Incomplete Block Designs

Randomized Block Design

• We want to compare t treatments• Group the N = bt experimental units into b

homogeneous blocks of size t.• In each block we randomly assign the t treatments

to the t experimental units in each block.• The ability to detect treatment to treatment

differences is dependent on the within block variability.

Comments• The within block variability generally increases

with block size.• The larger the block size the larger the within

block variability.• For a larger number of treatments, t, it may not be

appropriate or feasible to require the block size, k, to be equal to the number of treatments.

• If the block size, k, is less than the number of treatments (k < t)then all treatments can not appear in each block. The design is called an Incomplete Block Design.

Commentsregarding Incomplete block designs

• When two treatments appear together in the same block it is possible to estimate the difference in treatments effects.

• The treatment difference is estimable.

• If two treatments do not appear together in the same block it not be possible to estimate the difference in treatments effects.

• The treatment difference may not be estimable.

Example• Consider the block design with 6 treatments

and 6 blocks of size two.

• The treatments differences (1 vs 2, 1 vs 3, 2 vs 3, 4 vs 5, 4 vs 6, 5 vs 6) are estimable.

• If one of the treatments is in the group {1,2,3} and the other treatment is in the group {4,5,6}, the treatment difference is not estimable.

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Definitions

• Two treatments i and i* are said to be connected if there is a sequence of treatments i0 = i, i1, i2, … iM = i* such that each successive pair of treatments (ij and ij+1) appear in the same block

• In this case the treatment difference is estimable.

• An incomplete design is said to be connected if all treatment pairs i and i* are connected.

• In this case all treatment differences are estimable.

Example• Consider the block design with 5 treatments

and 5 blocks of size two.

• This incomplete block design is connected.

• All treatment differences are estimable.

• Some treatment differences are estimated with a higher precision than others.

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Analysis of unbalanced Factorial Designs

Type I, Type II, Type III

Sum of Squares

Sum of squares for testing an effect

modelComplete ≡ model with the effect in.

modelReduced ≡ model with the effect out.

Reduced Completemodel modelEffectSS RSS RSS

Type I SS

• Type I estimates of the sum of squares associated with an effect in a model are calculated when sums of squares for a model are calculated sequentially

Example

• Consider the three factor factorial experiment with factors A, B and C.

The Complete model

• Y = + A + B + C + AB + AC + BC + ABC

A sequence of increasingly simpler models

1. Y = + A + B + C + AB + AC + BC + ABC

2. Y = + A+ B + C + AB + AC + BC

3. Y = + A + B+ C + AB + AC

4. Y = + A + B + C+ AB

5. Y = + A + B + C

6. Y = + A + B

7. Y = + A

8. Y =

Type I S.S.

2 1model modelABCSS RSS RSS I

3 2model modelBCSS RSS RSS I

4 3model modelACSS RSS RSS I

5 4model modelABSS RSS RSS I

6 5model modelCSS RSS RSS I

7 6model modelBSS RSS RSS I

8 7model modelASS RSS RSS I

Type II SS

• Type two sum of squares are calculated for an effect assuming that the Complete model contains every effect of equal or lesser order. The reduced model has the effect removed ,

The Complete models

1. Y = + A + B + C + AB + AC + BC + ABC (the three factor model)

2. Y = + A+ B + C + AB + AC + BC (the all two factor model)

3. Y = + A + B + C (the all main effects model)

The Reduced models

For a k-factor effect the reduced model is the all k-factor model with the effect removed

2 1model modelABCSS RSS RSS II

2modelABSS RSS Y A B C AC BC RSS II

3modelASS RSS Y B C RSS II

2modelACSS RSS Y A B C AB BC RSS II

2modelBCSS RSS Y A B C AB AC RSS II

3modelBSS RSS Y A C RSS II

3modelCSS RSS Y A B RSS II

Type III SS

• The type III sum of squares is calculated by comparing the full model, to the full model without the effect.

Comments

• When using The type I sum of squares the effects are tested in a specified sequence resulting in a increasingly simpler model. The test is valid only the null Hypothesis (H0) has been accepted in the previous tests.

• When using The type II sum of squares the test for a k-factor effect is valid only the all k-factor model can be assumed.

• When using The type III sum of squares the tests require neither of these assumptions.

An additional Comment

• When the completely randomized design is balanced (equal number of observations per treatment combination) then type I sum of squares, type II sum of squares and type III sum of squares are equal.

Example

• A two factor (A and B) experiment, response variable y.

• The SPSS data file

Using ANOVA SPSS package

Select the type of SS using model

ANOVA table – type I S.S

Tests of Between-Subjects Effects

Dependent Variable: Y

11545.858a 8 1443.232 45.554 .000

61603.201 1 61603.201 1944.440 .000

3666.552 2 1833.276 57.865 .000

809.019 2 404.509 12.768 .000

7070.287 4 1767.572 55.792 .000

760.361 24 31.682

73909.420 33

12306.219 32

SourceCorrected Model

Intercept

A

B

A * B

Error

Total

Corrected Total

Ty pe I Sumof Squares df

MeanSquare F Sig.

R Squared = .938 (Adjusted R Squared = .918)a.

ANOVA table – type II S.S

Tests of Between-Subjects Effects

Dependent Variable: Y

11545.858a 8 1443.232 45.554 .000

61603.201 1 61603.201 1944.440 .000

3358.643 2 1679.321 53.006 .000

809.019 2 404.509 12.768 .000

7070.287 4 1767.572 55.792 .000

760.361 24 31.682

73909.420 33

12306.219 32

SourceCorrected Model

Intercept

A

B

A * B

Error

Total

Corrected Total

Ty pe IISum ofSquares df

MeanSquare F Sig.

