VII. Introduction to 2k Factorial Based Experiments

A. An Overview of Experimental Design

In my experience, the absolute best way to teach the text's material through the section on the 2k full factorial designs is through simple, hands on experiments planned, conducted, and analyzed in class.

Two good experiments to conduct in class are

• the catapult, and

• the paper helicopter.

A friend of mine has conducted experiments building cars out of legos, which also works.

There are other examples which will work.

The key is to make this material real to the student.

Conducting an experiment that they plan, execute, and analyze together is very important.

Start the process as a 22 factorial.

Have the students determine the two factors and their specific levels.

In the course of planning the experiment discuss the concepts:

• experimental unit and experimental error,

• observational unit and observational error, (Be sure to explain the difference between the two!)

• randomization,

(Emphasize that randomization does not eliminate systematic bias; rather, it fairly distributes any systematic bias over the entire experiment.)

• replication, and

(Emphasize that replication allows the estimation of experimental error, which is the basis for formal statistical tests.)

• local control of error.

Discuss all of these concepts within the specific context of your experiment.

B. The 22 Factorial

After you conduct the experiment, set up the appropriate analysis.

Some key points are:

1. The concept of the “design variable”.

Let x1 be the design variable for Factor A; thus,

In a similar manner, let x2 be the design variable associated with Factor B; thus,

level high itsat isA Factor if 1

level low itsat isA Factor if 1-1

x

level high itsat is BFactor if 1

level low itsat is BFactor if 1-2

x

2. The basic structure of the 22 design

The basic 22 factorial in design variables is simply all of the possible combinations of the two levels and is given by

x1 x2

-1 -1 1 -1 -1 1 1 1

This design forms a square in terms of the design variables.

3. The model

where

• yi is the ith response,

• β0 is the y-intercept, , which in this case is the overall mean response,

• β1 is the regression coefficient associated with x1,

• β2 is the regression coefficient associated with x2

• β12 is the interaction coefficient, and

• εi is a random error

The interaction term allows the model to adjust the impact of the Factor A for the specific level used of Factor B, and vice versa.

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4. The concept of “effects”

The main effect due to Factor A is simply

average response for the - average response for the high level of Factor A low level of Factor A

Similarly, the main effect due to Factor B is

average response for the - average response for the high level of Factor B low level of Factor B

Note:

• If an effect is positive, then going from the low to the high level increases the response.

• If an effect is negative, then going from the low to the high level decreases the response.

A naming convention often proves useful when discussing effects.

Let a denote the average of the responses when A is at its high level and B is at its low level.

Similarly, let b denote the average of the responses when B is at its high level and A is at its low level.

By following this convention, ab is the average of the responses when both A and B are at their high levels.

By convention, we let (1) represent the average of the responses when all the factors are at their low levels.

With this convention, we can summarize the results of a 22 factorial experiment by the following.

x1 x2 -1 -1 (1) 1 -1 a -1 1 b 1 1 ab

In terms of our convention, the main effect for Factor A is

In a similar manner, the main effect for Factor B is

2

)1(

2 ofeffect

babaA

2

)1(

2 ofeffect

aabbB

5. The concept of interaction

We can best illustrate the concept of an interaction through an interaction plot.

Suppose the following summarizes the results from a 22 factorial experiment. x1 x2 y -1 -1 (1) 2 1 -1 a 5 -1 1 b 4 1 1 ab 7

The two lines are parallel.

Since the effect of one factor does not depend upon the specific level used of the other, we say they have no interaction.

Suppose the following summarizes the results from another 22 factorial experiment. x1 x2 y -1 -1 (1) 2 1 -1 a 2 -1 1 b 4 1 1 ab 7

In this case, there seems to be no effect due to Factor A when B is at its low level, but there seems to be a positive effect due to A when B is at its high level.

So, the effect of A does seem to depend upon the level of B, and the lines are not parallel.

