Mathematics and Statistics –
Design and Analysis of Experiments
Week 8-10: Factorial Designs
Blocking-Confounding Fractional Factorial Designs
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Design of Engineering Experiments Blocking & Confounding in the 2k
• Text reference, Chapter 7
• Blocking is a technique for dealing with
controllable nuisance variables
• Two cases are considered – Replicated designs
– Unreplicated designs
Blocking a Replicated Design
• This is the same scenario discussed previously in Chapter 5
• If there are n replicates of the design, then each replicate is a block
• Each replicate is run in one of the blocks (time periods, batches of raw material, etc.)
• Runs within the block are randomized
Blocking a Replicated Design
Consider the example from Section 6-2; k = 2 factors, n = 3 replicates
This is the “usual” method for calculating a block sum of squares
3B2 y2 SSBlocks = i − … i=1 4 12
ANOVA for the Blocked Design Page 305
Confounding in Blocks
• Now consider the unreplicated case
• Clearly the previous discussion does not apply,
since there is only one replicate
• To illustrate, consider the situation of Example 6.2, the resin plant experiment
• This is a 24, n = 1 replicate
Experiment from Example 6.2
Suppose only 8 runs can be made from one batch of raw material
The Table of + & – Signs, Example 6-4
ABCD is Confounded with Blocks
(Page 310)
Observations in block 1 are reduced by 20 units…this is the simulated “block effect”
Effect Estimates
The ABCD interaction (or the block effect) is not considered as part of the error term
The reset of the analysis is unchanged from the original analysis
Another Illustration of the Importance of Blocking
Now the first eight runs (in run order) have filtration rate reduced by 20 units
The interpretation is harder; not as easy to identify the large effects
One important interaction is not identified (AD)
Failing to block when we should have causes problems in interpretation the result of an experiment and can mask the presence of real factor effects
Confounding in Blocks
• More than two blocks (page 313)
– The two-level factorial can be confounded in 2,
4, 8, … (2p, p > 1) blocks
– For four blocks, select two effects to confound,
automatically confounding a third effect
– See example, page 314
– Choice of confounding schemes non-trivial; see Table 7.9, page 316
• Partial confounding (page 316)
General Advice About Blocking
• When in doubt, block
• Block out the nuisance variables you know about, randomize as much as possible and rely on randomization to help balance out unknown nuisance effects
• Measure the nuisance factors you know about but can’t control (ANCOVA)
• It may be a good idea to conduct the experiment in blocks even if there isn’t an obvious nuisance factor, just to protect against the loss of data or situations where the complete experiment can’t be finished
Design of Engineering Experiments The 2k-p Fractional Factorial Design
• Text reference, Chapter 8
• Motivation for fractional factorials is obvious; as the number of factors becomes large enough to be “interesting”, the size of the designs grows very quickly
• Emphasis is on factor screening; efficiently identify the factors with large effects
• There may be many variables (often because we don’t know much about the system)
• Almost always run as unreplicated factorials, but often with center points
Why do Fractional Factorial Designs Work?
• The sparsity of effects principle
– There may be lots of factors, but few are important
– System is dominated by main effects, low-order interactions
• Theprojectionproperty
– Every fractional factorial contains full factorials in fewer
• Sequentialexperimentation
– Can add runs to a fractional factorial to resolve difficulties (or ambiguities) in interpretation
The One-Half Fraction of the 2k
• Section 8.2, page 321
• Notation: because the design has 2k/2 runs, it’s referred to as a 2k-1
• Consider a really simple case, the 23-1
• Note that I =ABC
The One-Half Fraction of the 23
For the principal fraction, notice that the contrast for estimating the main effect A is exactly the same as the contrast used for estimating the BC interaction.
