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## PowerPoint Slideshow about ' FACTORIAL EXPERIMENTS' - allistair-emerson

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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

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

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)]

- marginal effect of A when B is a “+”

...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
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