CRD in R

Experimental Designs are part of ANOVA in statistics. They are a predefined algorithms that help us in analyzing the differences among group means in an experimental unit. Completely Randomized Design (CRD) is one part of the Anova types.

Completely Randomized Design -

The three basic principles of designing an experiment are replication, blocking and randomization. In this type of design , blocking is not a part of the algorithm. The samples of the experiment are randomly with replications are assigned to different experimental units. Lets consider an example -

Experiment — Adding more baking powder to cakes increases the heights of the cake.

Lets see with CRD how the experiment will be analyzed.

As shown in the above figure the baking powder is divided 4 different tablespoons(tbsp) and four replicate cakes heights (respectively for A,B,C,D) were made with each tbsp in a random order. Then the results of tbsp is compared to see if actually the height is affected by baking powder.

The replications are just permutations of the different cakes heights respectively for A,B,C,D.

Lets see the above example in R language- The heights of each cake are randomly noted for each tbsp

tbsp 0.25 0.5  0.75  1 
     1.4  7.8  7.6  1.6
  #(A,B,C,D)
     2.0  9.2  7.0  3.4
  #(A,B,D,C)
     2.3  6.8  7.3  3.0
  #(B,A,D,C)
     2.5  6.0  5.5  3.9  #(B,A,C,D)
##  the randomization is done directly by the program

The replications of the cakes are done with below code -

R

treat<- rep(c("A", "B", "C", "D"), each = 4)
fac <- factor(rep(c(0.25, 0.5, 0.75, 1), each = 4))
treat

Output-

[1] A A A A B B B B C C C C D D D D
Levels: A B C D

Creating dataframe-

R

height <- c(1.4, 2.0, 2.3, 2.5,
          7.8,9.2, 6.8, 6.0, 
          7.6, 7.0, 7.3, 5.5,
          1.6, 3.4, 3.0, 3.9)
exp <- data.frame(treat, treatment = fac, response = height)
mod <- aov(response ~ treatment, data = exp)
summary(mod)

Output-

Df Sum Sq Mean Sq F value   Pr(>F)    
treatment    3  88.46  29.486   29.64 7.85e-06 ***
Residuals   12  11.94   0.995                     
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

For every experiment significance is either 0.05 or 0.01 which is given by the person taking experiment. For this example lets consider the significance as 5% i.e 0.05

We should see the value of Pr(>F) which is 7.85e-06 ,i.e, < 0.05. Thus reject the hypothesis . If the value is >0.05 then accept the hypothesis.

For our example as Pr<0.05 reject the hypothesis.

Lets consider one more example-

Experiment — Adding rocks to water increases the height of water in the container.

Lets see this experiment in figure as follows-

Consider that if adding four rocks to 500ml,600ml and 700ml respectively , increases the height of water correspondingly. For example — adding 6 rocks to 500 m water has 7 ms height increased.

rocks four six eight
      5    5.3    6.2 [500 600 700]
      5.5    5    5.7   [600 500 700]
      4.8    4.3   3.4  [700 600 500]

Lets code -

R

rocks<- rep(c("four", "six", "eight"), each = 3)
rocks
fac <- factor(rep(c(500,600,700), each = 3))
fac
[1] "four"  "four"  "four"  "six"   "six"   "six"   "eight" "eight" "eight"
[1] 500 500 500 600 600 600 700 700 700
Levels: 500 600 700

Creating dataframe-

R

height <- c(5, 5.5, 4.8,
			
          5.3, 5 , 4.3,
      
            4.8 , 4.3 , 3.4)
exp1 <- data.frame(rocks, treatment = fac, response = height)
mod <- aov(response ~ treatment, data = exp1)
summary(mod)
Df Sum Sq Mean Sq F value Pr(>F)
treatment    2  1.416  0.7078   2.368  0.175
Residuals    6  1.793  0.2989

Here 0.175>>0.05 thus hypothesis is accepted.

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Also published in geeksforgeeks.org

https://www.geeksforgeeks.org/completely-randomized-design-with-r-programming/