Factorial Designs Outlines:  2 2 factorial design  2 k factorial design, k>=3  Blocking and confounding in 2 k factorial design

2 2 factorial design  The experiment consists of 2 factors- High (+) and Low(-)  The design can be represented as a square with 2 2 =4 runs  Let the letters (1), a, b, and ab also represent the totals of all n observations taken at these design points

2 2 factorial design  Main effect of A, B and Interaction effect AB Contrast of A, B, ABUsed to calculate SS

2 2 factorial design  Sum of Square (SS)

2 2 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

2 2 factorial design  Effect of A, B, AB

2 2 factorial design  SS of A, B, AB

2 2 factorial design  Model Adequacy Checking

2 k factorial design, k>=3  The experiment consists of k factors, each factor consists of 2 level (+,-)  For example k=3;

2 k factorial design, k>=3  Main & Interaction effects

2 k factorial design, k>=3  Main & Interaction effects The value in the brackets are “Contrast”

 Effects  SS 2 k factorial design, k>=3

 Ex. Consider the surface roughness experiment. This is a 2 3 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: 2 k factorial design, k>=3

 Single replication of 2 k 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 (+,-)

2 k factorial design, k>=3

 Main and Interaction effects 2 k factorial design, k>=3

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

Blocking a replicated 2 k design  Suppose that the 2 k factorial design has been replicated n times. Each replicate is run in one block.  The effect of block should be considered. Block 1 (1) a b ab Block 2 (1) a b ab Block n (1) a b ab

Blocking a replicated 2 k design  Ex. Consider the chemical process. Suppose that only four experimental trials can be made from a single batch of raw material. factorTreatment combination replicatetotal AB123 --A low, B low A high, B low A low, B high A high, B high

Blocking a replicated 2 k design  Low effect of blocks

Blocking and confounding in 2 k design  Blocking: It is often impossible to run all the observations in a 2 k 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 2 2 factors: there are 4 treatments

Blocking and confounding in 2 k design  Contrast these 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 are identical. 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 2 k design  Consider 2 3 design divided into 2 blocks

Blocking and confounding in 2 k design  Consider 2 4 design divided into 2 blocks

Blocking and confounding in 2 k design  Consider 2 3 design divided into 2 blocks with 4 replicates R1 R2 R3 R4