Q 546. Suppose you want to conduct an experiment with five factors, each with two levels. Will you prefer to do a Full Factorial or a Fractional Factorial design? Provide details how your choice will impact the time, resources, and complexity of the experiment?

 

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For 5 factors with 2 level in Full factorial need 32 runs . I would prefer full factorial if i need 3 factors interaction without any aliasing and if  i have no resource or raw material constraint. However in practical situations we have limited resource and we are interested till 2 factor interaction  (un-aliased) hence I will choose fractional factorial DoE with 16 trials where main factors aliased with four factor interaction , 2 factor interactions are aliased with 3 factor interactions so it will give good resolution even with fractional factorial DoE (considering 3 factor interaction are rarely exist in system).

Using fractional factorial experiments  will save time , manpower , resources, without much compromising resolution of design. 

It would depend on whether we are doing a “Screening Design” or an “Optimization Design.” 

If the criticality of the 5 factors is yet not established, then we would go ahead with a Resolution V fractional factorial design.  Out of the 5 factors, the factors with significant main effects can be further considered for a full factorial optimization design.

 

Half Factorial - Screening Design: If the significance of the given 5 factors is questionable and are not yet validated as critical Xs, we can use a resolution V design to screen out non-critical factors.  Half factorial experiments for screening are used primarily in two scenarios:

1)     The existing historical data is inconclusive.

2)     There is no historical data available at all.

 

Impact on Time, Resources, and Complexity:  A design summary of a fractional-factorial experiment with 5 factors and 2 levels with and without replication is shown below.

Without replication

With replication

A fractional factorial design even with replication would require 32 runs which is almost half of a full factorial.  Complexity also would be comparatively less as we would only be focusing primarily on the main effects and not the interactions.  Since the effort is less too, fewer resources would be deployed.

 

Full Factorial - Optimization Design – If all of these 5 factors are found to be critical, then we may want to optimize their behavior towards the response variable by conducting a full factorial experiment with replication.  The following is the design summary for same:

Impact on Time, Resources, and Complexity:  A design summary of a full factorial experiment with 5 factors and 2 levels with replication is shown below.

In comparison to a fractional factorial design, a full factorial experiment would be more time consuming as the number of runs would be more.  We would have to conduct 64 runs with replication for a full factorial experiment.  If any of these 5 factors warrant an addition of a centre point to rule out curvilinearity, then a few more runs would be added to it.  Complexity would increase as interaction effects are also to be studied along with main effects and would necessitate utilization of more resources due to a sizeable number of experiments.  We need to also factor in the Scope, Time, Cost and Resource constraints while conducting a full factorial experiment.  Most of the R&D departments with higher risk appetite usually proceed with full factorial as they have to always come up with a robust design.

 

 

When conducting an experiment with five factors, each with two levels, we have a total of 2^5 = 32 possible combinations of levels. A Full Factorial design would require testing all 32 combinations, while a Fractional Factorial design would only test a subset of these combinations.

If time and resources are limited, a Fractional Factorial design would be preferred over a Full Factorial design. A Fractional Factorial design would allow us to test the main effects of each factor and their interactions while reducing the number of experiments required. This can significantly decrease the time and resources needed for the experiment.

For example, a commonly used Fractional Factorial design for 5 factors with two levels each is the 2^5-2 design, also known as the "half-fraction" design. This design only requires testing 16 out of the 32 possible combinations, while still allowing us to estimate the main effects and two-way interactions of all five factors.

The complexity of the experiment would also be reduced with a Fractional Factorial design. With a Full Factorial design, it can be difficult to keep track of all the possible combinations and their results, making it harder to analyze the data. A Fractional Factorial design simplifies the experimental setup and analysis, making it easier to interpret the results.

Overall, if time and resources are limited, a Fractional Factorial design would be a more practical approach than a Full Factorial design for an experiment with five factors and two levels each.

Benchmark Six Sigma Expert View by Venugopal R

Fractional factorial designs help to significantly reduce the number of experimental runs without compromising on the decision making outcome.

 

For 5 factors, each at 2 levels, we will have to conduct 32 trials as a Full Factorial design. It is a recommended practice to perform a replication – hence the total number of experiments with replication will become 64.

However, if we look the above table, which gives the options for Factorial designs, for 5 factors we have an option of performing a ‘Resolution V’ design.

 

Resolution V designs are considered to be as good as ‘Full factorial’, since we will not be missing out on any main effect or two-factor interactions.

