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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