FIV\C'lTONJ\L l•'AC'l'ORIJ\L l1E~~IGN~; by Walter T. Federer and B. Leo Raktoe Cornell University and University of Guelph January, 1980 Abstract A generalized treatment of fractional factorial designs is presented. Precise definitions are given, and within this framework the subject of fractional replication of factorials is formulated. Construction and optimality properties of fractional replicates are discussed in some detail. i~ In the Biometrics Unit Mimeo Series, Cornell University, Ithaca, New York. FRACTIONAL FACTORIAL DESIGNS by Walter T. Federer and B. Leo Raktoe Cornell University and University of Guelph BU-701-M Introduction January-, 1980 Ever since Fisher [1926] introduced the notion of "factorial experimen- tation" a tremendous development of ideas in this area has taken place. In factorial experimentation (originally called "complex experimentation" by Fisher), several factors may be studied simultaneously instead of experlinent- ing with them one at a time. For example, in an agricultural experiment we may assess the effects of nitrogen and :phosphate fertilizers on the yield of wheat by carrying out an experiment with various combinations of levels of the two fertilizers. If the experimenter specified k1 levels of the nitrogen fertilizer and k2 levels of the phosphate fertilizer and all the k1 · k2 combinations are used in the experiment, then such an experiment is called a "complete factorial". If fewer than the k1 · k2 combinations are used, then the tenn "fractional factorial" has been used in the literature for such an experiment. Yates [1935] provided the first comprehensive approach to complete fac- torials and also presented some ideas in fractional factorials. It was Fisher [1942], however, who systematically constructed classes of fractional factorials, where each of the factors had the same prime number of levels. These designs came about as a by-product of the construction of "confounded designs". The fonnal approach to fractional factorial designs is due to Finney [1945]. Since then numerous authors have made contributions in -2- resolving some of the ensuinG problems. Many problems in factorial theory turn out to have a geometric, alge- braic or combinatorial flavor. As a consequence mathematical structures, such as finite groups, finite rings, finite fields and finite geometries, can be successfully used in elucidating and resolving many issues. Fisher [19!~2, 1945] used finite Abelian groups and Bose [1947] relied heavily on finite Euclidean and finite projective geometries in the construction and enumeration of "regular fractions" of symmetrical prime powered factorials. More recently, a general algebraic-combinatorial theory of fractional fac- torials has been developed by Pesotan, Raktoe and Federer [1975]. This theory relied on some invariance results of Srivastava, Raktoe and Pesotan [1976] and several unsolved problems associated with it have been reported by Raktoe and Pesotan [1974]. In the sections below a systematic discussion is presented on the most important aspects of fractional factorials. Factorial Arrangements and Fractional Factorial Designs In this section use is made of the notation and definitions which were developed by Raktoe, Hedayat and Federer [1973] in an unpublished monograph. Since then, these authors have used it in two papers, namely, Hedayat, Raktoe and Federer [1974] and Federer, Hedayat and Raktoe [1975]. A distinction will be made between sets and collections. In a set there is a listing of distinct elements, while in a collection repetitions are allowed. In many scientific investigations experimenters are interested in studying the effects of t controllable variables. Such variables will be called factors and they will be denoted by F1' F2, • • ·, Ft . For each factor there will be a specified range of values of interest to the