1. Correlation coefficient and path coefficient analysisBy
Rajesh Ranjan and Amit Kumar Gaur
PAU, Ludhiana ,Punjab

2. What is correlation?
Correlation is a statistical device which help in analyzing the covariation of two or more variables. It helps us in determining the degree of relationship between two or more variables. But it does not tell us about cause and effect relationship. Correlation analysis consist of two simple steps:
Determining whether a relationship exists and, if it does, measuring it.
Testing whether it is significant.

3. Why we study correlation?
To find the nature of relationship between two or more variables.
To estimate the value of one variable if the value of another is given.
To reduce the range of uncertainty. The prediction based on correlation analysis is likely to be more valuable and near to reality.

4. What are the reasons for correlation between variables?
It may be due to pure chance, especially in a small variables in a sample. But in the universe there may not be any relationship between the variables.
Both the correlated variables may be influenced by one or more other variables e.g. a high degree of correlation between the yield/acre of rice and tea may be due to the fact that both are related to amount of rainfall.
Both the variables may be mutually influencing each other so that neither can be designated as cause and effects e.g. price and demand.

5. Different types of correlations
There are three ways to classify the correlation
1. Type 1
Positive correlation
Negative correlation
No correlation
2. Type 2
Linear correlation
Non linear correlation
3. Type 3
Simple correlation
Multiple correlation
Partial correlation

6. Type1
Positive correlation
Negative correlation
No correlation
Positive correlation: If two related variables are such that when one increases (decreases), the other also increases (decreases)
Negative correlation: If two variables are such that when one increases (decreases), the other decreases (increases)
No correlation: If both the variables are independent.

7. Non-Linear correlation
Type 2
Linear correlation
Non-Linear correlation
Linear correlation: When plotted on a graph it tends to be a perfect line
Non-Linear correlation: When plotted on a graph it is not a straight line

8. Type 3 Simple correlation Multiple correlation Partial correlation
Simple correlation: In this only two variables are studied.
Multiple correlation: In this three or more variables are studied simultaneously.
Partial correlation: we recognize more than two variables but consider only two variables to be influencing each other and effect of other influencing variables being kept constant.

9. Type 1 Type 2 Graphical representation of type1 and type 2 correlation
Positive correlation
Negative correlation
No correlation
Type 2
Positive linear
Negative linear
Non linear

10. Interpretation of coefficient of correlation
When r = +1, it means there is perfect positive relationship between the variables.
When r = -1, it means there is perfect negative relationship between the variables.
When r = 0, it means there is no relationship between the variables.
When r is closer to -1 or +1 than relationship between the variables are also closer.

11. Properties of coefficient of correlation
The coefficient of correlation lies between -1 to +1.
The coefficient of correlation is independent of change of scale and origin of the variable x & y.
The degree of relationship between the two variables is symmetric 𝑟 𝑥𝑦 = 𝑟 𝑦𝑥 .

12. Coefficient of determination
It is a useful way of interpreting the value of coefficient of correlation between two variables
Coefficient of determination ( 𝐫 𝟐 ) =
For example If the value of r = 0.9 than r 2 =0.81 it means 81% of the variation in the dependent variable has been explained by the independent variable
Explained variance Total variance

13. Properties of coefficient of determination
Its range lies between 0 to 1
Represented by r2
The coefficient of determination is a measure of how well the regression line represents the data
If the regression line passes exactly through every point on the scatter plot, it would be able to explain all of the variation
The further the line is away from the points, the less it is able to explain

14. Methods of studying correlation
Scatter diagram method
Karl’s Pearson coefficient of correlation
Spearman’s coefficient of correlation
Concurrent deviation method

15. Scatter Diagram method
                                                                                                      
Scatter Diagram method
Scatter Diagram
Scatter Diagram

16. Merits and limitations of scatter diagram
It is simple and non mathematical methods of studying correlation between the variables
Making a scatter diagram usually is the first step in investigating the relationship between two variables
Limitations
In this method we can not measure the exact degree of correlation between the variables

