God I hate stats. In a factorial experiment , each possible combinations of the levels of the factors in each replicate of the experiment is investigated. An experiment with two factors (A, B) is represented by a flat sheet in 3-D surface plot. Factorial experiments are the only way to discover interactions between variables . A 2 k factorial design provides the smallest # of runs for which k factors can be studied in a complete factorial design. 2 2 Design (A,B factors): =-. a results for all a high b,ab are similar Main effect of A: = +--= +- + = +- -A yA yA a ab2n b 12n 12na ab b 1 Main effect of B: = +--= +- + = +- -B yB yB b ab2n a 12n 12nb ab a 1 AB interaction: == +( )- + = + - -AB ab 1 2n a b2n 12nab 1 a b For the above 3 equations, quantities in brackets are contrasts. Sums of squares formulas: = +- -SSA a ab b 124n = +- -SSB b ab a 124n = +( )- -SSAB ab 1 a b24n SS T calculated “as usual” with 4n-1 DF, SS E with 4(n-1) DF with subtraction. = + + + + SST yi2 1 a b ab24n Formulas for Two-Level Factorial with k factors Each at Two Levels and N total Trials V(Effect)=-σ2n2k 2 Coefficient = effect/2 = =-seeEffect 2σ2N2 σ2n2k 2 = =( )-seCoefficient 122σ2N2 12 σ2n2k 2 = = t ratio effectseeffect coefficientsecoefficient = σ2 MSE =SS E /(4n-4)
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