Factorial Designs

Outlines: 22 factorial design 2k factorial design, k>=3 Blocking and confounding in 2k factorial design

22 factorial design

The experiment consists of 2 factors- High (+) and Low(-) The design can be represented as a square with 22=4 runs

Let the letters (1), a, b, and ab also represent the totals of all n observations taken at these design points

22 factorial design

Main effect of A, B and Interaction effect AB

Contrast of A, B, AB Used to calculate SS

22 factorial design

Sum of Square (SS)

22 factorial design

Ex. An article in the AT&T Technical Journal (Vol. 65, March/April 1986, pp. 39–50) describes the application of two-level factorial designs to integrated circuit manufacturing. A basic processing step in this industry is to grow an epitaxial layer on polished silicon wafers. The wafers are mounted on a susceptor and positioned inside a bell jar. Chemical vapors are introduced through nozzles near the top of the jar. The susceptor is rotated, and heat is applied.

A deposition time and B arsenic flow rate. two levels of deposition time are short (-) and long (+) two levels of arsenic flow rate are 55% (-) and 59%(+) n=4 replications

22 factorial design

Effect of A, B, AB

22 factorial design

SS of A, B, AB

22 factorial design

Model Adequacy Checking

2k factorial design, k>=3

The experiment consists of k factors, each factor consists of 2 level (+,-)

For example k=3;

2k factorial design, k>=3

Main & Interaction effects

2k factorial design, k>=3

Main & Interaction effects

The value in the brackets are “Contrast”

Effects

SS

2k factorial design, k>=3

2k factorial design, k>=3

Ex. Consider the surface roughness experiment. This is a 23 factorial design in the factors feed rate (A), depth of cut (B), and tool angle (C), with n 2 replicates.

Main and Interaction effects

SS:

2k factorial design, k>=3

2k factorial design, k>=3

2k factorial design, k>=3

Single replication of 2k design Ex. Study the effects of Gap, pressure, C2F6 Flow rate and

power to the etch rate for silicon nitride

There are 4 factors, each factor has 2 level (+,-)

2k factorial design, k>=3

2k factorial design, k>=3

Main and Interaction effects

2k factorial design, k>=3

2k factorial design, k>=3

2k factorial design, k>=3

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The regression coefficient is one-half the effect estimate because regression coefficients measure the effect of a unit change in x1 on the meanof Y, and the effect estimate is based on a two-unit change from low to high.

Total average

Blocking a replicated 2k design

Suppose that the 2k factorial design has been replicated n times. Each replicate is run in one block.

The effect of block should be considered.

Block 1(1)abab

Block 2(1)abab

Block n(1)abab

Blocking a replicated 2k design Ex. Consider the chemical process. Suppose that only four

experimental trials can be made from a single batch of raw material.

factor Treatment

combination

replicate total

A B 1 2 3

- - A low, B low

28 25 27 80

+ - A high, B low

36 32 32 100

- + A low, B high

18 19 23 60

+ + A high, B high

31 30 29 90

Blocking a replicated 2k design

Low effect of blocks

Blocking and confounding in 2k design Blocking: It is often impossible to run all the observations in

a 2k factorial design under homogeneous conditions. Blocking is the design technique that is appropriate for this general situation.

Confounding: a useful procedure for running the 2k design in 2p blocks where the number of runs in a block is less than the number of treatment combinations, where p < k.

For 22 factors: there are 4 treatments

Blocking and confounding in 2k design

Contrastthese contrasts are unaffected by blocking since in each contrast there is one plus and one minus treatment combination from each block.

•two treatment combinations with the plus signs, ab and (1), are in block 1 and the two with the minus signs, a and b, are in block 2•the block effect and the AB interaction areidentical.•That is, the AB interaction is confounded with blocks.

If {1,b} and {a, ab} then the main effect of A have been confounded with blocks.Usually, we confound the highest order interaction with blocks!

Blocking and confounding in 2k design

Consider 23 design divided into 2 blocks

Blocking and confounding in 2k design

Consider 24 design divided into 2 blocks

Blocking and confounding in 2k design

Consider 23 design divided into 2 blocks with 4 replicates

R1 R2 R3 R4