FACTORIAL EXPERIMENTS

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# FACTORIAL EXPERIMENTS - PowerPoint PPT Presentation

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

BASIC APPROACH
• 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
• 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 2K EXPERIMENT

average of

responses for

treatment 1

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