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I am looking through the linear association between variable $X$ and $Y$ with some controlling variables $\mathbf{Z} =(Z_1, Z_2, Z_3, \dots)$

One approach is to do a regression $E(Y)= \alpha_0 +\alpha_1 X + \sum\beta_iZ_i$ and look at $\alpha_1$.

Another approach is to calculate the partial correlation $\rho_{XY.\mathbf{Z}}$

My first question is which one is more appropriate? (Added: From the comments, they are equivalent but different presentations)

For multivariate normal $(X,Y,\mathbf{Z})$, partial correlation $\rho_{XY . Z}$ would be a better choice as its value can tell us how well $X$ and $Y$ being associated, for example $\rho_{XY . \mathbf{Z}} = \pm 1$ can be interpreted as $X$ and $Y$ have a perfect linear relationship after controlling $\mathbf{Z}$.

How about if some $Z_i$'s are categorical? Is partial correlation still proper to measure the association between $X$ and $Y$ controlling for $Z_i$'s? (Added: It is acceptable by changing categorical variables to dummy and then control for them, just like how we handle them in regression). As when I learnt partial correlation, it was used for multivariate normally distributed variables; I am not sure whether it is still appropriate and meaningful when normality of data is violated (say, highly skewed) or even the continuity is not the case (say, one of our controlling variables is 'place of birth'). The calculation of partial correlation of $X$ and $Y$ controlling $Z$ involve Pearson's correlation $\rho_{XZ}$ and $\rho_{YZ}$ which does not make sense when $Z$ is categorical, which makes $\rho_{XY . Z}$ look weird.

Also, is there any robust version of partial correlation (like kendall's $\tau$/Spearman's rank correlation to Pearson's correlation)?

Raised from ssdecontrol: Regression "works" and "makes sense" with categorical predictors, but correlation is occasionally said to be inappropriate for categorical data. Since regression is partial correlation, we have an apparent paradox.

Thanks.

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    $\begingroup$ Partial correlation reflect regression coefficient (stats.stackexchange.com/q/76815/3277), so the choice between them is dictated purely by presentation considerations. Categorical predictors can be incorporated in a linear model by making them dummy (or some other contrast)variables. For your request of "robust" version, this question may be related: stats.stackexchange.com/q/40995/3277. $\endgroup$ – ttnphns Jan 13 '15 at 11:52
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    $\begingroup$ Partial correlation and beta coefficient are just different ways to present basically the same thing. Yes, you may use partial correlation with any predictors to be dummy. $\endgroup$ – ttnphns Jan 13 '15 at 12:06
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    $\begingroup$ You might take interest in partial Eta squared as the general measure of effect size in linear model (regression, ANOVA). It is SSeffect/(SSeffect+SSerror). For a continuous predictor (a covariate) it equals the square of the partial correlation coefficient. $\endgroup$ – ttnphns Jan 13 '15 at 12:17
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    $\begingroup$ @ttnphns i think this question could lead to some insightful answers. Regression "works" and "makes sense" with categorical predictors, but correlation is occasionally said to be inappropriate for categorical data. Since regression is partial correlation, we have an apparent paradox and an explanation is in order $\endgroup$ – shadowtalker Jan 13 '15 at 14:57
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    $\begingroup$ The only time conclusions based on partial correlation require the variables all to be multivariate normal is if you're testing for conditional independence using the inverse correlation or covariance matrix. If you're making no claims about independence, then a calculation of partial correlation is always legitimate. $\endgroup$ – goodepic Jan 14 '15 at 7:06

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