NOTE ON THE ANALYSIS OF A RANDOMIZED BLOCK DESIGN. Junjiro Ogawa University of North Carolina

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1 NOTE ON THE ANALYSIS OF A RANDOMIZED BLOCK DESIGN by Junjir Ogawa University f Nrth Carlina This research was supprted by the Office f Naval Research under Cntract N. Nnr-855(06) fr research in prbability statistics at Chapel Hill. Reprductin in whle r in part fr any purpse f the United States Gvernment is permitted. Institute f Statistics Mimegraph Series N. 218 January, 1959

2 NOTE ON THE ANALYSIS OF A RANDOMIZED BLOCK DESIGN' By Junjir Ogawa Institute f Statistics, University f Nrth Carlina Suppse that we are given b blcks each f which cntains k mre r less hmgeneus experimental units r plts k treatments r varieties t be cmpared by experiment. We shall say that the design is a rmized (cmplete) blck design if we assign k treatments at rm (fr instance take k cards with the numbers l,,k n them, shuffle them well lay them ut in a rw t determine the psitin f the first blck. Repetitin f this prcess will prduce the assignment f the treatments in the secnd blck s frth.) VA mathematically rigrus treatment f this arrangement is at present nt yet available. An apprximate test f varietal effects is pssible by treating the arrangement as a tw-way classificatin design ignring the variatin f sil fertility within the rws." [1] The purpse f the present nte is t give a justificatin f the usual analysis f the rmized blck design. Althugh the same argument can easily be extended t general incmplete balanced blck design, fr the sake f simplicity the simplest case, a rmized cmplete blck, is treated. 'This research was supprted by the Office f Naval Research under Cntract N. Nnr-855(06) fr research in prbability statistics at Chapel Hill. Reprductin in whle r in part is permitted fr any purpse f the United States Gvernment.

3 2 T begin with the explanatin f sme useful cncepts f design, we shall be cncerned with general incmplete blck design with v treatments, b blcks each f which cntains k plts. There are n =kb experimental units r plts n the whle. We number them in sme way but nce fr all; fr instance, the i-th experimental unit in the j-th blck bears the number (j - I) + i. The bservatin at i-th plt is dented by xi the whle bservatins are represented by an n-dimensinal vectr ~ whse i-th cmpnent is xi' is called an ~servatin vectr. We shall define the incidence vectrs f treatments as fllws: (1 ) t: al t: 1, if the plt f receives a2 the treatment a ~ =, where t: af = 0, therwise t: an the matrix is called the incidence matrix f treatments. The linear subspace which is generated by ~1 5 v is called treatment space. Likewise the incidence vectrs f blcks are defined by I 1'J a l 1, if the plt f belngs 1'J a2 t the blck a (3) ]a =, where 1'J a f = 0, therwise 1'J an the incidence matrix f blcks is defined by

4 the linear subspace which is generated by ]1 ~b is called the blck space. - The cmpnents f ~ prjected int the blck space with respect t the basis vectrs W is expressed as (5) N = (p'w is called the incidence matrix f the design. (6) N =lin II, where aa 1, if treatment a ccurs in blck a 0, therwise Nw frm the very definitins it is seen that r, ~I =, where r a sts fr the replicatin f treatment a. Thus if in particular then (8) T* - 1 qxpl - r ' = r = r v * 1 HI B = k are idemptents they are the prjectin peratrs. Evidently the treatment space the blck space has the intersectin which is generated by the vectr 1 whse cmpnents are all unity. The prjectin peratr n this intersectin is where G is the n x n matrix whse elements are all unity

5 4 In the special case which we are nw ging t discuss (10) v=k, r=b fur matrices (11) I, G, B = kb*, T = bt* are s-called lirelatinship matrices" f the design [2]. It is knwn that the linear clsure f the matric set {I,G,B,T} is a linear assciative algebra it is cmmutative. Indeed, the multiplicatin table is as fllws: I B T G B kb G kg T G bt bg G kg bg ng The decmpsitin f the unit element int rthgnal idemptents is given by 1 (1 1 ( I = ng + j(b'. - ~) + bt - ~) + (1 - i(b - bt + ~) hence ( ) 1 _ (1 1) (1 1) I - ~ - j(b - fig + bt - ~ + (I - j(b - bt + fig) The meaning f (12) is: ~B - ~ is the prjectin peratr f the 1 1 cntrast space int blck space, bt - fig is the prjectin peratr int treatment space I - ~B - ~T + ~ is the prjectin peratr int errr space, i.e., where ( 1. 1) 1 I 1 I - -kb -.:.(3 x =,.. 'If\T{ x - - II x =is - I x, n - K - n n B = L\' x - 1 f x - n L f e blck i f=1

