1

Blocking & Confounding in the 2k Factorial Design

• Text reference, Chapter 7

• Blocking is a technique for dealing with controllable nuisance variables

• Two cases are considered– Replicated designs– Unreplicated designs

2

Blocking a Replicated Design

• This is the same scenario discussed previously (Chapter 5, Section 5-6)

• 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

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

2 23...

1 4 12

6.50

iBlocks

i

B ySS

Section 6-2

4

ANOVA for the Blocked Design Page 267

Analysis of variance table [Partial sum of squares]

Sum of Mean FSource Squares DF Square Value Prob > FA 208.33 1 208.33 53.19 < 0.0001B 75.00 1 75.00 19.15 0.0024AB 8.33 1 8.33 2.13 0.1828Pure Error 31.33 8 3.92Cor Total 323.00 11

Section 6-2

5

Confounding in Blocks

• Now consider the unreplicated case

• Clearly the previous discussion does not apply, since there is only one replicate

• To illustrate, consider the situation of Example 6-2, Page 228

• This is a 24, n = 1 replicate

6

Confounding in Two Blocks

• A single replicate 22 design

• Each raw material is only enough for two runs – needs two materials (blocks)

A possible design

7

Confounding in Two Blocks

• Estimating effects

A = ½[ab + a – b – (1)]

B = ½[ab + b – a – (1)]

AB = ½[ab + (1) – a – b]

8

Example of Confounding for a 23 design in Two Blocks

9

Other Methods of Constructing the Blocks

• Use of a defining contrast

L = 1x1 + 2x2 + 3x3 + …+ kxk

xi: level of the ith factor in a particular treatment combination (0 or 1)

i: exponent appearing on the ith factor in the effect to be confounded (0 or 1)

• Treatment combinations that produce the same value of L (mod 2) will be placed in the same block

10

Other Methods of Constructing the Blocks

• Example: a 23 design with ABC confounded with blocks

1 = 1; 2 = 1; 3 = 1

x1 A; x2 B; x3 C;

L = x1 + x2 + x3

(1) 000: L = 0 = 0 a 100: L = 1 = 1

ab 110: L = 2 = 0 b 010: L = 1 = 1

ac 101: L = 2 = 0 c 001: L = 1 = 1

bc 011: L = 2 = 0 abc 111: L = 3 = 1

11

Estimation of Error

• Example: a 23 design, must be run in two blocks with ABC confounded, four replicates

12

• It would be better if blocks are designed differently in each replicate, to confound a different effect in each replicate – partial confounding

13

Example of Unreplicated Design (Ex. 7-2)

• Response: filtration rate of a resin

• Factors: A = temperature, B = pressure, C = mole ratio/concentration, D= stirring rate

• One batch of raw material is only enough for 8 runs. Two materials are required.

• ABCD is chosen for confounding.

• L = x1 + x2 + x3 + x4

14

Example 6-2

Suppose only 8 runs can be made from one batch of a raw material

15

Construction and analysis of the 2k factorial design in 2p incomplete blocks (p<k).

The Table of + & - Signs, Example 6-2

16

ABCD is Confounded with

Blocks (Page 272)

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

17

Effect Estimates

Term Effect SumSqr

A 21.625 1870.56 B 3.125 39.0625 C 9.875 390.062 D 14.625 855.563 AB 0.125 0.0625 AC -18.125 1314.06 AD 16.625 1105.56 BC 2.375 22.5625 BD -0.375 0.5625 CD -1.125 5.0625 ABC 1.875 14.0625 ABD 4.125 68.0625 ACD -1.625 10.5625 BCD -2.625 27.5625 ABCD 1.375 7.5625

Example 6-2

18

The ANOVA

The ABCD interaction (or the block effect) is not considered as part of the error term

The rest of the analysis is unchanged from Example 6-2