Math 558 Lecture #4
Bread Rise Experiment1
In a bread rise experiment, the experimenter wants to examine three different rise times (35 minutes, 40 minutes, and 45 minutes) and tests four replicate loaves of bread at each rise time. The following can be used to create the CRD.
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set.seed(300)
f <- factor( rep( c(35, 40, 45 ), each=4))
fr <- sample( f, 12 )
loaf <- 1:12
plan <- data.frame( Loafs=loaf, time=fr )
write.csv( plan, file = "Plan1.csv", row.names = FALSE) getwd()
1Lawson, pg 18
CRD generated in R
loaf time height 1 45
10 40 11 35 12 45
Linear Model for CRD
The linear model for a completely randomized design is given below
Yiw = μi + εiw
Here, the subscript i represents the ith treatment applied to the plot w where w = 1, 2, 3..ri(number of times treatment i is replicated. Also
i ∈ τ = {1, 2, ..t}. The experimental errors εiw are mutually independent due to randomization and are assumed to follow the normal distribution.
Statistical Analysis
Statistical Analysis
Statistical Analysis
Statistical Analysis
Linear Model for CRD
The above model can be written as
Yiw = μ+τi + εiw
Here τi = μi − μ is the effect of the ith treatment. The responses are normally distributed. i. e Yiw ∼ N(μ + τ1, σ2). Equivalently
εiw ∼ N(0, σ2). Let us consider the case of equal replication
(r1 = r2 = ..ri = ..rt). The sample mean of the data for the ith treatment is
1 ri y ̄=r y
i. ∑ iw i w=1
RCD: Statistical Analysis
The overall mean is
The least square estimators are obtained by minimizing the error sum of squares.
SSE = ∑ ∑(yiw −μi)2
The resulting estimators are
for i = 1,2,..t
μˆ = y ̄ i i.
1t y ̄=t y ̄
.. ∑ i. i=1
Linear Model in the matrix form
Let us consider a CRD with three treatments, each with four replications. Then the effects model in the matrix form can be written as
Y = Xβ + ε
y11 y
12 y13
y14 y 21
1100 ε11 1100 ε
y22 1 = y23 1
y24 y31 y32 y33
0 τ1 ε22
× + (1)
ε31 ε32 ε33
12 1100 ε13
1100 ε14 1010 μ ε
1 1 1 1
0 τ2 ε23 0 τ3 ε24
y34 1001 ε34
Statistical Analysis
The matrix β is the matrix of model parameters. The least squares estimators of the model parameters are obtained by solving the normal equations X′ Xβ = X′ y The matrix X′ X is singular therefore, cannot be inverted. The R command lm solves this problem by dropping the column that corresponds to the first treatment. The matrix that will be
1 0 0 1 0 0
1 0 0 1 0 0
1 1 0
1 1 0 used in the calculations is therefore, X1 = 1 1 0
1 1 0 1 0 1 1 0 1 1 0 1
Statistical Analysis
Solving the above system of equations with X1 we have (X′ X)−1X′ y = βˆ
μˆ + τˆ 1
Where βˆ = τˆ − τˆ 21
τˆ − τˆ 31
The first treatment is treated as a control and all other treatments are compared to it.
Statistical Analysis
Rise Time 35
Loaf Heights 4.4, 5.0,5.5,6.75 6.5, 6.5, 10.5,9.5 9.75,8.75,6.5,8.25
Table: Bread Rise Experiment
Statistical Analysis
Statistical Analysis
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