In this post, we will learn about Partial Correlation and will perform on a data as Partial Correlation Example. In multiple correlations there are more than 2 variables, (3 variables and above) also called multivariable, in partial correlation there involved 3 or more variables, partial correlation is defined as the degree of the linear relationship between any two variables, in a set of multivariable data, by keeping the effect of all other variables as a constant.
Table of Contents
Partial Correlation Formula
For three variables say $X_1, X_2, X_3$ then the partial correlation measures the relation between $X_1$ and $X_2$ by removing the influence of $X_3$ is the partial correlation $X_1$ and $X_2$. And is given as
$$r_{12 \cdot 3}= \frac{ r_{12} – r_{13} r_{23}} {\sqrt{(1-r_{13}^2)(1- r_{23}^2)} }$$
If we want to find the partial correlation between $X_1$ and $X_3$ then
$$r_{13\cdot 2}= \frac{ r_{13} – r_{12} r_{32}}{ \sqrt{(1- r_{12}^2)(1- r_{32}^2)}}$$
If we want to find the partial correlation between $X_2$ and $X_3$ then
$$r_{23\cdot 1}= \frac{r_{23} – r_{21} r_{31}}{\sqrt{(1- r_{21}^2)(1- r_{31}^2)}}$$
Partial Correlation Graphical Representation
Partial correlation is a statistical measure of relationship between two variables while controlling for (excluding or eliminating) the effects of one or more additional variables. For three variables, say $X, Y,$ and $Z$ is
Partial Correlation is used when researchers want to determine the strength and direction of relationship between two variables without the influence of other variables. This is particularly useful in multivariate analysis where multiple variables may be interrelated. The partial correlation coefficient ranges from $-1$ to $+1$, with $-1$ indicating a perfect negative correlation, $+1$ indicating a perfect positive correlation, and 0 indicating no correlation.
Partial Correlation Example
For Partial Correlation Example, consider the following data with some basic computation.
| $X_1$ | $X_2$ | $X_3$ | $X_1X_2$ | $X_1X_3$ | $X_2X_3$ | $X_1^2$ | $X_2^2$ | $X_3^2$ | |
|---|---|---|---|---|---|---|---|---|---|
| 7 | 4 | 1 | 28 | 7 | 4 | 49 | 16 | 1 | |
| 12 | 7 | 2 | 84 | 24 | 14 | 144 | 49 | 4 | |
| 14 | 8 | 4 | 112 | 56 | 32 | 196 | 64 | 16 | |
| 17 | 9 | 5 | 153 | 85 | 45 | 289 | 81 | 25 | |
| 20 | 12 | 8 | 240 | 160 | 96 | 400 | 144 | 64 | |
| Total | 70 | 40 | 20 | 617 | 332 | 191 | 1078 | 354 | 110 |
First compute $r_{21}, r_{13}, r_{23}, r_{12}, r_{31}$, and $r_{32}$.
\begin{align}
r_{12} &= \frac{n\Sigma (x_1 x_2 ) – (\Sigma x_1)(\Sigma x_2 )} {\sqrt{\left[n\Sigma x_1 ^2 -(\Sigma x_1)^2\right] \left[n \Sigma x_2^2 – (\Sigma x_2 )^2\right]}}\\
&= \frac{5(617)-(70)(40)} {\sqrt{\left[5 (1078)-(70)^2\right]\left[5(354)-(40)^2\right]} } = 0.987\\
r_{13} &= \frac{n\Sigma(x_1 x_3 ) – (\Sigma x_1)(\Sigma x_3 )}{\sqrt{\left[n\Sigma x_1^2 – (\Sigma x_1 )^2\right]\left[n \Sigma x_3^2 – (\Sigma x_3 )^2\right]}}\\
&= \frac{5(332)-(70)(20)}{\sqrt{\left[5 (1078)-(70)^2\right]\left[5(110)-(20)^2\right]}}= 0.959\\
r_{23} &= \frac{n\Sigma(x_2 x_3 )-(\Sigma x_2 )(\Sigma x_3 )}{\sqrt{\left[n\Sigma x_2^2 -(\Sigma x_2 )^2\right]\left[n\Sigma x_3^2 -(\Sigma x_3 )^2\right]}}\\
& = \frac{5(191)-(40)(20)}{\sqrt{\left[5(354)-40^2\right]\left[5(110)-20^2\right]}}= 0.971\\
r_{12\cdot 3} &= \frac{r_{12} – r_{13} r_{23} } {\sqrt{(1 – r_{13}^2) (1 – r_{23}^2) }}\\
& = \frac{0.987-(0.959)(0.971)} {\sqrt{(1-(0.959)^2)(1-(0.971)^2)}}\\
&=\frac{0.05659}{0.0681} = 0.8305
\end{align}
Partial correlation is commonly used in statistical analysis, especially in fields like psychology, social sciences, and any area where multivariate relationships are analyzed.
