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CONFOUNDING IN FACTORIAL EXPERIMENTS
AND FRACTIONAL FACTORIALS
SEEMA JAGGI AND P.K.BATRA
Indian Agricultural Statistics Research Institute, Library Avenue, New Delhi-110 012 seema@iasri.res.in
- Introduction When the number of factors and/or levels of the factors increase, the number of treatment combinations increase very rapidly and it is not possible to accommodate all these treatment combinations in a single homogeneous block. For example, a 2^5 factorial would have 32 treatment combinations and blocks of 32 plots are quite big to ensure homogeneity within them. A new technique is therefore necessary for designing experiments with a large number of treatments. One such device is to take blocks of size less than the number of treatments and have more than one block per replication. The treatment combinations are then divided into as many groups as the number of blocks per replication. The different groups of treatments are allocated to the blocks.
There are many ways of grouping the treatments into as many groups as the number of blocks per replication. It is known that for obtaining the interaction contrast in a factorial experiment where each factor is at two levels, the treatment combinations are divided into two groups. Such two groups representing a suitable interaction can be taken to form the contrasts of two blocks each containing half the total number of treatments. In such case the contrast of the interaction and the contrast between the two block totals are given by the same function. They are, therefore, mixed up and can not be separated. In other words, the interaction has been confounded with the blocks. Evidently the interaction confounded has been lost but the other interactions and main effects can now be estimated with better precision because of reduced block size. This device of reducing the block size by taking one or more interaction contrasts identical with block contrasts is known as confounding. Preferably only higher order interactions, that is, interactions with three or more factors are confounded, because their loss is immaterial. As an experimenter is generally interested in main effects and two factor interactions, these should not be confounded as far as possible.
The designs for such confounded factorials can be called incomplete block designs. However usual incomplete block designs for single factor experiments cannot be adopted as the contrasts of interest in two kinds of experiments are different. The treatment groups are first allocated at random to the different blocks. The treatments allotted to a block are then distributed at random to its different units.
When there are two or more replications, if the same set of interactions are confounded in all the replications, confounding is called complete and if different sets of interaction are confounded in different replications, confounding is called partial. In complete confounding all the information on confounded interactions are lost. But in partial
confounding, the confounded interactions can be recovered from those replications in which they are not confounded.
Advantages of Confounding It reduces the experimental error considerably by stratifying the experimental material into homogeneous subsets or subgroups. The removal of the variation among incomplete blocks (freed from treatments) within replicates results in smaller error mean square as compared with a RBD, thus making the comparisons among some treatment effects more precise.
Disadvantages of Confounding
- In the confounding scheme, the increased precision is obtained at the cost of sacrifice of information (partial or complete) on certain relatively unimportant interactions.
- The confounded contrasts are replicated fewer times than are the other contrasts and as such there is loss of information on them and they can be estimated with a lower degree of precision as the number of replications for them is reduced.
- An indiscriminate use of confounding may result is complete or partial loss of information on the contrasts or comparisons of greatest importance. As such the experimenter should confound only those treatment combinations or contrasts which are of relatively less or of importance at all.
- The algebraic calculations are usually more difficult and the statistical analysis is complex, especially when some of the units (observations) are missing.
- A number of problems arise if the treatments interact with blocks.
1.1 Confounding in 2^3 Experiment Although 2^3 is a factorial with small number of treatment combinations but for illustration purpose, this example has been considered. Let the three factors be A, B, C each at two levels.
Factorial Effects →→→→ Treat. Combinations ↓↓↓↓
A B C AB AC BC ABC
(a) + - - - - + + (b) - + - - + - - (ab) + + - + - - - (c) - - + + - - + (ac) + - + - + - - (bc) - + + - - + - (abc) + + + + + + +
The various factorial effects are as follows: A = (abc) + (ac) + (ab) + (a) - (bc) - (c) - (b) - (1) B = (abc) + (bc) + (ab) + (b) - (ac) - (c) - (a) - (1) C = (abc) + (bc) + (ac) + (c) - (ab) - (b) - (a) - (1) AB = (abc) + (c) + (ab) + (1) - (bc) - (ac) - (b) - (a) AC = (abc) + (ac) + (b) + (1) - (bc) - (c) - (ab) - (a) BC = (abc) + (bc) + (a) + (1) - (ac) - (c) - (ab) - (b) ABC = (abc) + (c) + (b) + (a) - (bc) - (ac) - (ab) - (1)
In the above arrangement, the main effects A, B and C are orthogonal with block totals and are entirely free from block effects. The interaction ABC is completely confounded with blocks in replicate 1, but in the other three replications the ABC is orthogonal with blocks and consequently an estimate of ABC may be obtained from replicates II, III and IV. Similarly it is possible to recover information on the other confounded interactions AB (from I, III, IV), BC (from I, II, IV) and AC (from I,II, III). Since the partially confounded interactions are estimated from only a portion of the observations, they are determined with a lower degree of precision than the other effects.
