FRACTIONAL FACTORIAL NURNABI MEHERUL ALAM M.Sc. (Agricultural Statistics), Roll No. 443 I.A.S.R.I, Library Avenue, New Delhi- Chairperson: Dr. P.K. Batra Abstract: Fractional replication can be defined as the technique by which useful information can be obtained with a reasonable degree of precision from less than a full replicate of a factorial experiment. In this method there is always some loss in information, but we use a fraction of replication instead of full replication in such a way that loss of information is minimum. This type of experiment is used when a complete replication is considered too large or is not feasible. Here, our main interest is to estimate the main effects and two factor interactions. In this write up the concept of fractional factorial, their use and some methods of construction will be discussed. Key words: Fractional Replication, Alias, Defining Contrast, Half and Quarter Replicate, Blocking, Resolution Plan.. Introduction In factorial experiments when the total number of treatment combinations becomes large than total number of plots and also the block size increase considerably which makes the experiment expensive and more time is needed to carry out the experiment. Also it becomes difficult to maintain the homogeneity within the blocks. In addition nonexperimental error may increase. For example, treatment labeling may be interchanged; the plot number may be wrongly noted etc. More over large factorial experiment leads to increase (i) Error degrees of freedom (ii) Experimental resources required (iii) Non-experimental type of error. To overcome these problems, Finney (945) proposed to conduct fractional replication in large factorial experiments. When the experiment is very large, we consider only a fraction of all possible treatment combinations and the underlying experimental strategy is called fractional factorial plan or fractional replication. However the reduction in experiment is not without paying a price. It achieves estimation of lower order effects, under the assumption that higher order effects with which it is aliased is/ are non-existent.. Fractional Replication The technique by which useful information can be obtained with a reasonable degree of precision from less than a full replicate of a factorial experiment is known as fractional replication. Such a plan aims at drawing valid statistical inferences about the relevant factorial effects through an optimal utilization of the available resources under consideration. One or two factors suspected of possibly having significant first-order interactions could be assigned in such a way as to avoid having them aliased.
3. Some Preliminaries Suppose there are three factors A, B and C at two levels each and only four treatment combinations symbolized by a, b, c and abc are to be used, out of the 8 treatment combinations, a, b, c, ab, ac, bc, abc. This is called the ½ fraction of 3 experiment. Let y, y, y 3 and y 4 are yields of one plot of each of these treatments The relation between response and effects in a fraction of 3 experiment is given as follows: Treatment effects Combination μ A B C AB AC BC ABC () + - - - + + + - a + + - - - - + + b + - + - - + - + c + - - + + - - + ab + + + - + - - - ac + + - + - + - - bc + - + + - + + - abc + + + + + + + + Then, under the model, the expected values of these four observations are E (y ) = μ + A - B - C - AB - AC + BC + ABC E (y ) = μ - A + B - C - AB + AC - BC + ABC E (y 3 ) = μ - A - B + C + AB - AC - BC + ABC E (y 4 ) = μ + A + B + C + AB + AC + BC + ABC where μ is the general mean. Clearly all the eight parameters cannot be estimated from these four observations. It can be seen that E {(y 4 + y y -y 3 )/4} = A+BC; E {(y 4 - y +y -y 3 )/4} = B + AC; E {(y 4 - y y +y 3 )/4} = C+ AB; E {(y 4 + y +y +y 3 )/4} = μ + ABC. Each main effect can be estimated except for a disturbance from the interaction of the other pair. The general mean can be unbiasedly estimated if the three-factor interaction is of negligible magnitude. This means that the general mean and the main effects are estimable through the half fraction if three factor and two factor interactions are assumed to be absent. In other word, we can say that μ is completely confounded with ABC and the main effects are confounded with the interaction of other two factors. In any fractional factorial designs linear combination of two or more factors are estimated by an observational contrast. 4. Alias All factorial effects that appear in the expectation of an observational contrast are said to be aliased with each other. Here A is aliased with BC, B is aliased with AC and C is aliased with AB. The general mean is aliased with three-factor interaction ABC. We express it as