Since the effect of one factor depends on the specific level used of the other, we say that the two factors interact.

The key for testing for interaction is a test for parallelism.

Consider the effect of the Factor B at the low level of A which is given by

Δ1 = b - (1) .

The effect of Factor B at the high level of A is

Δ2 = ab - a .

If the two factors do not interact, then the effect of B does not depend upon A; thus, the effect of B is the same at the high and low levels of A.

On the other hand, if A and B have a positive interaction, then we expect the response when both A and B are at their high levels to be larger than we expect from the main effects alone.

Consequently, if A and B have a positive interaction, then the effect of B at the high level of A is larger than the effect at the low level of A, or Δ2 > Δ1.

We define the interaction between A and B by

2

)1(2

)]1([][2

and between n Interactio 12

baab

baab

BA

We can obtain the numerators for estimating the main effects and the interaction more easily by looking at our model

We see that the interaction really involves the product of the design variables associated with the two factors.

This model also points out the importance of the overall mean, which in this case is the y-intercept β0.

We find the overall mean by averaging all the responses.

Let I stand for the intercept.

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The table of contrasts simply rewrites the table for our design to include the intercept and the interaction terms.

Consider the column for x1.

If we use the -1's and +1's in this column to combine the average response for each treatment combination, we obtain

-(1) + a - b + ab

which is the numerator for the effect of A.

I x1 x2 x1 x2 1 -1 -1 +1 (1) 1 1 -1 -1 a 1 -1 1 -1 b 1 1 1 +1 ab

In a similar manner we can use the columns for x2 and x1 x2 to obtain

-(1) - a + b + ab

(1) -a - b + ab

which are the numerators for the main effect of B and the AB interaction, respectively.

For two-level factorial designs, the denominator for estimating effects and interactions will always be one-half of the number of distinct factorial treatment combinations.

In the case of the 22, we have 4 different factorial treatment combinations, so our denominator is 2 for estimating the effects.

We always use the total number of distinct treatment combinations in our denominator to estimate the intercept.

The appropriate estimate of the regression coefficient corresponding to a particular effect is

Because of this relationship, we rarely calculate effects by hand.

However, the Table of Contrasts is important for conceptual reasonswhen we discuss fractional factorial designs.

7. Finally, perform a complete and thorough analysis, including residual analysis, of your experimental data using the software package of choice.

2

effect est. t coefficien regression est.

C. The General 2k Factorial Design

Repeat the 22 experiment only now use four factors of the students‘ choosing.

Do not replicate the experiment.

By not replicating the experiment,

1. we can illustrate how to use a normal probability plot to identify important factors,

2. we can discuss the main effects principle, and

3. we should be able to illustrate “hidden replication”.

The normal probability plot of the effects actually tests the hypotheses

0 s' theof oneleast at :

0 s' theall :0

aH

H

If we use a 2k factorial design, under this null hypothesis all the β's will follow a normal distribution with the same mean (0) and the same variance.

Thus, if the null hypothesis is true (nothing is important), all of the estimated effects should form a straight line when plotted on a normal probability plot.

Effects which fall off the straight line are then considered important.

The main effects principle simply states that main effects tend to be more important than two-factor interactions, which tend to be more important than three-factor interactions.

Hidden replication observes that if one of the factors along with all of the interactions involving this factor are unimportant (really 0), then the 24 design we ran is actually a replicated 23 design!

We thus have replication, and we can perform formal statistical tests.

D. Half Fractions of the 2k Design

1. The Basic Idea

Even for a moderate number of factors, the 2k factorial can require an excessive number of runs.

A possible solution: Fractional Factorial Designs.

We shall consider the simplest fractional factorial, the half fraction within the context of a specific example.

An engineer must study the effect of

• Argon flow rate (Factor A),

• Deposition rate (Factor B), and

• Deposition temperature (Factor C)

on the uniformity of the thickness of a silicon wafer.

