used in the regression. The standard error of the regression slope is: $$e_a = \frac{e_y}{\sum \left(x_i - \overline{x_i} \right)^2},
        \label{unc_slope}$$ and the standard error of the intercept is: $$e_b = e_y \: e_a \sqrt{\frac{\sum x_i^2}{n}}.
        \label{unc_intercept}$$ $e_a^2$ and $e_b^2$ are the diagonal terms in the covariance matrix of the multivariate normal distribution of regression slope and intercept from which Monte Carlo values are drawn. Slope and intercept values are strongly negatively correlated, which is captured by their covariance when performing the linear regression. The off-diagonal terms in the covariance matrix of the multivariate normal distribution constrain the random sampling of slope and intercept values to avoid sampling unrealistic combinations.

    ![image](f04){width="16cm"}

    An example of this sampling is shown in Figure \[Ex\_regression\] for two projects of different regression strengths. We sample 500 slope and intercept values from a multivariate normal distribution centered around the best-fit parameters, and with the covariance matrix derived from the standard errors of slope and intercept and their covariance. As shown in Figure 4, the low standard errors found for the leftmost regression relationship constrain the possible slope and intercept values that can be sampled, while the high standard errors in the rightmost regression relationship allow for a 