Half Fraction of a 2^k Factorial Experiment lecture - 37

Construction

 of 

Fraction Factorial Experiment 

An allowable fraction may be 1/ 2, 1/4 and 1/8   fraction of the overall experiment depending on the value of k. a fraction may be denoted as follow:

Where p is the fraction index.

The table of contrast may be used to setup the different fractions.

In 2^(4-1) fractional factorial experiment, the number of factors k = 4 and fraction index: p = 1. The number of runs (level combination) = 8.

Resolution of a Design

The resolution of a design indicates its power and ability to separately estimates effects of the factors and interactions denoted by Roman digits.

If some main effects are aliased or confounded with 4 factors interaction, the resolution of the experiment will 4 + 1 = 5 and represented by V. if some main effects are confounded with two factor interactions the resolution of the experiment will 2 + 1 = 3 and represented by III. The resolution 2 does not exist because it confounded the main effects.

Resolution
III	IV	V
Main effects and two factors’ interactions are confounded or aliased.	·         The main effects are aliased with three interactions, and ·         Two factors’ interactions are aliased with some other two factors interactions.	·         The main effects are aliased with four interactions, and ·         Two factors’ interactions are aliased with three factors interactions.

Half Fraction of a 2^k Factorial Experiment

When the fraction index is equal to one that’s  p = 1 of 2^k factorial experiment, then it is called half fraction factorial experiment. One higher order interaction is confounded in 2^(k-1) factorial experiment.

Let the half fraction of  2^4 factorial experiment is 2^4 / 2^1 treatment combinations. The higher order interaction is ABCD. 

The defining contrast is:

I = ABCD

The alias structure is

A X I

A = BCD

B = ACD

C = ABD

D = ABC

AB = CD

AC =BD

AD = CD

same for I = - ABCD

ANOVA Table:

SV	df
A	1
B	1
C	1
D	1
Error	3
Total	7

Example:

A fractional factorial experiment is performed in a single replicate I = ABCD as a defining relation. The treatment combination and observed responses are as under:

(1)	a(d)	b(d)	ab	c(d)	ac	bc	abc(d)
45	100	45	65	75	60	80	96

1.      Give the aliases structure of the design.

2.      Using Yates algorithm estimates the effects.

Solution: The given experiment is 2^4 factorial experiment. The half fraction is .2^(4-1) = 8

 The defining relation is I = ABCD.

The aliases structure is obtained as:

I = ABCD

A = BCD

B = ACD

C = ABD

AB = CD

AC = BD

BC = AD

ANOVA Table for 2^(4-1) design:

SV	df	SS	MS	F	F tab
A	1	722	722	1.54	10.13
B	1	4.50	4.5	0.01	10.13
C	1	392.0	392	0.83	10.13
D	1	544.5	544.5	1.16	10.13
Error	3	1408.5	496.4
Total	7	3071.5

Example: 

An experiment was conducted to enhance the yield of oil extraction from peanuts. A  design was selected. Interestingly, the negative half fraction was used; . The data is given below:

Test the significance at 5 %.

Solution: 

The 2^(5-1) with  E = - ABCD 

 Applying Yates algorithm

ANOVA Table:

SV	df	SS	MS	F	F tab
AB	1	400.00	400.00	0.13	59.44
AC	1	25.00	25.00	0.008
BC	1	1.00	1.00	0.0003
AD	1	12.25	12.25	0.004
BD	1	0.25	0.25	0.00008
CD	1	56.25	56.25	0.018
ABCD	1	210.25	210.25	0.069
Error	8	24114	3014.25
Total	15	136.50

Example: 

Consider the table of contrasts from a  design. The four factors investigated are A, B, C, D.

a.      What are the values of k and p?

b.      What is the defining relation? What is the resolution of this design?

c.      What are the aliasing relations?

Solution:

a. There are 4 factors so, k = 4 and p = 1

.b. By inspection of the table we can see that D=ABC. The defining relation is I=ABCD.

 The main effects are aliased with three factors interactions so the resolution is IV.

c. The aliasing relations are:

D = ABC

I = ABCD

A = BCD

B = ACD

C = ABD

AB = CD

AC =BD

AD = BC

• Read More: Quarter Fraction of Factorial Experiment