4. Blocking Factors

Design and Analysis of Experiments

Dr. Tai-Yue Wang Department of Industrial and Information Management

National Cheng Kung UniversityTainan, TAIWAN, ROC

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Experiments with Blocking Factors

Dr. Tai-Yue Wang Department of Industrial and Information Management

National Cheng Kung UniversityTainan, TAIWAN, ROC

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Outline

The Randomized Complete Block Design The Latin Square Design The Graeco-Latin Square Design Balanced Incomplete Block Design

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In some experiment, the variability may arise from factors that we are not interested in.

A nuisance factor ( 擾亂因子 )is a factor that probably has some effect on the response, but it’s of no interest to the experimenter … however, the variability it transmits to the response needs to be minimized

These nuisance factor could be unknown and uncontrolled use randomization

The Randomized Complete Block Design

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If the nuisance factor are known but uncontrollable use the analysis of covariance.

If the nuisance factor are known but controllable use the blocking technique

Typical nuisance factors include batches of raw material, operators, pieces of test equipment, time (shifts, days, etc.), different experimental units


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Many industrial experiments involve blocking (or should)

Failure to block is a common flaw in designing an experiment (consequences?)


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We wish determine whether or not four different tips produce different readings on a hardness testing machine.

One factor to be consider tip type Completely Randomized Design could be

used with one potential problem the testing block could be different

The experiment error could include both the random and coupon errors.

The Randomized Complete Block Design-example

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To reduce the error from testing coupon, randomize complete block design(RCBD) is used


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Each coupon is called a “block”; that is, it’s a more homogenous experimental unit on which to test the tips

“complete” indicates each testing coupon (BLOCK) contains all treatments

Variability between blocks can be large, variability within a block should be relatively small

In general, a block is a specific level of the nuisance factor


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A complete replicate of the basic experiment is conducted in each block

A block represents a restriction on randomization

All runs within a block are randomized Once again, we are interested in testing the

equality of treatment means, but now we have to remove the variability associated with the nuisance factor (the blocks)


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Suppose that there are a treatments (factor levels) and b blocks

The Randomized Complete Block Design– Extension from ANOVA

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Suppose that there are a treatments (factor levels) and b blocks


1,2,...,

1, 2,...,ij i j ij

i ay

j b

0 1 2 1: where (1/ ) ( )

b

a i i j ijH b

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A statistical model (effects model) for the RCBD is

The relevant (fixed effects) hypotheses are

1,2,...,

1, 2,...,ij i j ij

i ay

j b

0 1 2 1: where (1/ ) ( )

b

a i i j ijH b


jiH oneleast at :1

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Or


0 oneleast at :

0:

1

210

i

a

H

H

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Partitioning the total variability


2.. . .. . ..

1 1 1 1

2. . ..

2 2. .. . ..

1 1

2. . ..

1 1

( ) [( ) ( )

( )]

( ) ( )

( )

a b a b

ij i ji j i j

ij i j

a b

i ji j

a b

ij i ji j

T Treatments Blocks E

y y y y y y

y y y y

b y y a y y

y y y y

SS SS SS SS

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The degrees of freedom for the sums of squares in

are as follows:

Therefore, ratios of sums of squares to their degrees of freedom result in mean squares and the ratio of the mean square for treatments to the error mean square is an F statistic that can be used to test the hypothesis of equal treatment means

T Treatments Blocks ESS SS SS SS

1 1 1 ( 1)( 1)ab a b a b


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Mean squares


2

1

2

2

1

2

2

1

1

E

a

ij

Block

a

ii

treatment

MSE

b

aMSE

a

bMSE

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F-test with (a-1), (a-1)(b-1) degree of freedom

Reject the null hypothesis if

F0>F α,a-1,(a-1)(b-1)


E

Treatments

MS

MSF 0

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ANOVA Table


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Manual computing:


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Meaning of F0=MSBlocks/MSE? The randomization in RBCD is applied only to

treatment within blocks The Block represents a restriction on

randomization Two kinds of controversial theories

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Meaning of F0=MSBlocks/MSE? General practice, the block factor has a large effect

and the noise reduction obtained by blocking was probably helpful in improving the precision of the comparison of treatment means if the ration is large

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The Randomized Complete Block Design– Example

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To conduct this experiment as a RCBD, assign all 4 pressures to each of the 6 batches of resin

Each batch of resin is called a “block”; that is, it’s a more homogenous experimental unit on which to test the extrusion pressures

The Randomized Complete Block Design– Example

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The Randomized Complete Block Design– Example—Minitab

StatANOVATwo-way

Vascular-Graft.MTW

Two-way ANOVA: Yield versus Pressure, Batch

Source DF SS MS F PPressure 3 178.171 59.3904 8.11 0.002Batch 5 192.252 38.4504 5.25 0.006Error 15 109.886 7.3258Total 23 480.310

S = 2.707 R-Sq = 77.12% R-Sq(adj) = 64.92%

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Vascular-Graft.MTW

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Basic residual plots indicate that normality, constant variance assumptions are satisfied

No obvious problems with randomization No patterns in the residuals vs. block Can also plot residuals versus the pressure

(residuals by factor) These plots provide more information about

the constant variance assumption, possible outliers

The Randomized Complete Block Design– Example —Residual Analysis

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Vascular-Graft.MTW

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The Randomized Complete Block Design– Example —No Blocking

StatANOVAOne-way

Vascular-Graft.MTW

One-way ANOVA: Yield versus Pressure

Source DF SS MS F PPressure 3 178.2 59.4 3.93 0.023Error 20 302.1 15.1Total 23 480.3

S = 3.887 R-Sq = 37.10% R-Sq(adj) = 27.66%

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The Randomized Complete Block Design– Example—No Blocking-Residual

Vascular-Graft.MTW

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化學品樣品

類別 1 2 3 4 5 yi . yi . 1 1.3 1.6 0.5 1.2 1.1 5.7 1.14 2 2.2 2.4 0.4 2.0 1.8 8.8 1.76 3 1.8 1.7 0.6 1.5 1.3 6.9 1.38 4 3.9 4.4 2.0 4.1 3.4 17.8 3.56 y j. 9.2 10.1 3.5 8.8 7.6 39.2 1.96

y j. 2.30 2.53 0.88 2.20 1.90 y.. y..

