The below mentioned article provides a study note on correlation.
Correlation:
In Biostatistics, sometimes we study two characters or variables on the same sample and try to find out the existence of any kind of relationship between these two characters. For example, different concentrations of pesticide and their effect on germination, panicle length and number of grains.
Definition:
Correlation is the relationship which can reveal whether the change in one variable would cause change in the other or not. Such relationship between the two sets of characters or variables can be expressed quantitatively by the degree of relationship, called Correlation Coefficient.
Kinds of Correlation:
There are 3 different kinds of correlations:
Positive, Zero and Negative correlations.
(a) Positive Correlation:
When the values of two variables change together in the same direction then the relationship is called positive correlation.
This type of correlation may be perfect positive or moderately positive:
(i) Perfect Positive Correlation:
When both the variables increase and decrease in the same proportion then it is perfect positive correlation.
(ii) Moderately Positive Correlation:
In this case, two variables are positively correlated but the changes do not occur in the same proportion. The coefficient value lies between + 1 and 0.
(b) Negative Correlation:
If one variable increases (or decreases) and the other decreases (or increases) then the relationship is called negative correlation.
Such as size and number of fruits/plant are negatively correlated.
This negative relation may also be of two kinds:
(i) Perfect Negative Correlation:
This kind of relationship is really very rare in case of biological situation, such as increase in temperature decreases the lipid content of the cell.
(ii) Moderately Negative Correlation:
In this relationship, the variables are negatively correlated but not very perfectly, such as increase in post harvesting period decreases the viability of seeds. Here also the coefficient value lies in between 0 and – 1.
(c) Zero Correlation:
If the two variables have no correlation, i.e., there is no consistency on value of observation, in such cases the two values of variables are called with zero correlation.
Coefficient of Correlation:
When the two variables have any direct relationship then the degree of relationship between these two variables is expressed by quantitative expression which is called Coefficient of Correlation. This quantitative measure expresses the degree of closeness of the linear relationship between the two variables.
The correlation coefficient is designated by the letter ‘r’ and it is also called as Karl Pearson’s Coefficient of Correlation which is calculated by the following formula:
Steps to calculate the value of ‘r’:
(a) Two series are made by x and y variable.
(b) Mean of both the series are calculated, x and y.
(c) The deviation of each observation is calculated as dx and dy.
(d) Squaring of the deviations are noted.
(e) The deviations of both the variables are multiplied.
(f) All the data are summed up according to formula to calculate ‘r’.
Properties of Correlation Coefficient (Fig. 13.1):
1. Correlation coefficient lies between – 1 and + 1, i.e., -1 ≤ r ≤ +1
2. If r = + 1, the correlation is perfect and positive, if it is less than + 1 then moderately positive.
3. If r = – 1, the correlation is perfect and negative, if it is higher than – 1 then moderately negative.
4. If r = 0, there is no correlation between the variables.
5. The coefficient of correlation is not affected by change and scale of origin.
Example 1:
Find out the correlation coefficient between the two attributes of 5 plants.
The value of r = 0.986 gives us the view that the two attributes like height of the plant and number of leaves are positively correlated and the value is as near about 1, so it can be concluded that the relationship is towards perfect positive relation.
Example 2:
The effect of pesticide, ‘Nuvan’ is tested on germination of Phaseolus seeds. Find out the correlation coefficient.
The data is arranged in the following table to calculate ‘r’:
The value of r = – 0.9677 denotes that the two variables, i.e., pesticide concentration and germination percentage are negatively correlated, though it is not perfectly negative but approaching towards perfect negative correlation.
Significance of Correlation Coefficient:
The calculated correlation coefficient should be checked from the correlation coefficient (r) table for the degree of freedom (number of pairs of observation minus one), i.e., (n – 1) at 0.05 to 0.001 probability level.
In Example 1, the ‘r’ value obtained is 0.986, the table r value at (10 – 1) = 9 degree of freedom at 0.001 probability level is 0.847. As the observed value is higher than the table value, so the ‘r’ value is highly significant.
In Example 2, the ‘r’ value is – 0.9677, the table ‘r’ value at (18 – 1) = 17 degree of freedom at 0.001 probability level is 0.693. As the observed value is much higher than the table value so, the ‘r’ value or the correlation is highly significant.