Math 558 Lecture #16
library(daewr)
data(“drug”)
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Note: Details on “daewr” (design and analysis of experiments with R) package are posted on my courses.
modelR<-aov(rate∼ dose+rat, data = drug) summary(modelR)
Df Sum Sq dose 4 0.4602 rat 9 1.6685
Mean Sq 0.11505 0.18538
F value 13.78 22.20
Pr(>F) 6.53e-07 *** 3.75e-12 ***
Residuals 36 0.3006 0.00835
the F-test shows that there is a significant difference in treatment factor levels. To interpret the differences in treatment factor levels, comparisons of means should be made.
After F-test
When the null hypothesis
H0 :μ1 =μ2 =….μt
is rejected, this indicates that at least two of the treatment means are different and equivalently at two treatment effects are different which does not necessarily means that all treatment effects are significantly different from each other. The experimenter needs to do further investigation here to test exactly which treatments effects differ. That is why the study of contrasts and comparisons is very important. Some possible tests can be
1. H0:μ1−μ2=0
2. H0:μ2−μ4=0
3. H0 :μ1 = (μ2 +μ3 +μ4 +μ5)/4
In context of our example the first hypotheses addresses the question ” Does the drug dose of 0.5 creates any difference in the behaviour of rats as compared to no dose (control)?”
The second hypothesis addresses the question “Is the effect of drug dose of 0.5 significantly different from dose 1.0”
The third hypothesis addresses the question “is the average of all drugs significantly different from the control?” We will come back to this topic later.
Now we will check the relative efficiency of our block design as compared to randomized complete block design with the same treatments with b replications each.
Relative Efficiency RCBD vs CRD
Note that the relative efficiency of design I compared to design II is defined in terms of the number of trials of design II required to achieve the same result as one trial of design I. The RE index gives the number of such trials for designs II.
MSE(RCBD) = SSE = 0.00835(from the table) (b−1)(t−1)
MSE(CRD) = SSB + SSE t(b−1)
= 1.6685 + 0.3006 5(10 − 1)
Relative Efficiency
Design I = RCBD and Design II = CRD
RE = (dfI + 1)(dfII + 3)MSE(II) (dfI + 3)(dfII + 1)MSE(I) = (36 + 1)(45 + 3)0.043758
(36 + 3)(45 + 1)0.00835 =5.2413
This means that if we don’t remove block to block variability (which is rat to rat variability in this case )from the overall error it would take approximately five times as many trials to have the equivalent variances for treatment means if each rat had been used for only one trial in a CRD design. In other words we need to increase the replication in CRD five times to get the variances equivalent to RCBD in this experiment.
Relative Efficiency
The reduction in variance due to blocking is given by 1 − MSE(RCBD)
for our design
1− 0.00835 ≈0.8 0.043758
Which mean that there is approximately 80% reduction in variance due to blocking.
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