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THE RANDOMIZEDCOMPLETE BLOCK DESIGN

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Example

Suppose we want to test whether four different laboratories

produce the same result on analyzing a given sample.

As the result may be affected by the sample, each sample is

measured by the four laboratories.

Results are

Laboratory 1 2 3 4

1 9.3 9.4 9.6 10

2 9.4 9.3 9.8 9.9

3 9.2 9.4 9.5 9.7

4 9.7 9.6 10 10.2

Sample

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Laboratory 1 2 3 4

1 9.3 9.4 9.6 10

2 9.4 9.3 9.8 9.9

3 9.2 9.4 9.5 9.7

4 9.7 9.6 10 10.2

Sample

Data entry in R

Each row represents a result of one lab in one sample.

If replicate exists, then we will have more that one row with

the same lab and sample (which is not the case here).

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Example

Each sample contains uncontrolled effects that we are

interested in separate from the main effect, i.e. the results of

each laboratory.

Thus, the samples are BLOCKS and their variability will be

subtracted from the error.

Block may differ substantially one from the other.

They do not represent random samples of a given population.

This is a randomized complete block design because each

sample is tested by all the laboratories.

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Analysis with block effect

Analysis without block effect

Data analysis

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The importance of randomization

Four different brands of tires:A,B,C,D

Do they exihbit the sametread loss after 20000 milesof driving?

As the tires muts be tried indifferent cars, we decide touse four sets of tires in thesame car.

The experimental design is

indicated in the table 1. This iscompletely confoundeddesign.

A complete random design isindicated in 2

Table 1. Cars

1 2 3 4

A B C D

A B C D

A B C D

A B C D

Table 2. Cars

1 2 3 4

C A D A

A A C D

D B B B

D C B C

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Completely randomized

Brand A is never used on car 3

Brand B is never used on car 1

Any variation within Brand A may reflect variation

between cars 1,2 and 4.

Table 2. Cars

1 2 3 4

C A D A

A A C D

D B B B

D C B C

Brands

A B C D

171 142 121 131

142 143 122 111

132 133 103 113

134 84 94 94

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Data entry

Brands

A B C D

171 142 121 131

142 143 122 111

132

133

103

113

134 84 94 94

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ANOVA

Results are not significative.

There are no differences due to the brand of the tire.

The error term may reflect a variability between cars.

The completely randomized design averaged out the car

effects, but it did not elimiante the variance among cars.

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Completely randomized block design

The four brands are tested in each car randomly.

Cars are BLOCKS in this design.

We analyze this as a two-way ANOVA

Note that the numbers are the same, but know theycome from a different design.

Table 2. Cars

1 2 3 4

B D A C

C C B D

A B D B

D A C A

Brands

A B C D

CAR 1 17 14 12 13

2 14 14 12 11

3 13 13 10 11

4 13 8 9 9

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Data entry

Brands

A B C D

CAR 1 17 14 12 13

2 14 14 12 113 13 13 10 11

4 13 8 9 9

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ANOVA

The effect of the car is significative.

As we substract the car variability from the error term,

the estimated error variance has decreased.

The four brands show difference in tread loss

(p=0.007).

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Comparisons between brands

Brand A shows a higher tread loss than C

and D, but not B.

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THE LATIN SQUARE DESIGN

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The latin square design

This is an appropriate design when we want to block two

different sources of variation.

The two block adds uncontrolled noise to the data and may

confound the main effect.

For instance, consider that five procedures (A,B,C,D,E) are to be

tested by five operators in five equivalent samples. Since both

samples and operators introduce some variability, we use a

latin square design

Samples 1 2 3 4 5

1 A B C D E

2 B C D E A

3 C D E A B

4 D E A B C

5 E A B C D

Operators

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Data entry

Samples 1 2 3 4 5

1 A (24) B (20) C (19) D (24) E (24)

2 B (17) C (24) D (30) E (27) A (36)

3 C (18) D (38) E (26) A (27) B (21)

4 D (26) E (31) A (26) B (23) C (22)5 E (22) A (30) B (20) C (29) D (31)

Operators

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The latin square design

Samples 1 2 3 4 5

1 A (24) B (20) C (19) D (24) E (24)

2 B (17) C (24) D (30) E (27) A (36)

3 C (18) D (38) E (26) A (27) B (21)

4 D (26) E (31) A (26) B (23) C (22)5 E (22) A (30) B (20) C (29) D (31)

Operators

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The latin square design