1. 2 n Factorial Experiment 4 Factor used to Remove Chemical Oxygen demand from Distillery Spent Wash R.K. Prasad and S.N. Srivastava (2009). “Electrochemical degradation of distillery spent wash using catalytic anode: Factorial design of experiments,” Chemical Engineering Journal, Vol. 146, pp. 22-29.

2. Data Description Response: Y = % Chemical Oxygen Demand Removed from Distillery Spent wash Factors and Levels:  A: Current Density (mA/cm 2 ) – 14.285, 42.857  B: Dilution (%) – 10, 30  C: Time (hrs) – 2, 5  D: pH – 4, 9 Experimental Runs: 16 – All 2 4 Combinations of levels of A,B,C,D

3. Data – Normal Order For the Label, any factor at its high level appears in lower case form. (1) Corresponds to the case when all factors are at their low levels.

4. Table of Contrasts - I Create a Column for the intercept (I), one for each Main Effect and each Interaction (A,…,D, AB,…,CD, ABC,…,BCD, ABCD), and one for the response (y). If there were multiple replicates per treatment, replace y with the mean of those r replicates Create a row for each experimental run (treatment), using the Labels from the previous slide. For the Intercept Column, put +1 in each row For all Main Effects, Put +1 if that factor was at its high level, -1 if at its low level (Note: Books use +/-)

5. Table of Contrasts - II For Interactions, multiply the coefficients in each row for the Main Effects that make up that Interaction. For Row 1 and Column AB: A has coefficient -1, B has -1, so AB has (-1)(-1) = +1 For Row 1 and Column ABC: (-1)(-1)(-1) = -1 For Row 1 and Column ABCD: (-1)(-1)(-1)(-1) = +1 An Interaction will have a coefficient of +1 if it has an even number of its Main Effects at their low levels, -1 if an odd number.

6. Table of Contrasts - III Create 4 Rows below this “matrix”: Contrast, Divisor, Effect, Sum of Squares

7. Table of Contrasts - IV

8. Analysis of Variance Notes: Factor D (pH) has by far the largest effect on the outcome. With all mean effects and interactions, there are no error degrees of freedom, and no tests can be conducted Consider dropping interactions with small sums of squares to obtain an error term (Authors dropped: AB, AC, BC, and BCD)

9. Regression Approach

10. Further Model Reduction (Simplification) When testing the effects after removing the Interactions with the smallest effects, we find BD, ACD, and ABCD all have P-values that are > 0.10. Now we remove them for a simpler model.

11. Normal Probability Plot of Factor & Interaction Effects Under hypothesis of no main effects or interactions, estimated effects should be approximately normally distributed with mean 0. Construct a normal probability plot of estimated effects Clearly, several effects fall well away from central line

12. A Simple Test for Effects & Interactions Method described by Lenth (1989):  Obtain s0 = 1.5*median(|Effects|)  Compute: pseudo standard error: PSE = median(|Effects|*Indicator(|Effect| < 2.5*s0))  Compute Simultaneous Margin of Error: SME = t(.05/(2*Cm),d)*PSE where m = # of Effects, Cm=m(m-1)/2, d=m/3  Consider effect significant if |Effect| > SME Based on this criteria, only pH main effect is significant. When not making adjustment for multiple tests (ME), 3 effects are significant or very close