Chapter 8

(AST301) Design and Analysis of Experiments II

Author

Md Rasel Biswas

8 Two-Level Fractional Factorial Designs

8.1 Introduction

As the number of factors in a \(2^{{k}}\) factorial design increases, the number of runs required for a complete replicate of the design rapidly outgrows the resources of most experimenters.
For example, a complete replicate of the \(2^6\) design requires 64 runs. In this design only 6 of the 63 degrees of freedom correspond to main effects, and only 15 degrees of freedom correspond to two-factor interactions.
There are only 21 degrees of freedom associated with effects that are likely to be of major interest. The remaining 42 degrees of freedom are associated with three-factor and higher interactions.

If the experimenter can reasonably assume that certain high-order interactions are negligible, information on the main effects and low-order interactions may be obtained by running only a fraction of the complete factorial experiment.
A major use of fractional factorials is in screening experiments — experiments in which many factors are considered and the objective is to identify those factors (if any) that have large effects.
The factors identified as important are then investigated more thoroughly in subsequent experiments.

The successful use of fractional factorial designs is based on three key ideas:

The sparsity of effects principle: when there are several variables, the system will be driven primarily by some of the main effects and low order interactions
The projection property: Fractional factorial designs can be projected into larger designs in the subset of significant factors
Sequential experimentation: It is possible to combine the runs of two or more fractional factorials to assemble sequentially a larger design to estimate the factor effects and interactions of interest

8.2 The one-half fraction of the \(2^k\) design

Consider a situation in which three factors, each at two levels, are of interest, but the experimenters cannot afford to run all \(2^3=8\) treatment combinations.
They can, however, afford four runs.
This suggests a one-half fraction of a \(2^3\) design.
Because the design contains \(2^{3-1}=4\) treatment combinations, a one-half fraction of the \(2^3\) design is often called a \(2^{3-1}\) design.

The four treatment combinations can be selected according to the plus sign of a factorial effect, which is known as a generator or a word.
Thus, if \({ABC}\) is the generator of the \(2^{3-1}\) design, then we select the four treatment combinations \(a, b, c,\) and \(abc\) as our one-half fraction.
We call \({I}={ABC}\) as the defining relation for the design.
In general, defining relation will always be the set of all columns that are equal to the identity column \(I\).

The estimates of the main effects of \(A, B\), and \(C\) are \[ \begin{aligned} & {[A]=\frac{1}{2}(a-b-c+a b c)} \\ & {[B]=\frac{1}{2}(-a+b-c+a b c)} \\ & {[C]=\frac{1}{2}(-a-b+c+a b c)} \end{aligned} \]
The estimates of the interactions \({AB}, {BC}\), and \({AC}\) are \[ \begin{aligned} & {[B C]=\frac{1}{2}(a-b-c+a b c)} \\ & {[A C]=\frac{1}{2}(-a+b-c+a b c)} \\ & {[A B]=\frac{1}{2}(-a-b+c+a b c)} \end{aligned} \]

Thus, \([{A}]=[{BC}],[{B}]=[{AC}]\), and \([{C}]=[{AB}]\) consequently, it is impossible to differentiate between \({A}\) and \({BC}, {B}\) and \({AC}\), and \({C}\) and \({AB}\).
In fact, when we estimate \({A}, {B}\), and \({C}\) we are really estimating \({A}+{BC}, {B}+{AC}\), and \({C}+{AB}\). Two or more effects that have this property are called aliases.
In our example, \({A}\) and \({BC}\) are aliases, \({B}\) and \({AC}\) are aliases, and \({C}\) and \({AB}\) are aliases.

