Biometrika Trust

The Design of Optimum Multifactorial Experiments Author(s): R. L. Plackett and J. P. Burman Source: Biometrika, Vol. 33, No. 4 (Jun., 1946), pp. 305-325 Published by: Biometrika Trust Stable URL: http://www.jstor.org/stable/2332195 . Accessed: 01/04/2011 12:26Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at . http://www.jstor.org/action/showPublisher?publisherCode=bio. . Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.

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[ 305 ]

THE

DESIGN

OF OPTIMUM

MULTIFACTORIAL

EXPERIMENTS

BY R. L. PLACKETT ANDJ. P. BURMAN1. INTRODUCTION

A problem in in which often occurs thedesign an experiment physical industrial research of or is thatofdetermining suitabletolerances the cofmponents a certain for more of assembly; the in generally ascertaining effect quantitative qualitative of of or alterations thevarious characteristic the complete It of components uponsomemeasured assembly. is sometimes to shouldbe; butit is to themoregeneral case whenthis possible calculatewhatthiseffect is notso thatthemethods givenbelowapply. In sucha case it might appearto be bestto the and of varythe components independently studyseparately effect each in turn.Such a procedure, either labouroraccuracy, while carry a complete however, wasteful is of to out factorial experiment to makeup assemblies all possiblecombinations then com(i.e. of of wouldrequire assemblies, Ln whereL is thenumber values(assumed ponents) of constant) at which each component appear. For L equal to 2 thisnumber largefor n can is moderate and quiteimpracticable n greater for L is than,say,10. For larger thesituation evenworse. is the factorial which Whatis required a selection N assemblies from complete will of design enablethecomponent effects be estimated with sameaccuracy ifattention been the as had to concentrated varying single on a the are component throughout N assemblies. Designs given below for L = 2 and all possibleN < 100 except N = 92 (as yet not known),and for L= 3,4,5,7 whenN = Lr (for allr). The following results have beenobtained: (a) Wheneach component appearsat L values,all maineffects be determined may with themaximum N precision possible using assemblies, and only L2 divides and certain if, if, N, further conditions satisfied. are (b) For L = 2, thesolution theproblem for of is practical purposes complete. designs In oftheform = Lr,theeffects certain N of the interactions between components also be may estimated withmaximum precision. The precision naturally increases withthe number assemblies of measured, and to this extentdependson thejudgement theexperimenter. of Before explaining procedure the in someintroductory are necessary theassumptions remarks detail, on madeand themethod of least squares.2. EXPERIMENTAL EFFECTS WHEN L = 2

in Each component the assembly it appearsat twovaluesthroughout; willbe convenient to call one ofthemthenominal and theother extreme, the where former the usuallyrefers to theactualnominal valueand thelatter an extreme thetolerance to of rangefor comthe for ponentin question(the same extreme each appearanceof the component a given in Denote the measurable experiment). characteristics the components the assembly of in (onepercomponent) xl, x2,..., xnand themeasured by assembly characteristic y. byThen y = Y(X1,x2,**., Xn),

where functional the relationship in general is unknown. Supposethatthenominal value ofxi is x?and ofy,yl. Thus y x? ...,I x?).

3061 component is

Design of optimum multifactorial experimentsml = [y(xI,,X2, X3, ..., Xn) y(x4, X2 X3, . * Xn)].2n.

Suppose also that the extremevalue of xi under considerationis x,. Then the main effect of wherethe total numberof possible assemblies is 2n. In each of the two summationsabove, the indiceson the xi (i * 1) rangeover all possible sets of values. Similarly, M2,M3, ... mnare defined. For brevitythe above equation will be written: ml = [-Py(x') _y(A?)]/2n, and in generalEy (xi,x ... 4k) will representthe function evaluated with xixj ... xk,taking y the values shownand summedover all possible sets ofvalues ofthe variables that have been suppressed.The main effect a componentis thusseen to be the mean effect themeasured of on assembly characteristicwhich that componentwould produce if acting on its own. Proceeding further definethe interaction we between components1, 2, 3, ..., p asm(123)...= pY(X1

X2.

p') - 22y(xIx2 ...xp_ xpo)+ y(xI X2* ... Xp-2Xp?_1Xp?) + * + (-1)P

wherethe innersummationis as explained above; the outer extends over the pC1,pC2,etc., selectionsof 1, 2, 3, etc., indices 0 available. The nature of an interactionhas been discussed by Fisher (1942) and others,and our definition accords with the usual one. If main effects regardedas being ofthe first are orderofsmall quantitiesand if the function y maybedifferentiated, first the approximationto m(123 p) isM(123...p) = (aPy/aXlaX2...

Py(XX02 ... xpO)]/2n,

axp) (x'- X)

(X2-x2)

wherethe coefficient mi is + 1 accordingas the ith componentis at extremeor nominalin of thejth assembly; the coefficient M(123 p)is + 1 accordingas the numberofplus ones among of the coefficients m1m2 mpis odd or even. In doing this the signs of the odd-orderinterof ... actions (involvingan even numberof factors.) have been reversed,but the notation is convenient,forthen the coefficients Yi are all minus one. It is assumed that Yi is always one in of the selected assemblies,and this is no real restriction upon the design. 3.LEAST SQUARES AND PRECISION

the derivativebeingaveragedoverthevalues it takes all 8et8ofvalues ofthe remaining for components. This shows that when the variables are measured on a continuousscale we may validly neglect all the interactionsabove a certain order,fora (p - 1)th orderinteraction (one in p components)is of thepth orderof smallness. But the justification this assumpfor tion when some of the xi are qualitative and not quantitative (and it is frequently made) mustbe foundin considerations outsidethe data whichthe experiment provides,in commonsense or philosophical grounds. The grand mean M = Zy(x1,X2, ..., xn)/2nwherethe summationis over all possible sets of values of the components. In thejth assembly of an actual experiment,some components will be at nominal and some at their extreme values. If the true value of the assembly characteristic then y., it is foundon solving the above equations that is ... +ajnmn+aj (1) n+lM(12)+ ... +aj,2nm(123 ). Yj = M+ajlml+aJ2m2+

... (Xp-Xp),

The purposeoftheexperiment to estimatethoseofthe quantitiesm as may not be assumed is negligiblefroma set of measurements r2,..., rN. For this we must solve a set of N linear r1, equations represented (1). The equations always involve M, and therefore estimate q by to of the m's it is necessaryto make at least (q + 1) measurements.If exactly (q + 1) assemblies

R. L. PLACKETT

AND J.

