The bivariate correlation is defined as

Here, cov(xi, y) is the covariance between the scalar QoI and the ith input parameter, and V(xi) and V(y) are the variances of xi and y, respectively. Note that for a UQ study with more than one QoI, the correlations between QoIs and input parameters are computed for each QoI. The correlation result, ranging from −1 to 1, measures the linear relationship between xi and y, where a correlation equals 1 when y linearly grows with xi. Correlation is equal to −1 when y linearly decreases with xi.

The rank correlation is defined as the bivariate correlation between the rank values of xi and y. The rank correlation computes the monotonic relationship between xi and y. A rank correlation equals 1 when the QoI monotonically grows with xi and it equals −1 when the QoI monotonically decreases with xi.

Note that the bivariate correlation does not take into account the possible effect that other input parameters might have on the QoI. For instance, healthcare funding and disease rate could be positively correlated according to their bivariate correlation. Such an observation is contradictory because healthcare visits increasing with healthcare funding is ignored. On the other hand, partial correlation can be calculated to determine the linear relationship between xi and y where all the linear effects from the other input parameters are removed:

Here, x\i denotes all parameters but xi. The rank partial correlation is defined as the bivariate correlation between the rank values of xi and y. In terms of linear regression, the partial correlation can be interpreted as the correlation between the residuals resulting from the linear regression of xi with x\i and y with x\i.