We indicate this by the notation \([{A}] \rightarrow {A}+{BC},[{B}]\) \(\rightarrow {B}+{AC}\), and \([{C}] \rightarrow {C}+{AB}\).
The alias structure for this design may be easily determined by using the defining relation \({I}={ABC}\).
Multiplying any column (or effect) by the defining relation yields the aliases for that column (or effect).
In our example, this yields that the alias of \(A\) is \(BC\). \[{A} . {I}={A} . {ABC}={A}^2 {BC} = BC\]
Similarly, we find that (B and AC) and (C and AB) are aliases

This one-half fraction, with \({I}=+{ABC}\), is usually called the principal fraction.
Now suppose that we had chosen the other one-half fraction, that is, the treatment combinations associated with minus in the \({ABC}\) column, known as the complementary or alternate fraction.
The defining relation for this design is \({I}=-{ABC}\).
The alternate fraction gives us \[ \begin{aligned} {[A]^{\prime} } & \rightarrow A-B C \\ {[B]^{\prime} } & \rightarrow B-A C \\ {[C]^{\prime} } & \rightarrow C-A B \end{aligned} \]

Thus, when we estimate \({A}, {B}\), and \({C}\) with this particular fraction, we are really estimating \({A}-{BC}\), \({B}-{AC}\), and \({C}-{AB}\).

In practice, it does not matter which fraction is actually used. The two one-half fractions form a complete \(2^3\) design.

Suppose that after running one of the one-half fractions of the \(2^3\) design, the other fraction was also run. Thus, all eight runs associated with the full \(2^3\) are now available.

We may now obtain de-aliased estimates of all the effects by analyzing the eight runs as a full \(2^3\) design in two blocks of four runs each.
This could also be done by adding and subtracting the linear combination of effects from the two individual fractions. \[ \begin{aligned} & \frac{1}{2}\left([A]+[A]^{\prime}\right)=\frac{1}{2}(A+B C+A-B C) \rightarrow A \\ & \frac{1}{2}\left([A]-[A]^{\prime}\right)=\frac{1}{2}(A+B C-A+B C) \rightarrow B C \end{aligned} \]

Thus, for all three pairs of linear combinations, we would obtain the following:

Furthermore, by assembling the full \(2^3\) in this fashion with \({I}=+{ABC}\) in the first group of runs and \({I}=-{ABC}\) in the second, the \(2^3\) confounds \({ABC}\) with blocks

Design resolution

The preceding \(2^3−1\) design is called a resolution III design
In such a design, main effects are aliased with two-factor interactions.

A design is of resolution \(R\) if no \(p\)-factor effect is aliased with another effect containing less than \(R-p\) factors

We usually employ a Roman numeral subscript to denote design resolution;
Thus, the one-half fraction of the \(2^3\) design with the defining relation \(I = ABC\) (or \(I = −ABC\)) is a \(2^{3-1}_{III}\) design.

Design resolutions describe how much the effects in a fractional factorial design are aliased with other effects.
When we do a fractional factorial design, one or more of the effects are confounded, meaning they cannot be estimated separately from each other.
Usually, we want to use a fractional factorial design with the highest possible resolution.

Resolution III designs.

These are designs in which no main effects are aliased with any other main effect, but main effects are aliased with two-factor interactions and some two-factor interactions may be aliased with each other.
The \(2^{3−1}\) design in Table 8.1 is of resolution III (\(2^{3-1}_{III}\))

Resolution IV design.

These are designs in which no main effect is aliased with any other main effect or with any two-factor interaction, but two-factor interactions are aliased with each other.
A \(2^{4−1}\) design with \(I = ABCD\) is a resolution \(IV\) design (\(2^{4-1}_{IV}\))

Resolution V design.

These are designs in which no main effect or two-factor interaction is aliased with any other main effect or two-factor interaction, but two-factor interactions are aliased with three-factor interactions.
A \(2^{5-1}\) design with \(I = ABCDE\) is a resolution \(V\) design (\(2^{5-1}_{V}\))

In general, the resolution of a two-level fractional factorial design is equal to the shortest number of letters in any word in the defining relation

Higher the resolution better the design (why?)

We never want a resolution II design, because a design would alias two main effects. Thus minimum acceptable resolution in III.
Resolution III designs have some main effects aliased to two-factor interactions. If we believe that only main effects are present and all interactions are negligible, then a resolution III design is sufficient for estimating main effects.
Resolution III designs are called main effects design for this reason.
If we believe that some two factor interactions may be non negligible but all three way and higher interactions are negligible then a resolution IV is sufficient for main effects.
Low resolution fractional factorials are often used as screening designs, where we are trying to screen many factors to see if any of them has an effect. This is usually an early stage of investigation.

