Special Topics In Design A Brief Survey

2k Full Factorial Designs Used to screen through several potential factors to determine a subset of important factors and interactions Obtains estimates of effects of each factor and interaction as well as sums of squares for the ANOVA (very few if any df for error) All factors are set at -1/+1 for low and high levels of the factor Normal probability plot of estimated effects reveals important factors

2k Full Factorial – Experiment in Bath Oil Dispersion Factor A – Rapeseed Lecithin Ethanol Soluble Fraction (-1=8, +1=12) Factor B – Anhydrous Ethanol (-1=5, +1=15) Factor C – Polysorbate 80 (-1=1, +1=3) Response – Dispersion Degree of Bath Emulsions

The 3 large negative effects (AB, A, ABC) fall far from the “line” formed by the smaller (in absolute value) effects

2k-p Fractional Factorial Designs As the number of factors k increases, the total number of runs for the full factorial increases multiplicatively to 2k Usually researchers are interested in main effects and effects corresponding to lower order interactions Goal is to choose a fraction of the full factorial design that can obtain estimates of main effects and lower-order interactions (which will necessarily be “confounded” with higher order interactions Makes use of Incomplete Block Designs to obtain the fraction of the full replicate. Defining contrasts (available in general design textbooks and on the web) are available to select a design for general k and p

2k-p Fractional Factorial Designs Depending on k and p, the level of confounding will depend on their levels. The “Resolutions” of designs are typically reported: Resolution III – Main Effects are confounded with 2-factor interactions, and pairs of 2-factor interactions are confounded Resolution IV – No main effects are confounded with other main effects or 2- factor interactions, pairs of 2-factor interactions are confounded Resolution V – No pairs of main effects or 2-factor interactions are confounded, pairs of 2-factor and 3-factor interactions are confounded Defining Contrasts: Factor labels can be raised to powers (0 mod 2): A1 = A, A2 = A0 = I Set I equal to the effects used to create incomplete blocks Each effect will be confounded with 2p-1 other effects These can be obtained by multiplying the effects through “defining relation”

2k-p Fractional Factorial Examples Example 1 – k=4, p=1 (Half-replicate, with 8 runs) Confound 4-Way Interaction with 2 Blocks of Size 8 I=ABCD (or could use I=-ABCD) Confounded Effects: A=AI=AABCD=A2BCD=BCD, B=ACD, C=ABD, D=ABC, AB=CD, AC=BD, AD=BC Bold runs below are used in experiment ((1),cd,bd,bc,ad,ac,ab,abc) This Design is Resolution IV

2k-p Fractional Factorial Examples Example 2 – k=5, p=2 (Quarter-replicate, with 8 runs) I = ABE = CDE = ABCD (Cochran & Cox (1957), Plan 6A.2, page 277 Note: This Design is Resolution III Runs with A=+1: 26, 32, 19, 21 Runs with A = -1: 2, 8, 11, 13 Runs with BE = +1: 26, 32, 19, 21 (B and E are both +1 or -1) Runs with BE = -1: 2, 8, 11, 13 Effects and Sums of Squares can be obtained for: A/BE, B/AE, C/DE, D/CE, E/CD, AC/BD, and AD/BC

Response Surface Designs Goal: Find values of numeric predictor variables that optimize the output variable of a process. Second-Order Model contains all main effects, quadratic terms, and 2-way interactions in a regression equation. Model with k predictors:

Response Surface – Central Composite Design Factors are set at 3 equally spaced levels, coded as -1/0/+1 2k factorial design is set up at each combination of +/-1 2k points are set where one factor is set at +/-a and all other factors are set at 0. a is often set at 1.414, 1.682, or 2 c points are set at the center, with each factor is set at 0 F-test for Lack-of-Fit is conducted due to replication at the center point Total sample size ≡ 2k + 2k + c Designs are often replicated in blocks

Example – Damaged Rice in Bioethanol Production Experiment with 3 Factors in utilizing damaged rice in Bioethanol Factors: Malt Extract Concentration, Yeast Inoculum Concentration, Fermentation Time Responses: Ethanol Concentration, Yeast Cell Yield, Utilized Reducing Sugars

Results of Regression – Y=Ethanol Concentration

Plots of the Response Surface

Response Surface – Box-Behnken Design Design has k factors, each at 3 levels coded as -1/0/+1 For all k(k-1)/2 pairs of factors, set up a 22 design with coded levels -1/+1, with all other “excluded” factors being set to level 0 Center points added for a Lack-of-Fit test Example: Experiment in making cordyceps rice wine with k=3 factors: Liquid-to-Solid Ratio, Koji Addition, Temperature

Mixture Designs Goal: Optimize the response across a set of mixtures among k ingredients, where proportions of ingredients always sum to 1 Similar to Response Surface Designs, Mixture Designs are fit based on linear regression, with the restriction that the Xs sum to 1 Four commonly fit models (all are fit without an intercept based on the full set of predictors):

Example – Wax Ratio in Lipstick Formulations Experiment used k=3 types of wax in lipstick manufactured with Sweet Almond Oil Formulations (Waxes): Ozokerite, Beeswax, and Candelilla Wax. Responses: Breaking Point (g) and Softening Point (ᵒC) Experimental Runs: 16 Mixtures of the Waxes

Summary Results for 4 Models fit in EXCEL Total and Regression Sums of Squares and R2 and F are incorrect due to fitting regression through origin. They are corrected for each model

Mixture Plot from R mixexp and daewr Packages