Recently I've learnt about something called partial correlation coefficient (denoted as $\rho_{i,j|1...i,j...n}$ or in short, say $\rho_{i,j}$), which is like Pearson correlation between variables $X_i$ and $X_j$, but with assumption that all other variables in $X$ are fixed, i.e. we can say it is "conditional correlation" conditioned on all variables except $X_i, X_j$. One can compute it as $$\rho_{i,j} = \frac{-C_{i,j}}{\sqrt{C_{ii}C_{jj}}},$$ where $C_{i,j}$ is an algebraic complement of element $i,j$ in matrix $C = corr(X)$.

The question is where and when is it used? I understand intuitively how to interpret this coefficient, but never seen such thing before in any data analysis case (like, in Medium posts or textbooks) and just wondering why.

  • $\begingroup$ It's used in time series analysis, for instance. Or in any situation where you may have spurious correlations and you want to estimate the correlation between two random variables, after removing the contribution of all other variables. For instance in correlation network analysis. $\endgroup$
    – piplustwo
    Apr 18, 2023 at 9:18
  • $\begingroup$ @piplustwo do you have any example of resource where and how it is used? $\endgroup$ Apr 18, 2023 at 11:20
  • $\begingroup$ sachaepskamp.com/dissertation/Chapter2.pdf this is a nice R package for ppcor network analysis cran.r-project.org/web/packages/qgraph/qgraph.pdf $\endgroup$
    – piplustwo
    Apr 18, 2023 at 12:28


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