This phenomena is called aliasing and it occurs in all fractional designs
Aliases can be found directly from the columns in the table of + and – signs
Aliasing in the One-Half Fraction of the 23 A = BC, B = AC, C = AB (or me = 2fi)
Aliases can be found from the defining relation I = ABC by multiplication:
AI = A(ABC) = A2BC = BC
BI =B(ABC) = AC
CI = C(ABC) = AB
Textbook notation for aliased effects:
[A]→A+BC, [B]→B+AC, [C]→C+AB
The Alternate Fraction of the 23-1
• I = -ABC is the defining relation
• Implies slightly different aliases: A = -BC, B= -AC, and C = -AB
• Both designs belong to the same family, defined by I = ABC
• Supposethatafterrunningtheprincipalfraction,the alternate fraction was also run
• The two groups of runs can be combined to form a full factorial – an example of sequential experimentation
Design Resolution
Design Resolution
• ResolutionIIIDesigns: – me = 2fi
• Resolution IV Designs:
– 2fi = 2fi 23−1 III
• ResolutionVDesigns:
– 2fi = 3fi – example
Construction of a One-half Fraction The basic design; the design generator
Projection of Fractional Factorials
Every fractional factorial contains full factorials in fewer factors
The “flashlight” analogy
A one-half fraction will project into a full factorial in any k – 1 of the original factors
Example 8.1
Example 8.1
Interpretation of results often relies on making some assumptions Ockham’s razor
Confirmation experiments can be important
Adding the alternate fraction – see page 322
The AC and AD interactions can be verified by inspection of the cube plot
Confirmation experiment for this example: see page 332
Use the model to predict the response at a test combination of interest in the design space – not one of the points in the current design.
Run this test combination – then compare predicted and observed.
For Example 8.1, consider the point +, +, -, +. The predicted response is
Actual response is 104.
Possible Strategies for Follow-Up Experimentatio n Following a Fractional Factorial Design
The One-Quarter Fraction of the 2k
The One-Quarter Fraction of the 26 (An Example of the 26-2 Design)
Complete defining relation: I = ABCE = BCDF = ADEF
Alternative One-Quarter Fraction of the 26 Another example of a 26-2 design
• Uses of the alternate fractions E=ABC, F=BCD
• Projection of the design into subsets of the original six variables
• Any subset of the original six variables that is not a word in the complete defining relation will result in a full factorial design
– Consider ABCD (full factorial)
– Consider ABCE (replicated half fraction) – Consider ABCF (full factorial)
The General 2k-p Fractional Factorial Design
• Section8.4,page340
• 2k-1 = one-half fraction, 2k-2 = one-quarter fraction,
2k-3 = one-eighth fraction, …, 2k-p = 1/ 2p fraction
• Add p columns to the basic design; select p
independent generators
• Important to select generators so as to maximize
resolution, see Table 8.14
• Projection – a design of resolution R contains full
factorials in any R – 1 of the factors
• Blocking
Table 8.14 (Page 342)
2k-p Fractional Factorial Designs
The General 2k-p Design: Resolution may not be Sufficient
Main effects aliased with the 2fis
Resolution III Designs: Section 8.5, page 351
• Designs with main effects aliased with two- factor interactions
• Used for screening (5 – 7 variables in 8 runs, 9 – 15 variables in 16 runs, for example)
• A saturated design has k = N – 1 variables
• See Table 8.19, page 351 for a
Resolution III Designs
A saturated design with n=8 runs (m=n-1=7 factors)
Resolution III Designs
• Sequential assembly of fractions to separate aliased effects (page 354)
• Switching the signs in one column provides estimates of that factor and all of its two-factor interactions
• Switching the signs in all columns dealiases all main effects from their two-factor interaction alias chains – called a full fold-over
• Defining relation for a fold-over (page 356)
• Be careful – these rules only work for Resolution III
• There are other rules for Resolution IV designs, and other methods for adding runs to fractions to dealias effects of interest
• Example 8.7, eye focus time, page 354
• These are a different class of resolution III design
• The number of runs, N, need only be a multiple of
• N = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, …
• The designs where N = 12, 20, 24, etc. are called nongeometric PB designs
• See text, page 357-358 for comments on construction of Plackett-Burman designs
Plackett- Example: The Plackett- for 11 factors
This is a nonregular design
Projection of the 12-run design into 3 and 4 factors
All PB designs have projectivity 3 (contrast with other resolution III fractions)
• The alias structure is complex in the PB designs
• For example, with N = 12 and k = 11, every main
effect is aliased with every 2FI not involving itself
• Every 2FI alias chain has 45 terms
• Partial aliasing can potentially greatly complicate interpretation if there are several large interactions
• Use carefully – but there are some excellent opportunities
Resolution IV and V Designs (Page 366)
A resolution IV design must have at least 2k runs. “optimal” designs may often prove useful.
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