Due to time and budget constraints, she can only afford 4 runs.

Ideally, she should run a 23 experiment, but that experiment requires 8 runs, and she can only afford 4.

The basic question is how to select four runs from the basic 23.

Consider the Table of Contrasts for the full 23 experiment.

The one effect least likely to be present is the three-factor interaction.

I x1 x2 x3 x1x2 x1x3 x2 x3 x1 x2 x3 1 -1 -1 -1 +1 +1 +1 -1 1 1 -1 -1 -1 -1 +1 +1 1 -1 1 -1 -1 +1 -1 +1 1 1 1 -1 +1 -1 -1 -1 1 -1 -1 1 +1 -1 -1 +1 1 1 -1 1 -1 +1 -1 -1 1 -1 1 1 -1 -1 +1 -1 1 1 1 1 +1 +1 +1 +1

Consider using the four treatment combinations which have +1 in the x1 x2 x3 column as our design; thus, our design is

The resulting table of contrasts follows.

1. The column for the three-factor interaction (x1 x2 x3) is the same as the intercept (I) column.

In this situation, we say that the three-factor interaction, ABC, is aliased with the intercept.

x1 x2 x3 1 -1 -1

-1 1 -1 -1 -1 1 1 1 1

I x1 x2 x3 x1x2 x1x3 x2 x3 x1 x2 x3 1 1 -1 -1 -1 -1 +1 +1 1 -1 1 -1 -1 +1 -1 +1 1 -1 -1 1 +1 -1 -1 +1 1 1 1 1 +1 +1 +1 +1

By choosing the three-factor interaction as the basis for cutting the full factorial in half, we have lost our ability to estimate it.

2. The column for x1 is the same as the column for x2 x3.

As a result, Factor A is aliased with the BC interaction.

Consequently, our estimate of the main effect of A is the same as the estimate for the BC interaction.

If this estimate is important, we have no statistical basis for determining if it is due to the main effect of A or to the interaction of B with C.

3. In a similar manner, Factor B is aliased with the AC interaction, and Factor C is aliased with the AB interaction.

As a result, all of the main effects are aliased with the two-factorinteractions.

A similar problem arises if we use the treatment combinations where x1 x2 x3 = -1.

How can we get around this problem?

Suppose that the engineers propose the following model

which assumes that only the main effects are important.

With this model and the proposed design, if an estimated effect appears significant, we conclude that the corresponding main effect and not the interaction is important.

In those cases where this assumption is reasonable, we may use the proposed design.

On the other hand, if one of the interactions is truly important, we mistakenly conclude that a main effect is important when it is not.

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If we use x1 x2 x3 = 1 to select the treatment combinations, the resulting design is called the positive half fraction of the 23.

If we use x1 x2 x3 = -1 to select the treatment combinations, the resulting design is called the negative half fraction of the 23.

In either case, since we used x1 x2 x3 to select the treatment combinations, we say the ABC is the defining interaction for the design.

Geometrically, the positive fraction is the following:

In general, we denote a half (either positive or negative) fraction ofthe 23 by 23-1 where

• 2 indicates the number of levels for each factor,

• the exponent 3 indicates the number of factors, and

• the exponent -1 indicates a half (2-1) fraction.

The total number of treatment combinations is 23-1 = 4.

2. The Alias Structure

We can always use the table of contrasts to determine which effects are aliased with one another.

A quicker method uses modulo 2 arithmetic (clock arithmetic with only two numbers: 0 and 1).

In modulo 2 arithmetic, 1 + 1 = 0.

To obtain the “alias structure,” we first note that since ABC is the defining interaction, it is aliased with the intercept or the identity.

We thus write I = ABC, which is read “ABC is aliased with the intercept.”

To determine with what A is aliased, we add A to the defining interaction, ABC, using modulo 2 arithmetic; thus,

A + ABC = BC ,

In a similar manner, we obtain

B + ABC = AC C + ABC = AB .

