# Goodness-of-fit value from orthogonal distance regression

I have two sets of data, $x$ and $y$, with poissonian distributions. I want to check if the relation between $x$ and $y$ is a proportionality $y = ax + b$, so I used some algorithms to do Bivariate Correlated Errors and intrinsic Scatter (BCES). It is the first time I use it.

These algorithms return the parameters values, their errors, and a $2\times2$ covariance matrix.

I am trying to read tons of numbers and definitions, often ambiguous and/or conflicting, and anyway not easy-to-understand.

My goal is to understand how good is the fit, that is if the line fits the data points. Something like the $\chi^2$ goodness-of-fit. Why the algorithms return the covariance matrix (instead of a goodness-of-fit value)? Is it possible to infer one from the other?

• (1) It's not particularly useful to talk about Poisson errors; better to simply refer to the Poisson distribution of the observed response. (2) $y=ax+b$ isn't simple proportionality, that would be $y = ax$. (3) when you say errors, do you mean standard errors or something else? (4) I am not sure TLS (in its usual form) is suitable for Poisson data May 2 '14 at 10:10
• Your inputs to whatever function you used took account of the different variability for the Poisson counts? How large are the counts, typically? May 2 '14 at 10:15
• Glen_b, Thank you for all these clarifications. (1) ok, got it. (2) it is simple proportionality, except for an offset then :) (3) Yes, I mean standard errors. (4) Do you know anything suitable for Poisson data? However, I am using an algorithm based on Bivariate Correlated Errors and intrinsic Scatter (BCES) that is made appositely for astronomical data, so I think it is suitable. May 2 '14 at 10:20
• I don't know of a specific program, but it should be possible simply to write the corresponding model and calculate likelihood. I will look at the paper and see if I can follow it. May 2 '14 at 10:24
• Okay the paper's actually pretty clear; it's doing MoM. If you're happy with MoM estimates it should be fine. May 2 '14 at 11:13