Stat Today: More Chapter 3

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

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

One-at-a-Time Experiments Have discussed the factorial layout for experimentation –all level combinations –m replicates –performed in random order Another obvious strategy is call the “one-factor-at-a-time approach” 1identify the most important factor, 2investigate this factor by itself, keeping other factors fixed, 3decide on optimal level for this factor, and fix it at this level, and 4move on to the next most important factor and repeat 2-3

Example Suppose have 2 factors A and B, each with 2-levels (-1,+1)

Comments Disadvantages of the on-factor-at-a-time approach –less efficient than factorial experiments –interactions may cause misleading conclusions –conclusions are less general –may miss optimal settings

Assessing Effect Significance For replicated experiments, can use regression to determine important effects Can also use a graphical procedure The graphical procedure can be used for replicated and replicated factorial experiments

Normal and Half-Normal Probability Plots Graphical method for assessing which effects are important are based on normal probability plots (a.k.a normal qq-plots) Let be the sorted (from smallest to largest) effect estimates Plot, where represents the cumulative distribution function of the standard normal (N(0,1)) distribution

Normal and Half-Normal Probability Plots That is, we plot the quantiles of our sample of effects versus the corresponding quantiles of the standard normal If no effect is important then the sample of effects appear to be a random sample from a normal distribution…we observe: Otherwise:

Normal and Half-Normal Probability Plots Why does this work?

Normal and Half-Normal Probability Plots Half-Normal Plots

Example: Epitaxial Growth Layer Experiment