17. Karl Pearson’s Correlation Coefficient
Karl Pearson ( ) British mathematician and statistician
The extent to which two variables vary together is called covariance and its measurement is the correlation coefficient
r = Covariance ( X, Y)
SD (X) . SD (Y) or
r = /N[ (X – X) (Y – Y)]
1/N(X – X)2 1/N(Y – Y)2
An alternative computational equation is given below.
r = N(XY) – (X) (Y)
[N X2 – (X)2] [NY2 – (Y)2]
Where N= No. of Pairs

18. r12 = 0.926 Yield (X) Nitrogen applied (Y) X2 Y2 XY e.g. 16.2 262.44
262.44
31.5 40
992.25
1600
1260
30.6 60
936.36
3600
1836
39.4 80
6400
3152
12.9
166.41 25
625
1000
31.9
1914
37.5
3000
18.9
357.21
36.1
1444 38
2280
40.3
3224
∑X=358.3
∑Y=540
∑X2=
∑Y2 =34800
∑XY =19110
r12 = 0.926

19. Test for Significance of Observed Correlation Coefficient
Null Hypothesis Ho: ρ = 0
Alternative Hypothesis H1: ρ ≠ 0 ( Two tailed test)
Test Statistics
t = r √ n ~ tα/2, n-2 d.f
√ 1- r2
Where r is the sample correlation coefficient.
If t cal ≤ tα/2, n-2 d.f we don’t enough evidence to reject Ho
Example : Calculated value of r is 7.75 at 10 d.f at 1% level of significance
( t = 3.169)
Conclusion : No- Significant Correlation exists between yield and Nitrogen applied

20. Merits and limitations of Karl Pearson’s Correlation Coefficient
It is most popular method used for measuring the degree of relationship.
It helps us to find the exact degree of correlation
Limitations
The correlation coefficient always assumes linear relationship regardless of the fact whether assumption is correct or not
Takes more time to computes correlation coefficient

21. Spearman’s Coefficient of correlation
A method to determine correlation when the data is not available in numerical form and as an alternative, the method of rank correlation is used.
Thus when the values of the two variables are converted to their ranks, and there from the correlation is obtained, the correlations known as rank correlation.
This method was developed by British psychologist Charles Edward Spearman in 1904

22. Where, m= number of times common ranks are repeated x Rank y𝐷 2
𝑟 𝑠 = 1- 6 { +
1 12
( 𝑚 3 - m) +
1 12 ( 𝑚 3 - m) + ………}
𝑁 3 -N
Where, m= number of times common ranks are repeated x
Rank y
Var 1 3 1 4
Var 2 7 6
Var 3
Var 4 2
Var 5 5
Var 6 36
Var 7 9 62
( 𝑅 𝑥 − 𝑅 𝑦 ) 2
𝐷 2 2

23. By using the formula 𝑟 𝑠 = 1- 6 x 62 / 7 3 -7 = 1-1
By using the formula 𝑟 𝑠 = 1- 6 x 62 / = Hence, 𝑟 𝑠 =

24. Merits and limitations of Spearman’s coefficient of correlation
This method is simpler to understand and easier to apply as compared to the Karl pearson’s method
This method can be used with great advantage where the data are of a qualitative in nature
Limitations
This method should not be applied where N exceeds 30 because the calculations become tedious and require a lot of time

25. Concurrent deviation method
Steps involved in this methods are
Find out the direction of change of x variable as compared with the first value whether, second value is increasing or decreasing or constant, and denote this column by 𝐷 𝑥
If it is increasing put a (+) sign and if decreasing (–) sign and if constant than put 0
Similarly we can do it for y variable and denote this column by 𝐷 𝑦
Multiply 𝐷 𝑥 with 𝐷 𝑦 and find value of C, i.e. is number of (+) sign

26. 𝑟 𝑐 = ± 2𝐶−𝑛/𝑛 x 𝐷 𝑥 y 𝐷 𝑦 𝐷 𝑥 x 𝐷 𝑦 60 65 55 - 40 + 50 35 56 75 30 63
𝑟 𝑐 =
± 2𝐶−𝑛/𝑛 x
𝐷 𝑥 y
𝐷 𝑦
𝐷 𝑥 x 𝐷 𝑦 60 65 55 - 40 + 50 35 56 75 30 63 70 80 20
C= 8
± 2𝐶−𝑛/𝑛
6/10
𝑟 𝑐 = = =
0.774