P.

BURMAN

307

there a uniqueset ofm's satisfying equations;ifmorethan(q+ 1) are the are measured, is and the best estimates therewill be no unique solution measured are, as is well known, of obtained themethod leastsquares.Thisobtainstheset of mn's by which minimizes

S

(rj-M-aj1m1-aj2M2

ap,

* ..)2

interactions muchlessthan 2n,all high-order beingneglected, that thenumberofassemblies so N may be made much smallerthan forthe completefactorialdesign. As already stated,thegreaterthenumberofassembliesmeasured,thegreatertheprecision withwhichcomponenteffects may be estimated. On account of errorsof measurementand the the neglectof certaineffects minimumS0 ofS is not zero. In fact Sol(N - q - 1) provides an unbiased estimate 82 of o-2, the variance of error of each measurement (assumed the same for all assemblies). The error variance in the estimation of an effectmi is of the formo-2/t, whereti is called the precisionconstant. It depends only on the design and can be increased indefinitely increasingN. Our object is to find of the experiment, by designswhich maximize all the ti simultaneouslyforgiven N. They will be called optimum has a t-distribution the null hypothesis-that the true on designs. The ratio of Mi to 8/4ti of to value of mi is zero. The effect increasingthe precisionis, first, increasethe power of the t-testin detectingany departureof mi fromzero; secondly,to increase the accuracy of its estimation. In the designsgiven at the end of this paper, forL = 2 all main effects may be estimated with maximum precisionN (given N assemblies), that is, the standard errorof = aiVN providedN is a multipleof 4. In cases whereN = 2rcertaininteractions may also The choice ofN (subject to N > q + 1) will depend be estimatedwiththe maximumprecision. wishes to minimizethe effect his experimenital on the extent to which the experimenter of error. 4. REQUIREMENTS FOR OPTIMUM DESIGNS (ANY L) I. Consider now the case of n componentseach of which may take L values. If interactions are neglected,the truevalues yi may be assumed linearfunctions certainconstants of representing the main effects, was proved rigorouslyforthe case L = 2. In general let as due to the jth component at its lth value. The true value of the xj(j) representthe effect measurementon the ith assembly is

in whosetruevalue is yj. Normally + 1) is where is themeasurement thejth assembly r; (q

yi

X

xj(,) -=12,...,j=

1,2,...,n

N ,

1= 1,2,...,ILwhere 1 representsthe value at which the jth componentappears in the ith assembly. We now introducecertain new variables in termsof which to express the x1(,),as the primary is of interest inthechange assemblycharacteristic caused by certainchanges thecomponents. in Let Q be a non-singularL x L matrixwhose firstcolumn consists entirelyof ones, such that Q = OD, where 0 is orthogonaland D diagonal. The conditionon the first column of Q impliesthat dl = VL. - xl(l =Q-lX = Let U=

1

[t2

X1(2)

J?t1,_

LXl(L)

308+ + u, = (x1(l) x1(2)

Design of optimummultifactorial experiments

+ of *-. xl(L))/L= mean effect component1 and U2,U3, ..., UL are constants which determinethe effect changesin this componentupon the assembly characteristic. of will be used later. Therefore The orthogonality property X1 = QU=

u1+a12u2U1+a22u2

+ a13u3 +*+a23u3

+alLUL],

+ ... +a2LUL

Ul+aL2U2+aL3U3+

... +faLLULJ

where the aij are certainconstants. Similarly,introducevariables v1, V2, ponent 2 and write:x2=

V3, ...,

VL forcom-

x2() X2(2)

v1+al22 v1+a22v2

+al33 +a23V3

+ ... + alLVL + ...+a2L

VL

X2(L)I

vl+aL2V2+aL3V3+

...

+ aLLVL

of wherev1is the mean effect component2. And so on. Hence

Y=

yY2

=AX,

where X=

[M1

M = ul + v, + ... to n terms.

VL::I and rij = rji.

U2

that

column consistingof ones, A is a matrixwith N rows and n(L - 1) + 1 columns,the first and the remainderconsistingof the elementsaij belongingto Q. The columnsfall into sets and the rows of the submatrix to (corresponding the components)of (L - 1) afterthe first, consist of repetitionsof the rows of Q. At this stage renumberthe formedby such a set of suffices the aij so that A may be written(aij). II. The vector Y is known. Solving the equations by least squares (assuming N>n(L1)+ 1) gives the so-called normal equations A'Y = A'AX = CX say, i.e. variance of a singleobservationy, it is proved in text-books - C-'A' Y. If 0-2 iS the error X(Ok) 2var iS=

|

kk 211 C 1,

I

whereOk

to t n x n matrix.It is required minimizeCkk 1/1C 1,i.e. to maximize = | C I/I I by suitCkk Iable choice of design. whereri Write cjjlckc 3 = r.j and the matrix R = [rij],

the kthelementof X and Ckk the cofactorofCkk in C = [ci1],C beinga symmetric

Now

r=

E ariarj

/(

=

a 2 2i) (

ar;)

If (ali, a2i, as aNj) and (a1j,a2j, ..., aNj) be interpreted the co-ordinatesof two points = cos PiOPj, where0 is the originand hence r2. 1. Pi and PI in a Euclidean N-space, thenrij

R. L. PLACKETT Now t = j R j Skil Rkk where Sk j,=

AND J.

