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Chapter 8
Factorial Experiments
Factorial experiments involve simultaneously more than one factor and each factor is at two or more levels. Several factors affect simultaneously the characteristic under study in factorial experiments and the experimenter is interested in the main effects and the interaction effects among different factors.
First, we consider an example to understand the utility of factorial experiments.
Example: Suppose the yield from different plots in an agricultural experiment depends upon
- (i) variety of crop and (ii) type of fertilizer. Both the factors are in the control of the experimenter.
- (iii) Soil fertility. This factor is not in the control of the experimenter.
In order to compare different crop varieties
- assign it to different plots keeping other factors like irrigation, fertilizer, etc. fixed and the same for all the plots.
- The conclusions for this will be valid only for the crops grown under similar conditions with respect to the factors like fertilizer, irrigation etc.
In order to compare different fertilizers (or different dosage of fertilizers)
- sow single crop on all the plots and vary the quantity of fertilizer from plot to plot.
- The conclusions will become invalid if different varieties of the crop are sown.
- It is quite possible that one variety may respond differently than another to a particular type of fertilizer.
Suppose we wish to compare
- two crop varieties – a and b , keeping the fertilizer fixed and
- three varieties of fertilizers – A, B and C.
This can be accomplished with two randomized block designs ( RBD ) by assigning the treatments at random to three plots in any block and two crop varieties at random.
The possible arrangement of the treatments may appear can be as follows.
With these two RBDs ,
- the difference among two fertilizers can be estimated
- but the difference among the crop varieties cannot be estimated. The difference among the crop varieties is entangled with the difference in blocks.
On the other hand, if we use three sets of three blocks each and each block having two plots, then
- randomize the varieties inside each block and
- assign treatments at random to three sets.
The possible arrangement of treatment combinations in blocks can be as follows:
Here the difference between crop varieties is estimable but the difference between fertilizer treatment is not estimable.
Factorial experiments overcome this difficulty and combine each crop with each fertilizer treatment. There are six treatment combinations as aA, aB, aC, bA, bB, bC. Keeping the total number of observations to be 18 (as earlier), we can use RBD with three blocks with six plots each, e.g.
Now we can estimate
- the difference between crop varieties and
- the difference between fertilizer treatments. Factorial experiments involve simultaneously more than one factor each at two or more levels.
bB bA bC bC bB bA bA bC bB
and
bB aB aB bB bB aB
and
bA aC aB bB aA bC aA aC bC aB bB bA bB aB bA aC aA bC
aA aB aC aC aA aB aB aC aA
aC bC bC aC aC bC
aA bA bA aA bA aA
In order to make an experiment, there are 4 different combinations of values of current and voltage.
- Current = 5 Ampere and Voltage = 200 Volts, denoted as C V 0 0 (^) a b 0 0
- Current = 5 Ampere and Voltage = 220 Volts, denoted as C V 0 1 (^) a b 0 1.
- Current = 10 Ampere and Voltage = 200 Volts, denoted as C V 1 0 (^) a b 1 0
- Current = 10 Ampere and Voltage = 220 Volts, denoted as C V 1 1 (^) a b 1 1
The responses from those treatment combinations are represented by a b 0 0 (^) (1), ( a b 0 1 ) ( ), b
( a b 1 0 ) ( ) a and ( a b 1 1 ) ( ab ), respectively.
Now consider the following:
I. (^ C V^ o^ o^ )^2 ^ (^ C Vo^1 ): Average effect of voltage for the current level C 0
: (^ a b^ o^ o^ )^2 (^ a bo^1^ )^ (1)^ 2 ( ) b
II. (^ C V^1^ o^ )^2 ^ (^ C V^1 1 ) :Average effect of voltage for the current level C 1
: (^ a b^1^ o^ )^2 (^ a b^1 1^ )^ ( ) a^^ 2 (^ ab )
Compare these two group means (or totals) as follows:
Average effect of V 1 level – Average effect at V 0 level
( ) ( ) (1) ( ) 2 2 Main effect of voltage = Main effect of.
b ab a
B
^
Comparison like ( C o V 1 ) - ( C o V o ) ( a ) - (1): indicate the effect of voltage at current level C o and ( C 1 V 1 ) - ( C 1 V 0 ) ( ab ) - (b): indicate the effect of voltage at current level C 1.