 

The number of experimental runs required for Resolution V is 16.

 

We may also perform replication, and then the number of runs will be 32. Hence, we need to conduct only half the number of runs as compared to Full Factorial.

 

By opting for the Fractional Factorial (Resolution V), we will have the following benefits.

1. Reduced number of runs – Lower effort, cost, time and resources
2. We can afford a replication for the same number of runs equal to a full factorial without replication
3. We will not lose any information on the most important outputs viz. Main Effects or Two Factor Interaction Effects.
4. For Full Factorial with 5 factors & 2 levels each, we will have:

         5 main effects

         10 two factor interactions

         10 three factor interactions

          5 four factor interactions

          1 five factor interactions

 

Total number of terms will be 31.

 

For Resolution V design, we will have to deal with:

          5 main effects

          10 two factor interactions

The higher order interactions will be aliased with the above terms and we will not be considering them.

 

Hence the total number of terms will be 15

 

Note: For a Resolution V design, the 2 Factor Interactions will be confounded with 3 Factor Interactions. Main effects will be confounded with 4-factor interactions. However, since interactions involvring 3 or more factors are very rare, we can safely consider these effects as due 2 Factor Interactions / main effects.

 

Thus, with lesser of terms in the ANOVA, the complexity of the analysis will also be less.

Design Of Experiment is a statistical tool first developed by Sir R A Fisher, Ronald Aylmer in 1920, while estimating the impact of different input variable, designated as Factors, on single output variable designated as Levels. Factorial design can be used to interpolate the main effects (effect of dependent variable on independent variable) and interaction effects (effect of interaction between dependent variables on the independent variable). As such, design of experiment is classified as:

Full Factorial Design of Experiment A full factorial design involves all possible factor combinations in a design of experiment, and, most importantly, varies the factors (input variables) simultaneously rather than one factor at a time.

A confounding variable is a variable, that has effect on supposed cause and the supposed effect. A confounding variable is corelated to independent variable, and shares causal relationship with dependent variable. Higher day Temperature may be considered as a confounding variable having association with ice cream eating tendency of people and sun cream usage by people.

Confounding effect refers to eliminating the block effect of treatment factor and considering treatment effects being contributed by estimation of main effects and interaction effects from linear combination of the experimental observations.

So, in a 2³-factorial design of experiment, the estimation for three-way interaction (ABC interaction) is eliminated, by confounding with the block. The confounding effect is generally observed, when full factorial designs are run in blocks and, the block size is smaller than the number of different treatment combinations.

Blocking in Design of Experiments refers to arranging data samples/units in groups or blocks that are similar to one another.

Replication in Design of Experiments refers to repeating experimental situation by replicating the experimental unit. Replication allows estimate of variance for each experimental unit. This permits experimenter to reduces variability in experimental results, increasing experimentation design significance and the confidence level of experimental output.

Trail design of experiment for “five factor into two level” Full Factorial Design of Experiment

 

 

 

Trail	Factor A	Factor B	Factor C	Factor D	Factor E	Notation
1	-	-	-	-	+	e
2	+	-	-	-	-	a
3	-	+	-	-	-	b
4	-	-	+	-	-	c
5	-	-	-	+	-	d
6	+	+	+	-	+	abce
7	-	+	+	+	+	bcde
8	+	+	-	+	+	abde
9	+	-	+	+	+	acde
10	+	+	+	+	-	abcd
11	+	+	-	-	-	ab
12	+	-	-	+	-	ad
13	+	-	+	-	-	ac
14	+	-	-	-	+	ae
15	-	+	+	-	-	bc
16	-	+	-	+	-	bd
17	-	+	-	-	+	be
18	-	-	+	+	-	cd
19	-	-	+	-	+	ce
20	-	-	-	+	+	de
21	+	+	+	-	-	abc
22	+	-	+	+	-	acd
23	+	-	+	-	+	ace
24	+	+	-	-	+	abe
25	+	+	-	+	-	abd
26	-	+	+	+	-	bcd
27	-	+	+	-	+	bce
28	-	+	-	+	+	bde
29	+	+	+	+	+	abcde
30	+	-	-	+	+	ade
31	-	-	+	+	+	cde
32	-	-	-	-	-	-

 