P.Sk

BURMAN

309

Ear

2

and so

must be fixed otherwiset may be

increased indefinitely. This is equivalent to fixingthe scale of measurement, the preceding section having dealt with the choice of origin at the mean. Eliminate the pth row and columnfromI R I and IRkk; by pivotal condensation:multiplythepth columnby rpj (p * k) and subtractfromthejth Xcolumn all j * p. A row of zeros appears in the pth row except for in the diagonal place wherethereis a one. The determinants have been reduced in ordbrby one, and the second is still a principalminorof the first. Thus rpj Irij = rij- riprjp (remembering = rjp)=

I(1-r2p )'(1-r2

)1 rij.p I

defining p in this manner,where the suffices appearing afterthe dot representcolumns ri that have been eliminated. Therefore taking out factorsfromrows and columnst = Sk H (1-r i .) r11|i#P=

/H

Sk(l -r')

| rij.p I/1[rij.p]kkI

i#p,k

(1

r*p) [rij.p]kk

Now

rij.p= (cos PiOPj-cos Pi OPp cos Pj OPp)/sin Pi OPp sin Pj OPp,

which is the formulaforthe cosine of the projectionof angle Pi OPj on to the (N- 1)-space orthogonalto OPP. Therefore r2jpS1 (itj) and

i

=

1.

The method has obtaiineda ratio of two determinantsof the same type as before,and the process is repeated, step by step, until that in the numeratoris of the form1rqk.pIp2 ...

rqk.PlP2

.

!

and the denominatoris 1. Row and column pl, P2' ..., are eliminated in turn (no p being equal to k), and so 2 t = Sk(1 -r 2 ) ( 1-rk2. ) (1-rk3.1 2) * (1 -r q*pp2,p 2)X This is a maximum only when rk, = 0 forall p * k and all k. For equal precisionSk must A'A = C = tI. Hence the designs forwhich the be constantforall k and t = Sk. Therefore maximum precision is attained are those which correspondto columns of an orthogonal matrix (apart froman arbitrarymultiplier). At this point it is convenientto prove the formulaforthe errorvariance. Let A be the non-square matrixwith orthogonalcolumnsof the equations: Y = AX. Introduce further columns U so that (A, U) is a square orthogonalmatrix,and corresponding dummyvariables whose column vector is Z. The least squares solutionof the above equations is XO given by 1 A'AXo = AT'Y,therefore (1) tIXo=A'Y, Xo -tA'Y. -t

The equations[A, U]

[z] ==

Y have a uniquesolution, on multiplying and by

so the resultingvalue of X

as before.The residual vector E = Y - AXo = UZ. Sum of squares of residuals = E'E = Z'U'UZ = tZ'Z = tfz..

(2)

310

Design of optimummultifactorial experiment8

III. Consider now any pair of componentsf and y. Suppose they appear togetherat their lth and l'th values respectivelywl times. This definesan L x L matrix W = [w11j]. The scalar productof a columnof A belongingto f by a columnof A belongingto g is zero by the orthogonality A. Let these correspondto the uth and vthcolumnsof Q and revert of to the old suffices aiJcorresponding Q, i.e. Q = [aij], wherei,j = 1, 2, ..., L. of to in dummy sufficesequals Then a, w . al,v QIWQ

NO 0 O O

0 O... ...

... ... ...

01.O O

The N appears because the first column forf is the same as the first column forg, equal to the first column of A which consistsentirelyof ones. Now Q = OD where 0 is orthogonaland D diagonal, therefore Q'WQ = DO'WOD, i.e.O'WO-D-lN 0]D-1 01.o-

=

[N/d21

o

Therefore

W= OO'WOO' = -14L

~

-NIL O1 [l/L

I[1L other terms= [N/L2 N/L2...

1 VL ... 1/VL terms other

N/L2]

N/L2

N/L2

NIL22

LNIL2 2 NIL

NIL2

Sum of termsin lth row is the numberof replicationsof the lth value off. Therefore (i) Each componentis replicated at each of its values the same numberof times. (ii) Each pair of componentsoccur togetherat every combination of values the same number of times. (iii) The numberof assemblies is divisible by the square of the numberof values. The converse-that underthese conditionsthe matrixA is orthogonal-can be proved by these steps. The actual matrix Q chosen is unimportant,and the design can be reversirig specifiedby means of a rectangulararraywithN rows and n columnscontainingL different letters (a, b,c, ..., k) representing the L values of each component. The problem is then a purely combinatorialone. If N - KL2, the maximum numberof columnsn is (KL2-1)/(L-1)

factorial designs.

or its integralpart since KL2 > n(L

-

1) + 1. We propose to call designs of this type multi-

R. L. PLACKETT

AND J.

P.

BURMAN

311the other columns

Returningto the case of L = 2, it is necessary to obtain an orthogonal4K x 4K matrix A whose firstcolumn consists entirelyof ones. Choosing Q=-

ofA consistof equal numbersof + 1 and -1. The signsmay be changed down the lengthof certaincolumnswithoutspoilingthe designso that thefirst apart from corner the row element consists entirelyof - 1. Then apart fromthis row (in futurecalled the basic row) and the first column the design consistsof a square matrixwith 2K plus and (2K - 1) minus ones in each column and (by orthogonality) row,and such that each pair of columnscontains a pair of plus ones in the same row K times. The estimatesof componenteffects obtained from are equation (1) of ? 4 (I): 1

Xo = iKAlY

(heret=4K).

