1. 1 FACTORIAL EXPERIMENTS THIS APPROACH HAS SIGNIFICANT ADVANTAGES AND DISADVANTAGES

2. 2 BASIC APPROACH  Access to the system or simulation  k control-able independent variables (Factors)  Each has an on/off, hi/low, present/absent  CAUTION: These are not conditions or cases, they are decision-able

3. 3 THE GOAL  We seek unbiased estimates of the marginal EFFECT of the “HI” setting for each Factor Isolated In conjunction with other Factors  Independence of effect is NOT assumed  We’re going to collect data according to the design, then produce all the answers at the end

4. 4 3-FACTOR 2 K EXPERIMENT average of responses for treatment 1

5. 5 ESTIMATING AN EFFECT  eA is the effect of varying factor A  eA is the average of treatments that vary only in the setting of A 1&2, 3&4, 5&6, 7&8  the Variance of eA requires all of the variances, covariances, 3-factor variances NONE of which we assume to be 0 (negligible)

6. 6 SINGLE-FACTOR ESTIMATION  Note the connection between the terms in the expression and the signs (+/-) on the table

7. 7 TWO-FACTOR ESTIMATION  eAB is half the distance between... marginal effect of A when B is a “+”  (1/2)*[(R1-R2) + (R5-R6)] marginal effect of A when B is a “-”  (1/2)*[(R3-R4) + (R7-R8)]

8. 8...more TWO-FACTOR ESTIMATION  the signs are the vector product of columns A and B!  eAB = eBA  Higher-order combinations are built the same way averages and mid-points vector products

9. 9 DISCUSSION  eA is the AVERAGE of the effect of A over the equally-weighted mixture of the hi’s and low’s of the other factors  Is eA significant?  Is eA an unbiased estimate of the Truth?  Could you do a cost/benefit analysis with this sort of analysis?

10. 10 ONE CURE  Let Ri j be the jth observation of response to the ith treatment  Treat the eA j as a sample, build a confidence interval, do univariate analysis  Not available to traditional experimental statisticians

11. 11 FRACTIONAL FACTORIAL  3 factors require 8 treatments!!?  5 factors would require 32! supports up to 5-way effect measurement high-order effects can often be assumed negligible  2 k-p factorial design “confounds” effects of order k-p+1, k-p+2,...,k

12. 12 DESIGN TABLE  2 4-1 design  D’s column is the same as AxBxC  eABC is confounded with eD  more than two settings: Latin Squares  more control on confounding: Blocked Experiment