FRACTIONAL REPLICATION M.L.Agarwal Department of Statistics, University of Delhi, Delhi -. In a factorial experiment, when the number of treatment combinations is very large, it will be beyond the resources of the investigator to experiment with all of them. For such cases, Finney () proposed a method in which only a fraction of treatment combinations will be experimented with. In fractional factorial, though the size of the experiment is reduced, information on certain higher order interactions is sacrificed. The crucial part of the specification of the fractionally replicated design is the suitable choice of the defining or identity relationship. The nonestimable effects or interactions for the selected fraction of treatment combinations, when equated with I are called the identity relation. After selecting a fraction of treatment combinations, one can easily note that any contrast of the selected treatment combinations represents more than one effect or interaction and all effects or interactions represented by the same treatment combinations are called aliases. In aliases, by assuming that other interactions are negligible when compared with one of them, in which he is interested, the experimenter can estimate it by the corresponding contrast of the selected treatment combinations. The reader is requested to observe the resemblance between the methods of constructing confounded plans and the methods of constructing fractional replicate plans and also note that both methods can be together employed advantageously in some circumstances. / k Replicate of N Factorial Experiments Let us consider a factorial experiment in three factors n, p, k, each at two levels. The eight complete treatment combinations will be, n, p, np, k, nk, pk, npk. Instead of experimenting with all eight-treatment combinations, let us consider the four treatment combinations, np, nk, and pk. We observe that these four treatment combinations occur with the negative sign in the NPK interaction. If [np] denotes the total yield of r plots receiving the treatment combination np, etc., we note that (r) - {[] +[np] +[nk] +[pk]} represents µ - / NPK, (r) - {-[] +[np] +[nk] -[pk]} represents / (N - PK), (r) - {-[] +[np] -[nk] +[pk]} represents / (P - NK), (r) - {-[] -[np] +[nk] +[pk]} represents / (K - NP), where µ is the general mean. If we assume that the two- and three-factor interactions are negligible, the above four orthogonal functions of the treatment combinations can be used to estimate the general mean µ and the main effects N, P and K, respectively. This state of affairs will be represented by I NPK, () which is known as the identity relation, and N PK, P NK, K NP, ()
which are called alias sets. We see that the alias sets are obtained from the identity relation by multiplying both sides with the main-effect symbols and denoting the square of a symbol by unity. We remark that the experiment could have been conducted with the treatment combinations n, p, k and npk, and in that case also it is impossible to separate the mean µ from the NPK interaction, the N effect from the PK interaction, the P effect from the NK interaction, and the K effect from the NP interaction. Hence we have identity relationship and alias sets given in () and (). Thus we note that the treatment combinations with positive signs in the NPK interaction or the treatment combinations with negative sign in the NPK interaction have given rise to the same state of affairs. Conventionally we select the set of treatment combinations including the control one. In general let us consider the problem of constructing a / k fraction of a n factorial experiment,. Such an experiment will be denoted by n-k. Of the total of ( n -) effects and interactions in the full factorial experiment, k - will be inseparable from the mean, and the remaining n - k will be mutually inseparable in sets of k, there being ( n-k- -) such sets. The treatment combinations will be selected to be of the same sign as the control in the interactions X, X,..., X k will also be inseparable from the mean, and the identity relationship will become I = X = X = X X = X =X X =X X = X X X = etc. () The alias sets of an effect or interaction Y are the generalized interactions of Y with X, X, X X, etc. As a further illustration, consider a fraction of the factorial experiment (, ) obtained by confounding ABCD, ABEF, and CDEF. The treatment combinations in fractions are, ab, cd, abcd, ef, abef, cdef, abcdef, bce, ace, bde, ade, bcf, acf, bdf, adf. The defining contrast is I ABCD ABEF CDEF, () and the alias sets in this case are A BCD BEF ACDEF, C ABD ABCEF DEF, E ABCDE ABF CDF, AB CD EF ABCDEF, AD BC BDEF ACEF, AF BCDF BE ACDE, CF ABDF ABCE DE, ACF BDF BCE ADE. B ACD AEF BCDEF, D ABC ABDEF CEF, F ABDDF ABE CDE, AC BD BCEF ADEF, AE BCDE BF ACDF, CE ABDE ABCF DF, ACE BDE BCF ADF, Design Resolution: A design is of resolution R if no p-factor effect is aliased with another effect containing less than R-p factors. We usually employ a Roman numeral subscript to denote design resolution; thus, the one-half fraction of the design with the - defining relation I = ABC (or I = -ABC) is a design. III
Designs of resolution III, IV and V are particularly important. The definitions of these designs and an example of each follows:. Resolution III designs. These are designs, in which no main effects are aliased with any other main effect, but main effects are aliased with two-factor interactions and two-factor interactions may be aliased with each other. The - design considered earlier with treatment combinations n, p, k, npk is of resolution - III ( III ).. Resolution IV designs. These are designs in which no main effect is aliased with any other main effect or with any two-factor interactions, but two-factor interactions are aliased with other A - design with I = ABCD is of resolution - IV ( IV ).. Resolution V designs. These are designs in which no main effect or two-factor interaction is aliased with any other main effect or two-factor interaction, but twofactor interactions are aliased with three-factor interactions. A - design with I - = ABCDE is of resolution V ( V ). In general, the