1. 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. 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 Blocks6.5023.25 A(Concentration) 208.33 1 50.320.0004 B(Catalyst) 75.00 1 18.120.0053 AB 8.33 1 2.010.2060 Error 24.84 64.14 Total 323.00 11

3. 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)45 2+---a71 3-+--b48 4++--ab65 5--+-c68 6+-+-ac60 7-++-bc80 8+++-abc65 9---+d43 10+--+ad100 11-+-+bd45 12++-+abd104 13--++cd75 14+-++acd86 15-+++bcd70 16++++abcd96

4. 4 Prof. Indrajit Mukherjee, School of Management, IIT Bombay The Table of + & - Signs, Example 6-4 ABABCACBCABCDADBDABDCDACDBCDABCD 1--+-++--++-+--+ a+----++--++++-- b-+--+-+-+-++-+- ab+++--------++++ c--++--+-++--++- ac+--++----++--++ bc-+-+-+--+-+-+-+ abc+++++++-------- d--+-++-++-+-++- ad+----+++-----++ bd-+--+-++++--+-+ abd+++----+-++---- cd--++--+++-++--+ acd+--++--+---++-- bcd-+-+-+-+++-+-+- abcd++++++=+-++++r++ Contrast constant for the 2 4 design

5. 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. 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 A10.8121.6251870.56326.3 B1.563.12539.06250.55 C4.949.875390.06255.49 D7.3114.625855.562512.03 AB0.0620.1250.0625<0.01 AC-9.06-18.1251314.06318.48 AD8.3116.6251105.56315.55 BC1.192.37522.56250.32 BD-0.19-0.3750.5625<0.01 CD-0.56-1.1255.06250.07 ABC0.941.87514.06250.2 ABD2.064.12568.06250.96 ACD-0.81-1.62510.56250.15 BCD-1.31-2.62527.56250.39 Block (ABCD)-18.6251387.56319.51

7. 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 Blocks1387.56251 A 1870.5625 1 89.76<0.0001 C 390.0625 1 18.720.0019 D 855.5625 1 41.050.0001 AC 1314.0625 1 63.05<0.0001 AD 1105.5625 1 53.05<0.0001 Error 187.5625 920.8403 Total 711.4375 15

8. 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 81 25 1121 71 131 28 3411 45 95 1 68 12611 60 2711 60 13811165 79 123 6101 180 16111 145 51211184 1413 1175 151411186 101511170 416111176