27. Merits and limitations of Concurrent deviation method
It is simplest of all the methods
When the number of items is very large this method may be used to form a quick idea about the degree of relationship before making use of more complicated methods
Limitations
This method does not differentiate between small and big changes. e.g. If x increases from 100 to 101 the sign will be + and if y increases 60 to 160 the sign will be +. Thus both get equal weight when they very in same direction
The results obtained by this method are only a rough indicator of the presence or absence of correlation

28. PATH ANALYSIS
This was given by Sewell Wright in 1921.
If the cause and effect relationship is well defined, it is possible to represent the whole system of variables in the form of a diagram , known as path diagram
Path analysis is a method of splitting correlations into different components for interpretation of effects
Let Yield ‘Y’ of barley is the function (effect) of various components ( casual factors) like number of ears per plant ( 𝑥 1 ) , ear length ( 𝑥 2 ) and 100-grain weight ( 𝑥 3 ) etc. a
𝑥 1
r 𝑥 1 𝑥 2 b Y
𝑥 2
r x1 𝑥 3 c
r 𝑥 2 𝑥 3
𝑥 3 h
Some other undefined factors designated by R R

29. Definition
Path coefficient can be defined as ratio of standard deviation due to a given cause to the total standard deviation of the effect.
If Y is the effect and 𝑥 1 is the cause, the path coefficient for the path from cause 𝑥 1 to the effect Y is σ 𝑥 1 / σY
A set of simultaneous equations can be written directly from the Path diagram and the solution of these equations provides information of the direct and indirect contributions of the casual factors to the effect
Y = 𝑥 𝑥 2 + 𝑥 3 +R
Correlation between 𝑥 1 and Y i.e r ( 𝑥 1 , Y) is defined as
r ( 𝑥 1 , Y) = Cov ( 𝑥 1 , Y)
σ 𝑥 1 . σY

30. Correlation between 𝑥 1 and Y i.e r ( 𝑥 1 , Y) is defined as
r ( 𝑥 1 , Y) = Cov ( 𝑥 1 , Y)
σ 𝑥 1 . σY
By putting the value of Y in above equation, we get
r ( 𝑥 1 , Y) = Cov ( 𝑥 1 , 𝑥 1 + 𝑥 2 + 𝑥 3 +R)
= Cov ( 𝑥 1 ,x1) /( σ 𝑥 1 . σY) + Cov ( 𝑥 1 , 𝑥 2 ) /( σ 𝑥 1 . σY)
+ Cov ( 𝑥 1 , 𝑥 3 ) /( σ 𝑥 1 . σY) + Cov ( 𝑥 1 ,R) /( σ 𝑥 1 . σY)……………(1)
Where Cov ( 𝑥 1 , 𝑥 1 ) = V(x1)
Cov( 𝑥 1 ,R) =0 ( Assumed)
Cov ( 𝑥 1 , 𝑥 2 ) = r(x1, 𝑥 2 ) σx1 . σ 𝑥 2
Thus the equation 1 becomes:
r ( 𝑥 1 , Y) = V( 𝑥 1 )/σ 𝑥 1 .σY + r( 𝑥 1 , 𝑥 2 ) σ 𝑥 1 .σ 𝑥 2 / σ 𝑥 1 .σY +r( 𝑥 1 , 𝑥 3 )σ 𝑥 1 .σ 𝑥 3 / σ 𝑥 1 .σY
= σ 𝑥 1 /σY + r( 𝑥 1 , 𝑥 2 ) σ 𝑥 2 /σY +r( 𝑥 1 , 𝑥 3 )σ 𝑥 3 /σY ……………….(2)

