value of $\varepsilon_y$ is added in quadrature when $y$ is used as a proxy for $x$.

The bivariate fits are intended to establish the physical relationships among the various parameters and also to fit residuals. The actual mass estimation equations that we use will be based on multivariate fits of the general form $$\label{eq:twoparameters}
z = a + b\left( x - x_0 \right) +c\left(y - y_0\right),$$ where the parameters are as described above, plus an additional observed parameter $y$ that has median value $y_0$. Similarly to linear fits, the plane fitting minimizes the quantity $$\label{eq:chi2plane}
    \chi^2=\sum_{i=1}^N \frac{[a + b (x_i-x_0) + c (y_i-y_0) - z_i]^2}
    {(b \Delta x_i)^2 + (c \Delta y_i)^2 + (\Delta z_i)^2 + \varepsilon_z^2},$$ with $\Delta x_i$, $\Delta y_i$ and $\Delta z_i$ as the errors on the variables $(x_i,y_i,z_i)$, and $\varepsilon_z$ as the sigma of the Gaussian describing the distribution of intrinsic scatter in the $z$ coordinate; $\varepsilon_z$ is iteratively adjusted so that the $\chi^2$ per degrees of freedom $\nu=N-3$ has the value of unity expected for a good fit. The observed scatter is $$\label{eq:twoparmDeltadef}
\Delta = \left\{ \frac{1}{N-3} \sum_{i=1}^N
\left[y_i - a - b\left(x_i - x_0\right)
-c\left(y_i - y_0\right)\right]^2 \right\}^{1/2}.$$

Masses Based on 
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The $R$–$L$ 