Lecture 8 Last day: 3.1 Today: Next day: , Assignment #2: Chapter 2: 6, 15 (treat tape speed and laser power as qualitative factors), 27, 30, 32, and 36….DUE IN 1 WEEK

Example - Epitaxial Layer Growth In IC fabrication, grow an epitaxial layer on polished silicon wafers 4 factors (A-D) are thought to impact the layer growth Experimenters wish to determine the level settings of the 4 factors so that: the process mean layer thickness is as close to the nominal value as possible the non-uniformity of the layer growth is minimized

Nominal-The-Best Procedure In the previous example, the aim was to adjust the settings of the factors so that the process was on target Also, want the deviations from the target to be as small as possible Can quantify the deviations from the target using a loss function

Quadratic Loss Let t be the system target value, and y be the system response The loss due to deviations can be expressed as The expected loss is:

Two-Step Procedure Quadratic loss suggest a 2-step procedure for hitting the target and reducing the expected loss 1Select levels of some factors to minimize V(y) 2Using factors not used in 1, adjust E(y) as close to t as possible Need adjustment factors in step 2 Will come back to this!

Full Factorial Designs at 2 Levels Notation/terminology: 2 k experiment, where –k is the number of factors –each factor has two levels: low, high (denoted by -1, +1) Each replicate has a run-size of 2 k Table 3.1 shows a 2 4 experiment with 6 replicates Experiment is performed as a completely randomized design

Notes Reasons for use: Two-level factors enable the largest no. of factors to be included in experiment, for a given no. of runs Responses often assumed to be monotonic functions of the factors. So, testing at low and high values will detect a factor’s effect 2-level factorial experiments can be used in a screening mode -- find factors with large effects; eliminate those with small effects. Then, follow up these experiments with subsequent experiments Experimental results are (relatively) easy to analyze and interpret

Computing Factorial Effects Experiment is run to see which factors are significant Suppose for the moment, there was only 1 factor with 2 levels Could compare average at the high level to the average at the low level Suppose there are several factors

Computing Factorial Effects Can measure the joint effect of two factors (say A and B)

Interaction Plots Can visualize the joint effect of two factors using an interaction plot Display average at the [(-1,-1); (-1,+1); (+1,-1); (+1,+1)] level combinations of 2 factors

Interaction Plot (from Lecture 5)

Interpretation:

Variance of an effect estimate:

Example: Epitaxial Layer Growth Data from Table 3.1

Example: Epitaxial Layer Growth

Common Assumptions Can address relative importance of effects and their relationships Hierarchical Ordering Principle: Lower order terms are more likely to be important than higher order terms Effect Sparsity: Number of relatively important effects is small Effect Heredity: For an interaction to be significant, at least one of its parents should be significant

Using Regression for the Analysis Can compute factorial effects directly as before Or can use linear regression to estimate effects Model: Regression Estimates:

Using Regression for the Analysis Consider an unreplicated 2 4 factorial design Effect estimate of factor A Regression estimate of factor A