2 k factorial designs l Examine the response for all possible combinations of k factors at two levels l Are carried out as replicated or unreplicated experiments l Aim to elucidate the linear effect of the response over the range of the factor levels chosen

Design matrix and thickness data for a single crystal layer growth experiment Polished silicon wafers are mounted on a susceptor, which is spun inside a metal bell jar. The jar is injected with chemical vapours through nozzles. A: susceptor-rotation method (continuous or oscillating) B: nozzle position (2 or 6) C: deposition temperature (1210 or 1220 o C) D: deposition time (low or high)

Design matrix and planning matrix l The planning matrix is obtained by randomising the run order l Full or restricted randomisation depending on the presence of factors that may be hard to change l Randomisation “eliminates” the effect of lurking variables Example: room humidity as a lurking variable

Key properties of 2 k designs l Balance means that each factor level appears in the same number of runs l Two factors are orthogonal if all their level combinations appear in the same number of runs

Design matrix and thickness data for a single crystal layer growth experiment Polished silicon wafers are mounted on a susceptor, which is spun inside a metal bell jar. The jar is injected with chemical vapours through nozzles. A: susceptor-rotation method (continuous or oscillating) B: nozzle position (2 or 6) C: deposition temperature (1210 or 1220 o C) D: deposition time (low or high)

Main effects plot for the single crystal layer growth experiment

Interaction effects The A  B interaction effect can be defined as where denotes the average of the z -values with both A and B at the + level

Interaction effects – alternative definitions The A  B interaction effect can also be defined as where the expression can be interpreted as the conditional main effect of B given that A is at the + level.

Interaction effects – alternative definition The A  B interaction effect measures the difference between the averages of two groups, one whose product of the levels of A and B is + and the other whose product of the levels is -.

Interaction effects plot for the single crystal layer growth experiment A-against-B plot

Synergistic and antagonistic effects l An A-against-B plot is called synergistic if l An A-against-B plot is called antagonistic if

Interaction effects plot for the single crystal layer growth experiment C-against-D and D-against-C plots C: deposition temperature D: deposition time

Interaction between three factors The A  B  C interaction effect can be defined in three equivalent ways:

Interaction between k factors The A 1,A 2,…,A k interaction effect can be defined as:

Estimators of factorial effects – general form Estimators of factorial effects measure the difference between the averages of two groups, one whose product of the factor levels is + and the other whose product of the levels is -.

Estimation of factorial effects Regression can be used to compute factorial effects. Assume that the experimental errors are independent and normally distributed with mean 0 and variance  2.Then

Fundamental empirical principles for factorial effects l Hierarchical ordering principle Lower order effects are more likely to be important than higher order effects Effects of the same order are equally likely to be important l Effect sparsity principle The number of relatively important effects in a factorial experiment is small l Effect heredity principle In order for an interaction to be significant, at least one of its parent factors should be significant

The “one-factor-at-a-time” approach l Identify the most important factor l Investigate this factor by itself and determine its optimal value l Identify the next most important factor l ………..

Factorial effects for the single crystal layer growth experiment

Planning matrix for a 2 3 design Injection moulding process that is producing an unacceptable percentage of burned parts P: injection pressure R: screw rpm S: injection speed

Drawbacks of the “one-factor-at-a-time” approach l Requires more runs (replicates) for the same precision in the effect estimation l Cannot estimate some interactions l Does not conduct a comprehensive search over the whole experimental region l Can miss optimal settings of factors

Arrangement of factorial designs in blocks The block effect is confounded with the three-factor interaction effect.

Arranging a 2 3 design in four blocks of size two The blocks I,II,III and IV are defined according to (B 1,B 2 ) = (-,-), (-,+), (+,-), (+,+), respectively B 1 x B 2 = 12 x 13 = 1 x 1 x 2 x 3 = I x 2 x 3 = 23 Alternative blocking schemes?

Assumption for all blocking schemes The block by treatment interactions are negligible, i.e., the treatment effects do not vary from block to block ( In the previous blocking scheme, B 1 x 1 = 2 and B 1 x 2 = 1)

Arranging a 2 k design in 2 q blocks of size 2 k-q l We need q independent variables B 1, B 2, …, B q for defining 2 q blocks. l Select the factorial effects v 1, v 2, …, v q that shall be confounded with B 1, B 2, …, B q. l Define the remaining block effects by multiplying the B i ’s l The 2 q -1 products of the B i ’s and the column I form the so-called block-defining contrast subgroup

Minimum aberration blocking scheme l Denote by g i (b) the number of i-factor interactions that are confounded with block effects l A blocking scheme b 1 is said to have less aberration than scheme b 2 if g r (b 1 ) < g r (b 2 ) where r is the smallest number such that g r (b 1 )  g r (b 2 ) Appendix 3A

A formal test of effect significance without s 2 Define a pseudo standard error (PSE) where Set Individual error rate (IER) and experiment-wise error rate (EER) Lenth (1989)

A formal test of effect significance with s 2 Studentized maximum modulus statistic Tukey, 1953, 1994