We can summarize the entire alias structure by the following table.

We often stop the number of rows once every main effect and interaction appear in the table.

The alias structure in this form tells us the largest model we can estimate from the experiment.

I = ABC A = BC B = AC C = AB

AB = C AC = B BC = A

I = ABC A = BC B = AC C = AB

3. Using a 22 to Generate a 23-1

Note that the 23-1 design requires 4 different treatment combinations, just like the 22.

Consider the following table of contrasts for the 22 factorial design.

If we can assume that the two-factor interaction is unimportant, then we can use the x1 x2 column to determine the settings for a third factor, C, whose design variable is x3.

In so doing, we make x3 = x1 x2, which is equivalent to aliasing C with the AB interaction.

The defining interaction is C + AB which is ABC even under modulo 2 arithmetic.

The resulting design is the positive half fraction of the 23.

I x1 x2 x1x2 1 -1 -1 +1 1 1 -1 -1 1 -1 1 -1 1 1 1 +1

This approach emphasizes the projection property of half fractions.

If one of the factors is actually unimportant, then the half fraction becomes a full factorial in the other factors.

For example, consider the positive half fraction of the 23.

If Factor C is unimportant, then the design becomes the following 22.

Half fractions readily extend for higher numbers of factors.

In each case, one uses the highest order interaction to generate the design.

Together, generate a 24-1 design.

x1 x2 1 -1

-1 1 -1 -1 1 1

4. Design Resolution

Design resolution refers to the length (number of letters) in the smallest defining or “generalized” interaction and tells the user some critical information about the alias structure.

• For a Resolution III design, the smallest defining or generalized interaction has three letters.

As a result, at least one main effect is aliased with a two-factor interaction.

We typically use Resolution III designs for screening since they are the smallest designs available.

• For Resolution IV designs, the smallest defining or generalized interaction has four letters.

As a result, the smallest interaction aliased with a main effect is a three-factor.

At least some of the two-factor interactions are aliased with each other.

Engineers use Resolution IV designs when they believe that some but not all of the two-factor interactions may be important.

• For Resolution V designs, the smallest defining or generalized interaction has five letters.

The smallest interaction aliased with a main effect has four factors, and the two factor interactions are aliased with three factor or higher interactions.

Resolution V designs are very important for process optimization.

By convention, we note the resolution of a fractional factorial design by a subscript.

For example, denotes a Resolution III, half fraction of a 23 factorial design.

Similarly, denotes a Resolution IV, quarter fraction of a 26 factorial design.

132

III

262

IV

5. Factorial Based Designs and Traditional Engineering Experimentation

Engineers not well trained in statistics often pursue two basic approaches:

• one factor at a time, and

• “shotgun”.

In general, we should not pursue either of these strategies.

With one factor at a time experimentation, we run one treatment combination with every factor at a specified level, usually the low level.

We then run a series of treatment combinations where we use the high level for one factor and the low level for the others.

If we have k factors, we run a series of k treatment combinations, allowing each factor to be at its high level once.

The experiment thus consists of k+1 total treatment combinations.

One factor at a time experimentation suffers from several major drawbacks:

• it cannot estimate interactions;

• it does not cover the entire experimental region; and

• it produces less precise estimates of the effects than a corresponding factorial or fractional factorial design.

Efficient model estimation requires us to spread our design points evenly on the boundary of the region of interest defined by our model.

Problems occur because the one-factor approach provides no information in the upper right hand quadrant.

The shotgun approach randomly selects points over the region of interest.

It gets its name because the pattern looks like a shotgun blast.

This approach also has several drawbacks:

• it tends to use many more treatment combination and thus tends to waste resources;

• it rarely covers the experimental region well; and

• it produces less precise estimates of the effects.

Some engineers pursue a shotgun approach as an expression of their creativity.

In general, a better way to express creativity is in judicial selection of the appropriate factors and their ranges for the experiment.