Thus they can be evaluated by addition and subtraction with only one division. This simplicity appears in the illustrativeexample given in ?? 9 and 10. The dummyvariables zi are similarlyevaluated and the estimated errorvariance is2 -

4K-n-i 1

4K

_Z2

5. METHODS OF SOLUTION

Certain methods of constructing orthogonalmatriceswith elementsplus or minus one are known (Paley, 1933). They depend upon the theoryof finite fields,an outline of which will now be given. A fieldF is definedas a set of quantities which is closed with respect to two operations, addition and multiplication(i.e. if a, b in F, so are a+ b,ab). These quantities satisfythe followinglaws: (i) a+b = b+a. (ii) a+(b+c) = (a+b)+c. (iii) a(b+c) = ab+ac. ab=

ba.

a(bc) = (ab) c.

(iv) There is an x such that a + x = b foreverya, b. From these it may be proved: (a) There is a unique quantity 0 such that a + 0 = a forall a. (b) The quantity x in (iv) is unique. (c) a . 0 = 0. Finally, we add (v) There is a y such that ay = b forevery b, all a * O, to our axioms. Hence as before:(d) There is a unique quantity 1 such that a. 1 = a forall a. (e) There is a unique quantitya-1 such that a. a-' = 1 (a * 0). (f ) The quantityyin (v) is unique. y = a-lb. Consider the integers 0, 1, 2, ..., (p-1) where p is prime, and write a = b if (a-b) is divisible by p. Then this set of integersformsa finitefieldas may be easily shown. For example, whenp = 5, the numbersin the fieldare 0, 1, 2, 3, 4. 2+4= 6 = 1, 2+3 = 5 = 0, 2.3 = 4.4= 1. Hence 2 and 3 are reciprocalsand 4 is its own reciprocal.This fieldis called the Galois fieldof orderp, GF(p). Now suppose x is a numberalgebraic over GF(p), that is, x satisfiesan algebraic equation with coefficients GF(p). Then it definesan algebraic extension of GF(p), namely, all in polynomialsin x with coefficients GF(p). If x satisfiesan equation irreduciblein GF(p) in and of degreen, thereare pn distinctpolynomialsin x. They are of the form f(x) = ao+ c4x+ *** +a.- xn- (ao,a1, ...,aa1 in GF(p)).

312

Design ofoptimum multifactorial experiments

Such an algebraic extensionis, in fact,a field. Moreover,all fieldsof degreen over GF(p) may be shown to be equivalent. There is a member of the extended field cx such that 1, x,a2, a3, ... q.Xq-2 (q = pn) constitutethe non-zeroelements of the fieldand aq-1 = 1; an extension of Fermat's theorem. Any one of these equivalent fieldsis denoted by GF(p'). We shall now requirevarious simple theorems.DEF. If a non-zeroelementofa finite fieldis a perfect square (a residue of the field.All othernon-zeroelementsare non-Q.R.'S.-

b2)it is called a quadratic

TH. 1. The numbersof Q.R.'S and non-Q.R.'sare equal (p > 2). For every b = au, a = b2=a2u = (A integral). a2u-A(q-1) But (q- 1) is even (p > 2). Hence only even powersof acare Q.R.'S. Thereforethereare 1(q- 1) Q.R.'s. and -(q- 1) non-Q.R.'s. We now definethe Legendre functionx(a):

x(o) = O, X(a) = + 1 when a is aQ.R.=-1 when a is a non-Q.R. Th. 1 states that . X(a) = O.(summationover whole field).a

TH. 2. X(a) x(ab). This is trivial when a = 0 or b = 0. Otherwisea = a),b = av and ab = au+v is a Q.R. if and only if (u + v) is even, i.e. u and v of same parity. This proves the result. X(b) = TH.3. X(-1)=+1ifq=4t+l} forintegralt.

Por aq-L = +1. Therefore ai(-l) -+ 1 =-1 since powers of a are distinctup to aq-1. Hence -1 is a Q.R. if and only if I(q-1) is even = 2t and q = 4t+ 1.

Tia. 4.

Y

X( -

il)

x(i-

i2) =-1=

(summation overallj in GF(pn); p > 2; il t i2),byTh. 2._____

X(j-il)X(j-i2) Put ]Expression=Putu=uovE

Ex{(j-il)(j-i2)} j.

(i1+i2)2

2

',2=p0

x(u2 - u) (j is summed over whole fieldso u will be also).

Expression = Z x{u2(v2 - 1)} (u is summed over whole fieldso v will be also)v

(uo4s0).

=

z2X(u2)X(v2-1)v

~

=

EX(V2_1)v

by Th. 2.=

Nowif v2- 1 = x2, v2_x2 If v+x = y, v-x = y-. Therefore

1

(v-x) (v+x)

1.

v - 2(Y+y-1)

x = 2(y-y1).=

Hence the numberof values of v forwhich x(v2- 1) for which v = I(y+y-1)

+ 1 or 0 is the numberof values of v

R. L. PLACKETT

AND J.

P.

BURMAN

313

Nowif (y-w)(1-yw) = 0. y2W+W = yw2+y, y+Y 1 w+w-1 Therefore= y ory-1.y andy-1aredistinct w unless = ? 1,v = + I when y Hence every oftheI (q- 1)reciprocal (y, corresponds to one pairs y-1) a distinct ofv. value Thusthere I (q-3) values v for are of which - 1) = + 1 (excluding= + 1). v X(V2X(V2- 1) = X(0) = 0.