31. r ( 𝑥 1 , Y) = σ 𝑥 1 /σY + r( 𝑥 1 , 𝑥 2 ) σx2/σY +r( 𝑥 1 , 𝑥 3 )σ 𝑥 3 /σY ……………….(2)
Where as per definition,
σ 𝑥 1 /σY =a, the path coefficient from 𝑥 1 to Y
σ 𝑥 2 /σY =b, the path coefficient from 𝑥 2 to Y
σ 𝑥 3 /σY =c, the path coefficient from 𝑥 3 to Y
Thus
r ( 𝑥 1 , Y) = a + r( 𝑥 1 , 𝑥 2 ) b +r( 𝑥 1 , 𝑥 3 ) c …………………..(3)
The correlation between 𝑥 1 and Y may be partitioned into three components
Due to direct effect of 𝑥 1 on Y which amounts to ‘a’
Due to indirect effect of 𝑥 1 on Y via 𝑥 2 which amounts to r( 𝑥 1 , 𝑥 2 ) b
Due to indirect effect of 𝑥 1 on Y via 𝑥 3 which amounts to r( 𝑥 1 , 𝑥 3 ) c

32. Similarly one can work out the equations for r( 𝑥 2 ,Y), r( 𝑥 3 ,Y) and r(R,Y).
We thus finally get a set of simultaneous equations as given below
r ( 𝑥 1 , Y) = a + r( 𝑥 1 , 𝑥 2 ) b +r( 𝑥 1 , 𝑥 3 ) c …………………………………(.A)
r ( 𝑥 2 , Y) = r( 𝑥 2 , 𝑥 1 ) a + b + r( 𝑥 2 , 𝑥 3 ) c …………………………………(B)
r ( 𝑥 3 , Y) = r( 𝑥 3 , 𝑥 1 ) a + r( 𝑥 3 , 𝑥 2 ) b + c …………………………………..(C)
r ( R, Y) = h
The residual effect can be obtained by the following formula
h 2 = 1- a b c 2 -2r( 𝑥 1 𝑥 2 )ab-2r( 𝑥 1 𝑥 3 )ac-2r( 𝑥 2 𝑥 3 )bc
Considering only the first three factors i.e. 𝑥 1 , 𝑥 2 and 𝑥 3 , the simultaneous equations given above can be presented in matrix notation as
r 𝑥 1 𝑥 1 r 𝑥 1 𝑥 2 r 𝑥 1 𝑥 3
r 𝑥 2 𝑥 1 r 𝑥 2 𝑥 2 r 𝑥 2 𝑥 3
r 𝑥 3 𝑥 1 r 𝑥 3 𝑥 2 r 𝑥 3 𝑥 3
r 𝑥 1 Y
r 𝑥 2 Y
r 𝑥 3 Y a b c =
A = B.C
C = B-1A

33. Path analysis
Let us consider 4 characters and the correlations among them are as follows. Here 4 stands for Y. x1 stands for ears/plant x2 for ear length x3 for 100 grain weight 𝑟 12 = 𝑟 23 =
𝑟 13 = 𝑟 24 =
𝑟 14 = 𝑟 34 = a x1
𝑟 12 b
𝑟 13 Y x2 c
𝑟 23 x3 h
Some other undefined factors designated by R R

34. Note that 𝑃 14 =a, 𝑃 24 =b and 𝑃 34 =c Matrix method
𝑟 14 = 𝑃 𝑟 12 𝑃 𝑟 13 𝑃 34
𝑟 24 = 𝑟 21 𝑃 𝑃 𝑟 23 𝑃 34
𝑟 34 = 𝑟 31 𝑃 𝑟 23 𝑃 𝑃 34
Note that 𝑃 14 =a, 𝑃 24 =b and 𝑃 34 =c
Matrix method
A=B.C
Here, the value of A and B are known. We have to find the value of C vector
C= 𝐵 −1 A
𝑟 14 𝑟 24 𝑟 34
𝑟 11 𝑟 12 𝑟 13 𝑟 21 𝑟 22 𝑟 23 𝑟 31 𝑟 32 𝑟 33
𝑃 14 𝑃 24 𝑃 34 =