Construction and Analysis of the One-Half Fraction

A one-half fraction of the \(2^k\) design of the highest resolution may be constructed by

writing down a basic design consisting of the runs for a full \(2^{k-1}\) factorial and
then adding the the \(k^\text{th}\) factor by identifying its plus and minus levels with the plus and minus signs of the highest order interaction \(ABC\cdots(K-1)\)

Constructing one-half fractions

E.g. the \(2^{3-1}_{III}\) fractional factorial design of the highest resolution can be obtained by writing down the full \(2^2\) factorial as the basic design and then equating the factor \(C\) to \(AB\)

Note that any interaction effect could be used to generate the column for the \(k\)th factor.
However, using any effect other than \({ABC}\)….(K-1) will not produce a design of the highest possible resolution.
Another way to view the construction of a one-half fraction is to partition the runs into two blocks with the highest order interaction \({ABC} \ldots {K}\) confounded. Each block is a \(2^{{k}-1}\) fractional factorial design of the highest resolution.

Sequences of fractional factorials

Fractional factorial designs could be less expensive and efficient in experimentation, if the runs can be made sequentially
If the interest is in investigating \(k=4\) factors (i.e. \(2^4=16\) runs) then it is preferable to run a \(2^{4-1}_{IV}\) fractional design, analyze the results, and then decide on the best set of runs to perform the next.
If necessary we can always run the alternate fraction and get the complete \(2^4\) design, where both half-fractions are considered as blocks

Projection of Fractions into Factorials

Any fractional factorial design of resolution \(R\) contains complete factorial designs (possibly replicated factorials) in any subset of \(R − 1\) factors.

This is an important and useful concept
Because the maximum possible resolution of a one-half fraction of the \(2^k\) design is \(R = k\), every \(2^{k−1}\) design will project into a full factorial in any (\(k − 1\)) of the original \(k\) factors.
Furthermore, a \(2^{k−1}\) design may be projected into two replicates of a full factorial in any subset of \(k − 2\) factors, four replicates of a full factorial in any subset of \(k − 3\) factors, and so on.

EXAMPLE \(8.1\)

Consider the filtration rate experiment in Example 6.2. The original design is a single replicate of the \(2^4\) design.
In that example, we found that the main effects \({A}\), \({C}\), and \({D}\) and the interactions \({AC}\) and \({AD}\) were different from zero.
We will now return to this experiment and simulate what would have happened if a half-fraction of the \(2^4\) design had been run instead of the full factorial.

We will use the \(2^{4-1}\) design with \(I=A B C D\).

Using the defining relation, we note that each main effect is aliased with a three-factor interaction; that is, \({A}={A}^2 {BCD}={BCD}, {B}={AB}^2 {CD}={ACD}, {C}={ABC}^2 {D}=\) \({ABD}\), and \({D}={ABCD}^2={ABC}\).
Furthermore, every two-factor interaction is aliased with another two-factor interaction. These alias relationships are \({AB}={CD}, {AC}={BD}\), and \({BC}={AD}\).
The four main effects plus the three two-factor interaction alias pairs account for the seven degrees of freedom for the design.

\(\begin{aligned} {[A]=} & \frac{1}{4}(-45+100-45+65-75 \\ & \quad+60-80+96)=19.00 \rightarrow A+B C D \\ {[A B]=} & \frac{1}{4}(45-100-45+65+75-60-80+96) \\ = & -1.00 \rightarrow A B+C D\end{aligned}\)

Because factor B is not significant, we may drop it from consideration. Consequently, we may project this design into a single replicate of the \(2^3\) design in factors \({A}, {C}\), and \({D}\).
Based on the above analysis, we can now obtain a model to predict filtration rate over the experimental region. This model is

\(\begin{aligned} \hat{y}= & \hat{\beta}_0+\hat{\beta}_1 x_1+\hat{\beta}_3 x_3+\hat{\beta}_4 x_4+\hat{\beta}_{13} x_1 x_3+\hat{\beta}_{14} x_1 x_4 \\ \hat{y}= & 70.75+\left(\frac{19.00}{2}\right) x_1+\left(\frac{14.00}{2}\right) x_3+\left(\frac{16.50}{2}\right) x_4 \\ & +\left(\frac{-18.50}{2}\right) x_1 x_3+\left(\frac{19.00}{2}\right) x_1 x_4\end{aligned}\)

There are two large effects associated with two-factor interactions, \({AC}+{BD}\) and \({AD}+{BC}\).
In Example \(8.2\), we used the fact that the main effect of \(B\) was negligible to tentatively conclude that the important interactions were \({AC}\) and \({AD}\).
However, we can always isolate the significant interaction by running the alternate fraction, given by \({I}=-{ABCD}\).

