5. blocking and confounding (ch.7 blocking and...

Hae-Jin Choi School of Mechanical Engineering,

Chung-Ang University

5. Blocking and Confounding

(Ch.7 Blocking and Confounding Systems for Two-Level Factorials )

DOE and Optimization 1

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Why Blocking?

Blocking is a technique for dealing with controllable nuisance variables

Sometimes, it is impossible to perform all 2k factorial experiments under homogeneous condition

Blocking technique is used to make the treatments are equally effective across many situation

DOE and Optimization

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What is Blocking?

Each set of non-homogeneous conditions define a block and each replicate is run in one of blocks.

If there are n replicates of the design, then each replicate is a block

Each replicate is run in one of the blocks (time periods, batches of raw material, etc.)

Runs within the block are randomized


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

2 23...

1 4 126.50

iBlocks

i

B ySS=

= −

=

∑Chemical Processing

Concentration (A) Catalyst (B)

Filtration rate (response)


ANOVA for the Blocked Design


Confounding


In may case, it is impossible to perform a complete replicate of a factorial design in one block

Block size smaller than the number of treatment combinations in one replicate.

Confounding is a design technique for arranging experiments to make high-order interactions to be indistinguishable from(or confounded with) blocks.

Confounding in the 2k factorial Design


1 [ ( )]21 [ ( )]2

1 [ ( ) ]2

A a b

B b a

AB a b

= + − −

= + − −

= + − −

ab 1

ab 1

ab 1

A and B are Unaffected by blocks. One plus and one minus from each block -> block effect is cancelled out

AB is Confounded with blocking Same sign from each block -> block effect is not cancelled out

With two factors and two blocks



1 [ ( )]21 [ ( )]2

1 [ ( ) ]2

A a b

B b a

AB a b

= + − −

= + − −

= + − −

ab 1

ab 1

ab 1

A and B are Unaffected by blocks. One plus and one minus from each block -> block effect is cancelled out

AB is Confounded with blocking Same sign from each block -> block effect is not cancelled out

With two factors and two blocks


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With three factors and two blocks


How to assign the blocks in 2k factorials?

10 Confound with High-order Interaction term DOE and Optimization

Other method for construct the blocks

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Linear combination with

L= a1x1+a2x2+ … + akxk where xi = level of the ith factor ai = the exponent appearing on the ith factor in the effect to be confounded

Example Confounded with ABC in 23 Factorial Design (a1=1, a2=1, a3=1)

(1) : L = 1(0) + 1(0) + 1(0) = 0 -> Block 1 a : L = 1(1) + 1(0) + 1(0) = 1 -> Block 2 ac : L = 1(1) + 1(0) + 1(1) = 2 = 0 -> Block 1 abc : L= 1(1) + 1(0) + 1(0) = 3 = 1 -> Block 2

Aa1Ba2Ca3


Example of an Unreplicated 2k Design (repeated) A chemical product is produced in a pressure vessel. A factorial

experiment is carried out in the pilot plant to study the factors thought to influence the filtration rate of this product .

The factors are A = temperature, B = pressure, C = mole ratio, D= stirring rate

A 24 factorial was used to investigate the effects of four factors on the filtration rate of a resin

Experiment was performed in a pilot plant


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The Table of + & - Signs

Confound with interaction effect ABCD


ABCD is Confounded with Blocks

Observations in block 1 are reduced by 20 units…this is the simulated “block effect”

14 DOE and Optimization

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Effect Estimates

‘Block (ABCD)’ = ‘original ABCD’- 20 = 1.375-20 = -18.625 Or ‘Block (ABCD)’ = ӯblock1 - ӯ block2


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


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Without blocking, what happen??

Now the first eight runs (in run order) have filtration rate reduced by 20 units


The interpretation is harder; not as easy to identify the large effects

One important interaction is not identified (AD)

Failing to block when we should have causes problems in interpretation the result of an experiment and can mask the presence of real factor effects


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Confounding in More than Two Blocks

More than two blocks (page 313) The two-level factorial can be confounded in 2, 4, 8,

… (2p, p > 1) blocks For four blocks, select two effects to confound,

automatically confounding a third effect See example, page 313 Choice of confounding schemes non-trivial; see Table

7.9, page 316


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General Advice About Blocking

When in doubt, block

Block out the nuisance variables you know about, randomize as much as possible and rely on randomization to help balance out unknown nuisance effects

Measure the nuisance factors you know about but can’t control

It may be a good idea to conduct the experiment in blocks even if there isn't an obvious nuisance factor, just to protect against the loss of data or situations where the complete experiment can’t be finished


5. blocking and confounding (ch.7 blocking and...

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