I = ABC This relation is known as defining contrast or the identity relationship. 5. Defining Contrast The interactions, which are confounded for obtaining the fraction, are called defining contrast or the identity group of interactions. Generally in n factorial experiments aliases are obtained as generalized interaction of factorial effect with the defining contrast by, i) In defining contrast interpreting I as unity. ii) Multiplying each effect on both sides of the identity relationship and deleting the letter whose power is two. I ABC A = A BC = BC B = AB C = AC C = ABC = AB Here A is aliased with BC and so on. Instead of taking the blocks containing the treatments a, b, c and abc we can also take the block containing the other four treatment combinations viz. ab, ac, bc and. In this case it can be easily seen that E (y ) = μ + A + B - C + AB - AC - BC - ABC E (y ) = μ + A - B + C - AB + AC - BC - ABC E (y 3 ) = μ - A + B + C - AB - AC + BC - ABC E (y 4 ) = μ - A - B - C + AB + AC + BC ABC Here also the eight parameters cannot be estimated. We can easily find out the following relationships E {(y 4 + y 3 -y y )/4} = A+BC; E {(y 4 -y 3 -y +y )/4} = B - AC; E {(-y 4 +y 3 - y +y )/4} = C- AB; E {(y 4 + y +y +y 3 )/4} = μ + ABC. It is seen that this half replicate also reduces to same state of affairs and do not make much difference. It can be noticed that the four treatment combinations chosen for the fraction are solution of the equation x + x + x 3 = (mod ) for the first case and x + x + x 3 = (mod ) for choosing the second treatment combinations. where x i denotes the level of i th factor. The observation implies that the fraction is obtained by confounding ABC within a replication. The name of defining contrast came from the above idea. Instead of choosing ABC as defining contrast one could also choose AB as the same. But in that case A will be aliased with B. For this reason even under the assumption of absence of all interactions, this function is not capable of providing unbiased estimates of all main effects. It is best to take higher order interaction components as defining contrast. Because we can assume higher order interactions to be absent, as they are generally of less interest to the experimenter. 3
6. Utility of Fractional Factorials In order to study the effect of micronutrients, viz., Boron, Copper, Iron, Magnesium, Manganese, Molybdenum, Zinc, and the major nutrient, Potash, on the yield of paddy, an experiment was planned at the State Agriculture Research Station, Bhubaneshwar (Orissa). Each factor was to be tested at two levels. While there was no prior information that the factors are independent, it was believed that interactions with three or more factors would have negligible magnitudes. For this experiment, a full factorial would have involved testing of 8 =56 treatment combinations in a single replicate. However, since the interactions involving 3 or more factors could be assumed negligible, a fractional factorial design was call for. The actual design adopted was a resolution V design involving 64-treatment combination. 7. Some Examples of Fractional Factorial 7. Half Replicate of 5 Factorial Here out of 3 possible treatment combinations 6 treatment combinations will be taken. The obvious choice of the defining contrast is the five-factor interaction ABCDE. Here we get the alias of main effects I = ABCDE A = A BCDE = BCDE B = AB CDE = ACDE C = ABC DE = ABDE D = ABCD E = ABCE E = ABCDE = ABCD That means all main effects are aliased with four factor interactions. Similarly, for twofactor interactions, AB is aliased with CDE; AC is aliased with BDE etc. So every twofactor interaction is aliased with three factor interactions. If the effects of all 3 or more factors interaction are assumed to be negligible and we consider all main and two factor interactions then error degrees of freedom become. Because out of the total 5 degrees of freedom, 5 d.f. is for main effects and the remaining d.f. is for two factor interactions. For best result we should take some two-factor interactions instead of taking all and remaining two factor interactions as error. Because if we take all the two factor interactions in error, then the experiment can be used to estimate only the 5 d.f. and the error degrees of freedom will become. But if we consider that the treatment combinations AB, AC and BC have effects then 5 d.f. is for main effects, 3 d.f. is for these two factor interactions and remaining 7 d.f. will go in the error part. To get the design we take the6 treatment combination having all odd number of treatment combinations or the even (4, or none) number of treatment combinations. So the block content of the half replicate of the 5 design will be abcde, abc, abd, abe, acd, ace, ade, bcd, bce, bde, cde, a, b, c, d, e or abcd, abce, abde, acde, bcde, ab, ac, ad, ae, bc, bd, be, cd, ce, de, (). 4
7. Half Replicate of 6 Factorial It requires 3 units. If we take six factor interaction ABCDEF as defining contrast, then A is aliased with BCDEF, B is aliased with ACDEF, AB is aliased with CDEF, AC is aliased with BDEF, ABC is aliased with DEF and so on. That all main effects are aliased with five factor interactions, all two factor interactions are aliased with four factor interactions and all three factor interactions are aliased with three factor interactions. If we consider the effects of all 3 or more factor interactions are negligible then out of total 3 degrees of freedom, 6 will be for main effects, 5 will be for two factor interactions and remaining d.f. is due to error. Because there are 6 main effects, 5 two-factor interactions and pair of three factor interactions. The half fraction of 6 factorial design will consist of the following treatment combinations: abcdef, abcd, abce, abcf, abde, abdf, abef, acde, acdf, acef, adef, bcde, bcdf, bcef, bdef, cdef, ab, ac, ad, ae, af, bc, bd, be, bf, cd, ce, cf, de, df, ef, (). 