experimenter. These sets of values will be called levels of the factors and they will be -3- indicated by G1, G2, · • ·, Gt . A factor will be called quantitative if the underlying levels of interest are real nQ~bers and qualitative if the levels are specified qualities rather than real munbers. Denote the cardinality of G. by k., and throughout the development assume that the G.'s are finite ll l sets. The ::;ets of levels G1, G2, · · ·, Gt are potential levels and it is not necessarily true that all of them will be used in a particular experiment. t Let G be the Cartesian product of the G.'s, i.e., G= X, where the l i=l symbol X denotes the Cartesian product. The set G together with the F.'s l is often referred to as the k1 X k2 X • • • X kt factorial or k1 X k2 X • • • X kt crossed classification. An element of G is called a treatment and G itself is called the factor space or space of treatments. In the literature, the terms "treatment combinations", "assemblies", "runs" and subclasses" are also frequently used for treatments. t Let N be the cardinality of G, i.e., N = TI k., and let G be indexed by i=l l the index set [ l, 2, .•. , N} . Then a factorial arrangement or factorial design with parameters k1, k2, · · ·, kt' m, n, r 1, r 2, · • ·, rN is defined to be a collection of n treatments such that the jth treatment of G has multiplicity r.~O, m is the number of non-zero r.'s, and I: r.=n>O. A factorial J J j=l J arrangement is denoted by the symbol FA(~, k2, • · ·, kt ; m ; n ; r 1, r 2, rN) or simply by FA if everything is clear from the context. In the discipline of statistics the multiplicity r. is referred to as J the replication number of the jth treatment, i.e., how many times the jth treatment is repeated in the factorial arrangement. The definition of a factorial arrangement adopted here is in complete agreement with the defini- tion of a general t-way crossed classification with r. observations on the J jth treatment. A factorial arrangement is said to be a complete factorial arrangement -4- or a complete replicate if r. > 0 for all j = l, 2, · · ·, N. It is said to be a J minimal complete factorial arrangement if r. = l for all j J Note that a minimal complete factorial o.rranc;ement i~~ a sinr;le copy of the factor space G . A complete factorial arrancement such that r. = r for all j is said to J consist of £ complete replicates. A factorial arrangement is symmetrical if k. = s for all i = 1, 2, · · ·, t, 1 and otherwise it is asymmetrical or mixed. An FA is prime powered if k. = p~1 , 11 such that for each i, p. is a prime and u. is a natural number greater than 11 or equal to l It follows that a factorial arrangement can be symmetrical prime powered or mixed prime powered. A factorial arrangement is said to be an incomplete factorial arrange- ment or a fractional factorial design, or more simply, a fractional replicate, if some but not all r. 's are equal to zero. A fractional replicate is denoted J by FFA(k1, k2, ·· ·, kt; m; n; r 1, r 2, · • ·, rN) or by FFA if it is clear from the context. If the levels of the ith factor are made to correspond to the residue classes modulo k., i.e., G. = ( 0, 1, 2, · · ·, k. -1}, then under componentwise 11 1 addition modulo of the ki 's, it can be shown that G is an Abelian group. For the symmetrical prime powered st factorial each of the G.'s can be identi1 fied with the Galois field GF(s), where s =pu and p a prime. It can then be established that G is a t-dllnensional vector space over GF(s). From a geo- metric viewpoint such a vector space is known as a finite Euclidean geometry EG(t,s) of dlinension t over GF(s) Before proceeding further we present an example to illustrate the con- cepts defined so far. Example: An industrial experiment was planned to study the effect of both curing time and composition on the tensile strength of plastic