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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

FACTORIAL EXPERIMENTS

THIS APPROACH HAS SIGNIFICANT ADVANTAGES AND DISADVANTAGES

basic approach
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
the goal
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
3 factor 2 k experiment
3-FACTOR 2K EXPERIMENT

average of

responses for

treatment 1

estimating an effect
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)
single factor estimation
SINGLE-FACTOR ESTIMATION
  • Note the connection between the terms in the expression and the signs (+/-) on the table
two factor estimation
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)]
more two factor estimation
...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
discussion
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?
one cure
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
fractional factorial
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
design table
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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