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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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