Experimental Designs:
1. Treatments:
The objects of comparison, which an experimenter has to try in the field for assessing their values are known as treatments e.g. the varieties, manures, cultivation practices etc.
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2. Experimental Units:
Experimental material is divided into a number of ultimate units to which the treatments are applied and are known as experimental units.
3. Experimental Material:
This is the name given to the material on which the experiment is performed e.g., a field, soil, patients in a hospital etc.
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Basic Principles of Field Experimentation:
There are three basic principles of field experimentation which have been discussed below:
1. Replication:
The allocation of the treatments under investigation is known as replication.
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2. Randomization:
The allocation of the treatments to the different experimental units by a random process is known as randomization of treatments.
3. Local control:
The principle of making use of greater homogeneity in groups of experimental (e.g. blocks of a number of plots homogeneous within themselves) for reducing the experimental error as explained in the foregoing paragraph is known as local control. This is done by dividing the experimental material (e.g., field) into homogeneous groups of experimental units (e.g., blocks).
Condition for Application of Designs:
(i) C.R.D. (Completely Randomized Design) is apply when the experimental material is limited and homogeneous, such as the soil in the pot experiments.
(ii) R.B.D. is appropriate when the fertility gradient of the field is in one direction. It may be adopted upto 20 treatments without an appropriate loss of efficiency.
(iii) L.S.D. can replace the R.B.D., when the fertility gradient is in two directions instead of one. It may be adopted for number of treatments ranging from 5 to 8 or from 5 to 12 at the most.
(a) Completely Randomized Design:
This is the simplest type of design in which the whole experimental material is divided into a number of experimental units depending upon the number of treatments and the number of replications for each. After that the treatments are allotted to the units entirely by chance.
Advantages:
1. The method of analysis remains simple when the results from some units are rejected.
2. This design is specially useful in small experiments where, the supply of experimental material is scarce and homogeneous.
3. In this design any number of treatments and replicates may be used.
4. The statistical analysis of the data is very easy.
5. The relative loss of information due to missing data in smaller than that with any other design.
Application Conditions of C.R.D.:
1. C.R.D. used when the experimental material is limited in quantity and homogeneous.
2. Where it is expected that some of the units will be destroyed.
3. In small experiments where the increased accuracy from the alternative design is not sufficient to exceed in importance the loss of error degree of freedom.
(b) Randomized Block Design:
In this design the whole experimental material is divided into homogeneous groups, each of which constitutes a single replication. Each of these groups is further divided into the number of experimental units which are equal in all respects. The treatments are applied to these units by any random process.
Some important points are given below about this design:
1. In R.B.D., the number of blocks must be equal to the number of replications fixed for each treatment.
2. The number of plots in each block should be equal to the number of treatments.
3. Randomization of treatments in each block should be afresh.
4. The experimental error within each block are to be kept as small as practically possible and the variation from block to block as great as possible. In this way all the treatments which are assigned to one block experience the same type of environmental effects and therefore comparable.
Advantages of R.B.D.:
1. When the material is heterogeneous the residual variance can be reduced by choosing blocks or plots such that the plots within any block are fairly similar.
2. This design allows of any number of treatments and any number of replications but when the number of treatments is very large the efficiency of error control decrease.
3. If a part of the experimental material is damaged by some agricultural disaster like insect, floods, water logging etc.
4. By means of grouping more accurate results are usually obtained than with the C.R.D.
5. Although a reduction in the number of replications leads to a larger standard error, yet it furnishes a result of some value at least.
(c) Latin Square Design:
In this design it is essential that each row and each column should contain the same application of treatments and hence, the variation between the row means and between the column means can be assessed and eliminated from the error increasing the precision of the estimates.
i. The experimental design which simultaneously control the variation in two directions is known as Latin Square Design.
ii. The L.S.D. is very reliable to give precised results when the number of treatments is from 5 to 8 or at most 12. This design should not be used for less than 5 treatments.
The important points about L.S.D.:
1. No. of replications = No. of treatments
2. No. of rows = No. of column = No. of treatments
3. Randomization of treatments is done in such a way that each treatments occurs once and only once in each row and each column.
Advantages of L.S.D.:
This design gives precision high enough to reduce the standard error to less than 1%.