compounds. -5- Three times, 1 hour, ~1 hours and I~ hours were selected and f'our mixes, A, B, C and D were prepared. Observations were to be made on combinations of' curing times and cornpositions. This is a 3 X 4 f'actorial with the quantitative factor F1 =Curing time, the qualitative factor F2 =Composition, G1 = [1 hour, 2 hours, 4 hours} = [ 1, 2, I~}, and G2 =[A, B, C, D} • The set of treatment combinations is G ~ [ (l,A), (1, B), (1, C), (1, D), (2,A), (2, B), (2, C), (2, D), (4,A),(4,B),(4,c),(4,D)}. Note that in using G1 =(1,2,4} we have deleted units. Indeed, frequently we use labels to indicate the levels of' f'actors. For our example it is common to use the f'ollowing sets of' labels: G~ = [0,1,2} and G~ = { 0, 1, 2, 3} . The set of treatment combinations is then depicted as dj~ =[ (o, 0)' ( o, 1)' ( o, 2)' ( 0' 3)' (1, 0)' (1, 1)' ( 1, 2)' (1, 3)' ( 2, 0)' ( 2, 1)' ( 2, 2)' ( 2, 3)} with each element having the obvious real meaning, e.g., (2,0) = (4 hours, A) • Since k1 = 3 and k2 = 4 = 22 , all f'actorial arrangements in this example are mixed prime powered. The f'actor space G, or its equivalent representation G*, is a minimal complete factorial arrangement. The f'ollowing f'actorial arrange- ment in terms of' G~~ is complete but not minimal: FA(3,4;12;15;2,3,1,1,1,1,1,1,1,1,1,1) = [(o,o), (o,o), (o,l), (o,l), (o,1), (0,2), (0,3), (l,o), (1,1), (1,2), (1, 3), (2,o), (2,1), (2,2),(2,3)}. An example of a f'ractional replicate in terms of' G~~ is FA(3,4;5;6;l,l,o,o,o,o,o,l,l,2,o,o) = [ (o,o), (o,l), (1,3), (2,0), (2,1), (2,1)} . Finally note that G* under camponentwise addition modulo 3 and modulo 4 is an Abelian group of' order 12 . The Linear Model and Estimation of' Eff'ects for a Fractional Factorial Experiment In this section the linear model f'or analyzing data f'rom an experiment using a f'ractional f'actorial design is introduced. The approach adopted here can be f'ound in several places in the literature (e.g., Raktoe, Hedayat and Federer [1973]) and it conforms to the usual linear model notation found in Graybill [1976] and Searle [1971]. -6- Let D be a factorial arrangement. With each treatment g in D we associ- ate a random variable Yg , which is called an observation or response. We will assume the univariate case, i.e., Yg will be one-dimensional and assume values in a one-dimensional Euclidean set. Let YD be the n X 1 vector of ob- servations for the factorial arrangement D, where the components of YD are the Yg 's • In most settings a linear model is associated with minimal complete factorial arrangement n* in the following way: (i) and (1) where XTI* is a known NX N design matrix, IN is the identity matrix of order N, and p is the vector of N parameters consisting of N- 1 factorial effects and the mean. If :X]l = x1 ® x2 ® • • · ® Xt' where each Xi is a ki X ki orthogonal matrix with each entry in the first column equal to 1/~, and® denotes l. Kronecker product, then S is a vector of factorial effects under the product definition. This approach is especially applicable under the orthogonal polynomial and Helmert polynomial settings. For the symmetrical prime powered factorial the entries of ~~~ can be obtained from a fixed basic orthogonal matrix by using the geometric definition of effects. The model for any fractional factorial design D is induced by (1) in the sense that the design matrix XTI is read off from~*' taking repetitions of treatment combinations into account. The most practical partitioning of the parametric vector isS'= (f3j_!S2:f33), where Sl is an N1 X 1 vector to be estimated, s2 is an N2 X 1 vector not of interest and not assumed to be known, and s3 is an N3 X 1 vector of parameters -7- assumed to be known ('~;-Thich without loss of generality can be taken to be ?.ero), such thRt 1 s N1 s: N, 0 s: N2 s N- 1 and 0 :s; N3 = N- N1 - N2 s N- 1 • This partitioning explicitly leads to the following four cases: (i) N1 =N, N2 =N3 =o, (iii) N2 = 0, N3 ,f 0 and (ii) N2 = O, N3 f O, (iv) N2 f O, N3 = 0 (2) Case (i) may be viewed as a special case of (ii) by letting ~l exhaust ~ so that N3 = 0 Similarly, case (iv) can be considered a special case of (iii) by letting ~l and ~2 exhaust ~ so that N3 = 0 It thus suffices to analyze cases (ii) and (iii) in (2) above. ···Denote a parameter in ~ by the symbol ~i- ~i 13~ , where (x1, x2, · · ·, xt) t is an element of G= X G., i=l l G1 = [o, 1, 2, · · ·, ki -1} . Then ~1°~~ ··.