The average interaction effect of voltage and current can be obtained as
1
1 1 1 1
Average effect of voltage Average effect of voltage at current level at current level Average effect of voltage at different levels of current. ( ) ( ) ( ) ( ) 2 2 ( ) ( ) ( ) (1) 2 2 A
o
o o o o
I I
C V C V C V C V
ab b a
^
^
verage interaction effect.
Similarly
(^10)
(^1 1 )
( ) ( ) (^) (1) ( ) :Average effect of current at voltage level. 2 2 ( ) ( ) ( ) ( ) : Average effect of current at voltage level 2 2
o o o
o
C V C V (^) b (^) V
C V C V a ab (^) V
Comparison of these two as
0 1 0 1 1 1 0 0 1 0
Average effect of current Average effect of current at voltage level at voltage level ( ) ( ) ( ) ( ) 2 2 ( ) ( ) (1) ( ) 2 2 Main effect of current = Main effect of.
V V
C V C V C V C V
a ab b
A
^
^
Comparison like
1 0 0 0 0 1 1 0 1 1
( ) ( ) ( ) (1) : Effect of current at voltage level ( ) ( ) ( ) ( ) : Effect of current at voltage level
C V C V b V C V C V ab a V
The average interact effect of current and voltage can be obtained as
0 1
1 1 0 1 1 0 0 0
Average effect of current Average effect of current at voltage level at voltage level Average effect of current at different levels of voltage ( ) ( ) ( ) ( ) 2 2 ( ) ( ) ( ) (1) 2 2 Ave
V V
C V C V C V C V
ab a b
^
^
rage interaction effect Same as average effects of voltage at different levels of current. (It is expected the interaction effect of current and voltage is same as the interaction effect of voltage and cu
that
rrent).
Factorial effects Treatment combinations Divisor (1) ( a ) ( b ) ( ab ) M A B AB
The model corresponding to 22 factorial experiment is
yijk Ai B j ( AB ) ij ijk , i 1, 2, j 1, 2, k 1, 2,..., n
where n observations are obtained for each treatment combinations.
When the experiments are conducted factor by factor, then much more resources are required in comparison to the factorial experiment. For example, if we conduct RBD for three-levels of voltage V 0 (^) , V 1 and V 2 and two levels of current I (^) 0 and I 1 , then to have 10 degrees of freedom for the error
variance, we need
- 6 replications on voltage
- 11 replications on current. So the total number of fans needed is 40. For the factorial experiment with 6 combinations of 2 factors, the total number of fans needed are 18 for the same precision.
We have considered the situation up to now by assuming only one observation for each treatment combination, i.e., no replication. If r replicated observations for each of the treatment combinations are obtained, then the expressions for the main and interaction effects can be expressed as
A (^) r ab a b
B (^) r ab b a
AB (^) r ab a b
M (^) r ab a b
Now we detail the statistical theory and concepts related to these expressions.
Let Y * (^) ((1), a , b , ab ) 'be the vector of total response values. Then
' * *
' * *
' * *
A
B
AB
A (^) r Y (^) r Y
B (^) r Y (^) r Y
AB (^) r Y (^) r Y
Note that A, B and AB are the linear contrasts. Recall that a linear parametric function is estimable only when it is in the form of linear contrast. Moreover, A, B and AB are the linear orthogonal contrasts in the total response values (1), a , b , ab except for the factor 1/2 r.
