1 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Blocking a Replicated Design Consider the example from Section 6-2; k = 2 factors, n = 3 replicates.

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1 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Blocking a Replicated Design Consider the example from Section 6-2; k = 2 factors, n = 3 replicates This is the “usual” method for calculating a block sum of squares Chemical Process Experiment in Three Blocks Block 1Block2Block 3 (1)=28 a=36 b=18 ab=31 Block TotalsB 1 =113B 2 =113B 3 =113

2 Prof. Indrajit Mukherjee, School of Management, IIT Bombay ANOVA for the Blocked Design Page 267 Analysis of variance for the chemical process experiment in the three blocks Source of variation Sum of squares Degrees of freedom Mean squareF0F0 P-value Blocks A(Concentration) B(Catalyst) AB Error Total

3 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Experiment from Example 6.2 Suppose only 8 runs can be made from one batch of raw material Pilot Plant Filtration Value Experiment Run Number FactorsRun label Filtration Rate (gal/h) ABCD 1----(1) a b ab c ac bc abc d ad bd abd cd acd bcd abcd96

4 Prof. Indrajit Mukherjee, School of Management, IIT Bombay The Table of + & - Signs, Example 6-4 ABABCACBCABCDADBDABDCDACDBCDABCD a b ab c ac bc abc d ad bd abd cd acd bcd abcd++++++=+-++++r++ Contrast constant for the 2 4 design

5 Prof. Indrajit Mukherjee, School of Management, IIT Bombay ABCD is Confounded with Blocks (Page 279) Observations in block 1 are reduced by 20 units…this is the simulated “block effect” Block 1Block 2 (1)=25a=71 ab=45b=48 ac=40c=68 bc=60d=43 ad=80abc=65 bd=25bcd=70 cd=55acd=86 abcd=76abd=104 (b) Assignment of the 16 runs to two blocks (a)Geometric View A B C = Runs in Block 1 = Runs in Block 2 D - +

6 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Effect Estimates Effected Estimate for the Blocked 2k Design in Example Model Term Regression Coefficient Effect Estimate Sum of Squares Percent Contribution A B C D AB <0.01 AC AD BC BD <0.01 CD ABC ABD ACD BCD Block (ABCD)

7 Prof. Indrajit Mukherjee, School of Management, IIT Bombay The ANOVA The ABCD interaction (or the block effect) is not considered as part of the error term The reset of the analysis is unchanged from the original analysis Analysis of variance for Example Source of variation Sum of squares Degrees of freedom Mean squareF0F0 P-value Blocks A < C D AC < AD < Error Total

8 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Another Illustration of the Importance of Blocking Now the first eight runs (in run order) have filtration rate reduced by 20 units The Modified Data From Example Run OrderStd Order Factor A Temperature Factor B Pressure Factor C Concentration Factor D Stirring Rate Response Filtration Rate