8.3 The one-quarter fraction of the \(2^k\) design

One-quarter fraction of the \(2^k\) design could be useful when the number of factors is large.
This design contains \(2^{k-2}\) runs and is usually called a \(2^{k-2}\) fractional factorial.
The construction of the \(2^{k-2}\) fractional factorial design requires to write down the basic design with \(k-2\) factors first and then two additional columns are constructed from appropriately chosen interactions involving the first \(k-2\) factors.

The one-quarter fraction of the \(2^k\) design has two generators, if \(P\) and \(Q\) are the two generators then their generalized interaction \(PQ\) also acts as a generator
Complete defining relation is \(I=P=Q=PQ\)
As an example, consider \(2^{6-2}\) design with \(I=ABCE\) and \(I=BCDF\) as the design generators
- The complete defining relation for this design is \(I=ABCE=BCDF=ADEF (ABCE \times BCDF)\)
- This \(2^{6-2}\) design is a resolution \(IV\) design, why?

For the \(2^{6-2}\) design with \(I=ABCE=BCDF=ADEF\), the main effects are aliased with two- and five-factor interactions, e.g.
\[\begin{align*} A(I) &= A(ABCE) = A(BCDF) = A(ADEF) \\ A &= BCE = ABCDF = DEF \end{align*}\]
When we estimate the main effect of \(A\), we actually estimate \(A + BCE + DEF + ABCDF\)
Similarly, when we estimate 2-factor interaction \(AB\), we actually estimate \(AB + CE + ACDF + BDEF\)

The complete alias structure of this design is

To construct the design, first write down the basic design, which consists of the 16 runs for a full \(2^{6-2}\) \(=2^4\) design in \({A}, {B}, {C}\), and \({D}\).
Then the two factors \(E\) and \(F\) are added by associating their plus and minus levels with the plus and minus signs of the interactions \(A B C\) and \({BCD}\), respectively. This procedure is shown below

Another way to construct this design is to derive the four blocks of the \(2^6\) design with \({ABCE}\) and \({BCDF}\) confounded and then choose the block with treatment combinations that are positive on \({ABCE}\) and \(BCDF\).
This would be a \(2^{6-2}\) fractional factorial with generating relations \({I}={ABCE}\) and \({I}={BCDF}\), and because both generators \({ABCE}\) and \({BCDF}\) are positive, this is the principal fraction.

For this \(2^{6-2}\) design, there are three alternate fractions, which are \[\begin{align*} I&=ABCE, \;\; I=-BCDF;\\ I&=-ABCE, \;\; I=BCDF;\quad \text{and} \\ I&=-ABCE, \;\; I=-BCDF \end{align*}\]

EXAMPLE \(8.4\)

A quality improvement team has decided to use a designed experiment to study the injection molding process so that shrinkage can be reduced. The team decides to investigate six factors Mold temperature (A), Screw speed (B), Holding time (C), Cycle time (D), Gate size (E), and holding pressure (F) - each at two levels.

The objective of this experiment is of learning how each factor affects shrinkage and also, something about how the factors interact. The team decides to use the 16-run two-level fractional factorial design.

The only large effects are A (mold temperature), B (screw speed), and the AB interaction

The plot of the AB interaction shows that the process is very insensitive to temperature if the screw speed is at the low level but very sensitive to temperature if the screw speed is at the high level.
Based on this initial analysis, the team decides to set both the mold temperature and the screw speed at the low level

8.4 The general \(2^{k-p}\) fractional factorial design

A \(2^k\) fractional design containing \(2^{k-p}\) runs is called \(1/2^p\) fraction of the \(2^k\) design or \(2^{k-p}\) fractional factorial design
These designs require the selection of \(p\) independent generators and the corresponding defining relation consists of the initially chosen \(p\) generators and their \(2^p-p-1\) generalized interactions
The alias structure can be found by multiplying each effect column by the defining relation and generators should be chosen carefully so that effects of potential interest are not aliased with each other, and each effect will have \(2^p-1\) aliases
For moderately large values of \({k}\), we usually assume higher order interactions (say, third- or fourth-order and higher) to be negligible, and this greatly simplifies the alias structure.