The Randomized Complete Block Design– Other Example

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SS yy

abT ijji

2

1

5

1

4 2

2 2 22

13 16 3 439 2

2025 69

..

( . ) ( . ) ( . )( . )

.

SSy

b

y

abFactorsi

i

a

. ..

( . ) ( . ) ( . ) ( . ) ( . ).

2

1

2

2 2 2 2 257 8 8 6 9 17 85

39 220

18 04

SSy

a

y

abBlocksj

j

b

. ..

( . ) ( . ) ( . ) ( . ) ( . ) ( . ).

2

1

2

2 2 2 2 2 29 2 10 1 3 5 8 8 7 64

39 220

6 69

SS SS SS SSE T Blocks Factors 0 96.


Two-way ANOVA: 濃度 versus 化學品類別 , 樣品

Source DF SS MS F P化學品類別 3 18.044 6.01467 75.89 0.000樣品 4 6.693 1.67325 21.11 0.000Error 12 0.951 0.07925Total 19 25.688

S = 0.2815 R-Sq = 96.30% R-Sq(adj) = 94.14%

One-way ANOVA: 濃度 versus 化學品類別

Source DF SS MS F P化學品類別 3 18.044 6.015 12.59 0.000Error 16 7.644 0.478Total 19 25.688

S = 0.6912 R-Sq = 70.24% R-Sq(adj) = 64.66%


Blocking effect

Without blocking effect


Blocking effect

Without blocking effect

Source of Variation

Sum of Squares

自由度

df

變異數

MS

F0

化學品 18.04 3 6.01 75.13

集區 6.69 4 1.67

誤差 0.96 12 0.08

總和 25.69 19

Source of Variation

Sum of Squares

自由度

df

變異數

MS

F0

化學品 18.04 3 6.01 12.59

誤差 7.65 16 0.48

總和 25.69 19

The RCBD utilizes an additive model – no interaction between treatments and blocks

Treatments and/or blocks as random effects Missing values What are the consequences of not blocking if we

should have?

The Randomized Complete Block Design– Other Aspects

1,2,...,

1, 2,...,ij i j ij

i ay

j b

Sample sizing in the RCBD? The OC curve approach can be used to determine the number of blocks to run..see page 133

The Randomized Complete Block Design– Other Aspects

1,2,...,

1, 2,...,ij i j ij

i ay

j b

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The Latin Square Design

These designs are used to simultaneously control (or eliminate) two sources of nuisance variability

Those two sources of nuisance factors have exactly same levels of factor to be considered

A significant assumption is that the three factors (treatments, nuisance factors) do not interact

If this assumption is violated, the Latin square design will not produce valid results

Latin squares are not used as much as the RCBD in industrial experimentation

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The Latin Square Design

The Latin square design systematically allows blocking in two directions

In general, a Latin square for p factors is a square containing p rows and p columns.

Each cell contain one and only one of p letters that represent the treatments.

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A Latin Square Design – The Rocket

Propellant

This is a 5 5 Latin square design

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Statistical Analysis of the Latin Square Design

The statistical (effects) model is

The statistical analysis (ANOVA) is much like the analysis for the RCBD.

1,2,...,

1, 2,...,

1, 2,...,ijk i j k ijk

i p

y j p

k p

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The Standard Latin Square Design

A square with first row and column in alphabetical order.

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Other Topics

Missing values in blocked designs RCBD Latin square Estimated by

)1)(1(

2)( '...

'..

'..

'..

pp

yyyypy kji

ijk

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Other Topics

Replication of Latin Squares To increase the error degrees of freedom Three methods

1. Use the same batches and operators in each replicate

2. Use the same batches but different operators in each replicate

3. Use different batches and different operator

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Other Topics

Replication of Latin Squares ANOVA in Case 1

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Other Topics

Replication of Latin Squares ANOVA n Case 2

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Other Topics

Replication of Latin Squares ANOVA n Case 3

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Other Topics

Crossover design p treatments to be tested in p time periods using

np experiment units. Ex : 20 subjects to be assigned to two periods

First half of the subjects are assigned to period 1 (in random) and the other half are assigned to period 2.

Take turn after experiments are done.

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Other Topics

Crossover design ANOVA

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Graeco-Latin Square

For a pxp Latin square, one can superimpose a second pxp Latin square that treatments are denoted by Greek letters.

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Graeco-Latin Square

If the two squares have the property that each Greek letter appears once and only once with each Latin letter, the two Latin squares are to be orthogonal and this design is named as Graeco-Latin Square.

It can control three sources of extraneous variability.

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Graeco-Latin Square

ANOVA

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Graeco-Latin Square --Example

In the rocket propellant problem, batch of material, operators, and test assemblies are important.

If 5 of them are considered, a Graeco-Latn square can be used.

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4. Blocking Factors

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