L.S.D. control the variations due to two sources of variation.
Disadvantages of L.S.D.:
i. The analysis becomes very complicated when several plot yield are missing.
ii. An L.S.D. with two treatments is not possible.
iii. S.D. is less flexible in nature.
Test of Significance:
The statistical procedure for deciding whether the difference under study is significant or non-significant, is called the test of significance.
Null hypothesis (H0) is the hypothesis which is a test for possible rejection under assumption that it is true.
Alternate hypothesis:
Any hypothesis which is complementary to the null hypothesis is called alternate hypothesis. T his is denoted by H1.
Two Type of Error:
1. Type 1st error: Type 1st error is committed when we reject the hypothesis when in reality it is true. The probability of committing a type 1st error is denoted by a.
2. Type IInd error:
Type IInd error is committed by not rejecting the null hypothesis when it is false. The probability of committing a type IInd error is denoted by β.
Level of Significance:
The maximum probability of rejecting the hypothesis when it is true or in other words, the maximum probability of type f error is known as level of significance.
X2 (Chi-square)-test:
The value of the quantity of x2 is calculated by the help of the following expression:
X2 = Ʃ (O-E/E)2
Where, О = Observed frequency in a cell
E = Expected frequency of the same cell
S = Summation taken over all the cells Expected value calculated by the following equation:
E = RT x CT
Where, E = Expected frequency
RT = Rows total
CT= The column total for the column containing the cell.
N = The total number of observation.
Properties of x2Distribution:
Some properties are given below:
1. X2 ranges from 0 to α and is always positive.
2. Mean of the chi-square distribution is equal to V, and variance – 2V, where V stands for degree of freedom.
3. When there is a perfect agreement of the observed with the hypothetical distribution of frequencies, the value of X2 is zero.
Applications of x2-Test:
1. Testing the homogeneity of correlation coefficients.
2. Testing the independence of attributes.
3. Testing the expectation of a ratio.
4. Comparison with expectation of normal Binomial and Poisson distributions.
5. Application in genetic problem and detection of linkage in genetics.
6. Comparison of a sample variance with npopulation variance.
Tests of hypothesis:
Population:
Population is a group of similar objects living or non-living.
Sample:
Sample is a subset of population.
Census:
Complete enumeration of population.
Sampling:
Drawing the sample in a systematic way from the population.
Random sampling:
If every unit in the population base the equal chance of getting into sub group is random sampling.
Purposive sampling:
Selecting some units according to the preference of the investigator.
Sampling unit:
A finite object on which we record observations.
Tests of significance:
In applied investigations we are often interested in:
(i) Comparing some characteristics such as mean and variance of a group with a specific value.
For example µ with µ0, ϭ2 with ϭ02.
(ii) In comparing 2 or more groups with regard to the characteristic. For example p, with p2, a,2 with a22. In making such comparisons are cannot relay on the more numerical magnitude of the index of comparison such as the mean or variance. This is because each group is represented only by a sample of observations and if another sample were drain the numerical value would change. We are thus forced to draw inference in the presence of the sampling fluctuations which effect the observed differences between the groups clouding the real differences. Statistical science provides on objective procedure for distinguishing whether the observed differences is due to the real near sampling fluctuations. Such a procedure is called test of significance.
Parameter:
Characteristic of population or function of population values.
Statistic:
Characteristic of sample or function of sample values.
Hypothesis:
It is a definite statement about population parameters.
Null hypothesis [H0]:
A hypothesis of no differences.
Alternative hypothesis [H1]:
A hypothesis which is alternate to null hypothesis.
R.A. Fisher:
Null hypothesis is tested for possible rejection under the assumption that it is true. He is known as father of statistics.
It is a hypothesis of equality.
H1 is a complementary statement to H0.
Type I error:
Rejection the H0 when it is true.
Type II error:
Accepting the H0 when it is false.
Type I error is produces risk and type II error is consumer risk.
Size of type I error is fixed which is called as “level of significance where values are 51 and 1 %. It is the probability at which we allow type I error. Only 5 samples we may commit type I error. 1% level of significance means cut of 100 samples we may commit type I error in are sample. 5% level of significance means out of 100 samples.