35. B= 𝑩 −𝟏 = AS per equation, C= 𝐵 −1 A
− −0.516 −0.015 −
− −
𝑃 14 𝑃 24 𝑃 34
0.822 −0.004 −0.167
− −
− −0.2298 = =

36. 𝑃 14 =(1.0008)(0.822)+(-0.0276)(-0.004)+(0.0008)(-0.167)=0.8226
Where,
𝑃 14 =(1.0008)(0.822)+( )(-0.004)+(0.0008)(-0.167)=0.8226
𝑃 24 =( )(0.822)+(1.3636)(-0.004)+(0.7032)(-0.167)=
𝑃 34 =(0.0008)(0.822)+(0.7032)(-0.004)+(1.3629)(-0.167)=
𝑃 14 =
𝑃 24 =
𝑃 34 =
Residual effect:
1= ( 𝑃 𝑅4 ) 2 + ( 𝑃 14 ) 2 + ( 𝑃 24 ) 2 + ( 𝑃 34 ) 𝑃 14 𝑟 12 𝑃 𝑃 14 𝑟 13 𝑃 𝑃 24 𝑟 23 𝑃 34
1= ( 𝑃 𝑅4 ) 2 + (0.8226) 2 + (−0.1456) 2 + (−0.2298) 2 +2 (0.8226) (0.028) ( )+2( )(-0.516)( )
1= ( 𝑃 𝑅4 )
And hence,
𝑃 𝑅4 =
𝑃 𝑅4 =
(1.0000−0.7152)

37. Calculation of Direct and indirect effects
(a) Ears per plants( 𝑥 1 ) and grain yield ( 𝑥 4 )
Direct effects = 𝑃 =
Indirect effect via ear length ( 𝑥 2 ) = 𝑃 24 𝑟 =
Indirect effect via 100- grain weight ( 𝑥 3 ) = 𝑃 34 𝑟 =
Total(direct+ indirect) effects =
(b) Ears length ( 𝑥 2 ) and grain yield ( 𝑥 4 )
Direct effects = 𝑃 = Indirect effect via ears per plants ( 𝑥 1 ) = 𝑃 14 𝑟 =
Indirect effect via 100- grain weight ( 𝑥 3 ) = 𝑃 34 𝑟 =
Total(direct+ indirect) effects =
c) 100- grain weight ( 𝑥 3 ) and grain yield ( 𝑥 4 )
Direct effects = 𝑃 = Indirect effect via ears per plants ( 𝑥 1 ) = 𝑃 14 𝑟 =
Indirect effect via ear length ( 𝑥 2 ) = 𝑃 24 𝑟 =
Total(direct+ indirect) effects =

38. Direct(diagonal) and indirect effects on yield components on yield
Characters
Ears per plants
Ear length
100-grain weight
Genotypic correlation with yield
0.8226
0.0035
0.8220
0.0230
0.1186
0.0751

39. Interpretation of Path Analysis results
If the correlation coefficient between a casual factor and the effect is almost equal to its direct effect, than correlation explains the true relationship and a direct selection through this trait will be effective.
If the correlation coefficient is positive , but the direct effect is negative or negligible , the indirect effects seem to be cause of correlation. In such situations, the indirect casual factors are to be considered simultaneously for selection.
Correlation coefficient may be negative but the direct effects are positive and high. Under these conditions, a restricted simultaneous selection model is to be followed, i.e. restrictions are to be imposed to nullify the undesirable effects in order to make use of direct effects.
The residual effect determine how best the causal factors account for the variability of the dependent factor, the yield in this case. Its’ estimate being , the variables(ears per plant, ear length and 100 grain weight) explained only about 45% of the variability in the yield.

40. Conclusion
Correlation simply measures the association of characters but it doesn’t indicates the relative contribution of causal factors to seed yield
The component characters are themselves interrelated and often affect their direct relationship with seed yield
Path coefficient analysis permits the separation of the direct effects from indirect effects through other related characters by partitioning the correlation coefficient

41. References
A Simplified Introduction to Correlation and Regression by K. L. Weldon
The Correlation between Relatives on the Supposition of Mendelian Inheritance. By R. A. Fisher
Biometrical Genetics: The study continuous variation by Kenneth Mather and John L. Jinks
The Genetical Analysis of Quantitative Traits by Michael J. Kearsey and Harpal S. Pooni
Biometrical techniques in Plant Breeding by Singh and Narayan
Quantitative Genetics by Phundan Singh
Biometrical Methods in Quantitative Genetic analysis by Singh and Chaudhary

42. Thank You