~to is called the mean, and a factorial effect ~~ ~~ ... ~~t is said to be of order k if exactly k of the exponents are non-zero. A fractional factorial design D is said to be of resolution R if all factorial effects up to order k are estima- ble, where k is the greatest integer less than R/2, under the assumption that all factorial effects of order R- k and higher are zero. When R = 2r, then the design is known as a design of even resolution, and for R = 2r + 1 the design is said to be of odd resolution. Thus resolution 3 designs allow estimation of all main effects (i.e., effects of order 1) under the assumption that all interactions (i.e., effects of order 2 or higher) are zero. Designs of resolution 4 allow estimation of all main effects under the assumption that interactions of order 3 or higher (i.e., effects of order greater than or equal to 3) are zero without assuming that two-factor inter- actions (i.e., effects of order 2) are equal to zero. Note that designs of odd resolution belong to case (ii) and designs of even resolution belong to case (iii), respectively, of (2) above. -8- The model for any fractional factorial design D under case (ii) is given by: and (3) where the design matrix ~l is simply read off from ~* of (1), taking repetitions of treatment combinations into account. Similarly, the design matrix [~1 :~2 ] for a design D under case (iii) is obtained from~* as: and (4) The best linear unbiased estimator (BLUE) of ~l and the covariance for the two cases are given, respectively, by (5) and (6) below: and (5) and (6) In expression (6) A- denotes a generalized inverse of A • For brevity the covariances in either case will be written as M~1a2 , where ~l is known as the covariance matrix and MD itself is called the information matrix. An unbiased estimator of a2 is obtained by utilizing the BLUE's in (5) and (6), a s:xnviz., 2 = (YD- ~1s1 ) '(YD- 1 1 )/(n -N1 ) for case (ii), and for case (iii), ~2 = (YD- XDlSl- XD2I32) '(YD- ~lgl- ~2~2)/(n -Nl)' where I32 = (X:b2~2 ) - [JC02YD- X:b2XDl"~'l] • Under the assumption of normality, tests of hypotheses and confidence interval estimators for the vector ~l can be obtained as indicated, for example, in Graybill [1976]. -9- The problem of choosing an optimal fractional factorial design is dis- cussed in this section using optimality criteria developed by Kiefer [1959] and Hedayat, Raktoe and Federer [1974]. Let D be a class of competing fractional factorial designs in either setting (ii) or (iii) of the partitioning in (2) of the previous section. Assume that each design DE :Q is capable of providing an unbiased estimator for t3l • There are several optimality criteria based on the covariance MDmatrix -l of the BLUE of t3l • The most popular ones are based an the specMD . MDtruro or set of characteristic roots of -1 Denoting the roots of -l in increasing order of magnitude by A.1 , A.2, • • ·, A.N we have the following l functionals: (b) and (7) (c) A.l'\ =max( "-l' A.2, ••• ' A.Nl) A design which minimizes over :Q the criteria in (a), (b) and (c), respectively, is known as a d-optimal design, a-optimal design and e-optimal design. Statistical interpretations of these criteria are available and they may be found in Kiefer [1959]. It should be noted that one may base these optimality criteria