# Incomplete Block Designs. Randomized Block Design We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size.

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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. 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. 1212 2323 1313 4545 5656 4646

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 (i j and i j+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. 1212 2323 1313 4545 1414

Analysis of unbalanced Factorial Designs Type I, Type II, 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 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.

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

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 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. 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 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. = 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 Design A Balanced Incomplete Block Design (b = 15, k = 4, t = 6, r = 10, = 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, = 6 denotes summation over all blocks j containing treatment i.

Anova Table for Incomplete Block Designs Sums of Squares  y ij 2 = 234382  B j 2 /k = 213188  Q i 2 = 181388.88 Anova Sums of Squares SS total =  y ij 2 –G 2 /bk = 27640.6 SS Blocks =  B j 2 /k – G 2 /bk = 6446.6 SS Tr = (  Q i 2 )/(r – 1) = 20154.319 SS Error = SS total - SS Blocks - SS Tr = 1039.6806

Anova Table for Incomplete Block Designs

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