1.2 Construction of a Confounded Factorial Given a set of interactions confounded, the blocks of the design can be constructed and vice-versa i.e if the design is given the interactions confounded can be identified.
Given a set of interactions confounded, how to obtain the blocks? The blocks of the design pertaining to the confounded interaction can be obtained by solving the equations obtained from confounded interaction.
Example 1.1: Construct a 2^5 factorial in 2^3 blocks confounding interactions ABD, ACE and BCDE. Let x 1 , x 2 , x 3 , x 4 and x 5 denote the levels (0 or 1) of each of the 5 factors A,B,C,D and E. Solving the following equations would result in different blocks of the design:
For interaction ABD: x 1 +x 2 +x 4 = 0, 1 For interaction ACE : x 1 +x 3 +x 5 = 0, 1
Treatment combinations satisfying the following solutions of above equations will generate the required four blocks:
(0, 0) (0, 1) (1, 0) (1, 1)
The solution (0, 0) will give the key block (A key block is one that contains one of the treatment combination of factors, each at lower level).
There will be
5 3 =4 blocks per replicate. The key block is as obtained below:
A B C D E
1 1 1 0 0 abc 1 1 0 0 1 abe 1 0 1 1 0 acd 1 0 0 1 1 ade 0 1 1 1 1 bcde 0 1 0 1 0 bd 0 0 1 0 1 ce 0 0 0 0 0 (1)
Similarly we can write the other blocks by taking the solutions of above equations as (0,1) (1,0) and (1,1).
Given a block, how to find the interactions confounded? The first step in detecting the interactions confounded in blocking is to select the key block. If key block is not given, it is not difficult to obtain it. Select any treatment combination from the given block and multiply all the treatment combinations in the block by that treatment combination and we get the key block. From the key block we know the number of factors as well as the block size. Let it be n and k. We know then that the given design belongs to the 2n^ factorial in 2r^ plots per block. The next step is to search out a unit matrix of order r. From these we can find the interaction confounded.
Example 1.2: Given the following block, find out the interactions confounded: acde ad bcd bde e ab abec c
Since the given block is not the key block, multiply it by e and the following block is obtained: acd ade bcde bd (1) abe abc ce
This is the key block as it includes (1). It is seen that the factorial involves five factors and has been confounded in 2^3 (=8) plots per block. Hence, the given design is (2^5 , 2^3 ).
A B C D E 1 0 1 1 0 0 1 1 1 1 0 0 0 0 0 1 1 1 0 0
A B C D E
1 0 0 1(=α 1 ) 1(=β 1 ) 0 1 0 1(=α 2 ) 0(=β 2 ) 0 0 1 0(=α 3 ) 1(=β 3 )
The interaction confounded are A α^1 B α^2 C α^3 D, A β^1 B β^2 C β^3 E. Here ABD and ACE are independent interactions confounded and BCDE is obtained as the product of these two and is known as generalised interaction.
Source of Variation D.f Replication p- Blocks within replication
p(2n-r-1)
Treatments 2 n- Error By subtraction Total p2n-
The S.S. for confounded effects are to be obtained from those replications only in which the given effect is not confounded. From practical point of view, the S.S. for all the effects including the confounded effects is obtained as usual and then some adjustment factor (A.F) is applied to the confounded effects. The adjusting factor for any confounded effect is computed as follows: (i) Note the replication in which the given effect is confounded (ii) Note the sign of (1) in the corresponding algebraic expression of the effect confounded. If the sign is positive then A.F = [Total of the block containing (1) of replicate in which the effect is confounded] - [Total of the block not containing (1) of the replicate in which the effect is confounded] =T 1 -T 2 If the sign is negative, then A.F = T 2 - T 1 This adjusting factor will be subtracted from the factorial effects totals of the confounded effects obtained.