on the spectrum of the information matrix MD rather than on that of -l MD • Another criterion, which does not rely on the covariance matrix ~1, was developed by Hedayat, Raktoe and Federer [1974]. If in the settings (ii) and (iii) of (2) in the previous section the assumption that t33 =o is in doubt, then E[g1 ] = t31 +ADt33' where AD is known as the alias matrix of the design D relative to t3l and t33 (e.g., for case (iii) of (2) AD= (JCD1~1 )-1X!J1~) • .!. The norm 11~11 = [Trace(ADAD) ]2 was proposed by Hedayat et al. [1974] for the -10- selection of an optimal design and a design is said to be alias optimal if it minimizes //Anll over ~ . Apart from these criteria one may impose other desirable properties on D for the selection of a design, such as orthogonality (i.e., ~l is diagonal) or balancedness (i.e., ~-1 = ai + bJ, where J is a square matrix of order N1 all whose elements are l's). Orthogonality implies uncorrelatedness of the estimators of the elements of ~l and balancedness implies equal variances and equal covariances of the estimators. These concepts have been generalized to partial orthogonality and partial balancedness (e.g., see Srivastava and Anderson [1970]). If the mean is the first element in ~l' then the first element of the vector ~l +AD~3 is lmown as the generalized defining relationship of D relative to ~l and ~3 and the whole vector itself is called the aliasing structure of D relative to ~l and ~3 . The aliasing structure for a symmetrical prime powered fractional factorial design becomes tractable via group-theoretic techniques if the design is regular; i.e., it is a subspace (or coset of a subspace) when the complete set of treatment combinations is viewed as the t-dimensional vector space over the Galois field GF(s) (e.g., see Raktoe, Pesotan and Federer [1979]). For the 2t factorial the generalized defining relationship is lmown as the defining contrast (e.g., see Cochran and Cox [1957]). Construction of Fractional Factorial Designs The construction of an optimal design in either setting (ii) or (iii) of (2) is by no means a simple matter since it involves the design parameters ~' k2, • · ·, kt' m, n, r 1, r 2, .•• , rN' N1, N2, N3 and selection of the optimality criterion. Indeed, there is a formidable combinatorial problem associated with finding for the k1 X k2 X • • • X kt factorial all designs which -11- simply lead to unbiased estimation of ~l in (ii) or (iii) of (2), let alone obtaining the optimal ones. There is no unique method available for all factorials and depending on the nature of the factorial one may utilize various techniques to obtain useful designs. Raktoe, Hedayat and Federer [1973] list twenty-four methods in their unpublished monograph. These are: (1) Trial and error and/or computer methods; (2) Hadamard matrix methods; (3) Confounding techniques; (4) Group theory methods; (5) Finite geometrical methods; (6) Algebraic decomposition techniques; (7) Combinatorial-topological methods; ( 8) Fold-over techniques; (9) Collapsing of levels methods; (10) Composition (direct product and direct sum) methods; (11) Permutation of levels and/or factors methods. (12) Coding theory methods; (13) Orthogonal array techniques; (14) Partially balanced array techniques; (15) Orthogonal Latin square methods; (16) Block design techniques; (17) Weighing design techniques; (18) F-Square techniques; (19) Lattice design methods; (20) Graph-theoretical methods; (21) One-at-a-time methods; -12- (22) Inspection methods; (23) Patterned matrix methods; and (24) Cutting and adjoining matrix methods. In order to demonstrate the complexity of the problems, an illustration is given of the general combinatorial problem for case (ii) of (2) for reso- lution 3 designs in their simplest possible setting. Assume that the mean is also of interest of estimation so that a minimal (or saturated) resolu- tion 3 design calls for N1 =m =n =L:ki - t + 1 distinct treatment combinations for estimation of t31 since there are N1 =L:(ki - 1) + 1 parameters in t31 . We have seen that a necessary and sufficient condition for estimating t31 is that the rank of ~l (or of XDl~l) is equal tom . Denote the class of possible designs by D-m; then clearly its cardinality is equal to: ID-m /=CmN=(N!)I(m!)