Thereare twovaluesofv for which X(V2-1) = 0 (V + 1). I (q- 1) valuesofv forwhich Hence there are X(V2-1)--1. = Ex(v2-1) =-1. Therefore EX(i-i1)x(j-i2)i v

Applications I. Consider matrix = (aij)(i,j = 0,1,2,...,p) oforder + 1),where = 4t- 1. the A (p p = aoj = + 1, ajo ai1= X(j-i) (i*O,j*O, itj), Thescalar product 1stand(i + 1)throws of-

aii =-1.

aooajo+ ao0aii+ Ej=1

p

x(i-)

=1-1+0=0-

(Th.1).P +

Scalar product (ij + 1)thand (i2+ 1)throws of+ as10ai20 a1ias2il

t aI2i2aili2

x(j- i1) X(j -

i2)

withtheelements GF(pn) and theproof first) of runsexactly before. as

1-X(i1-i2)-X(i2-is)-i1 (Th.4) Osince = 4t-1 (Th.3). p Hence matrix is orthogonal. the A II. To construct oforder + 1 = 4t,we associatetherows columns A pn and (except the-

-

III. IfA is orthogonal A structed successive by doubling. Butan A oforder + 2(ppn1)

is alsoorthogonal hasdouble order A. and the of+

Hence an orthogonal matrix of order2h(pn A

1) (where = 4t- 1) or 2h can be conpn

IV. If pn = 4t+ 1, (pn+ 1) is notdivisible 4. by

= bio boj= +1 (i O j* O), bii= x(u - ui) (i * O)j * O) where is theelement GF(pn)associated of withthe (i + 1)throwand column B, of ui = boo 0. Scalar of product 1stand(i + 1)throws=pn

the B Consider matrix = (bij) (i,j = 0, 1,2, ...,pn) oforder(pn+ 1) [p1U 4t+ 1]. =

canbe obtained a slight by modification method. ofthe

+ boobio E x(u1-ui)= 0 (Th. 1).j=l

314

experinents nmultifactorial Design of oplimvioUm

Scalar product of (il + 1)th and (i2+ 1)th rows =1-1=0 Thus B is orthogonal. (Th.4).j=1

Nowreplace 1 bythesubmatrix C = +-1 bythesubmatrix-C= 0 by thesubmatrix D

+

1j].1

[[

I+jj-1

+1 -1

I * The newmatrixA thusformed is of order 1).2(pn+

and (2i2+ 2)throws.Thisis a (2 x 2) nmatrix M1,i2.

Consider the scalar products of the (2il + 1)th and (2il + 2)th rows with the (2i2+ 1)th= MA112 Epn

Now

C') C) (bi,j (b2i C') + (D) (bi2il + (bi02 C) (D')pft

[j * il, jii2]

i2]

= CC0E biiib + X(uil - ui2) (DC' + CD') 2J j=0(by Th. 2 and Th. 3 forp?l = 4t+ 1)=

[j* il, j

CC'0+X(U4

-Ui2)[i

0]

since E bij =0

pn

= bbi2i

0

of (orthogonality B) and the omittedtermsvanish

Thus by successivedoublingwe may obtain matricesoforder2h(p?,

and HenceA is orthogonal ofthe required. type

Finally, it is clear that the (2il + 1)th and (2i, + 2)th rows are orthogonalto each other.+ 1)

wherepxl= 4t+ 1.

V. Summing up we have: If N = 2h(p4+ 1) = 4K, wherep is an odd primeor zero, an orthogonalmatrixA can be constructedwith plus and minus ones. The matrices constructibleby these methods include all values of N = 4K up to 100 the excepting92. Those of order2rare structurally same as the completefactorialdesignin r factorsif theyhave been obtainedby successivedoubling.These will be called geometrical geometries.It is clear that iftwo columns designsbecause oftheirclose connexionwithfinite to and ifthe interactioncolumn corresponding themin main effects, of the design represent it the completefactorialcase is a dummyin the actual experiment, may be used to estimate the interaction.The conditionfora column to be the interactionbetweenp other columns is that it is + D, where D is a column vector whose ith element is the product of the ith elements of the originalp columns. So far interaction columns have only been found in the geometrical designs and in them every interaction between an arbitrary set of columns is a column of the design. It must also be mentionedthat the cyclic designs for N = 2robtainable by the methodof? 8 dependingon GF(2r) are in factmerelypermutations of the geometricdesigns. They are the formsused in the tables forconvenience.

R. L.6.

PLACKETT

AND J.

P.

BURMAN

315

CASE OF MORE THAN TWO LEVELS

for We nowprovideexperimental effects withmaximum designs determining componentprecisionwhen the number of values L is greater than 2. These solutions cover the cases wherethe numberof assemblies N = Lr, L being a prime or a power of a prime and r any successive columns of the design are positive integer.Two methods are given: in the first, formedby simple operations on the preceding columns; in the second, which is of more limitedapplication,thedesignis specified one column,all othersbeingcyclicpermutations by of this.7. MODIFIED FACTORIAL DESIGNS

