statistical design of experiments

Monday, Aug 13, 2007Dr. Gary Blau, Sean Han

Statistical Design of Experiments

SECTION V SCREENING


OBJECTIVES• Show how to screen or select the most

important main effects with fewer experiments.

• Show how to construct fractional factorial experiments by sacrificing interactions

• Understand the concept of confounding / aliases

• Learn how to write the mathematical model for each fractional factorial experiment


INTRODUCTION• Suppose you had 3 factors at 2 levels. A full

factorial experiment would be 23 = 8 experimental runs. However, you can only do 4 runs. Which runs would you choose?


INTRODUCTION• If we choose these four points to run,

• We have a balanced design. (Same number of values of each factor at the high and low levels.)


MAIN EFFECTS AND INTERACTIONS

• Now calculate the main effects:

• What are the interactions?


CONFOUNDING / ALIASING

• The main effect of A and the BC interaction are numerically the same

• The main effect of B and the AC interaction are numerically the same

• The main effect of C and the AB interaction are numerically the same


CONFOUNDING / ALIASING• What does this mean? We cannot distinguish between A and BC, that is

A is confounded with BC B is confounded with AC C is confounded with AB Or

A and BC are aliasesB and AC are aliasesC and AB are aliases


BUILDING A ½ FRACTION OF A 23 FACTORIAL FROM A 22 FACTORIAL EXPERIMENT

• Here is a full factorial in two factors. Set sign of C equal to sign of AB.



Two different fractions


MODEL FOR ½ FRACTIONAL FACTORIAL OF A 23 FACTORIAL EXPERIMENT



• Assume that: 1. D does not interact with A, B, and C 2. The ABC interaction is negligible

• Set the signs of D equal to the signs of the ABC interaction

• D = ABC is called a Design Generator



Notice the confounding pattern AB=CD, AC=BD, AD=BC and A=BCD, B=ACD, C=ABD, D=ABC.


MODEL FOR ½ FRACTIONAL FACTORIAL EXPERIMENT

• ½ Fraction of a 24 = 8 experimental runs

Mathematical ModelY = µ + (A or BCD) + (AB or CD) + (B or ACD) + (AC or BD) + (C or ABD) + (AD or BC) +

(D or ABC) + error



There is another ½ fraction of the full 24 experiment found by:

• Set D = - ABC, and calculate the signs accordingly

• D = - ABC is also called a Design Generator



Here is the other ½ fraction of the 24

factorial experiment :


BUILDING A ¼ FRACTION OF A 25 FACTORIAL FROM A 23 FACTORIAL EXPERIMENT

• Assume that:

1. ABC interaction is negligible2. BC interaction is negligible3. Other higher order interactions are negligible

• Set D = ABC and E = BC. These are Design Generators for the ¼ fraction. (May others exist. )


BUILDING A ¼ FRACTION OF A 25 FACTORIAL FROM A 23 FACTORIAL EXPERIMENT

• This is a ¼ fraction of a full 25 factorial experiment:


PRINCIPAL BLOCK• Principal Block is the fraction that contains the treatment

combination (1), i.e., with all factors at the low level.

• If one set of design generators is available, the other sets may be obtained by changing the signs of one or more design generators.

• In the example on the previous slide, the design generators D = ABC and E = BC produced one ¼ fraction.

• The design generators which produce all ¼ fractions are:

D = ABC E = BC, D = ABC E = -BCD =-ABC E = BC, D = -ABC E = -BC


RESOLUTION• We may identify the appropriateness of a

fractional experiment by its resolution:

• Resolution III – main effects are clear of each other, but at least one main effect is confounded with a 2-way interaction.

• Resolution IV – main effects are clear of each other and 2-way interactions, but at least one pair of 2-way interactions is confounded.

• Revolution V – main effects are clear of each other, of 2-way and 3-way interactions. 2-way interactions are clear of each other, but at least one 2-way interaction is confounded with a 3-way interaction.


RESOLUTION

• Normally, we want to estimate main effects and 2-way interactions (Res V)


DEFINING CONTRAST There is a way to determine confounding

without comparing columns of +’s and –‘s

Defining Contrasts Consider the full 23 factorial experiment.

Suppose the following design generator is used:

D = ABC (1)

This will give rise to a ½ fractional factorial of a 24 factorial experiment as seen earlier.


DEFINING CONTRAST Multiply both sides of equation (1) by D and set all

squared terms to unity (mod 2 arithmetic) D*D =ABC*D I = ABCD (2)

Equation (2) is called the defining contrast. It is extremely useful to determine the confounding that has resulted from adding the fourth factor D. As we have seen we are unable to separate all the effects estimated in a full factorial experiment with 4 factors and some of these effects have the same name or aliases. These aliases are readily determined from the defining contrast by multiplying both sides of the equation by the effect you are interested in estimating. Note the equation is based on mode 2 arithmetic.


DEFINING CONTRAST

For the design generator in Equation (1), multiplying both sides of Equation (2) by A gives:

A = A2 * BCD = BCD

So that A and BCD are aliases. The same procedure is followed for all of the other factors and interactions.