It is important to select the \(p\) generators for a \(2^{k-p}\) fractional factorial design in a way that we obtain the best possible alias relationships
A reasonable criterion is to select the generators such that the resulting \(2^{k-p}\) design has the highest possible resolution, e.g. consider a \(2^{6-2}\) design

Defining relation \(I=ABCE=ABCDF \rightarrow\) Resolution III

Defining relation \(I=ABCE=BCDF \rightarrow\) Resolution IV

Sometimes resolution alone is insufficient to distinguish between designs. For example, consider the three \(2{ }_{{IV}}{ }^{7-2}\) designs, which are based on different defining relations, but have the equal resolution \(IV\).
Sometimes resolution alone is insufficient to distinguish between designs. consider the three \(2{ }_{{IV}}{ }^{7-2}\) designs given below

All of these designs are of resolution IV, but they have rather different alias structures (we have assumed that three-factor and higher interactions are negligible) with respect to the two-factor interactions.
Clearly, design A has more extensive aliasing and design \({C}\) the least, so design \({C}\) would be the best choice for a \(2_{{IV}}{ }^{7-2}\).
The three word lengths in design A are all 4; that is, the word length pattern is \(\{4,4,4\}\).

For design \(B\) it is \(\{4,4,6\}\), and for design \(C\) it is \(\{4,5,5\}\)
Notice that the defining relation for design \({C}\) has only one four-letter word, whereas the other designs have two or three.
Thus, design \({C}\) minimizes the number of words in the defining relation that are of minimum length. We call such a design a minimum aberration design.

Minimizing aberration in a design of resolution \({R}\) ensures that the design has the minimum number of main effects aliased with interactions of order \(R-1\), the minimum number of two-factor interactions aliased with interactions of order \(R-2\), and so forth.

Table \(8.14\) presents a selection of \(2^{{k}-{p}}\) fractional factorial designs for \({k} \leq 15\) factors and up to \({n} \leq 128\) runs.

The suggested generators in this table will result in a design of the highest possible resolution. These are also the minimum aberration designs.

The alias relationships for all of the designs in Table \(8.14\) for which \({n} \leq 64\) are given in Appendix Table X(a-w).

The alias relationships presented in this table focus on main effects and two- and three-factor interactions. The complete defining relation is given for each design.

This appendix table makes it very easy to select a design of sufficient resolution to ensure that any interactions of potential interest can be estimated.

Analysis of \(2^{k-p}\) fractional factorials

The \(i^\text{th}\) effect is estimated by \[\begin{align*} l_i &= \frac{\text{contrast}_i}{(N/2)}, \end{align*}\] where \(\text{contrast}_i\) can be found from the plus and minus sign of the column \(i\) and \(N=2^{k-p}\) is the total number of observations
The \(2^{k-p}\) design allows only \(2^{k-p}-1\) effects (and their aliases) to be estimated

Exercise 8.12.

An article in Industrial and Engineering Chemistry (“More on Planning Experiments to Increase Research Efficiency,” 1970, pp. 60-65) uses a \(2^{5-2}\) design to investigate the effect of \({A}=\) condensation temperature, \({B}=\) amount of material \(1, {C}=\) solvent volume, \(D=\) condensation time, and \(E=\) amount of material 2 on yield. The results obtained are as Follows:

\[ \begin{aligned} & {e}=23.2 \quad {ad}=16.9 \quad {~cd}=23.8 \quad {bde}=16.8 \\ & {ab}=15.5 \quad {bc}=16.2 \quad {ace}=23.4 \quad \text { abcde }=18.1 \end{aligned} \]

Verify that the design generators used were I \(={ACE}\) and \({I}={BDE}\).
Write down the complete defining relation and the aliases for this design.
Estimate the main effects.

Prepare an analysis of variance table. Verify that the AB and AD interactions are available to use as error.
Plot the residuals versus the fitted values. Comment on the results.