The methods given in this section forthe constructionof multifactorial designs,although discovered independently,are neverthelessidentical with those used by Bose & Kishen (1940) to expressthe generalizedinteraction the purposeofconfounding for certaincontrasts in with block differences agricultural experiments. They construct their interactions directlyfromfinite projective geometrieswithoutusing, as we have done, the intermediate device of orthogonalsets of Latin squares. We shall, however,describe these methods, as to in theymay not be familiar experimenters thiscountry, especiallynot in the way in which we propose using them. Suppose a completefactorialexperimentis carriedout forr factorseach at L levels (i.e. in this case r componentsmeasured at L values) so that Lr assemblies are made. Let the levels be called 0, 1, 2, ..., (L - 1). Then the r main effects definer columnsof a design array (with Lr rows) containingthese L symbols. Each symbolappears the same numberof times in a column as any other. Each combination of symbols for two columns occurs equally frequently. We shall apply the termorthogonalto such a pair of columns. Now let A, B be two orthogonalcolumns. The interactionAB has (L - 1)2 degrees of freedom. Since each column of the array is associated with (L - 1) degreesof freedom, first-order a interactionis representedby a set of (L - 1) columns which will be called the termsof this interaction. Similarly,an interactionof the mth order (i.e. involving m + 1 factors) is representedby (L -1 )m columnsof the design array. Now an interaction between two factors is most naturally defined by the following conditions: (a) Each combination of levels of A and B correspondsto only one level within each term of AB. (b) The termsof AB are orthogonalto A and B and to one another. Condition (a) means that if, forinstance, in one assembly level 2 of A and level 5 of B occur together, and ifa termofAB is definedto appear at level 3 in this case, thenwhenever A and B occur at theselevels together again, thistermofA B appears at level 3. Since, owing to the complete factorial basis of the design, every combination of levels of ABC occurs equally often,each combinationof levels of A and B occurs equally oftenwith every level of C. But such a combinationoflevels ofA and B fixesthe level in a termofAB by condition (a). Hence each level in a termof AB occurs equally oftenwith every level of C. In other words,each interactiontermwill be orthogonalto the main effects connected with it. not Now for condition (b). If the rows and columns of a square L x L array correspond respectivelyto the L levels ofA and B, each cell may be filledup withthe level appropriate to a particularinteraction term. For any particulartermofAB such a square will be Latin, because, regardinga row, the level of A is fixed; all the levels of the interactiontermmust

316

Design of optimummnultifactorial experiments

occurequallyoften withthislevelofA and henceeach symbol rowof appearsoncein every thesquare;similarly appearsoncein every it column order in term thattheinteraction may be orthogonal B. Finally, to of the for superimposing Latinsquares twoterms thesameinteraction, each symbol belonging thefirst to term mustappearoncein thesamecellwitheach symbol the secondterm, order of in thatthesetwo maybe orthogonal: thusifconditions (a) and (b) are satisfied interaction the are terms founded upona completely orthogonal set of Latin squares. It remains showthatthe terms to are from two different interactions orthogonal. This follows becauseevery combination levelsoffour of factors ABCD occurs equallyfrequently, i.e. each combination levelsofA and B occursequallyoften of witheverycombination of levelsofC and D. The former correspond levelsin theterms AB; thelatter levelsin to to of terms CD. Hencea term A B is orthogonal a term CD. Similarly B andA C may of of to of A be dealtwith.This showsthatthe first order interaction terms maybe joinedto themain factors partofthe balanceddesign.Higherorder as interactions be regarded first as may orderinteractions = (AB) (C), it beingunderbetweenthoseof lowerorder, e.g. (ABC) stoodthat(L - 1) terms derived are from eachterm (AB) takenwith factor so that of the C, in this case therewill be (L - 1)2 termsforthe secondorderinteraction. This procedure buildsup thedesign an inductive by process, whentheinteraction the(r-1 )thorder and of has beenobtained, willbe complete. totaldegrees freedom theoriginal it The of factorial in design= Lr-1. Hence the number factors of that may be measured interactions if areneglectedLr_1

L-1'

It maybe remarked there nothing in thistreatment thecomplete that is new factorial of design exceptthemodification theusual Fisherinteractions thattheymaybe placed of so on thesamefooting maineffects. as If L is a prime number, cyclicLatin squaresexistforming orthogonal each square an set: isobtained writing first ofsymbols standard by the row in order, successive being rows obtained by shifting symbols the alongp places fromeach row to the next (p = 1,2,..., (L - 1)). In thiscase the appropriate column the designis formed follows: of as assuming first the rowin theorder 1,2, ..., (L - 1), ifx levelofA and y levelofB occurtogether, corre0, the levelfor sponding thisinteraction term (y+ px),thesymbols is being reduced withmodulus L. The squaresfor = 1,2, ..., (L- 1), giveall theinteraction p terms. WhenL is thenthpower a prime thenwe associatetheL levelsofa factor of p, withthe elements a Galoisfield, of GF(pn). Supposetheseelements be u0o,u to ,UL_1 where uU2, the uo is the zeroand u1l unityof the field. If thenux level ofA and uylevel of B occur together, corresponding the levelsforinteraction terms uy, UP, and the squaresfor are + ux, = U1, U2, ..., UL-1 give all the termspresent.The methodof constructing completely u,p orthogonal ofLatin squaresfrom sets Galoisfields givenin Stevens(1939). is For L = 6 it is known thatno pairoforthogonal Latinsquaresexistsso it is notamenable to thistreatment. design N = 9, L = 3 is givenbelow;theaccompanying refers The for key to thecolumn vectors therows labelledas ifbelonging a complete and are to factorial design.alb, a1b2 a,b, A 0 0 0 B 0 1 2 (AB)1 (AB)2 A B 0 1 0 0 a2b, 1 1 2 a2b2 1 2 1 a2b3 1 2 Key: (AB)1= (A)+(B), (AB)1 (AB)2 1 1 a3bl 0 2 a3b2 0 2 a3b3 (AB)2= (A) + 2(B). A 2 2 2 B 0 1 2 (AB)1 (AB)2 2 2 0 1 1 0

R. L. PLACKETT

AND J.

P.

