Fractional Replication in Confounding Lecture - 36

 

Confounding

of 

Fractional Replication

Lecture - 36

Fraction Replication Factorial Experiment

In 2^k factorial experiment, when k is small it easy and convenient to use full factorial experiment. When is large, like k = 8, 9, …., the use full factorial experiment is tedious and time consuming.  Facing such situation, the technique of fractional factorial experiment use. The fraction factorial experiment is used to obtain to useful information using less replicate. The following equation is used to construct fractional replication of factorial experiment.

Where:

K is number of factors; m is the number of level and is the number of fractions.

In factorial experiment due to the following reasons:

i.                    In most of the situations interest lies in the lower order interactions and high order interactions are assumed to be negligible.

ii.                  The available resources are not sufficient for even a single replication of the complete factorial experiment.

In case, the higher-order interactions are not of much use or much importance, then they can possibly be ignored. The information on main and lower-order interaction effects can then be obtained by conducting a fraction of complete factorial experiments. Such experiments are called as fractional factorial experiments.

The utility of such experiments becomes more when the experimental process is more influenced and governed by the main and lower-order interaction effects rather than the higher-order interaction effects. The fractional factorial experiments need less number of plots and lesser experimental material than required in the complete factorial experiments. Hence it involves less cost, less manpower, less time etc.

To illustrate the procedure, consider 2^3 factorial experiment. In 2^3 factorial experiment, we can estimate main effects and interaction effects as contrasts.

Consider the 2^3 factorial experiment. The table of coefficients is:

Re arrange the rows by the ABC column (that’s divide into 2 factions)

Defining Contrast and Aliases Contrast

Defining Contrast

 A treatment which cannot be estimated at all from the selected treatment combination is called defining contrast.

In the above example ABC is a defining contrast. It will have all positive or all negative signs in 2^3   factorial experiment.

Aliases Contrast

The contrast that estimates A is the same as the contrast that estimate BC, such contrast is called aliases contrast. 

Generator

A defining contrast that uses for spotting a complete factorial experiment into two fractions is called generator denoted by I.

In the above example of  factorial experiment, the generator I = ABC and use this for spotting the full factorial experiment into two fractions.

I = ABC

The generator may be used to identify the aliases of different effects easily.

Consider 2^3 factorial experiment, the table of coefficients is given below:

I = ABC

I = + ABC

I = - ABC

Alias structure of  2^(4-1) with I = ABC

ANALYSIS (Design Construction)

We can use these concepts in the construction of a design containing 1 / 2   replication of  2^k factorial experiment. To do this we select a contrast usually high order interaction to serves as defining contrast. Then we use either those combinations which carry +ive sign or those combination which carry -  ive sign under the defining contrast.

For example; consider 2^k factorial experiment and chose ABCD as a defining contrast.

ABCD = (a - 1) (b - 1) (c - 1) (d - 1) 

ABCD = (a - 1) (b - 1) (cd - c - d + 1)

ABCD = abcd + ab + ac + ad + bc + bd + cd + 1 - (abd + abd + acd + a + bcd + b + c + d )

Then we could either the set

0r the set

To give us 1 / 2  replication of full set.

Proper use of factorial replication depends on choosing the defining contrast in such a way that:

·         No main effect or lower order interaction has another main effect or lower order interaction as an alias.

·         Some contrast and either alias are unimportant (negligible), so they can be used as error.

  Normally we want to estimate the main effects and first order interactions. If we assume that second order interactions and high order interaction are negligible. The implication is that six factors are the minimum number for which a  1 / 2  replication of a 2^k factorial is a useful design.

Let we want to run a 1 / 2 replication of a 2^k factorial experiment. We could choose ABCDEF is a defining contrast. This would give us the following alias pattern.

ANOVA Table:

SV	df
Main effect	6
Two factors interaction	15
Error	10
Total	31

Example:

Give your plan and ANOVA table of 2^4 factorial experiment with 4 replicate, when ABCD is confounded in all replications.

Solution:

1.  ABCD is confounded in all replications. The treatment combinations satisfy the following equation.

X1 + X2 + X3 + X4 = 0, 1 (mod 2)

 Block I consist the treatment combination for which

X1 + X2 + X3 + X4 = 0  (mod 2)

Block II consist the treatment combination for which

X1 + X2 + X3 + X4 = 1  (mod 2)

4 replications are given below:

ANOVA Table:

SV	df
Block	7
Main effects	4
Two factors interaction	6
Three factors interaction	4
Error	42
Total	63

2. Now using partial confounding technique

ABC is confounded with replication - I:

ABC = (a - 1) (b - 1) (c - 1) (d +1)

ABC = (a - 1)(b - 1) (cd + c - d - 1)

ABC = (a - 1) (bcd + bc - bd - b - cd - c + d +1) 

ABC = abcd +abc - abd - ab - acd - ac + ad + a - bcd - bc + bd + b + cd - c - d - 1

ABC = ( abcd + abc  + ad + a + bd + b + cd + c) - (abd + ab+ ac + acd + bcd + bc + d +1)

ABD confounded with replication – II:

ABD = (a - 1) (b - 1) (c + 1) (d - 1)

ABD = (a - 1) (b - 1) ( cd - c - d -1)

ABD = (a - 1) ( bcd - bc - bd - b - cd + c + d + 1)

ABD = abcd - abc - abd - ab - acd + ac + ad + a - bcd + bc + bd + b + cd - c - d -1

ABD = (abcd + abd + ac + a + bc + b + cd + d) - (abc + ab + acd + ad + bcd + bd + c + 1)

ACD confounded with replication – III:

ACD = (a -1) (b + 1) (c – 1) (d – 1)

ACD = (a -1) (b + 1) (cd – c – d + 1)

ACD = (a -1) ( bcd – bc + b + cd – c – d + 1)

ACD = abcd – abc – abd + ab + acd – ac – ad + a – bcd + bc + bd – b – cd + c + d – 1

ACD = (abcd + ab + acd + a + bc + bd + c + d) – (abc + abd + ac + ad + bcd + b + cd + 1)

BCD confounded with replication – IV:

BCD = (a + 1) (b – 1) (c – 1) (d – 1)

BCD = (a + 1) (b – 1) (cd – c – d + 1)

BCD = (a + 1) (bcd – bc – bd + b – cd + c + d – 1)

BCD = abcd – abc – abd + ab – acd + ac + ad – a + bcd – bc + b – cd + c + d - 1

BCD = (abcd + ab + ac + ad + bcd + b + c + d) – (abc + abd + acd + a + bc + bd + cd + 1)

Placed one fraction in block I of replication I and second fraction in block II of replication I:

ANOVA Table:

SV	df
Block	7
Main effects	4
Two factors interaction	6
ABC(II, III, IV)	1
ABD(I, III, IV)	1
ACD(I, II, IV)	1
BCD(I, II, III)	1
Three factors interactions	1
Error	41
Total	63

•  Read More: Half Fraction of Factorial Experiment