Math 558 Lecture #15
Model for Randomized Complete Block Design
The model for the analysis of randomized complete block design can be written as an effects model given below.
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yij = μ + τi + bj + εij
i = 1,2..t j = 1,2,..b
Here μ is the overall mean τi represents the effect of the ith treatment
and bj represents the effect of the jth block. Where εij ∼ N(0, σ2). We can test the assumptions of Normality and homogeniety by using the standard tests on the residuals.
Statistical Analysis of the RCBD
The model given in slide 2 is the fixed effects model. We are interested in estimating the treatments effects not the block effects on the response.A question arises here that if we are only interested in the treatment effects do we need to test the block effects? The answer is a cautious yes. This is because we intend to remove from the experimental error, the variation due to blocks. Therefore, the test on block factors can be used to confirm that the block factor is effective and explains variation that would otherwise be part of the experimental error. However, we cannot make any stronger conclusions about the block factors unless they are randomly chosen from a population and have sufficient replication.
Analysis of Variance for RCBD
Source of Variation
Treatments Blocks Error Total
Sum of squares
SSTr SSB SSE SST
Degrees of Mean freedom squares
t-1 SSTr t−1
b-1 SSB b−1
MSTr/MSE MSB/MSE
(t-1)(b-1) N-1
SSE (t−1)(b−1)
Analysis of Variance for RCB
The degrees of freedom for the error (b − 1)(t − 1) is smaller than it would be in a completely randomized design with b replicates of each treatment (t(b − 1)). However, if the experimental units within the blocks are more homogeneous, the MSE should be smaller resulting in the greater power for detecting differences among the treatment levels. The estimated variance of the experimental units within each block is given by
MSE(RCBD) = SSE (b−1)(t−1)
Analysis of Variance for RCBD
Estimate of the variance of the experimental units ignoring the blocks can be made from RCBD Anova table by using the following formula
MSE(CRD) = SSB + SSE t(b−1)
As (b − 1) + (t − 1)(b − 1) = t(b − 1) Now if SSB is zero then
MSE(CRD) = SSB + SSE = SSE < SSE = MSE(RCBD) t(b−1) t(b−1) (b−1)(t−1)
Relative Efficiency of RCBD as compared to CRD
A commonly used index for comparing the efficiency of two different designs is the inverse ratio of the variance pr unit, i.e., the MSE’s. Since different designs may have different degrees of freedom for error, a correction factor, suggested by Fisher, which multiplies the inverse ratio of variances will give a better measure of the relative efficiency (RE).
Relative efficiency of Design I to Design II = (dfI + 1)(dfII + 3)MSE(II) (dfI + 3)(dfII + 1)MSE(I)
Example(Lawson 117)
Consider the data in in the next slide from Lim and Wolfe (1997), partially modified from Heffner et al. (1974). The effect of the drug d-amphetamine sulfate on the behavior of rats was the object of the experiment. The behavior under study was the rate at which water-deprived rats pressed a lever to obtain water. The response was the lever press rate defined as the number of lever presses divided by the elapsed time of the session. The treatment factor levels were five different dosages of the drug in milligrams per kilogram of body weight, including a control dosage consisting of saline solution. An experiment, or run, consisted of injecting a rat with a drug dosage, and after one hour an experimental session began where a rat would receive water each time after a second lever was pressed.
Rat Behaviour Experiment
Rat 0.0 0.5 1.0 1.5 2.0 1 0.60 0.80 0.82 0.81 0.50 2 0.51 0.61 0.79 0.78 0.77 3 0.62 0.82 0.83 0.80 0.52 4 0.60 0.95 0.91 0.95 0.70 5 0.92 0.82 1.04 1.13 1.03 6 0.63 0.93 1.02 0.96 0.63 7 0.84 0.74 0.98 0.98 1.00 8 0.96 1.24 1.27 1.20 1.06 9 1.01 1.23 1.30 1.25 1.24
10 0.95 1.20 1.18 1.23 1.05
The experimental unit in these experiments was not a rat, but the state of a single rat during one experiment or run, since an individual rat could be used in many experiments by repeatedly injecting it with different doses of the drug (after an appropriate washout period) and by observing the lever pressing behavior. Because there was wide variability in the lever pressing rate between rats, an RCB design was used, and a rat represented the blocking factor. Each rat received all five doses in a random order with an appropriate washout period in between.
In this case, the rat is represented by bi in the model yij = μ + τi + bj + εij
The experimental error, represented by εij, is the error of the state of rat i during the run when it received dose j. Describing the experimental units and the randomization process is an important step towards specifying the most appropriate model.
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