The AstroStat Slog

You would think that something like “measurement error” is a well-defined concept, and everyone knows what it means. Not so. I have so far counted at least 3 different interpretations of what it means.

Suppose you have measurements X={X_i, i=1..N} of a quantity whose true value is, say, X₀. One can then compute the mean and standard deviation of the measurements, E(X) and σ_X. One can also infer the value of a parameter θ(X), derive the posterior probability density p(θ|X), and obtain confidence intervals on it.

So here are the different interpretations:

1. Measurement error is σ_X, or the spread in the measurements. Astronomers tend to use the term in this manner.
2. Measurement error is X₀-E(X), or the “error made when you make the measurement”, essentially what is left over beyond mere statistical variations. This is how statisticians seem to use it, essentially the bias term. To quote David van Dyk
   

   For us it is just English. If your measurement is different from the real value. So this is not the Poisson variability of the source for effects or ARF, RMF, etc. It would disappear if you had a perfect measuring device (e.g., telescope).

3. Measurement error is the width of p(θ|X), i.e., the measurement error of the first type propagated through the analysis. Astronomers use this too to refer to measurement error.

Who am I to say which is right? But be aware of who you may be speaking with and be sure to clarify what you mean when you use the term!

There is a lesson that statisticians, especially of the Bayesian persuasion, have been hammering into our skulls for ages: do not subtract background. Nevertheless, old habits die hard, and old codes die harder. Such is the case with X-ray aperture photometry. Continue reading ‘Background Subtraction [EotW]’ »