othogonl block stuctue. nn. Sttist., : Kchlick D., Heing F., Mejz S. (24): Contol tetments in Youden Sue with Split Units. Foli Fc. Sci. t. Univ. Msyk. unensis, Mthemtic 5: Kchlick D., Mejz S. (996): Repeted ow-column designs with split units. Comp. Sttist. & Dt nlysis 2: Kchlick D., Mejz S. (23): Whole plot contol tetments in Youden sue with split units. Collouium iometyczne 33: Kchlick D., Mejz S. (26): Repeted Youden Sue with Split Units geneted by GDPID(2). XVIIth. Summe School of iometics, "Cuent Tends in iometicl Resech", G. J. Mendel Univesity of gicultue nd Foesty, Lednice, ugust Eds. J. Htmnn i J. Michlek: Kuiki S., Mejz S., Mejz I., Kchlick D. (29): Repeted Youden sues with subplot tetments in pope incomplete block design. iometicl Lettes 46(2): Mejz I., Mejz S. (996): Incomplete split plot designs geneted by GDPID(2). Clcutt Sttist. ssoc. ull. 46: 7-27.

18 42 S. Kuiki, I. Mejz, K. Ozw, S. Mejz Mejz S. (992): On some spects of genel blnce in designed expeiments. Sttistic, nno LII, 2: Mejz S., Kuiki S. (23): Youden Sue with Split Units. J.L. d Silv et l. (eds.), dvnces in Regession, Suvivl nlysis, Exteme vlues, Mkov Pocesses nd Othe Sttisticl pplictions, Studies in Theoeticl nd pplied Sttistics, DOI.7/ ; Spinge-Velg elin Heidelbeg: 3-. Mejz S., Kuiki S. Kchlick D. (29). Repeted Youden Sues with subplot tetments in goup-divisible design. Jounl of Sttistics nd pplictions 4: Pece S.C., Cliński T., Mshll T.F. de C. (974): The bsic contsts of n expeimentl design with specil efeence to the nlysis of dt. iometik 6:

Chapter 7. Kleene s Theorem. 7.1 Kleene s Theorem. The following theorem is the most important and fundamental result in the theory of FA s:

Chapter 7. Kleene s Theorem. 7.1 Kleene s Theorem. The following theorem is the most important and fundamental result in the theory of FA s: Chpte 7 Kleene s Theoem 7.1 Kleene s Theoem The following theoem is the most impotnt nd fundmentl esult in the theoy of FA s: Theoem 6 Any lnguge tht cn e defined y eithe egul expession, o finite utomt,