The sum of squares of a linear parametric function ' y is given by
( ' )^2
y .^ If there are^ r^ replicates,
then the sum of squares is
( ' )^2
y r
.^ It may also be recalled under the normality of^ y ’s, this sum of
squares has a Chi-square distribution with one degree of freedom ( 1 2 ). Thus the various associated sum
of squares due to A B , and AB are given by the following: ' (^) * (^22) ' ' (^) * (^22) ' ' (^) * (^22) '
A A A B B B AB AB AB
SSA Y ab a b r r SSB Y ab b a r r SSAB Y ab a b r r
Each of SSA, SSB and SSAB has 1 2 under normality of Y *.
The sum of squares due to total is computed as usual 2 2 2 2 1 1 1 4
r TSS (^) i j k yijk Gr
where 2 2 1 1 1
r G (^) i j k yijk
is the grand total of all the observations.
The TSS has 2 distribution with (2 2 r 1) degrees of freedom.
23 Factorial experiment:
Suppose that in a complete factorial experiment, there are three factors - A B , and C , each at two levels,
viz., a 0 (^) , a 1 (^) ; b 0 (^) , b 1 and c 0 (^) , c 1 respectively. There are a total of eight number of combinations:
0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1
a b c a b c a b c a b c a b c a b c a b c a b c
Each treatment combination has r replicates, so the total number of observations are N 23 r 8 r that are to be analyzed for their influence on the response.
Assume the total response values are Y * (^) (^) (1), a , b , ab , c , ac , bc , abc '.
The response values can be arranged in a three-dimensional contingency table. The effects are determined by the linear contrasts ' (^) effect^ Y * (^) ' effect (1), a , b , ab , c , ac , bc , abc
using the following table: Factorial effect Treatment combinations (1) a b ab c ac bc abc I A B AB C AC BC ABC
Note that once a few rows have been determined in this table, rest can be obtained by simple multiplication of the symbols. For example, consider the column corresponding to a , we note that A has
- sign, B has – sign , so AB has – sign (sign of A sign of B ). Once AB has - sign, C has – sign then
ABC has (sign of AB X sign of C ) which is + sign and so on.
The first row is a basic element. With this a 1' Y * can be computed where 1 is a column vector of all
elements unity. If other rows are multiplied with the first row, they stay unchanged (therefore we call it as identity and denoted as I ).Every other row has the same number of + and – signs. If + is replaced by
1 and – is replaced by -1, we obtain the vectors of orthogonal contrasts with the norm 8( 2 ).^3
If each row is multiplied by itself, we obtain I (first row). The product of any two rows leads to a different row in the table. For example
2 2
A B AB
AB B AB A
AC BC A C BB AB
The structure in the table helps in estimating the average effect. For example, the average effect of A is
A (^41) r^ ( ) a (1) ( ab ) ( ) b ( ac ) ( ) c ( abc ) ( bc )
which has the following explanation.
(i) Average effect of A at low level of B and C ( a b c 1 0 0 (^) ) ( a b c 0 0 0 ) ^ ( ) a^ r^ (1).
(ii) Average effect of A at high level of B and low level of C ( a b c 1 1 0 (^) ) ( a b c 0 1 0 ) ^ (^ ab )^ r^ ( ) b
(iii) Average effect of A at low level of B and high level of C ( a b c 1 0 1 (^) ) ( a b c 0 0 1 ) ^ (^ ac )^^ r^ ( ) c .
(iv) Average effect of A at high level of B and C ( a b c 1 1 1 (^) ) ( a b c 0 1 1 ) ^ (^^ abc )^ r^ (^ bc ).
Hence for all combinations of B and C ,the average effect of A is the average of all the average effects
in (i)-(iv).
