# Partial – Part and Multiple Correlation

Partial – Part and Multiple Correlation, earlier we discussed Biserial correlation and kendall coefficient of concordance, today we are going to look at following.

Highlights:

• Partial Correlation
• Part Correlation
• Multiple Correlation
• Coefficient of determination
• Coefficient of alienation

Partial correlation: This is the relationship among more than two variables such that the effect of the third variable is being removed from the other two variables. So when you have XY – Z, it means we are removing the effect of the third variable which is ‘Z’.

Now, the formula follows;

Where, r is Pearson product moment correlation coefficient which we discussed in one of our previous sessions.

rxy = means Pearson product moment correlation between x and y. also rxy means correlation between x and z and ryz means Pearson product moment correlation between y and z.

Example:

Suppose a researcher is interested in finding the relationship between student age and their achievement in mathematics over 20 while partially out the effect of their height.

Note the data given below is hypothetical.

Let the coding or assignment be;

X = age

Y =Achievement

Z = height

Now we find the respective correlation using Pearson product moment for XY, XZ and YZ.

Pearson Correlation for xy(age and achievement)

= (6 x 532 – 39 x 81)/Sqr((6 x 271 – 39 x 39)(6 x 1111 – 81 x 81))

= 0.26

This shows a low relationship between student age and achievement.

Pearson Correlation for xz(age and height)

=  (6 x 191 – 39 x 27)/Sqr((6 x 271 – 39 x 39)(6 x 139 – 27 x 27))

= 0.89

The correlation coefficient of 0.89 indicate a high relationship between student age and height.

Finally, correlation for yz(achievement and height)

= (6 x 363 – 81 x 27)/Sqr((6 x1111 – 81 x 81)(6 x39 – 27 x 27))

= – 0.09

There is an inverse relationship between achievement and height.

## Partial – Part and Multiple Correlation

Now using the partial equation formula

r (xy-z) = (0.26 – 0.89 x (-0.09))/Sqr((1-0.89 x 0.89)(1-(-0.09 x -0.09)

= 0.74

This shows a high positive relationship between x(age) and y(achievement) while controlling effect of z(height). It implies that removing the effect of height from the relationship between age and achievement, increases the relationship between age and achievement.

### Part Correlation

In part correlation, the effect of one variable is removed from either of the variable or it is the removing of the effect of the third variable from either of the variable.

Take note that in partial correlation, we remove the effect of the third variable from the other two variable but in part correlation, we remove the effect of the third variable form one of the other two variable.

The formula is given as;

Now using the previous correlation coefficients we calculated, our part correlation will be;

rx(y-z)  = (0.26 – 0.89 x (-0.09))/Sqr(1-(-0.09 x -0.09))

= 0.34

Multiple Correlation

In multiple correlation, we find or determine the relationship between three or more variables simultaneously or at the same time. So in multiple correlation, we do not remove the effect of any variables.

Using the previous coefficients

rxy = 0.26, rxz = 0.89, ryz = – 0.09

Rx(y.z) = Sqr((0.26×0.26+0.89×0.89)-(2×0.26×0.89x(-0.09)))/(1-(-0-09x-0.09))

= 0.95

This shows a high positive relationship between age, achievement and height. It means that students’ age, achievement and height are highly related.

Coefficient of Determination: This tells us the proportion of relationship between the variables.

C.O.D = R(squared)

=0.95 x 0.95

=0.90 = 90% .  it implies that 90% relationship exit among the three variables.

Coefficient of Alienation: This tells us the proportion or percentage of how the variables are not related.