Exercise 2.1: Analyse the following 2^3 factorial experiment in blocks of 4 plots, involving three fertilizers N, P, K, each at two level:
Replication I Replication II Replication III Block 1 Block 2 Block 3 Block 4 Block 5 Block 6 np 101
p 88
np 115
pk 75
n 53 npk 111
n 90
npk 95
k 95
nk 100
npk 76 (1) 75
pk 115
nk 80
pk 90
p 65 k 55
nk 75
p 100
n 80
np 92
k 82
Step 1: Identify the interactions confounded in each replicate. Here, each replicate has been divided into two blocks, one effect has been confounded in each replicate. The effects confounded are Replicate I → NP Replicate II → NK Replicate III → NPK
Step 2: Obtain the blocks S.S. and Total S.S.
S.S. due to Blocks =
Bi i
2
1
6
- C.F = 2506
Total S.S. = ∑ ( Obs.)
2
Step 3: Obtain the sum of squares due to all the factorial effects other than the confounded effects.
Treatment Combinations
Total Yield Factorial Effects Sum of Squares (S.S)= [ ]^2 / 2^3 .r (1) 255 G= n 223 [N]=48 (^) 96= SN^2 p 253 [P]=158 (^) 1040.17= SP^2 np 308 [NP]=66 - k 232 [K]=10 (^) 4.17= SK^2 nk 255 [NK]=2 - pk 280 [PK]=-8 (^) 2.67= SPK^2 npk 282 [NPK]=-108 -
Total for the interaction NP is given by
[NP] = [npk] - [pk] - [nk] + [k] + [np] - [p] - [n] + [1]
Here the sign of (1) is positive. Hence the adjusting factor (A.F) for NP which is to be obtained from replicate I is given by
A.F. for NP = (101+111+75+55) - (88+90+115+75) = 342- = -
Adjusted effect total for NP becomes: [NP*] =[NP]- (-26) = 66+ = 92
It can easily be seen that the total of interaction NP using the above contrast from replications II and III also gives the same total i.e. 92.
Similarly A.F. for NK = A.F. for NPK = -
Hence adjusted effect totals for NK and NPK are [NK] =- [NPK] = -
x 1 +x 2 +2x 3 = 2
Block I Block II Block III A B C A B C A B C 1 0 1 1 0 0 1 0 2 0 1 1 0 1 0 0 1 2 1 1 2 1 1 1 1 1 0 2 0 2 2 0 1 2 0 0 0 2 2 0 2 1 0 2 0 2 1 0 2 1 2 2 1 1 1 2 0 1 2 2 1 2 1 2 2 1 2 2 0 2 2 2 0 0 0 0 0 2 0 0 1
- Balanced Design A partially confounded design is said to be balanced if all the interaction of particular order are confounded in equal number of replication.
How to Construct a Balanced Factorial Design Example 4.1: Construct a (2^5 , 2^3 ) balanced design achieving balance over three and four factor interactions.
Solution: Total no. of treatment combination = 2^5 = 32 Number of blocks per replicate= 25-3^ = The number of 3 factor interactions = (^5 C 3 ) = 10 The number of four factor interactions = (^5 C 4 ) = 5 So the total degrees of freedom to be confounded = 10+5 = 15
Since the design is (2^5 , 2^3 ), the degrees of freedom that can be confounded per replicate = 2 5-3^ -1 = 3. So the number of replicates required = 15/3 = 5 Since the total number of three factor interactions is 10 and there are five replicates so each block confounds two three factor interactions such that generalized interaction is of four factor
Three factor interactions are ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE. Four factor interactions are ABCD, ABCE, ABDE, ACDE, BCDE.
Confounding ABD and ACE using equations x 1 + x 2 + x 4 = 0, 1 x 1 + x 3 + x 5 = 0, 1
would result in replication I consisting of four blocks. Here only the key block for each of the replication is given. Similarly Replication II confounds ACD, BCE and ABDE, Replication III confounds BCD, ABCE and ADE, Replication IV confounds ABCD, ABE and CDE, Replication V confounds ABC, BDE and ACDE.