Similarly, other main and interaction effects are as follows:
B b ab bc abc a c ac^ a^ b^ c r r C c ac bc abc a b ab a^ b^ c r r AB ab c abc a b ac bc a^ b^ c r r AC (^) r b ac abc
^ ^ ^
^ ^
^ ^
(^)
( ) ( ) ( ) ( ) (^ 1)(^ 1)(^ 1)
a ab c bc a^ b^ c r BC a bc abc b ab c ac a^ b^ c r r ABC abc ab b c ab ac bc a^ b^ c r r
^ ^ ^
^ ^
^ ^
Various sum of squares in the 23 factorial experiment are obtained as
2 n^ Factorial experiment:
Based on the theory developed for 22 and 2^3 factorial experiments, we now extend them for the 2 n
factorial experiment. Capital letters A B C , , ,... denote the factors. They are the main effect contrast for the factors A , B C , ,... AB AC BC , , ,...denote the first order or 2-factor interactions ABC ABD BCD , , ,... denote the second-order or 3-factor interactions and so on. Each of the main effect and interaction effect carries one degree of freedom. Total number of main effects = ^1^ n^ n. Total number of first-order interactions = ^2^ n
Total number of second-order interactions = ^3^ n and so on.
Standard order for treatment combinations:
The list of treatments can be expressed in a standard order. For one factor A , the standard order is (1), a. For two factors A and B, the standard order is obtained by adding b and ab in the standard order of one factor A. This is derived by multiplying (1) and a by b , i.e. b {(1), } a (1), a , b , ab. For three factors, add c , ac bc , and abc which are derived by multiplying the standard order of A and B by c , i.e. c {(1, a , b , ab } (1), a , b , ab , c , ac , bc , abc.
Thus the standard order of any factor is obtained step by step by multiplying it with an additional letter to preceding standard order.
For example, the standard order of A, B, C and D is 24 factorial experiment is (1), , , , , , , , {(1), , , , , , , } (1), , , , , , , , , , , , , , ,.
a b ab c ac bc abc d a b ab c ac bc abc a b ab c ac bc abc d ad bd abd cd acd bcd abcd
How to find the contrasts for main effects and interaction effect:
Recall that earlier, we had illustrated the concept in writing the contrasts for main and interaction effects.
For example, in a 22 factorial experiment, we had expressed
A a b a b ab
AB a b a b ab
Note that each effect has two component - divisor and contrast. When the order of factorial increases, it is cumbersome to derive such expressions. Some methods have been suggested to write the expressions for factorial effects. First, we detail how to write divisor and then illustrate the methods for obtaining the contrasts.
How to write divisor:
In a 2 n^ factorial experiment,
- the general mean effect has divisor 2 n^ and
- any effect (main or interaction) has divisor 2 n ^1. For example, in a 26 factorial experiment, the general mean effect has divisor 26 and any main effect or
interaction effect of any order has divisor 2 6 1 ^ 25.
If r replicates of each effect are available, then
- the general mean effect has divisor r 2 n and
- any main effect or interaction effect of any order has a divisor (^) r 2 n ^1.
How to write contrasts:
Method 1:
Contrast belonging to the main effects and the interaction effects are written as follows: ( 1)( 1)( 1)...( 1) ( 1)( 1)( 1)...( 1) ( 1)( 1)( 1)...( 1)
( 1)( 1)( 1)...( 1) ( 1)( 1)( 1)...( 1)
( 1)( 1)( 1)...( 1).
... ( 1)( 1)( 1)...( 1)
A a b c z B a b c z C a b c z
AB a b c z BC a b c z
ABC a b c z
ABC Z a b c z
For example, in a 23 factorial experiment, write
- rows for main and interaction effects and
- columns for treatment combinations in standard order.
- Take treatment combination (1) to have an even number (zero) of letter common with every interaction. This gives the following table
Factorial effect Treatment combinations (1) a b ab c ac bc abc I A B AB C AC BC ABC
Interaction
Even number of letters ( AB , ABCD ,…)
Count the number of letters common between treatment combinations and interactions
Count the number of letters common between treatment combinations and interactions
Even number of letters
Odd number of letters
Even number of letters
Odd number of letters
Odd number of letters ( A , ABC ,…)
Sums of squares:
Suppose 2 n^ factorial experiment is carried out in a randomized block design with r replicates. Denote the total yield (output) from r plots (experimental units) receiving a particular treatment combination by the same symbol within a square bracket. For example, [ ab ] denotes the total yield from the plots receiving the treatment combination ( ab ).