Blocking Fractional Factorials

Occasionally, a fractional factorial design requires so many runs that all of them cannot be made under homogeneous conditions.
In these situations, fractional factorials may be confounded in blocks.
Appendix Table X contains recommended blocking arrangements for many of the fractional factorial designs in Table 8.14. The minimum block size for these designs is eight runs

In assigning fractional factorials into blocks, we need to be careful about the effects to be confounded with blocks
The \(2^{6-2}_{IV}\) design with \(I=ABCE=BCDF=ADEF\) is \[\begin{align*} (1), df, cef, bef, a, abef, acf, bc, \\ adef, bde, cde, abce, abd, acd, bcdf, abcdef, \end{align*}\]
This fractional design contains 16 treatment combinations.
We want to conduct the experiment into two blocks. Which effect should we select to be confounded with blocks?

In selecting an interaction to confound with blocks, we note from examining the alias structure in Appendix Table IX(f) that there are two alias sets involving only three-factor interactions.

The table suggests selecting ABD (and its aliases) to be confounded with blocks.

Assume the effect \(ABD\) confounded with blocks, so the defining contrast \(L=x_1 + x_2 + x_4=0 \;\;(\text{mod} 2)\)

\[\begin{align*} (1), df, cef, bef, a, abef, acf, bc, \\ adef, bde, cde, abce, abd, acd, bcdf, abcdef, \end{align*}\]

Block-1 \((L=0)\) \[\begin{align*} (1), abef, adef, bde \\ abce, acd, bcdf, abcdef \end{align*}\]

Block-2 \((L=1)\) \[\begin{align*} df, cef, bef, a, acf, \\ bc, cde, abd \end{align*}\]

Notice that the principal block contains those treatment combinations that have an even number of letters in common with ABD.

Example 8.6

A five-axis CNC machine is used to produce an impeller for a jet turbine engine. The blade profiles are an important quality characteristic.

Specifically, the deviation of the blade profile from the profile specified on the engineering drawing is of interest.

An experiment is run to determine which machine parameters affect profile deviation.

The eight factors selected for the design are as follows: \[ \begin{aligned} & A=\text { x-Axis shift} \\ & B=\text { y-Axis shift } \\ & C=\text { z-Axis shift } \\ & D=\text { Tool supplier } \\ & E=\text { a-Axis shift } \\ & F=\text { Spindle speed }(\%) \\ & G=\text { Fixture height } \\ & H=\text { Feed rate }(\%) \end{aligned} \]

The profile deviation is measured using a coordinate measuring machine, and the standard deviation of the difference between the actual profile and the specified profile is used as the response variable.

The machine has four spindles. Because there may be differences in the spindles, the process engineers feel that the spindles should be treated as blocks.

The engineers feel confident that three-factor and higher interactions are not too important, but they are reluctant to ignore the two-factor interactions.

From Table 8.14, two designs initially appear appropriate: the \(2_{{IV}}^{8-4}\) design with 16 runs and the \(2_{{IV}}^{8-3}\) design with 32 runs. The experimenters decide to use the \(2_{{IV}}^{8-3}\) design in four blocks.

Because the response variable is a standard deviation, it is often best to perform the analysis following a log transformation.

Suppose that process knowledge suggests that the appropriate interaction is likely to be AD.

Following table shows the resulting analysis of variance for the model with factors A, B, D, and AD (factor D was included to preserve the hierarchy principle).

Notice that the block effect is small, suggesting that the machine spindles are not very different.

Normal probability plot of the residuals is suggestive of slightly heavier than normal tails, so possibly other transformations should be considered.

The \({AD}\) interaction plot shows that running \({A}\) at the low level ( 0 offset) and buying tools from supplier 1 gives the best results.

The projection of this design into four replicates of a \(2^3\) design in factors \(A, B\), and \(D\) is shown below.

The figure indicates that the best combination of operating conditions is A at the low level (0 offset), B at the high level (0.015 in offset), and D at the low level (tool supplier 1).

Exercise 8.45

Consider the design:

What is the generator for column E?
If \({ABC}\) is confounded with blocks, run 1 above goes in the block —-. Answer either \(+\) or \(-\).
What is the resolution of this design?
Find the estimates of the main effects.