Incomplete Block Designs
Randomized Block Design • We want to compare t treatments • Group the N = bt experimentalunits 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. 1 2 2 3 1 3 4 5 5 6 4 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 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. 1 2 2 3 1 3 4 5 1 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 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.
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 = m+ A + B+ C+ AB+ AC+ BC+ ABC
A sequence of increasingly simpler models Y = m+ A + B + C + AB + AC + BC + ABC Y = m+ A+ B + C + AB + AC + BC Y = m+ A + B+ C + AB + AC Y = m+ A + B + C+ AB Y = m+ A + B + C Y = m+ A + B Y = m+ A 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 Y = m+ A + B + C + AB + AC + BC + ABC (the three factor model) Y = m+ A+ B + C + AB + AC + BC (the all two factor model) 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 (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
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. • 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. • 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 • bk = rt • Both sides of this equation are found by counting the total number of experimental units in the experiment. • 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 • 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. A food processing company is interested in comparing the taste of six new brands (A, B, C, D, E and F) 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 yij2 = 234382 S Bj2/k = 213188 S Qi2 = 181388.88 Anova Sums of Squares SStotal =SS yij2 –G2/bk = 27640.6 SSBlocks =S Bj2/k – G2/bk = 6446.6 SSTr = (S Qi2 )/(r – 1) = 20154.319 SSError = SStotal - SSBlocks - SSTr = 1039.6806
Anova Table for Incomplete Block Designs
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