6 5 ( ( 1 1~) 1 ;M,' 1 I' 1 _ t'fn" - 14) bt - ~ ~ = b ~ ~ - n..! ~ - ':V~ - 1. x, where T, T = T - 1 E x f T k ~af=l, a-rc, a=l,,v. The essential difference between the rmized blck design tw-way classificatin design is that the incidence vectrs f treatment are rm in the frmer case whereas they are fixed vectrs in the latter case. If the plt effect can be ignred, the underlying mdel fr a rmized blck design is that the cnditinal distributin f the residual! =~ - g! - ~. 1-11' ~ 2 where g is the general mean 1 1 = (t", t k ) 2'= (b", bb) are treatment effects blck effects respectively, given ~ is (16) There will be n lss f generality by assuming that k b E t a = Lb a =0 a=l Obviusly the prbability f ~ is discrete l/(kl)b then a=l If we dente the n x k matrix whse elements are all unity by 'J,. (17) G =Jq>1 =~JI since ~T - tp = (~T - i) (~T - ~) = (f; - *J)~'~(~~I -~') = b(~ - 1J) (~I - 1JI) b n b n

7 1 1 1 _ ( 1 1 1)( bt- i(b+ ~ - I - bt- i(b + ~ 1 - bt- i(b +. ~), it fllws that (18) (~I - ~JI)~ = (~,- ~JI )!+ g (~I - ~J I )!. + (~I - ~J')q>.i + (~I - ~J' )'i'h 6 = {~, -!J')e + t b n - - (19) e Thus ( ( ( ) ~' bt- ~)~ =!l(bt- ~)! +2i'b ~' - r/')! + bi'i (21 ) 1 1 1) _ x' (1 - -bt - -kb+.:g n... x - -e' (1 - -bt - -kb +.:G)e n... Take an rthgnal matrix f the frm (22) p = let e* =P e I & r (2.3) then we have e = P'e* - -

8 7 I (J4> _!J) ~ b n I( 1 1 1) P I --T - -B +.:<3 _n b k n hence 1 Q since

9 (~, - 1J 1 ) (I -IT _lb+10) = (~, _lj' ) (I - 1ftl)- (~I _lj,)(~_lj)q>' b n b k n b n k b n b n _ (1 1) -;41' - -J' - =4>'ff l + -J'ft' - -I -.':G q>1 b n n nk b k n k _ ]* 1 ~ 1 1 _ - ~I - _JI - - 'l" + -.r'l" -;4>' + _JI - 0 b n n n b n ' where J* sts fr the k x b matrix whse elements are all unity 8 0 R Since ~T -.':G n I - -T - -B +.':G are rthgnal idemptents with b k n rank k -1 (b-l){k-l) respectively, Q R are rthgnal idemptents f rank k - 2 (b-l) (k-1) respectively. Finally since (26), e* n if we cnsider the cnditinal jint distributin f ~/~ ~ /~, given q>, then they are mutually independent, i.e., T/02 beys the nn-central chi-square distributin f d.f. k-l with nn-centrality parameter bt't/0 2 :t /0 2 beys the chi-square distributin f d. f.

10 9 (b-1) (b-1). Thus the cnditinal (given~) distributin f the statistic (28) is the nn-central F distributin f d.f. (k-l~ (k-l)(b-l» with nn-centrality parameter bl'!,/0 2 Cnsequently the abslute distributin f the abve statistic is the same. Thus this seems t ffer a way f justificatin f the traditinal treatment f this prblem prvided that we can ignre the plt effect. References [lj Mann, H.B., Analysis design f experimen~. Dver Publicatins, Inc., 1949, Chapt. VII, p. 76. [2J James, A.T., "The relatinship algebra f an experimental design," Ann. f Math. stat., vl. 28, 1957.