BURMAN

317

order are For r = 4, N = 81, ifthe main effects taken as A, B, C, D, (ABCD)1, then all first interactionsare determinable.8.CYCLIC SOLUTIONS WVHEN L IS A PRIME NUMBER

We shall again be concernedin this section with Galois fields:each elementof GF(pn) will be represented by a set of n ordered numbers, each number being 0, 1, 2, ..., (p - 1) wherep is prime. Consider the block B1 of elementswhichhave the integerr in positions s = 1,'2,3,...,n). (r=0,1,2,...,p-1; For example,in GF(32) the block having 2 in position1 is 20, 21, 22. We requireto showthat if the elements of this block are multipliedin succession by any element of the fieldother than 000 ... Ot(1 < t p - 1), thenthe elementsof the resulting block B2 have equal numbers of all possible r in position s. There are in fact p',-' elementsin B1, and we need to prove that B2 is subdivisible into p t-2 elementshaving 0 in positionspn-2 elementshaving 1 in positions.. wwwwXXX.... .. XXXs... . . . .. . . .. . .. . . .. .

ppn-2 elementshavingp

- 1 in positions.

membersof all There is no loss in generality we considers = 1, i.e. we refer if now to the first elements of the field.Take now all the elements having 1 in this position. Multiply this members blockA1 by any elementb ofthe field and obtain blockA2. Suppose in A2 thatrofirst membersare p-1. are 0, r1first membersare 1,..., and rp-1first Clearly E ri = pn-l.0p-1

Case 1. ro,r1, all * 0. rp ..l. member 0) by subtractingone element of A1 fromall Form the comnplete block C, (first otherelementsof A1. If Cl is multipliedby b we get the block C2 formedalso by subtracting one element of A2 fromall other elements of A2. We can form the elements of (C2 (first member0) by subtractingone of the ro elements of A2 (firstmember0) fromitselfand all member0). the otherro- 1 such elements. Hence thereare roelementsof (2 (first We can also form the elementsof C2 (first member0) by subtracting one of the r1elements member 1) fromitselfand all the otherr, - 1 such elements. Hence there are r of A 2 (first elementsof C2 (first member0). = r_ Hence rO = r, = r2= p This result must be true for all other firstmembers,since all elements of the field are obtainable fromthose in A1 by addition or subtraction. Case 2. One or more of ro,r1, ..., r1_l = 0. r2, = ... = rk. Suppose in fact that ri,rj, ...,rk+O. Exactly as above, we can show ri This leads to a contradiction sincep7l-1 not divisibleby a numberless thanp, unlesswe have is Case 3. All except one of ro,r1,..., r_1 = 0. Suppose that the firstmembersof all elements in A2 are w, where 1 w 2, we obtainbalancedincomplete blockswithparameters r = I+L+L2+ ...+Lh-1v = Lh,

b = L +L2 + 3k = Lh-1 AwLpm

+..

L',

+pL2+ = eLh-2

where = L

and pm,p a prime, h> 1.

322

Design ofoptimum multifactorial experimentsr=(N-1)/(L-1); v = N; b = rL; k = N/L; A = (N-L)/L(L-1);

The balanced incompleteblock designs formedfrommultifactorial designs,forwhich are in fact of a special kind and have been called by Bose (1942) afflne A resolvable. balanced incompleteblock design is resolvable if we can separate the b blocksinto r sets of n blocks each (b = nr) such that each variety occurs once among the blocks of a given set; and if in addition either (i) b + 1 = v + r or (ii) any two blocks belongingto different sets have the same numberof varieties in common,then the other is true and the design is called affine resolvable because of its relationto certainfiniteEuclidean geometries. Bose has shown (1) If a resolvablebalanced incompleteblock designis such that any two blocks belonging to different have the same numberof varietiesin common,then b + 1 = v + r. sets (2) If fora resolvable balanced incompleteblock design b + 1 = v + r then any two blocks sets belongingto different have the same numberof varieties in common. We have shown (3) If a resolvableincompleteblock design (i.e. one withr, v, b, k given but not necessarily balanced in the sense that every two varietiesoccur together the same numberof blocks) in has b + 1 = v + r and is such that any two blocks belongingto different have the same sets numberof varietiesin common,then it is balanced. To sum up, ifa resolvableincompleteblock designhas any two ofthe following properties: (i) anytwoblocksbelonging different have thesame numberofvarieties common, to sets in (ii) balance, (iii) b+1=v+r, then it has the third.The orthogonalmatrix methodwe have used to prove (3) can also be used to provide shortproofsof (1) and (2). Consequently a multifactorialdesign can be formedfroma balanced incomplete block design provided that the latter is resolvable with parameters r = (N-1)/(L-1); v = N; b = rL; k = N/L; A = (N-L)/L(L-1). Bose has pointed out that affineresolvable designs can be constructed from the affine geometryEG(h, pm) (our notation) by taking varieties as points and blocks as (h -1)-flats; thisconstruction givesall the multifactorial designsforL > 2 whichhave so farbeen obtained. The most general aspect of the multifactorialdesign is obtained by consideringeach assembly as a block and each value of each componentas a variety. We obtain a partially balanced incompleteblock design (Bose & Nair, 1939) with parameters:

r = N/L; v = L(N-1)/(L-1); A1= N/L2; n, = [(N-1)/(L-1)-1]p1 = [[(N -1)/(L -1) -2] L

b = N; L;

k = (N-1)/(L-1);A2=0;n2 =

(L-1);

(L -1)]

24

[[(N- -1)/(L -1) -1]L

O

AlthoughBose & Nair state that generalmethodsforthe construction partiallybalanced of incompleteblock designsare to appear, we have been unable to findthem,so that thisaspect of the multifactorial design does not yield more solutionsof the problem. 12.SUMMARY

Methods are developed to avoid the complete factorial experimentin industrial experimentationwhen the numberof factorsis so large that the standard procedureis impractic-

R. L. PLACKETT AND J. P. BURMAN

323

maineffects of the linearhypothesis, problem determining a able. By assuming simplified solutions all one. to is precision reduced a combinatorial Practically useful withmaximum for appearsat twolevels,but thesolutions more wheneach factor ofthishave beenfound to of The relationship thesesolutions someencountered limited. thantwolevelsare fairly blockshas beendiscussed. in balancedincomplete the and the methodof We are indebtedto Mr G. A. Barnardforsuggesting problem to our for approachby least squares; and to Dr Bronowski drawing attention the useful of out work carried as partoftheprogramme theMinistry was paperofR. E. A. C. Paley.The of of Supply,Researchand Development(S.R. 17), and appears by kindpermission the Scientific Chief Officer.TABLE OF DESIGNS

of is N, opposite thenumber L design given row A. Designsfor = 2. The first ofanycyclic A of ones: As assemblies. statedat theendof? 4 (III) thematrix hereconsists plusand minus row signs.Thereis alwaysa final ofminussigns belowbyplus and minus theseare denoted for -the basicassembly-tobe added. In thedesigns N = 28,52,76,100thesquareblocks in thethree lattercasestheextracolumn has are permuted themselves; amongst cyclically the element. The larger designsare grouped alternate apartfrom corner signsthroughout in fives convenience. forN=8.N=12. N=16. + + + - + - ++-+++--+_ ++++-+-++--+---

N=20. N =2.