7.3 Quarter Replicate of 6 Factorial The quarter replicate of 6 factorial is similar to any one block of ( 6, 4 ). Here independent interactions and one generalized interaction will be confounded. If we take the the defining contrasts as I ABCE ABDF CDEF As there are three defining contrast we will get three aliases for all main effects and two factor interactions. For example A is aliased with BCE, BDF and ACDEF. All main effects are aliased with two three-factor interactions and one four-factor interaction. For two-factor interactions the aliases are Two factor interactions AB AC AD AE AF CD CF Aliases CE, DF, ABCDEF BE, BCDE, ADEF BF, BDCE, ACEF BC, BDEF, ACDE BD, BCEF, ACDE EF, ABDE, ABCF DE, ABEF, ABCD Confounding ABCE and ABDF the block contents can be obtained as: A B C D E F 5
If only main effects are to be estimated and all other effects are considered as error, then out of total 5 d.f., 6 will be for main effects and remaining 9 will be error d.f. Here 9 d.f. for error are composed of the 7 aliase sets given above that involve two factor interactions, plus alias sets that contain only three factor interactions, namely: ACF = BCD = BEF = ADE and ACD = AFE = BCF = BDE. 8. Reduction of Block Size by Confounding Consider the example of half fraction of 6 factorial. Here we have seen that the block size become 3 which is too large. But if we want the design with small block size then the concept of reduction of block size by confounding comes. This can be obtained by confounding certain factorial effects with blocks. The principle may be illustrated by half fraction of 6 factorial with defining contrast as ABCDEF. If we want the block of 6 units out of 3 rearmament combinations, that means we want two blocks each of size 6, any factorial effects other than ABCDEF may be used to divide the 3 units into two blocks of 6 units. All interactions on this factorial effect on its alias will be lost, since both are confounded with blocks. So we have to find out such a pair of aliases whose effects can be sacrificed. Let us take the alias pair which will be confounded is ABC and DEF. Since this confounding do not disturb any other factorial effects. The plan is constructed in such a way that one block satisfies the equation a + b + c = and the second block satisfies a + b + c = given the condition that both the block satisfy the condition a + b + c + d + e + f =. The blocks are obtained in the following manner: () Take all the treatment combinations to get a half replicate of 6. () Divide these 3 treatment combination into two blocks such that one block contains only even number of treatments of the letter a, b and c. The second block will contain the rest of the treatment combinations. Obviously () satisfy the condition a + b + c + d + e + f = and () satisfy the condition a + b + c = for block and d + e + f = for block. The design obtained is as follows: Defining contrast: ABCDEF Confounded with blocks: ABC=DEF 6
Block Block () ad de ae df af ef bd ab be ac bf bc cd abde ce abdf cf abef adef acde bdef acdf cdef acef abcd bcde abce bcdf abcf bcef abcdef Analysis of variance: d.f. Blocks Main effects 6 Two-factor interactions 5 Error (Three factor interactions) 9 Total 3 For a design with block of size 8, we will have to take another alias pair because it will confound another one alias pair. If we take the alias pair ABD and CEF then the generalized with the alias pair ABC and DEF will confound the interactions CD and ABEF. It means one two-factor interaction will be confounded. To construct the plan, divide the blocks obtained in the design of block size 6 into halves that contain an odd and even number of letters c and d respectively. Defining contrast: ABCDEF Confounded with blocks: ABC, DEF, ABD, CEF, CD, ABEF Block : (), ef, ab, abef, acde, acdf, bcde, bcdf Block : de, df, ac, bc, abde, abdf, acef, bcef Block 3: ae, af, be,bf, cd, abcd, cdef, abcdef Block 4: ad, bd, ce, cf, abce, abcf, abef, bdef The analysis of variance will be as: d.f. Blocks 3 Main effects 6 Two-factor interactions (except CD) 4 Error 8 Total 3 7
9. Design Resolution Box and Hunter (96a, b) introduced the term Resolution as a means of classifying fractional factorial designs. According to them, a fractional factorial is said to be of Resolution R, if the smallest interaction in the identity group is an R-factor interaction. 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 3 design with the defining relation l ABC 3 (or l -ABC) is a design. III Design of resolution III, IV and V are particularly important. The definition 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 3- design considered 3 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 two-factor interactions, but two-factor interactions are aliased with other A 4-4 design with I=ABCD is of resolution IV ( IV ) 3. 