Trail	Factor A	Factor B	Factor C	Factor D	Factor E	Notation
1	-1	-1	-1	-1	1	e
2	1	-1	-1	-1	-1	a
3	-1	1	-1	-1	-1	b
4	-1	-1	1	-1	-1	c
5	-1	-1	-1	1	-1	d
6	1	1	1	-1	1	abce
7	-1	1	1	1	1	bcde
8	1	1	-1	1	1	abde
9	1	-1	1	1	1	acde
10	1	1	1	1	-1	abcd
11	1	1	-1	-1	-1	ab
12	1	-1	-1	1	-1	ad
13	1	-1	1	-1	-1	ac
14	1	-1	-1	-1	1	ae
15	-1	1	1	-1	-1	bc
16	-1	1	-1	1	-1	bd
17	-1	1	-1	-1	1	be
18	-1	-1	1	1	-1	cd
19	-1	-1	1	-1	1	ce
20	-1	-1	-1	1	1	de
21	1	1	1	-1	-1	abc
22	1	-1	1	1	-1	acd
23	1	-1	1	-1	1	ace
24	1	1	-1	-1	1	abe
25	1	1	-1	1	-1	abd
26	-1	1	1	1	-1	bcd
27	-1	1	1	-1	1	bce
28	-1	1	-1	1	1	bde
29	1	1	1	1	1	abcde
30	1	-1	-1	1	1	ade
31	-1	-1	1	1	1	cde
32	-1	-1	-1	-1	-1	-

  

Anova: Single Factor
SUMMARY
Groups	Count	Sum	Average	Variance
Factor A	32	0	0	1.032258
Factor B	32	0	0	1.032258
Factor C	32	0	0	1.032258
Factor D	32	0	0	1.032258
Factor E	32	0	0	1.032258
ANOVA
Source of Variation	SS	df	MS	F	P-value	F crit
Between Groups	0	4	0	0	1	2.430002
Within Groups	160	155	1.032258
Total	160	159

 

We replace trail data observation with “-“ sign replaced by “-1” and “+” sign by “+1” for better representation of presence of factor in design of experiment. Further to reach four blocks of data arrangement from 32 data observation points, confounding is shared for two higher effects ADE and BCE. After confounding, the data is categorized into four blocks, as per following scheme:

ADE	BCE	Block
-1	-1	1
1	-1	2
-1	1	3
1	1	4

 

The net impact of above changes is represented as follows:

 

Trail	Factor A	Factor B	Factor C	Factor D	Factor E	Notation	ADE	BCE	Block
1	-1	-1	-1	-1	1	e	1	1	4
2	1	-1	-1	-1	-1	a	1	-1	2
3	-1	1	-1	-1	-1	b	-1	1	3
4	-1	-1	1	-1	-1	c	-1	1	3
5	-1	-1	-1	1	-1	D	1	-1	2
6	1	1	1	-1	1	abce	-1	1	3
7	-1	1	1	1	1	bcde	-1	1	3
8	1	1	-1	1	1	abde	1	-1	2
9	1	-1	1	1	1	acde	1	-1	2
10	1	1	1	1	-1	abcd	-1	-1	2
11	1	1	-1	-1	-1	ab	1	1	4
12	1	-1	-1	1	-1	ad	-1	-1	1
13	1	-1	1	-1	-1	ac	1	1	4
14	1	-1	-1	-1	1	ae	-1	1	3
15	-1	1	1	-1	-1	bc	-1	-1	1
16	-1	1	-1	1	-1	bd	1	1	4
17	-1	1	-1	-1	1	be	1	-1	2
18	-1	-1	1	1	-1	cd	1	1	4
19	-1	-1	1	-1	1	ce	1	-1	2
20	-1	-1	-1	1	1	de	-1	1	3
21	1	1	1	-1	-1	abc	1	-1	2
22	1	-1	1	1	-1	acd	-1	1	3
23	1	-1	1	-1	1	ace	-1	-1	1
24	1	1	-1	-1	1	abe	-1	-1	1
25	1	1	-1	1	-1	abd	-1	1	3
26	-1	1	1	1	-1	bcd	1	-1	2
27	-1	1	1	-1	1	bce	1	1	4
28	-1	1	-1	1	1	bde	-1	-1	1
29	1	1	1	1	1	abcde	1	1	4
30	1	-1	-1	1	1	ade	1	1	4
31	-1	-1	1	1	1	cde	-1	-1	1
32	-1	-1	-1	-1	-1	-1	-1	-1	1

 

 

The above design is said to have Level V resolution, since generator term ABCDE is said to have five alphabets

A typical example of full factorial design of experiment is 2^k design, where 2 represents the levels or output independent variable and k represents factor or input dependent variable effect on 2 defined output conditions.

