Is there any way to conduct a partial correlation analysis if the DV is dichotomous?
There isn't really a dependent variable in partial correlation; at least, the dependence of the two main variables (whose unmediated relationship is of interest – call 'em
y) on the controlling variables is removed. The unexplained variance of
y, i.e., the residuals, are the subjects of partial correlation. Once you have the residuals from linear models of
y as predicted by your control variables, correlate them, and there you have your partial correlation.
A simple correlation between a continuous variable
z and a dichotomous variable
y is a point-biserial correlation. OLS regression of the same two variables will produce this same correlation as a slope coefficient if the variables are standardized. The residuals of
y regressed onto
z (doesn't matter if you standardize the variables – the next step will have the same effect) can then be correlated with the residuals of a linear model of
x regressed onto
z to find the partial correlation between
Here's a brief demonstration in r in case anyone else finds it useful (I wanted to check my answer):
d8a=data.frame(y=rbinom(99,1,.5),x=rnorm(99),z=rnorm(99)) #Simulated data cor(lm(scale(y)~scale(z),d8a)$residuals,lm(x~z,d8a)$residuals) #Base package method require(ggm);pcor(c('x','y','z'),var(d8a)) #Dedicated function for partial correlation
The correlation of the residuals of the linear models of
y as predicted by
z is the same as the result of the function dedicated to producing partial correlations. This is no surprise; this is all normal procedure for calculating partial correlations. The algebraic operations don't change when one variable is dichotomous. However, whether one should interpret a partial correlation the same way regardless of whether any variables are dichotomous is another question...(Comments welcome!)
BTW, this article may be of tangential interest:
Bay, K. S., & Hakstianz, R. (1972). Note on the equivalence of the significance test of the partial point-biserial correlation and the one-factor analysis of covariance for two treatment groups. Multivariate Behavioral Research, 7(3), 391–395.