MEASURES OF BLOCK DESIGN EFFICIENCY RECOVERING INTERBLOCK INFORMATION

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1 MEASURES OF BLOCK DESIGN EFFICIENCY RECOVERING INTERBLOCK INFORMATION Walte T. Fedee 337 Waen Hall, Biometics Unit Conell Univesity Ithaca, NY 4853 and Tey P. Speed Division of Mathematics & Statistics, CSIRO G.P.O. Box 965 Canbea, ACT 260, Austalia ABSTRACT. In evaluating goodness of a class of designs, eseaches have used a measue of design efficiency poposed by F. Yates in the thities. This measue consides only intablock infomation and does not make use of the infomation contained in the inteblock vaiance. The measues of efficiency poposed hee ae dependent upon the atio of the inteblock and intablock components of vaiance, i.e., aata 2 y. The efficiency of one block design to a second may not emain in$aiant with espect to this atio. Incomplete block designs which wee inefficient unde the intablock measue, now become quite efficient fo some atios of y. Likewise, the indications ae that inteblock infomation should always be ecoveed when analyzing data fom expeiments aanged in an incomplete block design. I. INTRODUCTION. In the mid-thities Yates (e.g. in 937) intoduced an efficiency facto fo patially confounded factoials and fo incomplete block designs. The facto is computed as the atio of the aveage vaiance of a diffeence between two adjusted means (o fo factoial effects) to the vaiance of a diffeence of two means fom an othogonal design such as a completely andomized o andomized complete block design assuming no change in the eo vaiance a 2 fo the two designs. It is common pactice in statistical liteatufe to pesent this efficiency facto fo designs and to discuss optimality of classes of designs in tems of the Yates efficiency facto, which only makes use of the intablock eo vaiance. No use is made of the infomation contained in the inteblock vaiance obtained fom the incomplete blocks (eliminating teatment effects) sum of squaes. A moe pope efficiency facto should make use of the infomation contained in both the intablock and the inteblock vaiances. Some measues accomplishing this ae pesented in this pape. BU-85-M in the Technical Repot Seies of the Biometics Unit, Conell Univesity, Ithaca, NY 4853.

2 II. BALANCED INCOMPLETE BLOCK DESIGN. A classical balanced incomplete block design (BIBD) consists of v teatments aanged in b incomplete blocks of size k, k<v, with epetitions of each teatment, and with each and evey pai of teatments occuing togethe in an incomplete block A times. The standad elations ae bk = v, A "" (k-)/(v-), and e = (-k )/(-v ) = v(k-)/k(v-). The facto e is the Yates efficiency facto. The usual esponse model assumed fo a BlED is () whee Yhij is the esponse fo the jth teatment in the ith incomplete block in the hth complete block, h=,...,; is,... b/; j,..,v; nhij is one if the jth teatment occus in the hith incomplete block and zeo othewise; ~ is a geneal mean effect; Ph is the hth complete block effect; ahi is the hith andom incomplete block effect distibuted with mean zeo and common vaiance ~a ~j is the jth teatment effect, and thij ae andom eo effects which wee distibuted with mean zeo and vaiance ~~. An analysis of vaiance is given in Table. TABLE. Analysis of vaiance fo a esolvable BIBD. Souce of vaiation Degee of feedom Expected value of mean squae Total Coection fo mean Teatment (ignoing incomplete block effects) Within teatments Blocks (eliminating teatment effects) bk = v v- v(-) b- Complete blocks - Incomplete blocks (elim. t.) b- Intablock eo v-v-b+ ~2 E ~2 t bk-v + ~2 b- a - k~2 + a ()"2 t Expected mean squae fo Ph= 0, i.e., no complete block effects. Intablock infomation o = /~~ Inteblock infomation o' /(0"~ + k~a)

