Incomplete Block Designs Randomized Block Design We want
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.
Comments regarding 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. 1 2 1 4 5 4 2 3 3 5 6 6 • 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.
Definitions • Two treatments i and i* are said to be connected if there is a sequence of treatments i 0 = i, i 1, i 2, … i. M = 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. 1 2 1 4 1 2 3 3 5 4 • This incomplete block design is connected. • All treatment differences are estimable. • Some treatment differences are estimated with a higher precision than others.
Analysis of unbalanced Factorial Designs Type I, Type III Sum of Squares
Sum of squares for testing an effect model. Complete ≡ model with the effect in. model. Reduced ≡ model with the effect out.
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 factorial experiment with factors A, B and C. The Complete model • Y = m + A + B + C + AB + AC + BC + ABC
A sequence of increasingly simpler models 1. Y = m + A + B + C + AB + AC + BC + ABC 2. Y = m + A+ B + C + AB + AC + BC 3. Y = m + A + B+ C + AB + AC 4. Y = m + A + B + C+ AB 5. Y = m + A + B + C 6. Y = m + A + B 7. Y = m + A 8. Y = m
Type I S. S.
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 = m + A + B + C + AB + AC + BC + ABC (the three factor model) 2. Y = m + A+ B + C + AB + AC + BC (the all two factor model) 3. Y = m + 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
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 (H 0) 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 kfactor 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
ANOVA table – type II S. S
ANOVA table – type III S. S
Incomplete Block Designs Balanced incomplete block designs Partially balanced incomplete block designs
Definition An 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 l is the same for each treatment pair. 2. if all treatment pairs i and i* appear together in exactly l blocks. • This ensures that each treatment difference is estimated with the same precision. • The value of l is the same for each treatment pair.
Some Identities Let b = the number of blocks. t = the number of treatments k = the block size r = the number of times a treatment appears in the experiment. l = 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) = l (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 Design A Balanced Incomplete Block Design (b = 15, k = 4, t = 6, r = 10, l = 6)
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:
Analysis for the Incomplete Block Design Recall that the parameters of the design where b = 15, k = 4, t = 6, r = 10, l = 6 denotes summation over all blocks j containing treatment i.
Anova Table for Incomplete Block Designs Sums of Squares SS yij 2 = 234382 S Bj 2/k = 213188 S Qi 2 = 181388. 88 Anova Sums of Squares SStotal = SS yij 2 –G 2/bk = 27640. 6 SSBlocks = S Bj 2/k – G 2/bk = 6446. 6 SSTr = (S Qi 2 )/(r – 1) = 20154. 319 SSError = SStotal - SSBlocks - SSTr = 1039. 6806
Anova Table for Incomplete Block Designs
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