# Error propagation in intersect of line and regression line: at each step or the full expression?

BACKGROUND

I am a PhD working with physics measurements, each with measurement uncertainties attached. A measure of central tendency is found using a model specific to my field. The output is a measure of central tendency ($$y$$) and an associated standard error ($$\delta y$$).

For the problem at hand I have two measures of central tendency plotted against some variable x that has no error attached to it. I also have a 1to1 line between y and x. This is where the problem starts.

PROBLEM

I wish to determine the point of intersect $$(x^*, y^*)$$ between the regression line and the 1to1 line. I also want to calculate the errors that go with those estimates.

So I have a pair of coordinates and their associated standard errors:

$$\begin{pmatrix} x_1 & y_1 \pm \delta y_1 \\ x_2 & y_2 \pm \delta y_2 \end{pmatrix}$$

And I have proportional line defined by:

$$y=1x$$

The first part is not too difficult. I find regression coefficients ($$y=\beta_1 x + \beta_0$$) and determine intersect:

• Step 1: calculate slope: $$\quad\quad\quad \beta_1 = \frac{y_2-y_1}{x_2-x_1}$$
• Step 2: calculate intercept: $$\quad\quad \beta_0 = y_2 - \beta_1 x_2$$
• Step 3: calculate $$x^*$$: $$\quad\quad\quad\quad x^* = \frac{\beta_0-0}{1-\beta_1}$$
• Step 4: calculate $$y^*$$: $$\quad\quad\quad\quad y^* = 1*x^* \quad or \quad y^* = \beta_1 x^* + \beta_0$$

The error-analysis part of it is the tricky bit. It seems to me that there are two main ways to go about error propagation here. I could propagate errors stepwise, starting with calculating error of $$\beta_1$$ ($$\delta \beta_1$$), then using that error estimate to calculate $$\delta \beta_0$$, then $$\delta x^*$$ and finally $$\delta y^*$$ (which is identical to $$\delta x^*$$). Alternatively, I could just reduce the steps to one fully substituted expression and do error propagation once.

$$x^* = \frac{\beta_0-0}{1-\beta_1} = \frac{y_2 - \beta_1 x_2}{1-\beta_1} = \frac{y_2 - \frac{y_2-y_1}{x_2-x_1} x_2}{1-\frac{y_2-y_1}{x_2-x_1}} \quad\quad \delta x^* = \sqrt{(\frac{\partial f}{\partial y_1}*\delta y_1)^2 + (\frac{\partial f}{\partial y_2}*\delta y_2)^2}$$

Which is better? Or am I giving you stats-guys a seizure by just reading this and an obvious third option is the way to go?