Correlation or Simple Correlation in linear form is mostly studied between two variables. However in statistics if one wants to study the relationships between three variables then there exists some common relation which is affected by the third variable. That is, the true relationship between two variables will be affected by third variable. In effect, one has to have a correlation measure of the two variables and keep the other constant to calculate the true relationship between the variables as third variable may affect and exaggerate or suppress the relationship between the two variables considered for study. The correlation between two variables keeping the other constant is called partial correlation in statistical terminology.  Partial correlation analysis is necessary when considering relationship in linear form between more than two variables. This is important because the correlation between the two studied separately may overstate or understate the true relationship between the two variables as the other may have some influence on the other two variables and overstate or understate the actual relationship between the studied variable. In the following discussion I will show the partial correlation analysis and its importance.

Meaning of Partial Correlation

Say one studies the relationship between three variables X, Y and Z consist of N subjects and finds that you have the following correlation as follows

X and Y correlation r = +0.50 and r2= 0.25

X and Z correlation r = +0.50 and r2 = 0.25

Y and Z correlation r = +0.50 and r2=0.25

 Note that there is a region which is common to all the three variables. The meaning of this is the relationship between two of the variables whether it is between x and y or x and z or y and z is affected by the third variable. Thus, the 0.25 variance overlap between X and Y approximately half is tied in with the overlap that exists between X and Z and Y and Z. In effect, partial correlation between the X and Y is the removal of the third variable Z and keeping constant so that it removes the correlation of X with z and correlation Y with Z.

In mathematical form, the partial correlation of r XY.Z or r XZ. Y or r YZ. X can be expressed as follows:

That is, r XY. Z = r XY – (r XZ) (r YZ)/square root of [1-r2 XZ]*square root of [1-r2YZ]

 r X Z. Y = r XZ – (r XY) (r YZ)/square root [1-r2XZ]*square root [1-r2YZ]

 r YZ. X = r YZ – (r XY) (r XZ)/square root [1-r2XY[* square root [1-r2 XZ]

By using this formulas r XY = 0.50 -0.50*0.50/sqrt [1-0.25]* sqrt [1-0.25]

That is r XY. Z = 0.33 and r2 XY .Z = 0.11. One can see the partial correlation in this instance is smaller than the correlation between X and Y not keeping Z constant and r2 is also smaller in this instant. That is the relation ship of X and Y keeping Z constant is different to that of the correlation between X and Y not keeping Z constant.

Example of Practical application of partial correlation

Say one has devised a measurement of Intelligence a scale and also has developed sub scales for comprehension, vocabulary and arithmetic ability. Then he wants to study say for subjects who has homogeneity in vocabulary to know the relationship between comprehension and arithmetic ability.  He has worked out for n subjects the correlation between these variables as follows in the past tests for intelligence sub-scale relationships.

Comprehension versus Arithmetic ability correlation r = +0.49 and r2 = 0.24

Comprehension and vocabulary correlation = r = +0.73 and r2 = 0.53

Arithmetic ability and Vocabulary r = +0.59 and r2 = 0.35

In this example the researcher if he calculates the partial correlation between comprehension and arithmetic ability keeping vocabulary constant then only he can infer whether comprehension is truly related to arithmetic ability as vocabulary ability as a relation ship with comprehension and with arithmetic ability and overstate the correlation between comprehension and arithmetic ability.

The partial correlation of comprehension and arithmetic ability keeping vocabulary constant can be worked out using the previous formulas as follows:

R CA. V = 0.49 – (0.73)* (0.50)/ sqrt [1-0.53]* sqrt[1-0.35]

That is, r C A. V = +0.11 and r2 = 0.01.

On the basis of the partial correlation of comprehension and arithmetic ability if the subjects are homogeneous in terms of vocabulary there is no substantial relationship between comprehension and arithmetic ability because the partial coefficient of determination which measures the partial correlation strength is too low.

This demonstrates the importance in actual reality the use of partial correlations if there is more then two variables relationships to be studied and to know the impact of the other variable on the variables under study and remove its effects. In addition, the partial correlation can be higher in some instances than in normal circumstances. In these situations partial correlation becomes more than the pure correlation due to the suppression of one variable on the other two so that he actual relationship is distorted and suppressed.

In essence partial correlation analysis is important when studying linear relationships between more than two variables and studying three variables instead to shed light on the impact of the third variable on the relationship under study.