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 5- design with I 4 V ABCD is of resolution V ( ). But this definition of Resolution applies only to regular fractional factorial designs. To solve this problem. Webb (968a) has generalized the definition of Resolution to cover all types of fractional factorials. According to him, a fractional factorial design is of resolution (R-) if it permits the estimation of all effects up to R-factor interactions, when all effects involving (R+) factor and more are assumed negligible. Further, a fractional factorial design is of resolution R if it permits the estimation of all effects upto (R-) factor interactions, when all interactions involving (R+) factor or more are assumed to be zero. More general definition of Resolution of an arbitrary fractional factorial experiment was provided by Dey and Mukarjee (999). According to them, a fractional factorial plan is said to be Resolution (f, t) if it permits the estimability of all effects with f factors or less under the assumption that all effects involving (t-) factors or more are of negligible magnitude. 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, 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 highest the resolution, the less restrictive the assumptions are required regarding which interactions are negligible in order to obtain a unique interpretation of the data. With designs of resolution three, and sometimes four, we seek to screen out the few important main effects from the many less important than others. For this reason, these designs are often termed main effects designs, or screening designs. 8
On the other hand, designs of resolution five, and higher, are used for focusing on more than just main effects in an experimental situation. These designs allow us to estimate interaction effects and such designs are easily augmented to complete a second-order design - a design that permits estimation of a full second-order (quadratic) model.. Orthogonal Fractional Factorial Plan The fractional factorial plan, which permits the estimation of all relevant effects with zero correlation is called orthogonal plan. Thus regular fractions are necessarily orthogonal. Some Methods of Construction. Using Hadamard Matrix By this method, a fractional factorial of n- factorial in n runs can be constructed. Take Hadamard matrix of order n in semi normal form. Delete first column of the matrix. Replace by. Consider remaining n- columns as factors each at level (-,) and rows as treatment combination. Example.: Consider 3 in 4 runs. First take a Hadamard matrix of order 4. H 4 = Deleting the first column and replacing by we get Here rows indicate the treatment combinations. So the treatment combinations in the block are abc b a c Example.: Consider 7 factorial in 8 runs. Start with a Hadamard matrix of order 8 as shown below: H 8 = 9
Deleting the first column and replacing by we get So the treatment combinations in the block are abcdefg bdf ade cdg abc beg afg cef By this method of construction we generally get saturated design. Definition.: In fractional factorial plan where error d.f. is is called saturated plan. In other words, where number of runs is equal to number of parameters to be estimated and as such no degrees of freedom is left for error is known as saturated plan..3 Using Latin squares By this method a fractional factorial of s s+ factorials in s runs can be constructed. When s prime or prime power, maximum number of MOLS (Mutually Orthogonal Latin Squares) of order s is s-. Let the symbols be denoted by,,,,s-. Superimpose all the Latin square in the compete set of MOLS one over other, so that we have an arrangement of s rows and s columns and each cell in the arrangement has s- symbols. Then number the rows and columns of this arrangement by the symbols,,,,s-. From this arrangement generate s combinations from the s cells by augmenting the cell entries by the row and column indices. Example.3: 3 4 factorial in 3 runs is constructed here. For s = 3, there will be 3-= MOLS as given below. L = L = Superimposing we get
So the 3 runs of the plan for a 3 4 experiment are:. Conclusions From the above discussion, it is seen that fractional factorials are useful:. in situations where some of the higher order interactions can be tentatively assumed to be zero.. in screening from a large set of factors, some factors with large effects. 3. in a sequential program of experiments. 4. when a complete replication is considered too large or is not feasible References Agarwal, M.L. (). Fractional replication. Efficient Experimental designs for Generation of Agricultural Technologies. Training manual, IASRI, New Delhi. Box, G.E.P. and Hunter, J.S. (96a). The k-p fractional factorial designs. Technometrics, 3, 3-35. Box, G.E.P. and Hunter, J.S. (96b). The k-p fractional factorial designs, Technometrics, 3, 449-456. Cochran, W.G., and Cox, G.M. (957). Experimental Designs. nd ed. New York: John Wiley& Sons, New York. Das, M.N and Giri, N.C (979). Design and Analysis of Experiments. d ed. New Age International (P) Limited, Publishers, New Delhi. Dey, A( 985). Orthogonal Fractional Factorial Designs, Wiley Eastern Limited Finney, D.J (96) The Theory of Experimental Design: The University of Chicago Press, Chicago, USA. Kempthorne, O (947). A simple approach to confounding and fractional experiments. Biometrika, 34, 55-7