COMPLETE RANDOMIZED DESIGN lecture - 01

 COMPLETE RANDOMIZED DESIGN

A Complete Randomized design is the simplest type of the basic designs and may be defined as; a design in which the treatments are assign to experimental units completely at random, that is randomization is done without any restriction.

A complete randomized design is considered to be most useful in situations where:

i. Experimental units are homogeneous.

ii.The experimental material is small, such as lab experiments etc.

iii. Some experimental units are likely to be destroyed or fail to respond.

 Experimental Lay Out

The layout of experiment is the actual placement of treatments on the experimental units. In C R D, the experimental units are homogenous, so the treatments are assigned randomly to the experimental units.

An example of the experimental lay out for a complete randomized design using four treatments (T1, T2, T3) and each is repeated 3 times, is given below:

 

CR Design Model Development

The general pattern of t treatments in r replications

 

Statistical Model

Let Yij denoted the yield of the ith observation on jth treatment, and then Yij may be represented by the following linear statistical model.

Yij = μ +τj + ϵij   i = 1, 2, ......, r   j =1, 2, .......t

Where:

 μ is the true mean effect, τj represents the effects of jth treatment and ϵij  denoted random error. Yij is the yield of j^th treatment in the i^th experimental unit.

In case of fixed effect and random effect models ϵij is independently identically follow normal distribution with mean is zero and constant variance.  

In case Fixed effect model:

∑τj = 0

Analysis

Suppose we have k treatment and experimental material is divided into n experimental units. We shall then assign to k- treatments at random to the n experimental units in such a way that the treatment τj ( j = 1, 2, …., t) and each time is replicated at the same number of times, then and r1 = r2 = ...=rt.

and r X t = n.

ANOVA table for CR Design

Hypothesis procedure 

 i.   H0 : T1 = T2 = T3   Vs.   H1 : T1 ≠ T2 ≠ T3  

ii. The significance level; α  

 iii. The test statistic: CR design

F = MST/ MSE

Which follow F distribution with t - 1 and n - t degree of freedom.

vi.  Reject H0, When  F ≥ Fα (t-1, n-t)df

v.   Computation

vi.   Remarks

Example: 

An experiment was conducted to compare the yields of three varieties of potatoes. Each variety was assigned at random to equal size of plots, four times. The yields were as follow:

Variety
A	B	C
23 26 20 17	18 28 17 21	16 25 12 14

Test the hypothesis that the three varieties of potatoes are not different in the yielding capabilities.

Solution: We set up our hypothesis as:

 i.   H0 : TA= TB = TC   Vs.   H1 : TA ≠ TB ≠ TC  

ii. The significance level; α = 0.05  

 iii. The test statistic: CR design

F = MST/ MSE

Which follow F distribution with 2 and 9 degree of freedom.

vi.  Reject H0, When  F ≥ Fα (2, 9)df = 4.26

v.   Computation

Variety
	A	B	C
	23 26 20 17	18 28 17 21	16 25 12 14
T. j	86	84	67	237
Sq.T. j	7396	7056	4489	18941

ANOVA table for CR Design

vi.   Remarks

   As F calculated value falls in the acceptance region, so we have not sufficient evidence to reject  thus we conclude the yield capabilities of the three varieties are identical.

•  Read More Estimation of missing value in CR Design
• Read More Post Hock Test in CR Design
• Read More Estimation of Parameters in CR Design
• Read More Expectation in CRD
• Read More: CRD subsampling