Total Run Count utilizing 2⁵ design = 5+10+10+5+1 =31+one minus term = 32 runs

Fractional Factorial Design of Experiment To overcome larger size of the design arising from Full Factorial run, fractional factorial design of experiment, is implemented in an experiment by researchers. Fractional Factorial design provide an alternative approach to screening of design of experiment, with a lower number of run count tested, for arriving at correct design of experiment by researcher. 

Fractional Factorial Design of Experiment refers to situation, when experimenter implements selected subset or "fraction" of the total runs, estimated for the full factorial design. The Factional Factorial design is implemented to generate confounding between the main effects and 2-way interactions. As result of confounding generated from limited runs of experiment, effects of other higher order interaction, cannot be studied independently and assumed to be negligible. This facilitates early estimation of main effects and interaction effects for the researcher.

The Fractional Factorial design of experiment are generalized by term l^k − p

Where, là is the number of levels in each treatment factor.

kà is the number of treatment factors.

pà is the number of interactions that are confounded.

The fraction of trials required is generalized using term 1/(l^p).

The effective HALF Fractional Factorial design of experiment for five factorial two level doe would have 2^5-1 = 16 runs.

The effective Quarter Fractional Factorial design of experiment for five factorial two level doe would have 2^5-2 = 8 runs.

HALF Fractional Factorial design of experiment for five factorial two level DOE

 

Treatment_Combination	I	A	B	C	D	E	ABCDE
A	+	1	-1	-1	-1	-1	1
B	+	-1	1	-1	-1	-1	1
C	+	-1	-1	1	-1	-1	1
D	+	-1	-1	-1	1	-1	1
E	+	-1	-1	-1	-1	1	1
abcde	+	1	1	1	1	1	1
AB	+	1	1	-1	-1	-1	-1
AC	+	1	-1	1	-1	-1	-1
AD	+	1	-1	-1	1	-1	-1
AE	+	1	-1	-1	-1	1	-1
BC	+	-1	1	1	-1	-1	-1
BD	+	-1	1	-1	1	-1	-1
BE	+	-1	1	-1	-1	1	-1
CD	+	-1	-1	1	1	-1	-1
CE	+	-1	-1	1	-1	1	-1
DE	+	-1	-1	-1	1	1	-1

 

Design Generators: E = ABCD, E = ABCD gives us the basis for the resolution of the design as V degree resolution.

Alias Structure

I + ABCDE, A + BCDE, B + ACDE, C + ABDE,  D + ABCE, E + ABCD

AB + CDE, AC + BDE, AD + BCE, AE + BCD, BC + ADE, BD + ACE, BE + ACD, CD + ABE, CE + ABD, DE + ABC

 

 

 

A 16-run 2⁵ Half Fractional factorial design can conveniently rewritten as:

Trail Run	Factor A	Factor B	Factor C	Factor D	Factor E=ABCD	Treatment Combination
1	-1	-1	-1	-1	1	e
2	1	-1	-1	-1	-1	a
3	-1	1	-1	-1	-1	b
4	1	1	-1	-1	1	abe
5	-1	-1	1	-1	-1	c
6	1	-1	1	-1	1	ace
7	-1	1	1	-1	1	bce
8	1	1	1	-1	-1	abc
9	-1	-1	-1	1	-1	d
10	1	-1	-1	1	1	ade
11	-1	1	-1	1	1	bde
12	1	1	-1	1	-1	abd
13	-1	-1	1	1	1	cde
14	1	-1	1	1	-1	acd
15	-1	1	1	1	-1	bcd
16	1	1	1	1	1	abcde

 

Anova: Single Factor
SUMMARY
Groups	Count	Sum	Average	Variance
Factor A	16	0	0	1.066667
Factor B	16	0	0	1.066667
Factor C	16	0	0	1.066667
Factor D	16	0	0	1.066667
Factor E=ABCD	16	0	0	1.066667
ANOVA
Source of Variation	SS	df	MS	F	P-value	F crit
Between Groups	0	4	0	0	1	2.493696
Within Groups	80	75	1.066667
Total	80	79

 

 

Pradeep is the winner for this answer. Well done!

 

Do review the answer by Mr Venugopal R, Benchmark Six Sigma's in-house expert