Replication I Replication II
A B C D E A B C D E 1 0 0 1 1 ade 1 0 0 1 0 ad 0 1 0 1 0 bd 0 1 0 0 1 be 0 0 1 0 1 ce 0 0 1 1 1 cde 1 1 0 0 1 abe 1 1 0 1 0 abd 1 0 1 1 0 acd 1 0 1 0 1 ace 0 1 1 1 1 bcde 0 1 1 1 0 bcd 1 1 1 0 0 abc 1 1 1 0 0 abc 0 0 0 0 0 (1) 0 0 0 0 0 (1)
Replication III Replication IV
A B C D E A B C D E 1 0 0 0 1 ae 1 0 0 1 1 ade 0 1 0 1 1 bde 0 1 0 1 1 bde 0 0 1 1 1 cde 0 0 1 1 0 cd 1 1 0 1 0 abd 1 1 0 0 0 ab 1 0 1 1 0 acd 1 0 1 0 1 ace 0 1 1 0 0 bc 0 1 1 0 1 bce 1 1 1 0 1 abce 1 1 1 1 0 abcd 0 0 0 0 0 (1) 0 0 0 0 0 (1)
Replication V
A B D C E 1 0 0 1 0 ac 0 1 0 1 1 bce 0 0 1 0 1 de 1 1 0 0 1 abe 1 0 1 1 1 abce 0 1 1 1 0 bdc 1 1 1 0 0 abd 0 0 0 0 0 (1)
- Fractional Factorials In factorial experiments, when the number of factors and/or levels of the factors are large, the total number of treatment combinations becomes so large that it is very difficult to organize an experiment involving these treatments even in a single replication as it is beyond the resources of the investigator to experiment with all of them. Economy of space and material may be attained by observing the response only on a fraction of all possible treatment combinations.
For example, even with seven factors each at three levels a complete factorial experiment would mean testing 2187 treatment combinations in a single replication. Such a large
Thus the main effect A and interaction BCD is estimated by the same contrast in the 1 2
(2^4 ) i.e. when only 8 combinations are used.
Aliases are two factorial effects that are represented by the same comparisons. Thus A and BCD are aliases. This is represented by A ≡ BCD. Similarly, we have other aliases
B ≡ ACD C ≡ ABD D ≡ ABC AB ≡ CD AC ≡ BD AD ≡ BC M ≡ ABCD
The four factor interaction ABCD in the fraction cannot be estimated at all. If all the 16 treatment combinations were available, ABCD may be computed.
The 8 treatment combinations which carry a negative sign in the above contrast are chosen. The eight treatment combinations chosen for the fraction are solutions of the equation
x 1 + x 2 + x 3 + x 4 = 0 (mod 2),
where xi denotes the levels of the ith^ factor for i =1,2,3,4. Thus the four factor interaction ABCD is confounded for obtaining the fraction. In fact, the fraction considered is nothing but one of the possible two blocks of size 8 each, obtained by confounding ABCD with the blocks.
The interaction(s) which is(are) confounded for obtaining the fraction is (are) said to form the identity group of interaction(s) or defining contrast(s). Once the identity group is given, the aliases relationships are easily obtained by taking the generalized interactions of the relevant effect with each of the members of the identity group. The outcome of using a half-replicate is
- loss of one factorial effect ABCD entirely
- each main effect is mixed with one of the 3-factor interactions, 2-factor effect is mixed with 2-factor interaction.
If the experiment shows an apparent effect of A, there is no way to know whether the effect is really due to A, due to BCD interaction or due to a mixture of the two. The ambiguity disappears if we assume that the higher order interaction effects are negligible. For getting unbiased estimates of some effects, assumptions regarding the absence of certain effects have to be made.
In a large number of real-life situations, fractional replication has been used effectively. In order to study the effect of micro-nutrients, 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 Agricultural Research Station, Bhubaneshwar (Orissa). Each factor was to be tested at two levels. For this experiment, a full factorial would have 2^8 = 256 treatment combinations in a single replicate. Assuming 3 or more factors interactions being negligible, a fractional factorial involving 64 treatment combinations can be used. Fractional factorials are useful in the situations where some of the higher order interactions can be assumed to be zero, in screening from a large set of factors, in a sequential programme of experiments.
