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FACTORIAL EXPERIMENTS

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

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FACTORIAL EXPERIMENTS

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  1. FACTORIAL EXPERIMENTS THIS APPROACH HAS SIGNIFICANT ADVANTAGES AND DISADVANTAGES

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

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

  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. ...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. 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. ONE CURE • Let Rij be the jth observation of response to the ith treatment • Treat the eAj as a sample, build a confidence interval, do univariate analysis • Not available to traditional experimental statisticians

  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 • 2k-p factorial design • “confounds” effects of order k-p+1, k-p+2, ...,k

  12. 24-1 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 DESIGN TABLE

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