3 Fo intablock contasts the vaiance of a diffeence between two estimated teatment effects is 2o~/e, when e (l-/k)/(-/v). Fo inteblock contasts the vaiance of a diffeence between two teatments effects is 2(a~ + kaa> (l-e) Fo the combined e.stimato of a diffeence between two teatment effects the vaiance is 2 {7 - e + a2 + ka2 3 2a 2 { + k }.--- ( 3) + k e ' whee = aalo~. Since the intablock contast vaiance is of the fom 2a~/e, it would be lofical to have the combined estimato vaiance in the same fo, i.e., 2ai/el, whee A second measue not involving e is * + k e k(l-e) el... = - + k + k (2) ( 4) * + k e (k+h k + +/y ( 5) 2 2 The latte measue of efficiency depends only upon = a~/oe and k; note that (4) is also a function of k and since e k/(k+l) fo v=k2 and k+. A compaison of the two measues is \iven in*table 2 fo k 3,7, and. Thee is little to choose between e and ez and it is suggested that ef be used as a measue of efficiency athe than e~. Note that as appoaches zeo the efficiency fo all k appoaches unity. When appoaches infinity, the Yates efficiency facto e is appoached fo all k. Fo small k, e is elatively low indicating an inefficient design. Howeve, ef indicates that designs with small k ae quite efficient if is /4 to /6, say. Fom these esults, it is suggested that inteblock infomation should always be ecoveed and that inefficiency of incomplete block design is not a poblem.

4 TABLE 2. Intablock - inteblock efficiencies fo vaious values of y fo k = 3, 7, and. k 3 7 y ef e~ ef e~ e* e~ 0 / / / / OG ke y + k y e* e~ = + k y + (k+)y Fo a andomized complete block design, the vaiance of a diffeence between two aithmetic means is 2 20' v - k e { + v - wheeas the vaiance of a diffeence between two adjusted teatment means fom a BIBD is 2a 2 e * (7) } ( 6)

5 Now (6) ~ (7), thei diffeence being v- k + k y + y v- + ke y v( v-k)(k-) y2 (v-)[v- + v(k-) ] (8) (8) is zeo when 0 and/o v = k. Equation (8) could be anothe measue of intablock - inteblock efficiency. Pehaps a moe appopiate measue would be a atio athe a diffeence to obtain (v-k) e~ /ef ( + ) v- The measue e~ would confom moe to the definition of efficiency oiginally pesented by Yates but would include both intablock and inteblock infomation. (9) III. OTHER BLOCK DESIGNS. A class of genealized N-ay designs wee discussed by Shafiq and Fedee (979, 983). Fo these designs the esponse model equation () is eplaced by Yghij V +Ph+ ahi + ~j + Eghij (0) whee g = O,..,nhij and when nhij = 0 thee is no esponse Yghij The othe symbols ae as defined in (). The above authos genealized the Yates efficiency facto fo this class of designs and they also poved that the Fishe inequality v~b holds fo this geneal class of balanced block designs. The efficiency facto ef fo the genealized balanced block design is e* = - c - A. (k + -) 2 whee c = ~ t nhij nhij is the numbe of times teatment j occus in block hi, and = E E nhij is the numbe of eplicates fo teatment j. h i Fo the class of esolvable incomplete block designs known as lattices, the aveage vaiance of a diffeence between two adjusted teatment means is 2 k==+l k+ --} o ()

6 ... 20'2 k-+ e ( ) ( + ) (2) k+ - + (l+ky)- As befoe, we may take e*.., k+ ( k (l+ky)- + ). (3) Note that is the numbe of confounding aangements and ~2 simple (double) lattice, 3 fo the tiple lattice, etc. fo the An intablock - inteblock measue of efficiency like e~ would be e* = 3 [ + ( - (l+ky)- k+ - + (l+ky)- k [ +-y] k+ )] (4) Note that v=k2 and is the numbe of geometical factoial effects confounded. Using the aveage vaiance of a diffeence between two adjusted teatment effects, we could constuct ef and e~ fo any class of incomplete block designs. The ideas in this pape may be used to constuct efficiency factos simila to ef and e~ fo cubic lattices, fo lattice squaes, and fo othe designs. IV. LITERATURE CITED. Shafiq, M. and W. T. Fedee (979). "Genealized N-ay balanced block designs.".b.iomet:.ika 66, pp Shafiq, M. and W. T. Fedee ( 983). "Geneal binay patially balanced designs." Ind.ian J. Ag.ic. St:at:.ist:., XXXV(2). Yates, F. (937). "The design and analysis of factoial expeiments." Impe.ial Bu. Soil Sci., J'ech. Comm. 35, pp. -95.