+ + - - + + + + - + - + --

--

+ + -

First nine rows

-+ + +

-

__

_+-+-

-

+

+-+++-+-+

N=32. N= 3. (Otaind__tial++++

--_+

+++__

+---+__+_+ _+-+--+_+

+-

+-+

= 40. N First Double designforN =20. + -++++---++++

elevn ++N=44.

++++--+--+ +_+__,++_-_++

++--++--++

+-+--+

-+-+-

,+-+++___++ -+++_-++-++-+

-+--++

-+++_

-

+++-_+++++ -++,+

N = 52.N=36+-+++ain+b+tr+l)+-+++

t

+ -

+-+-+-+

+-

+ -+-+-+++-+

+++-

+-+

-++-+--+

_+_-+-+_ + + + + +

++----+-+-++---+++

+-+-+-+++_

+-+-++ + -+

-+-+- + + + + +

N=32. N=44.

+

+ - + - -++ - - + - + - + -+ + ++ +++-

- + - + -+

+- + +-

++

-+ +

+

+ +- -

-

-

-++-+-+-++

++-+-++-+-+-

+--++-

++-+

-++-+

-++-+-+-+-

--++

+ -

+

+

-

+ + ++ -__

+ + _ + +++

+ _

+ _ -+

+

+

-+_

+

+

_++

++++

++

324N=60.++-++

experiments Design of optimum multifactorial+-++-+-++++ +++-

N = 56. Double designforN =28.+++++ ++

N =68. N=72. N=76.

N = 64. Double designforN =32.+ +-+ - +-+ +--+ + + +-+ -+ + ++ + +-+ --+ + +++ +--

+++++

++-++ +---++

++-+-++-++++++--+-+-+ +---+ +--+++---+-----

+

+

+-+++ +-

+-+

+-++-++-++-++-++-++-+++ ++ -+ ++ ++ ++ +-++-+ ++ ++ ++ ++

+-+-

-

-

-

++-++-++-++-++-+ ++-+-

d are rows given; obtain complete are to the the Thefirst three design squareblocks permute the The has column, element, alternate apartfrom corner signs. cyclically. firstN=80.N=84-+ ++++ --+-+ ++++---+++ --+ +-+ ++-+---++ -++-+++ +

N= N= N= N=

88. Double designforN = 44. 92. This designhas not yet been obtained. 96. Double designforN =48. 100.

+

-------+++++++

-------+++++++

R. L. PLACKETT AND J. P. BURMAN

325

designis is column givenbelowand the complete for B. Designs L = 3,5,7. The first (N-i )/(L-1) -1 timesand addinga rowofzeros.The it formed permuting cyclically by value the by A orthogonal matrix of? 4 (11)is obtained replacing component corresponding suppressed. column of symbols thedesignby therowsof Q (? 4 (J))withitsfirstN=9,Lz=3. N = 27, L = 3. N = 81, L = 3. N = 25, L = 5. N = 125, L = 5. N=

49, L

=

7.

01220211 00101 21120 11100 20212 21022 2 01111 20121 12120 20221 10201 10012 22021 00200 02222 10212 21210 10112 20102 20021 11012 00100 04112 10322 42014 43402 3313 02221 04114 13134 12021 10244 31402 00444 20322 32121 32404 22043 31230 40033 34014 41424 21430 34403 11241 03001 11302 33234 34231 01330 12243 2010 01262 21605 32335 20413 11430 65155 61024 54425 03646 634

REFERENCES blockdesigns.Sankhyd, 105. 6, BOSE,R. C. (1942). A noteon the resolvability balancedincomplete of K. in symmetrical factorial design. BOSE,R. C. & KISHEN, (1940). On theproblem confounding thegeneral of Sankhy&, 21. 5, balancedincomplete blockdesigns.Sanihyd, 337. 4, BOSE,R. C. & NAm,K. R. (1939). Partially 3rd FISHER, R. A. (1942). The DesignofExperiments, ed. Oliverand Boyd. F. R. TablesforBiological, and 2nd Agricultural MedicalResearch, ed., FISHER, A. & YATES, (1943). Statistical p. 14. Oliverand Boyd. HOTELLING, H. (1944). Some improvementsin weighing and other experimental techniques. Ann. Math. Stat. 15, 297. KISHEN, K. (1945). On the design of experimentsforweighingand making other types of measurement. Ann. Math. Stat. 16, 294. 33, LINDLEY, D. V. (1946). On the solution of some equations in least squares. Biometrika, 326. matrices.J. Math.Phys.12, 311. PALEY, R. E. A. C. (1933). On orthogonal STEVENS,W. L. (1939). The completelyorthogonalizedLatin square. Ann.Eugen.,Lond.,9, 82.

Addedin proof(1) Since this paper was written,one of the authors (J. P. B.) has obtained a design for L= 3, N = 18, n= 7, by trial and error. It is known that this is the largest value of n possible. (2) The designs given above forL = 2 provide what is effectively complete solution of a the experimentalproblem considered by Hotelling (1944) and Kishen (1945).