incomplete block designs. randomized block design we want to compare t treatments group the n = bt...
TRANSCRIPT
![Page 1: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/1.jpg)
Incomplete Block Designs
![Page 2: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/2.jpg)
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.
![Page 3: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/3.jpg)
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.
![Page 4: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/4.jpg)
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.
![Page 5: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/5.jpg)
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.
1
2
2
3
1
3
4
5
5
6
4
6
![Page 6: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/6.jpg)
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.
![Page 7: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/7.jpg)
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.
1
2
2
3
1
3
4
5
1
4
![Page 8: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/8.jpg)
Analysis of unbalanced Factorial Designs
Type I, Type II, Type III
Sum of Squares
![Page 9: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/9.jpg)
Sum of squares for testing an effect
modelComplete ≡ model with the effect in.
modelReduced ≡ model with the effect out.
Reduced Completemodel modelEffectSS RSS RSS
![Page 10: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/10.jpg)
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
![Page 11: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/11.jpg)
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 =
![Page 12: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/12.jpg)
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
![Page 13: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/13.jpg)
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 ,
![Page 14: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/14.jpg)
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
![Page 15: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/15.jpg)
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
![Page 16: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/16.jpg)
Type III SS
• The type III sum of squares is calculated by comparing the full model, to the full model without the effect.
![Page 17: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/17.jpg)
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.
![Page 18: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/18.jpg)
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.
![Page 19: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/19.jpg)
Example
• A two factor (A and B) experiment, response variable y.
• The SPSS data file
![Page 20: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/20.jpg)
Using ANOVA SPSS package
Select the type of SS using model
![Page 21: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/21.jpg)
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.
![Page 22: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/22.jpg)
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.
![Page 23: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/23.jpg)
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.
![Page 24: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/24.jpg)
Incomplete Block Designs
Balanced incomplete block designs
Partially balanced incomplete block designs
![Page 25: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/25.jpg)
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.
![Page 26: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/26.jpg)
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.
![Page 27: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/27.jpg)
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
![Page 28: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/28.jpg)
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.
![Page 29: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/29.jpg)
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)
![Page 30: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/30.jpg)
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.
1
t
Qii
![Page 31: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/31.jpg)
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
![Page 32: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/32.jpg)
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
![Page 33: Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649c875503460f9493f1a9/html5/thumbnails/33.jpg)
Next Topic: Designs for Estimating Residual Effects