I understand the concepts of partial and semi-partial correlation, but I am still not clear on when to use one over the other. Are there specific research questions that would determine that we need to control the third variable for only one vs for both?
Partial correlation explains the correlation between two continuous variables (let's say X1 and X2) holding X3 constant for both X1 and X2.
Semipartial correlation measures the strength of linear relationship between variables X1 and X2 holding X3 constant for just X1 or just X2. It is also called part correlation.
Partial correlation holds variable X3 constant for both the other two variables. Whereas, Semipartial correlation holds variable X3 for only one variable (either X1 or X2). Hence, it is called 'semi'partial
In practice the partial and semipartial r are the same in sign and tend to be similar in magnitude.