resolution of a two-level fractional factorial design is equal to the smallest number of letters in any word in the defining relation. Consequently, we could call the preceding design types three-letter, four-letter, and five-letter designs, respectively. We usually like to employ fractional designs that have the highest possible resolution consistent with the degree of fractionation required. The higher the resolution, the less restrictive the assumptions that are required regarding which interactions are negligible in order to obtain a unique interpretation of the data. Projection of Fractions into Factorials Any fractional factorial design of resolution R contains complete designs(possibly replicated factorials) in any subset of R- factors. This is an important and useful concept. For example, if an experimenter has several factors of potential interest but believes that only R- of them have important effects, then a fractional factorial design of resolution R is the appropriate choice of design. If the experimenter is correct, then the fractional factorial design of resolution R will project into a full factorial in the R- significant factors. Since the maximum possible resolution of a one-half fraction of the k design is R=k, every k- design will project into a full factorial in any (k-) of the original k factors. Furthermore, a k- design may be projected into two replicates of a full factorial in any subset of k- factors, four replicates of a full factorial in any subset of k- factors, four replicates of a full factorial in any subset of k- factors, and so on. Analysis of Fractional Factorials The analysis of fractional factorials is similar to the analysis of the full factorials. The treatment groups for each main effect or interaction are found by solving appropriate sets
of equations and then the sum of squares is obtained from the observation totals of these treatment groups by the usual method. For n factorials, the fractionally replicated designs can also be analysed by applying Yates algorithm. The only difference is that while writing the treatments, levels of k factors have to be ignored in the case of / k fraction. These k factors are so chosen that as a result of their suppression no treatment combination of the remaining n-k factors should have zeros only, or repeat. The other n-k factors are introduced one by one while writing the treatments, as in full replication. Here, n-k columns will be generated by following the same method as described in full factorial. An interaction corresponding to a contrast is also found similarly by considering only the (n-k) factors. The rest of the interactions which will contain the ignored factors also will form aliases of the above interactions. The fractions of n factorial can also be analysed on the same analogy. Here, also, while writing the treatments, factors are suppressed first and then they are written by introducing the factors one by one as described in full factorial. The operations and the correspondence of contrasts and interactions are also similar when the non-suppressed factors alone are considered. It is, however, not possible to write the aliases of such interaction components. But this does not create any serious problem. The linear and quadratic contrasts for a suppressed factor, L, come from contrasts involving those with which L is in alias. Illustration An exploratory trial on Cardamom was conducted in Madras state with seven factors each at two levels in one quarter of a replicate. The design was a confounded one with plots per block. The treatments were all combinations of presence and absence of Zinc, Copper, Boron, Iron, Manganese, Magnesium and Molybdenum denoted by A, B, C, D, E, F and G respectively. The plot size was six plants in a row. The doses in lb per acre were,,,,, and for Zinc, Copper, Manganese, Magnesium, Molybdenum, Boron and Iron, respectively. The layout plan and the yield of green pods in gr. per plot are given below.
Table Block Block ab ae cd bcde adf abdef bcf cef bdg fg befg abcdfg acdefg deg acg abceg be abcd acde bdf def acf abcef adg abdeg bcg ceg aefg cdfg bcdefg abfg The identity group of interactions for the above /( ) factorial is I=ABCDE = CDFG = ABEFG. The interactions confounded for the two blocks is are BCEF = ADF = BDEG = ACG. The data have been analysed by using Yates algorithm. For this purpose two factors have to be suppressed in the available treatment combinations. These two factors should be such that after suppression no treatment combination is repeated. In the present case E and G are such two factors. They have, thus, been suppressed. These have been shown in bracket while writing the treatment combinations in the table of analysis. The s.s. in the last column of Table corresponds to the treatment combinations of the non-suppressed factors only as written in the first column. Those three or four factor interactions which have either a main effect or two factor interaction in its alias group are shown along with the main effect or interaction in the last column. There are thus interactions none of which is in alias with a main effect or two factor interaction. One of them is confounded. The remaining five have been pooled together and used as main effect or two factor interaction. One of them is confounded. The remaining five have been pooled together and used as error sum of squares. The error mean square has thus come out as.. When tested against this mean square only the mean square for BG comes out significant. The average yield in the BG table are the following b b
g g.... The table indicates that application of Copper or Molybdenum alone proved better while their combined application depressed the yield. Table Treat. Obsn. S.S. a(e) b(e) ab c(eg) ac(g) bc(g) abc(eg) d(eg) ad(g) bd(g) abd(g) cd acd(e) bcd(e) abcd f(g) af(eg) bf(eg) abf(g) cf(e) acf bcf abcf(g) df(e) adf bdf abdf(e) cdf(g) acdf(eg) bcdf(eg) abcdf(g) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -............................... A B AB C AC BC ABC=DE D AD BD ABD=CE CD ACD=BE BCD=AE ABCD=E F AF BF ABF=EG CF ACF BCF ABCF DF ADF(Con) BDF ABDF CDF=G ACDF=AG BCDF=BG ABCDF=EF