Complete Randomized Design (Sub Samples) Lecture - 06

 

Complete Randomized Design

(Sub Samples)

The sub sampling technique is used, when the experimenter takes repeated observations on the same experimental units. Traditionally, when several observations are taken on a single experimental unit, these observations are combined by total or average and consider as a single observation on the experimental unit. Now if the observations with in experimental units are not combined, then the sub sampling technique is used.

In sub sampling, there are two sources of random variation associated with any observation. One is the random variation is among experimental units and the other is random variation among observation with in experimental units.

For example, consider an experiment where 3 feeding rations are to be compared. Each ration is randomly assigned to each of 5 pens and each pen contains 4 animals. In this case the pen is the experimental unit and the observations made on individual animals within a pen are sub samples. There are two sources of random variations associated with any observation made on each animal. One is the random variation from pen to pen within treatments, and the other is random variation among animals within pens. If the experimenter collects data on a pen basis, for example, weighs all animals in a pen together and expresses the result as total body weight or average body weight per animal, the appropriate ANOV falls in the category of a one-way ANOV without sub samples. The conclusions regarding treatment effects will be the same if individual animal data are analyzed. The additional information on animal variation can be useful in planning experiments with respect to more efficiency of allocation animals and pens to treatments.

Statistical Model

The CR design sub sample is represented by the following linear statistical model:

Yijk = μ + τj + ϵij + δijk   i =1, 2,..., r  j= 1, 2, ..., t  k =1, 2, ..., n

Where:

Yijk: The yield i^th observation of j^th treatment in k^th sub sample.

 μ: Population mean.

τj : is the treatment effect of j^th treatment.

 ϵij: Random error (variation among experimental units)

δijk: Sampling error (variation in a sample).

Statistical Analysis

Let Yijk is the yield of i^th observation of j^th treatment in k^th sub sample. Then it can arrange as:

The sub samples total can be arranged as:

 

ANOVA Table:

Example: 

The following data represent the result of three fertilizers to homogeneous experimental units and taking 3 measurements on a single experimental unit. The results are given below:

Observation	Fertilizer
	F1	F2	F3
1	18 18 15	23 20 16	22 23 20
2	21 16 24	20 14 24	20 15 22
3	18 15 12	17 13 16	19 21 18

Perform an appropriate design and test the significance of three fertilizers at 5 %.

Solution: 

The experimental units are homogeneous and 3 observations per experimental unit is collected, so CRD sub sampling is appropriate in this case.

The statistical model for this experiment is 

Yijk = μ + τj + ϵij + δijk   i =1, 2, 3  j= 1, 2, 3  k =1, 2, 3

Setup of hypothesis as:

i.    H0 : μF1 = μF2 =μF3 Vs.  H1 : μF1  ≠ μF2  ≠μF3

ii. The significance level; α = 0.05

iii. The test statistic:

iv. Reject  H0 , when F >= 5.14

v. Computation:

ANOVA Table:

vi. Comments: The effect of three fertilizers is insignificant.

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