The Partial Correlation Coefficient measures the relationship between any two variables, where all other variables are kept constant i.e. controlling all other variables or removing the influence of all other variables. Partial correlation aims to find the unique variance between two variables while eliminating the variance from the third variable. The partial correlation technique is commonly used in “causal” modeling of fewer variables. The coefficient is determined in terms of the simple correlation coefficient among the various variables involved in multiple relationships.
Assumptions for computing the Partial Correlation Coefficient
The assumption for partial correlation is the usual assumption of Pearson Correlation:
- Linearity of relationships
- The same level of relationship throughout the range of the independent variable i.e. homoscedasticity
- Interval or near-interval data, and
- Data whose range is not truncated.
We typically conduct correlation analysis on all variables so that you can see whether there are significant relationships amongst the variables, including any “third variables” that may have a significant relationship to the variables under investigation.
This type of analysis helps to find the spurious correlations (i.e. correlations that are explained by the effect of some other variables) as well as to reveal hidden correlations – i.e. correlations masked by the effect of other variables. The partial correlation coefficient $r_{xy.z}$ can also be defined as the correlation coefficient between residuals $dx$ and $dy$ in this model.
Suppose we have a sample of $n$ observations $(x1_1,x2_1,x3_1),(x1_2,x2_2,x3_2),\cdots,(x1_n,x2_n,x3_n)$ from an unknown distribution of three random variables. We want to find the coefficient of partial correlation between $X_1$ and $X_2$ keeping $X_3$ constant which can be denoted by $r_{12.3}$ is the correlation between the residuals $x_{1.3}$ and $x_{2.3}$. The coefficient $r_{12.3}$ is a partial correlation of the 1st order.
\[r_{12.3}=\frac{r_{12}-r_{13} r_{23}}{\sqrt{1-r_{13}^2 } \sqrt{1-r_{23}^2 } }\]
The coefficient of partial correlation between three random variables $X$, $Y$, and $Z$ can be denoted by $r_{x,y,z}$ and also be defined as the coefficient of correlation between $\hat{x}_i$ and $\hat{y}_i$ with
\begin{align*}
\hat{x}_i&=\hat{\beta}_{0x}+\hat{\beta}_{1x}z_i\\
\hat{y}_i&=\hat{\beta}_{0y}+\hat{\beta}_{1y}z_i\\
\end{align*}
where $\hat{\beta}_{0x}$ and $\hat{\beta_{1x}}$ are the least square estimators obtained by regressing $x_i$ on $z_i$ and $\hat{\beta}_{0y}$ and $\hat{\beta}_{1y}$ are the least square estimators obtained by regressing $y_i$ on $z_i$. Therefore by definition, the partial correlation between of $x$ and $y$ by controlling $z$ is
\[r_{xy.z}=\frac{\sum(\hat{x}_i-\overline{x})(\hat{y}_i-\overline{y})}{\sqrt{\sum(\hat{x}_i-\overline{x})^2}\sqrt{\sum(\hat{y}_i-\overline{y})^2}}\]
The coefficient of partial correlation is determined in terms of the simple correlation coefficients among the various variables involved in a multiple relationship. It is a very helpful tool in the field of statistics for understanding the true underlying relationships between variables, especially when you are dealing with potentially confounding factors.
Reference
Yule, G. U. (1926). Why do we sometimes get non-sense correlations between time series? A study in sampling and the nature of time series. J. Roy. Stat. Soc. (2) 89, 1-64.
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