1Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Blocking a Replicated DesignConsider the example from Section 6-2; k = 2 factors, n = 3 replicates

This is the “usual” method for calculating a block sum of squares

2 23...

1 4 126.50

iBlocks

i

B ySS

Chemical Process Experiment in Three BlocksBlock 1 Block2 Block 3(1)=28 (1)=28 (1)=28a=36 a=36 a=36b=18 b=18 b=18ab=31 ab=31 ab=31

Block Totals B1=113 B2=113 B3=113

2Prof. Indrajit Mukherjee, School of Management, IIT Bombay

ANOVA for the Blocked DesignPage 267

Analysis of variance for the chemical process experiment in the three blocks

Source of variation

Sum of squares

Degrees of freedom

Mean square F0 P-value

Blocks 6.50 2 3.25

A(Concentration)208.33

1 208.33 50.32 0.0004

B(Catalyst)75.00

1 75.00 18.12 0.0053

AB8.33

1 8.33 2.01 0.2060

Error24.84

6 4.14

Total323.00

11

3Prof. 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 ExperimentRun

NumberFactors Run

labelFiltration Rate

(gal/h)A B C D1 - - - - (1) 452 + - - - a 713 - + - - b 484 + + - - ab 655 - - + - c 686 + - + - ac 607 - + + - bc 808 + + + - abc 659 - - - + d 43

10 + - - + ad 10011 - + - + bd 4512 + + - + abd 10413 - - + + cd 7514 + - + + acd 8615 - + + + bcd 7016 + + + + abcd 96

4Prof. Indrajit Mukherjee, School of Management, IIT Bombay

The Table of + & - Signs, Example 6-4

A B AB C AC BC ABC D AD BD ABD CD ACD BCD ABCD1 - - + - + + - - + + - + - - +a + - - - - + + - - + + + + - -b - + - - + - + - + - + + - + -ab + + + - - - - - - - - + + + +c - - + + - - + - + + - - + + -ac + - - + + - - - - + + - - + +bc - + - + - + - - + - + - + - +abc + + + + + + + - - - - - - - -d - - + - + + - + + - + - + + -ad + - - - - + + + - - - - - + +bd - + - - + - + + + + - - + - +abd + + + - - - - + - + + - - - -cd - - + + - - + + + - + + - - +acd + - - + + - - + - - - + + - -bcd - + - + - + - + + + - + - + -abcd + + + + + + = + - + + + +r + +

Contrast constant for the 24 design

5Prof. 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 1 Block 2(1)=25 a=71ab=45 b=48ac=40 c=68bc=60 d=43ad=80 abc=65bd=25 bcd=70cd=55 acd=86

abcd=76 abd=104(b) Assignment of the 16

runs to two blocks

(a) Geometric ViewA

BC = Runs in Block 1 = Runs in Block 2

D - +

6Prof. Indrajit Mukherjee, School of Management, IIT Bombay

Effect EstimatesEffected Estimate for the Blocked 2k Design in Example

Model TermRegression Coefficient

Effect Estimate

Sum of Squares

Percent Contribution

A 10.81 21.625 1870.563 26.3B 1.56 3.125 39.0625 0.55C 4.94 9.875 390.0625 5.49D 7.31 14.625 855.5625 12.03AB 0.062 0.125 0.0625 <0.01AC -9.06 -18.125 1314.063 18.48AD 8.31 16.625 1105.563 15.55BC 1.19 2.375 22.5625 0.32BD -0.19 -0.375 0.5625 <0.01CD -0.56 -1.125 5.0625 0.07ABC 0.94 1.875 14.0625 0.2ABD 2.06 4.125 68.0625 0.96ACD -0.81 -1.625 10.5625 0.15BCD -1.31 -2.625 27.5625 0.39

Block (ABCD) -18.625 1387.563 19.51

7Prof. Indrajit Mukherjee, School of Management, IIT Bombay

The ANOVA

The ABCD interaction (or the block effect) is not considered as part of the error termThe reset of the analysis is unchanged from the original analysis

Analysis of variance for ExampleSource of variation

Sum of squares

Degrees of freedom

Mean square F0 P-value

Blocks 1387.5625 1A

1870.56251 1870.5625 89.76 <0.0001

C390.0625

1 390.0625 18.72 0.0019

D855.5625

1 855.5625 41.05 0.0001

AC1314.0625

1 1314.0625 63.05 <0.0001

AD1105.5625

1 1105.5625 53.05 <0.0001

Error187.5625

9 20.8403

Total711.4375

15

8Prof. 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 Order Std OrderFactor A

TemperatureFactor B Pressure

Factor C Concentration

Factor D Stirring

Rate

Response Filtration

Rate8 1 -1 -1 -1 -1 25

11 2 1 -1 -1 -1 711 3 -1 1 -1 -1 283 4 1 1 -1 -1 459 5 -1 -1 1 -1 68

12 6 1 -1 1 -1 602 7 -1 1 1 -1 60

13 8 1 1 1 -1 657 9 -1 -1 -1 1 236 10 1 -1 -1 1 80

16 11 -1 1 -1 1 455 12 1 1 -1 1 84

14 13 -1 -1 1 1 7515 14 1 -1 1 1 8610 15 -1 1 1 1 704 16 1 1 1 1 76