The simplest design which enables us to take care of variability among the units is the Randomised Block Design (RBD). This is the simplest design using all three principles (randomisation, replication, local control). This design has many advantages over other designs. This design is mostly used in real life situations.
Suppose we want to compare the effects of t treatments each of with r replicates. Then we need n=rt experimental units. First, we divide the experimental units into r homogeneous blocks (groups). Then assign the treatments at random to the units of a block.
The model we consider here is,
Select “Data Analysis” from “Data” from Toolbar > Select “Anova: Two-Factor Without Replication” and click on “OK” > In “Input Range” select the data only. You can choose “Alpha” as your desired level of significance. Select “Output options”. Click on “OK” > You get the table.
data <- data.frame( block = c("b1", "b2", "b3"), T1 = c(41,65,76),
T2 = c(45,67,72), T3 = c(44,66,76),
T4 = c(45,66,77), T5 = c(46,76,64)) #Loading data
expmnt <- melt(car.noise, id.var="block") #reshaping for use
summary(aov(yield~block+treatment,data = expmnt)) #result of ANOVA
> data <- data.frame( block = c("b1", "b2", "b3"), T1 = c(41,65,76),
+ T2 = c(45,67,72), T3 = c(44,66,76),
+ T4 = c(45,66,77), T5 = c(46,76,64)) #Loading data
> expmnt <- melt(car.noise, id.var="block") #reshaping for use
> names(expmnt)<-c("block","treatment","yield") #renaming
> summary(aov(yield~block+treatment,data = expmnt)) #result of ANOVA
Df Sum Sq Mean Sq F value Pr(>F)
block 2 2368.1 1184.1 46.013 4.09e-05 ***
treatment 4 6.9 1.7 0.067 0.99
Residuals 8 205.9 25.7
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Missing Plot Technique
Sometimes w.r.t. some experimental design we see that some of the observations in the layout of the experiment are missing. In the field experiment, it may happen due to the attack of pests or negligence of the observer. Again sometimes observations are so suspicious and it is better to treat them as missing. In such a situation, the convention is to estimate the missing observations in terms of available yields in order to have a complete layout. Once we get the complete layout, we carry out the usual analysis. Several techniques are proposed by different statistician of which the procedure, proposed by Yates, is the popular one and it is referred to as Yates Missing Plot Technique.
Let there are k missing observations in a layout of an experimental design consisting of t treatments where we are to test whether the effects of the treatments are equal or not. At first, we express the error sum of squares in terms of available yield and the missing observations. Now we determine the estimate of the missing observations minimizing error SS, E(x1,x2,…,xk) w.r.t. x1,x2,…,xk . Evidently, the estimates are obtained by the following system of equations,