Math 558 Lecture #14
Example Lawson page 117
An experiment was conducted to test the effect of different water solutions on the life of cut flowers. The treatment factor was the liquid to fill the vase. The levels were:
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1. Tap water
2. Tap water with one spoonful of sugar added
3. Tap water with one cup of carbonated water
4. Tap water with one cup of 7-up
The experimental units were single flowers and the response was the time in days until the flower wilted. The experimenter wanted the conclusions of her study to apply to many types of flowers, so she used an RCB design. The natural blocks were the flowers.
2. Carnation
Randomized block designs in R
Randomized block designs can be easily created using the function design.rcbd from the package agricolae (de Mendiburu, 2012b) . By default this function labels the experimental units as plots, and uses integers as the block numbers. The seed argument in the function call is for the randomization, and running the code with the same seed will result in the same randomized list. The function “levels” renames the blocks.
Randomized block designs in R
install.packages(“agricolae”) library(agricolae)
treat<-c(1,2,3,4)
design <- design.rcbd(treat, 4, seed = 10) rcb <- design$book1
levels(rcb$block) <- c("carnation", "daisy", "rose", "tulip") rcb
1creates a design lay to be used in the field
Randomized Complete Block Design
block treat carnation 1 carnation 3 carnation 4 carnation 2
daisy 4 daisy 3 daisy 1 daisy 2
rose 3 rose 2 rose 4 rose 1 tulip 2 tulip 1 tulip 4 tulip 3
Block subspace
During our introductory lectures we used the notation Ω to represent the set of all experimental units. We will use the same notation to represent he set of all blocks.
Let Ω is the set of b blocks of the size k. We will represent the block containing the experimental unit w by B(w). Now let us define the block subspace VB as
VB ={v∈V :vα =vβifB(α)=B(β)
This means that VB is the set of all the vectors which take constant value on each block for j = 1,2,..b. Let vj be a vector in VB such that an entry invj is1iftheplotwisinblockjand0otherwise.
Block subspace
The orthogonal basis for VB is {v1, v2.......vb} as vj.vj = k
vj.vi =0fori̸=j
Example 4.10 Text Table in the slide shows a block design for three treatments in two blocks of size four. The treatment vectors uA, uB, uC and the block vectors v1 and v2 are shown. Treatment A occurs twice in both clocks, so uA.vi = 2 for i = .1,2
Block subspace
Plot Block T uA uB uC v1 v2 11A10010 21A10010 31B01010 41C00110 52C00101 62A10001 72B01001 82A10001
Table: Caption
Orthogonal Block Design
Theorem .1
Given a block design for t treatments in b blocks of size k, let sij be the number of times that treatment i occurs in block j, for i = 1, ..., t and
j = 1, ..., b. Then the block design is orthogonal if and only if sij = ri /b for i = 1,...,tand j = 1,...,b.
Orthogonal Block Design
Definition A complete-block design has blocks of size t, with each treatment occurring once in each block.
Corollary Complete-block designs are orthogonal
Question Is the design given in slide 8 orthogonal?
Note sij = ui.vj
Discussion
Discussion
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