(N-m)!. Denote by D-m,m the class of designs in D-m which allow estimation of t31 and by !?m 0 the singular (viz., / ~1 / = 0) class ' of designs. The cardinality of D-m, 0 (and hence that of D-m,m) is not known in general at present. For the 2t factorial it has been enumerated for t:::; 7, which can be found in Raktoe [1979]: 2 2t / !?m/ = ct+l I!?m, ol I!?m,ml I!?m, ol I c~~l I!?m,ml I c~t+l 4 0 4 0 1 3 70 12 58 .1714 .8286 4 4368 1360 3008 .3114 .6886 t 56 7 906192 350000 556192 .3862 .6138 621216192 255036992 366179200 .4105 .5895 1429702652400 571462430224 858240222176 -3997 .6003 It may be shown that the proportion of singular designs goes to zero as t -13- goes to infinity for the 2t factorial. Using Helmert polynomials and corresponding non-normalized column- wise orthogonal matrices in (l) for the ~X k2 X • • • X kt factorial, one may deduce under the determinant criterion that for a saturated resolution III desic;n DE D-m : t II (k. ~ )2 ~ Idet X~, x_ll ~ t rr (k. !)~m t rr k~kt i=l l -~~-~ i=l l i=l l ( 8) The lower bound in (7) is attained by the one-at-a-time design D~~ = [ (ooo .•. o), (100·. ·O), (2oo ... o), · • ·, (k1 -1oo •.. o), .•• , (OlO· •. o), (o2o ••. o), ···,(Ok2 -10···0),···,(000···01),(000···02),···,(000···0kt -1)}, which is a least d-optimal resolution 3 design. The upper bound is attained if and only if an orthogonal design can be constructed. For the 2t factorial the bound is achieved whenever a Hadamard matrix of order t + l exists. A necessary condition for this is that t + l = 0 (mod. 4). Hadamard matrices have been constructed for all t + 1 ~ 200 so that minimal d-optimal resolution 3 designs are available for all 2t factorials, where t = 4s -l and s = 1, 2, ..• , 50 . · For t + 1 f 0 (mod. 4) other techniques have been used apart from co~pu4er methods. Examples of minimal d-optimal resolution III designs for the 2t factorial are: t d-optimal design 2 [(00),(10),(01)} 3 {(000),(110),(101),(011)} 4 [(oooo),(lllO),(llOl),(lOll),(olll)} 5 ((OOOOO),(lllOO),(llOlO),(llOOl),(lOlll),(Olll)} 6 ((OOOOOO),(lllOOO),(llOllO),(llOlOl),(lOllOO),(lOlOll),(Olllll)} 7 {(OOOOOOO),(lllOOO),(llOOl10),(10lOlOl),(l0010l1),(0l100l1), (0101101),(0011110)} -14- These designs are such that the spectrum of the information matrix is invari- ant under the group of level permutations. Some of the references listed below deal with constructions of other types of fractional factorial designs, such as resolution 4 and 5 designs which are not necessarily minimal. References and Selected Further Reading 1. Addelman, S. (1963). Techniques for constructing fractional replicate plans. J. Am. Stat. Assn. 58, 45-71. 2. Addelman, S. (1972). Recent developments in the design of factorial experiments. ~·Am. Stat. Assn. 67, 103-lll. 3. Anderson, D. A. and Federer, w. T. (1975). Possible absolute determinant values for square (0,1)-matrices useful in fractional replication. Utilitas Mathematica 7, 135-150. 4. Anderson, D. A. and Thomas, A. M. (1979). Resolution IV fractional factorial designs for the general asymmetric factorial. Communications in Statistics A8, 931-943. 5. Bose, R. C. (1947). Mathematical theory of the symmetrical factorial design. sankhya 8, 107-166. 6. Bose, R. C. and Srivastava, J. N. (1964). Mathematical theory of fac- torial designs. I. Analysis. II. Construction. Bull. Intern. Inst. Stat. 40, 780-794. 7. Box, G~ E. P. and Hunter, J. S. (1961). The 2k-p fractional factorial designs. Technometrics 3, 311-351. 