DEFINING CONTRAST• Consider a situation where you have 5 factors, A, B, C,

D, and E, and you only have time and raw materials enough for 8 experimental runs. You know that the BC interaction and the ABC interaction are negligible. In order to determine what 8 experiments to run, we postulate the design generators:

Design Generators: D = ABC; E = BC

• The defining contrasts are obtained from the design generators as follows: D * D = ABC * D

I = ABCD (3) E * E = BC * E

I = BCE (4)• Equations (3) and (4) are defining contrasts.


DEFINING CONTRAST• Another defining contrast is obtained by

multiplying all the previously found defining contrasts together:

I=ABCD*BCE=AB*BC*CDE=ADE (5)

• Therefore, the entire set of defining contrasts is:I = ABCD = BCE = ADE (6)

• Note: Given p design generators, there are 2p – 1 members in the entire set of defining contrasts.


ALIASES• Alias are effects that are confounded with

other effects.

• Rule: the effects confounded with any given effect in a fractional experiment are found by multiplying the defining contrast by the given effect. The whole set of comparisons is found by multiplying the defining contrasts by main effects, interactions, etc., until all effects have been accounted for.


ALIASES• Defining contrasts: I = ABCD = BCE = ADE

• To determine all aliases, multiply defining contrasts by all effects:

A*I =A*ABCD =A*BCE =A*ADEA = BCD = ABCE = DE

B*I =B*ABCD =B*BCE =B*ADEB = ACD = CE = ABDE


ALIASES Continue this until all effects and interactions

are accounted for.

The list of all aliases for this design is:A = BCD = ABCE = DEB = ACD = CE = ABDEC = ABD = BE = ACDED = ABC = BCDE = AEE = ABCDE = BC = ADAB = CD = ACE = BDEAC = BD = ABE = CDE


GUIDELINES• Select design generators for which the high level interactions

are negligible.

• The number of factors in the resulting defining contrasts should be about the same.

• Look at the worst case; find all aliases of main effects and 2-way interactions using the defining contrast containing the least number of factors.

• To find all aliases, multiply every factor and interaction by every defining contrast.

• Assign physical quantities to the factors in which any interactions are confounded with main effects or each other and decide whether you will be able to meaningfully interpret the results after the experiments are run.


ROLLER COMPACTION FRACTIONAL FACTORIAL EXAMPLE

• The following five factors were studied at two levels in a Roller Compaction process to determine the maximum dissolution (%):

Factors - Levels +A Roll Speed 4 rpm 6 rpmB Screw Speed 5 rpm 20 rpmC Roll Force 3 tons 12 tonsD Force/inch 2.5 tons/inch 4 tons/inchE throughput 2 kg/h 2 kg/h



• A full factorial experiment in 5 factors calls for 25 = 32 experimental runs.

• Because of time and raw material constraints, we are limited in this case to 8 experimental runs

• What is the largest run factorial design on which we can build the fractionated design?

2? ≤8



• The standard error for this type of reaction was known to be about 1%.

• The design generators that were used are:

D = - AC and E = AB



• Build the fractionated design on a full 23 factorial experiment



• Defining ContrastsI = - ACD = ABE = - BCDE

• Effects and AliasesA, -ABCDE, -CD, +BEB, -CDE, -ABCD, +AEC, -BDE, -AD, +ABCED, -BCE, -AC, +ABDEE, -BCD, -ACDE, +ABBC, -DE, -ABD, +ACEBD, -CE, -ABC, +ADE



• This experiment is of RESOLUTION III because main effects are confounded with two factor interactions.

• What is the mathematical model?

Y = µ + (A, -ABCDE, -CD or +BE) + (B, -CDE, -ABCD or +AE) + (C, -BDE, -AD or +ABCE) + (D, -BCE, -AC or +ABDE) + (E, -BCD, -ACDE or +AB) + (BC, -DE, -ABD or +ACE) + (BD, -CE, -ABC or +ADE) + error



• For the purpose of analysis, we apply the signs directly to the observations, e.g.,

Effect of A = (-59.1+57-58.6+63.9-67.2+71.6 -79.2+76.9)/4


JMP ANALYSIS


ANALYSIS

TermEstimate(t ratio) Prob>|t|

Intercept 66.6875 0.0002A -0.6625 0.5758B -2.9625 0.0976C -7.0375 0.0196*D -0.1375 0.9032E -0.0875 0.9382

Since we know the standard error is 1, the estimate will be equal to the t ratio.

The significant factor is C at .05 level.


TIPS Conducting Fractional Factorial

Experiments Sequentially

• Frequently used to eliminate unimportant factors

• A good test of levels selected for experimental region

• Additional factors may be introduced


SUMMARYWhen should you use fractional factorial experiments?

• When the number of factors are large• When the high accuracy given by a full factorial

is not necessary• When you know that certain interactions are

negligible• When a reliable estimate of experimental error

is already available• When you prefer to work in a sequential fashion• When you are screening the factors to identify

those that are most important.

statistical design of experiments

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