Introduction to Randomized Complete Block Design lecture - 07

 

Introduction to  Randomized Complete Block Design 

• Read More:Estimation of Missing Value In RCBD
• Read More: Post Hock Tests in RCBD
• Read More: Estimation of Parameters in RCBD
• Read More: Relative Efficiency of two Designs
• Read More: Expectation in RCBD
• Read More: RCBD Subsampling
• Read More:MCQ's on RCBD

lecture - 07

RCB design is used when there is a suspected or known one direction source of variation in the experimental material, like fertility, sand, and wind gradients, etc. This known or suspected source of variation is control and isolated by blocking the experiments.

Blocking means divide the experimental material into homogenous groups. The blocking is performed orthogonal of variability in the experimental material. The treatments are assigning randomly to the experimental units within each block. The treatments are assigned within the individual blocks at random with a separate randomization for each block.

Layout of RCB Design:

Suppose 4 treatments (A, B, C, D) are to be compared and each treatment is replicated 5 times but the fertility of the land is decreasing from north to south.

The step-by-step procedure used in the experiment with 4 treatments (A, B, C, D) and each treatment are replicated 5 times.

Step – 1:

To perform experiment, we needed 4 X 5 = 20 experimental units to perform the experiment.

Step – 2:

The fertility is decreasing from north to south, so blocks are performed east to west. Divide the experimental material into 5 equal sizes of blocks.

Step – 3:

Divide each block in 4 equal sizes of plots called experiment units to accommodate all treatments within a block.

Step – 4:

Using random numbers table and select 4 random numbers and assign from smallest to largest.

Random Number	0.723	0.987	0.321	0.459
Rank	3	4	1	2
Sequence Number	1	2	3	4
Treatment	A	B	C	D

The treatments assign to block – I as:

Repeat the same procedure for other blocks

Experiment & Model Development

It is desired to measure the effect of three different food supplements i.e., T1, T2, T3 on reproduction of three different kinds of parrots (i.e., Buggies, Non ring love birds and ring love birds) in 3 different captivities of homogeneous environment.

We required 9 pairs per captivity to assign T1, T2, T3. 

Total number of pairs required: 27

Step – 1: Number of blocks: 9 (Number of Pairs per Captivity).

Step – 2: Total number of pairs: 9 (Number of Pairs per Captivity) X 3 (Number of Captivity) = 27

Step – 3: Assign T1, T2, T3 in captivity 1, 2 and 3

Statistical Model

The statistical model for RCB Design:

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

Where:

Yij : is the yield of j^th treatment in the i^th block.

μ : is the overall mean.

ρi: is the effect of the i^th block.

τj : is the effect of j^th treatment.

In Fixed Effect Model:

The treatment and block effect are thought as deviation from overall mean “μ ”, so,

In Random Effect Model:

Model Assumptions:

·    i. The error term “ϵij ” is called random error and “ϵij ” is normally distributed with mean 0 and fixed variance.

ii.  It assumes that treatments have the same effect in every block, and the only effect of the block is to shift the mean response up or down.

Analysis of the model         

Suppose we have t number of treatment (i.e. T1, T2, ..., Tj,..., Tt) distributed over b number of blocks (i.e. B1, B2, ...., Bj, ...., Bb) and each block contain a complete set of treatments. The effect of treatment in various blocks in terms of yield is given below:

ANOVA Table for RCBD

SV	d.f	SS	MS	F
B / W Treatments	t-1	SST	MST	F1 = MST / MSE
B / W Blocks	b-1	MSB	MSB	F2 = MSB / MSE
Error	(t-1) (b-1)	SSE	MSE
Total	tb-1	SS Total

Example: 

The following data obtained from randomized complete block design with 3 treatments A, B, C and 3 blocks. Test the hypothesis at 5 % level of significance.

i.                    All 3 treatments are identical.

ii.                  The blocking having significant effect.

Block	Treatment
	A	B	C
1	5	12	15
2	7	10	14
3	8	16	18

Solution: State null & alternative hypotheses for treatments and blocks as:

${\mathrm{i. \; \; \; H}}_0 : {\tau }_1 = {\tau }_2 = {\tau }_3 Vs. \;$H1 : τ1  ≠ τ2  ≠ τ3

ii.     The significance level;  α  = 0.05

iii.     The test statistic:  RBD

 iv.     Critical regions:

F1 >= 6.91

F2 >= 6.91

v. Computation:

ANOVA Table

SV	d.f	SS	MS	F
b/w treatments	2	126	63	34.52
b/w blocks	2	24.67	12.375	6.81
Error	4	7.3	1.825
Total	8	158

F1 = MST / MSE = 34.52

F2 = MSB / MSE = 6.82

vi.  Remarks: 

The calculated  value falling in the rejection region and  value falling in the acceptance region. Thus it is concluded that the treatments effects are significant and blocking effects are insignificant.