8. Cochran, w. G. and Cox, G. M. (1957). Experimental Designs, 2nd edition, Wiley, New York. 9· Draper, N. R. and Mitchell, T. J, (1967). The construction of saturated 2~-P designs. Ann. Math. Stat. 38, 1110-1126. 10. Federer, w. T. and Balaam, L. N. (1972). Bibliography on Experiment and Treatment Design. Pre-1968. Oliver and Boyd, Edinburgh. 11. Federer, w. T., Hedayat, A. and Raktoe, B. L. (1975). Minimal unbiased designs for linear parametric functions. In A Survey of Statistical Design and Linear Models (J.N. Srivastava, editor), North Holland Publishing Co., 145-153. 12. Finney, D. J. (1945). The fractional replication of factorial experiments. Ann. Eugen. 12, 291-301. -15- 13. Fisher, R. A. (1926). The arrangement of field experiments. J. Ministry Ag. Sept. 33, 503-513. 14. Fisher, R. A. (1942). The theory of confounding in factorial experiments in relation to the theory of groups. Ann. Eugen. ll, 341-353. 15. Fisher, R. A. (191+5). A system of confounding for factors with more than two alternatives giving completely orthogonal cubes and higher powers. Ann. Eugen. 12, 283-290. 16. Graybill, F. A. (1976). Theory and Application of the Linear Model. Duxbury Press (Wadsworth), Belmont, California-.---- 17. Gulati, B. R. (1972). Construction of two-level saturated symmetrical factorial designs of resolution VI. Ann. Math. Stat. 43, 1652-1663. 18. Hedayat, A., Raktoe, B. L. and Federer, w. T. (1974). On a measure of aliasing due to fitting an incomplete model. Ann. Stat. 2, 650-660. 19. Hedayat, A. and Wallis, w. D. (1978). Hadamard matrices and their applications. Ann. Stat. 6, 1184-1238. 20. John, P. w. M. (1971). Statistical Design and Analysis of Experiments. Macmillan, New York. 21. Kiefer, J. C. (1959). Optimum experimental designs. J. Roy. Stat. Soc. B21, 272-304. 22. Margolin, B. H. (1969). Resolution IV fractional factorial designs. ~· Roy. Stat. Soc. B3l,514-523. 23. Margolin, B. H. (1972). Non-orthogonal main effect designs for asymmetrical factorial experiments. ~· Roy. Stat. Soc. B34, 431-440. 24. Pesotan, H., Raktoe, B. L. and Federer, w. T. (1975). On complexes of Abelian groups with applications to fractional factorial designs. Ann. Inst. Stat. Math. 27, 55-Bo. 25. Pesotan, H., Raktoe, B. L. and Worthley, R. G. (1978). On main effect foldover designs. ~· Stat. Plan. and Infer. 2, 277-291. 26. Plackett, R. L. and Burman, J. P. (1946). The design of optimum factorial experiments. Biometrika 33, 305-325. 27. Raktoe, B. L. (1979). How many degenerate simplices are generated by n + 1 vertices of the unit n-cube? Amer. Math. Mthly. 86, 49. 28. Raktoe, B. L. Hedayat, A. and-Federer, w. T. (1973). Concepts of Fractional Replication of Factorial Experiments. Monograph submitted for publication. 29. Raktoe, B. L. and Pesotan, H. (1974). On some unsolved algebraic problems arising from factorial design theory. Sankhya B46, 457-461. • -16- 30. Raktoe, n. L.• Pesotm1, H. m1d Federer, w. T. (1980). On aliasing and generalized defining l'clat.ionships of fractional factorial dcf::lt-71S. In press, Can. ~· of Stat. 7. 31. Searle, S. R. (1971). Linear Models. Wiley, New York. 32. Srivastava, J, N. and Anderson, D. (1970). Optimal fractional factorial plans for main effects orthogonal to two factor interactions: ~ series. J, Am. Stat. Assn. 65, 828-843. 33. Srivastava, J. N. and Chopra, D. V. (1975). On the comparison of cer- tain classes of balanced 28 fractional factorial designs of resolution V, with respect to the trace criterion. J, Indian Soc. Agric. Stat. 23, 124-131. -- 34. Srivastava, J, N., Raktoe, B. L. and Pesotan, H. (1976). On invariance and randomization in fractional replication. Ann. Stat. 4, 423-430. 35. Webb, S. R. (1968). Non-orthogonal designs of even resolution. Technametrics 10, 291-299. 36. Yates, F. (1935). Complex experiments. J. Roy. Stat. Soc. B2, 181-247.