Math 4220 Dr. Zeng

Chapter 6: THE 2k FACTORIAL DESIGN Symmetric factorial experiments (sometimes called symmetric factorials) are factorial experiments in

which all factors have the same number of levels. Such factorial experiments are denoted by nS ,

where S is the number of levels and n is the number of factors. We will first study the simplest of

symmetric factorials, namely 2n factorial experiments. 22 FACTORIAL EXPERIMENTS This is the simplest case.

Example: Chemical Process Consider an investigation into the effect of the concentration of the reactant and the amount of the catalyst on the conversion (yield) in a chemical process the objective of the experiment was to determine if adjustments to either of these two factors would increase the yield. Let the reactant concentration be factor A and let the two levels of interest be 15 and 25 percent. The catalyst is factor B, with the high level denoting the use of 2 pounds of the catalyst and the low level denoting the use of only 1 pound. The experiment is replicated three times, so there are 12 runs. The order in which the runs are made in random, so this is a completely randomized experiment.

“-” and “+” denote the low and high levels of a factor, respectively

• Low and high are arbitrary terms

• Geometrically, the four runs form the corners of a square

• Factors can be quantitative or qualitative, although their treatment in the final model will be different

Math 4220 Dr. Zeng

Estimating the main effect A, B and interaction effect AB

1[ab a b (1)]

2

1B [ab (1)]

2

1[ab (1) ]

2

An

b an

AB a bn

In this example, estimate the average effects as

1[90 100 60 80] 8.33

2(3)

1[90 60 100 80] 5.00

2(3)

1[90 80 100 60] 1.67

2(3)

A

B

AB

It is often convenient to write down the treatment combinations in the standard order (1), a, b, ab.

Effects (-1) a b ab

A -1 +1 -1 +1

B -1 -1 +1 +1

AB +1 -1 -1 +1

Next, consider determining the sums of squares for A, B, and AB.

2

2

2

1[ab a b (1)]

4

1SSB [ab (1)]

4

1[ab (1) ]

4

SSAn

b an

SSAB a bn

In this example, the sums of squares for A, B, and AB are

2

2

2

50208.33

4(3)

( 30)75

4(3)

108.33

4(3)

SSA

SSB

SSAB

Math 4220 Dr. Zeng

The total sum of squares is found in the usual way, that is, 22 2

2 ...

1 1 1 4

n

ijk

i j k

ySST y

n

In this example, SST=9398-9075=323 And the error sums of squares, SSE=SST-SSA-SSB-SSAB=323-208.33-75-8.33=31.34

2n FACTORIAL EXPERIMENTS n Factors, each at two levels (say high and low). Such factorial experiments are conducted using completely randomized or blocked designs. In conducting such an experiment in a CR design, we will use “r” replications. If we are running this experiment using an RCB design we will have “r ” blocks. NOTATION: Let capital Latin letters denote the factors (e.g. A,B,C).

Let small letters in combination denote the mean response of a particular treatment combination (as well as that treatment combination), where the absence of a small letter denote the low level of the factor associated with that letter and the presence of a small letter denote the high level of the factor associated with that letter.

e.g. Consider a 23 factorial. The means model is

Yijk ijk ijk

i level of factor A i 1 low, i 2 high j level of factor B j 1 low, j 2 high k level of factor C k 1 low, k 2 high

Math 4220 Dr. Zeng

The treatment combinations are

low low low low low high low high low low high high high low low high low high

high high low high high high

, , , , ,

, .

A B C A B C A B C A B C A B C A B C

A B C A B C

They are also denoted by 1 , c, b, bc, a, ac, ab, abc respectively.

Also we use the above notation to mean 111,112,121,122,211,212,221,222 respectively.

The ANOVA for the above 23 design in a completely randomized design is:

Source d.f.

A 1

B 1

AB 1

C 1

AC 1

BC 1

ABC 1

7. d.f. for SSTreatment

Error 8 r 1 Note: We have 8trt combinations, each replicated r times

Total 8r 1

The above SS can be obtained as squares of linear contrasts of treatment combination totals. The usual contrasts used are given below.

TREATMENT COMBINATIONS

Note:

means 1

means 1

1 a b ab c ac bc abc Division

EFFECT A 4 8

B 4 8

C 4 8

AB 4 8

AC 4 8

BC 4 8

ABC 4 8

For For

Computing Comp.

Effects SS

Math 4220 Dr. Zeng

Math 4220 Dr. Zeng

Example: Nitride etch process A 23 factorial design was used to develop a nitride etch process on a single-wafer plasma etching tool. The design factors are the gap between the electrodes, the gas flow and the RF power applied to the cathode. Each factor is run at two levels, and the design is replicated twice. The response variable is the etch rate for silicon nitride.

Compute the estimation of main effects and interaction effects. Also compute the sums of squares. Please verify your results with Table 6.5 A= B= C= AB= AC= BC= ABC=

Math 4220 Dr. Zeng

SSA= SSB= SSC= SSAB= SSAC= SSBC= SSABC=

Math 4220 Dr. Zeng

A SINGLE REPLICATE OF THE 2n DESIGN

Usually due to the limited resources, the number of replicates that the experimenter can employ maybe restricted. Available resources only allow a single replicate of the design to be run, unless the experimenter is willing to omit some of the original factors. This is called an unreplicated factorial. 24 DESIGN EXAMPLE: A chemical product. A chemical product is produced in a pressure vessel. A factorial experiment is carried out in the pilot plant to study the factors thought to influence the filtration rate of this product. The four factors are temperature (A), pressure (B), concentration of formaldehyde (C), and stirring rate (D). Each factor is present at two levels.

Math 4220 Dr. Zeng

Math 4220 Dr. Zeng

Remark: if the assumption of normality and equality of variance are violated. A data transformation of the response variable is often used (e.g. log transformation).