In a 2 2 factorial experiment, the factorial effect totals are
A ^ ^ ab ^ ^ b^ ^ a ^1 ab ^ = treatment total, i.e. the sum of^ r^ observations in which both the factors^ A^ and^ B^ are at the
second level.
a^ = treatment total, i.e., the sum of^ r^ observations in which factor^ A^ is at the second level and factor B is at the first level b^ = treatment total, i.e., the sum of^ r^ observations in which factor^ A^ is at the first level and factor^ B
is at the second level.
^1 =^ treatment total, i.e. the sum of^ r^ observations in which both the factors^ A^ and^ B^ are at the first
level.
(^1) ( ) ( ) ( ) (1) ' (^) (say).
r i i ab^ i b^ i a^ i A A
A y y y y y
where (^) A is a vector of +1 and =1 and yA is a vector denoting the responses from ab, b, a and 1.
Similarly, other effects can also be found.
Similarly, other effects can also be found. The sum of squares due to a particular effect is obtained as
Total yield^2 Total number of observations.
In a 2 2 factorial experiment in an RBD, the sum of squares due to A is ' 2 2
SSA A^ yA (^) r
In a 2 n^ factorial experiment in an RBD, the divisor will be r .2 n. If Latin square design is used based on
2 n^ 2 n Latin square, then r is replaced by 2 n^.
Example: Yates procedure for 23 factorial experiment. Treatment Yield (total from all r replicates) (1) (2) (3) (4) (5) (6) 1 (1)^ u 1 (^) (1) ( ) a v 1 (^) u 1 (^) u 2 w 1 (^) v 1 (^) v 2 [^ M ] a ( ) a (^) u 2 (^) ( ) b ( ab ) v 2 (^) u 3 (^) u 4 w 2 (^) v 3 (^) v 4 [ A ] b ( b ) (^) u 3 (^) ( ) c ( ac ) v 3 (^) u 5 (^) u 6 w 3 (^) v 5 (^) v 6 [ B ] ab ( ab ) (^) u 4 (^) ( bc ) ( abc ) v 4 (^) u 7 (^) u 8 w 4 (^) v 7 (^) v 8 [ AB ] c ( ac ) u 5 (^) ( ) a (1) v 5 (^) u 2 (^) u 7 w 5 (^) v 2 (^) v 1 [ C ] ac ( ac ) (^) u 6 (^) ( ab ) ( ) b v 6 (^) u 4 (^) u 3 w 6 (^) v 4 (^) v 3 [ AC ] bc ( bc ) (^) u 7 (^) ( ac ) ( ) c v 7 (^) v 6 (^) u 5 w 7 (^) v 6 (^) v 5 [ BC ] abc ( abc ) u 8 (^) ( abc ) ( bc ) v 8 (^) u 8 (^) u 7 w 8 (^) v 8 (^) v 7 [ ABC ] The sum of squares are obtained as follows when the design is RBD.
^2 3 ( ) Effect SS Effect (^) r..
For analysis of 2 n^ factorial experiment, the analysis of variance involves the partitioning of treatment sum of squares so as to obtain sum of squares due to main and interaction effects of factors. These sum of squares are mutually orthogonal, so Treatment SS = Total of SS due to main and interaction effects.
For example:
In 2 2 factorial experiment in an RBD with r replications, the division of degrees of freedom and the treatment sum of squares are as follows:
Source Degree of Sum of squares freedom
Replications r 1 Treatments 4 1 3
- A (^1) A ^2 / 4 r
- B (^1) B ^2 / 4 r
- AB (^1) AB (^) ^2 / 4 r
Error 3( r 1) Total 4 r 1
The decision rule is to reject the concerned null hypothesis when the related F - statistic Feffect F 1 (^) (1,3( r 1)).
