5. Blocking and Confounding (Ch.7 Blocking and Confounding Systems for Two-Level Factorials )
Hae-Jin Choi School of Mechanical Engineering, Chung-Ang University
DOE and Optimization 1 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 2 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
DOE and Optimization 3 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
Concentration (A) Catalyst (B)
3 22 Byi ... SSBlocks Chemical i1 4 12 Processing Filtration rate 6.50 (response)
DOE and Optimization 4 ANOVA for the Blocked Design
DOE and Optimization 5 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.
DOE and Optimization 6 Confounding in the 2k factorial Design
With two factors and two blocks
1 A[ab a b ( 1 )] 2 A and B are Unaffected by blocks. One plus and one minus from each block 1 B[ab b a ( 1 )] -> block effect is cancelled out 2 1 AB is Confounded with blocking AB[()]ab 1 a b 2 Same sign from each block -> block effect is not cancelled out
DOE and Optimization 7 Confounding in the 2k factorial Design
With two factors and two blocks
1 A[ab a b ( 1 )] 2 A and B are Unaffected by blocks. One plus and one minus from each block 1 B[ab b a ( 1 )] -> block effect is cancelled out 2 1 AB is Confounded with blocking AB[()]ab 1 a b 2 Same sign from each block -> block effect is not cancelled out
DOE and Optimization 8 Confounding in the 2k factorial Design
With three factors and two blocks
DOE and Optimization 9 How to assign the blocks in 2k factorials?
DOE and Optimization Confound with High-order Interaction term 10 Other method for construct the blocks
Linear combination with
L= a1x1+a2x2+ … + akxk where th xi = level of the i factor th ai = the exponent appearing on the i factor in the effect to be confounded
Aa1Ba2Ca3 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(1) + 1(1) = 3 = 1 -> Block 2
DOE and Optimization 11 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
DOE and Optimization 12 The Table of + & - Signs
Confound with interaction effect ABCD
DOE and Optimization 13 ABCD is Confounded with Blocks
Observations in block 1 are reduced by 20 units…this is the simulated “block effect”
DOE and Optimization 14 Effect Estimates
‘Block (ABCD)’ = ‘original ABCD’- 20 = 1.375-20 = -18.625 Or
‘Block (ABCD)’ = ӯblock1 - ӯ block2
DOE and Optimization 15 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
DOE and Optimization 16 Without blocking, what happen??
Now the first eight runs (in run order) have filtration rate reduced by 20 units
DOE and Optimization 17 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
DOE and Optimization 18 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
DOE and Optimization 19 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
DOE and Optimization 20