R Squared = .938 (Adjusted R Squared = .918)a.

ANOVA table – type III S.S

Tests of Between-Subjects Effects

Dependent Variable: Y

11545.858a 8 1443.232 45.554 .000

52327.002 1 52327.002 1651.647 .000

2812.027 2 1406.013 44.379 .000

1010.809 2 505.405 15.953 .000

7070.287 4 1767.572 55.792 .000

760.361 24 31.682

73909.420 33

12306.219 32

SourceCorrec ted Model

Intercept

A

B

A * B

Error

Total

Correc ted Total

Ty pe IIISum ofSquares df

MeanSquare F Sig.

R Squared = .938 (Adjusted R Squared = .918)a.

Incomplete Block Designs

Balanced incomplete block designs

Partially balanced incomplete block designs

DefinitionAn incomplete design is said to be a Balanced Incomplete Block Design.

1. if all treatments appear in exactly r blocks.• This ensures that each treatment is estimated with

the same precision• The value of is the same for each treatment pair.

2. if all treatment pairs i and i* appear together in exactly blocks.• This ensures that each treatment difference is

estimated with the same precision.• The value of is the same for each treatment pair.

Some IdentitiesLet b = the number of blocks.

t = the number of treatmentsk = the block sizer = the number of times a treatment appears in the experiment. = the number of times a pair of treatment appears together in the same block

1. bk = rt• Both sides of this equation are found by counting the

total number of experimental units in the experiment.

2. r(k-1) = (t – 1)• Both sides of this equation are found by counting the

total number of experimental units that appear with a specific treatment in the experiment.

BIB DesignA Balanced Incomplete Block Design(b = 15, k = 4, t = 6, r = 10, = 6)

Block Block Block 1 1 2 3 4 6 3 4 5 6 11 1 3 5 6

2 1 4 5 6 7 1 2 3 6 12 2 3 4 6

3 2 3 4 6 8 1 3 4 5 13 1 2 5 6

4 1 2 3 5 9 2 4 5 6 14 1 3 4 6

5 1 2 4 6 10 1 2 4 5 15 2 3 4 5

An Example A food processing company is interested in comparing the taste of six new brands (A, B, C, D, E and F) of cereal.

For this purpose: • subjects will be asked to taste and compare these

cereals scoring them on a scale of 0 - 100. • For practical reasons it is decided that each subject

should be asked to taste and compare at most four of the six cereals.

• For this reason it is decided to use b = 15 subjects and a balanced incomplete block design to assess the differences in taste of the six brands of cereal.

The design and the data is tabulated below:

Subject Taste Scores (Brands)

1 51 (A) 55 (B) 69 (C) 83 (D) 2 48 (A) 87 (D) 56 (E) 22 (F) 3 65 (B) 91 (C) 67 (E) 35 (F) 4 42 (A) 48 (B) 65 (C) 43 (E) 5 36 (A) 58 (B) 69 (D) 7 (F) 6 79 (C) 85 (D) 56 (E) 25 (F) 7 54 (A) 60 (B) 90 (C) 21 (F) 8 62 (A) 92 (C) 94 (D) 63 (E) 9 39 (B) 71 (D) 47 (E) 11 (F) 10 51 (A) 59 (B) 84 (D) 51 (E) 11 39 (A) 74 (C) 61 (E) 25 (F) 12 69 (B) 78 (C) 78 (D) 22 (F) 13 63 (A) 74 (B) 59 (E) 32 (F) 14 55 (A) 74 (C) 78 (D) 34 (F) 15 73 (B) 83 (C) 92 (D) 68 (E)

Analysis for the Incomplete Block Design

Recall that the parameters of the design where b = 15, k = 4, t = 6, r = 10, = 6

Block Totals j 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 G

Bj 258 213 258 198 170 245 225 311 168 245 199 247 228 241 316 3522

Treat Totals and Estimates of Treatment Effects

Treat Treat Total (Ti) j(i) Bj/k Diff = Qi Treat Effects (i)

(A) 501 572 -71 -7.89 (B) 600 578.25 21.75 2.42 (C) 795 624.5 170.5 18.94 (D) 821 603.5 217.5 24.17 (E) 571 595.25 -24.25 -2.69 (F) 234 548.5 -314.5 -34.94

)(ij denotes summation over all blocks j containing treatment i.

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Anova Table for Incomplete Block Designs Sums of Squares

yij2 = 234382

Bj2/k = 213188

Qi2 = 181388.88

Anova Sums of Squares

SStotal = yij2 –G2/bk = 27640.6

SSBlocks = Bj2/k – G2/bk = 6446.6

SSTr = (Qi2 )/(r – 1) = 20154.319

SSError = SStotal - SSBlocks - SSTr = 1039.6806

Anova Table for Incomplete Block Designs

S o u r c e S S d f M S F

B l o c k s 6 4 4 6 . 6 0 1 4 4 6 0 . 4 7 1 7 . 7 2 T r e a t 2 0 1 5 4 . 3 2 5 4 0 3 0 . 8 6 1 5 5 . 0 8 E r r o r 1 0 3 9 . 6 8 4 0 2 5 . 9 9

T o t a l 2 7 6 4 0 . 6 0 5 9

Next Topic: Designs for Estimating Residual Effects