Skip to main content# Statistical Analysis of Astronomical Measurement Errors

### Presentation #514.05 in the session “Education and Public Engagement III: TMT”.

Published onJan 11, 2021

Statistical Analysis of Astronomical Measurement Errors

Precise, repeated measurements of the redshift of a star or galaxy always exhibit variation even when there is no actual change in the redshift of the object being observed. SImilary, the redshift measured using spectroscopy typically will differ in both a systematic and random fashion from the corresponding measurement when photometry is used. These types of observed differences (in statistical terms referred to as *errors*) happen for all kinds of astronomical measurements. Besides inducing uncertainty, these differences complicate the determination of patterns and relationships. Among other things, regressing on an independent variable that includes random measurement error produces a biased regression coefficient. Statistical methodology used to assess the nature of measurement error has evolved over time and today can be easily handled using a measurement error model tailored to the study design. The parameters of the measurement error model describe the bias (systematic error) and the imprecision (random error). Depending on the design of the study and the measurements collected, the measurement error model may be simple, or if necessary, quite complex. Regardless of the particular model, the goal is to decompose the total observed error into its systematic and its random components. Failure to do this leads to confusion on how different measurement methods or techniques are actually related. The R package OpenMx provides a convenient way to construct measurement error models, estimate model parameters, and construct confidence intervals on any quantity of interest. Calibration equations can be easily deduced that account for bias and allow the measurements of one method to be translated into any other method. Path diagrams provide a way to visualize the measurement error model. Figure 1 shows a path diagram for a set of redshift measurements where there are two independent measurements of each redshift. Circles are used to denote the true but unknown redshifts which are represented by a latent variable. Squares are used to denote the observed measurements. Here, long arrows connect the true values to the observed measurements with short arrows denoting variances for error components Figure 2 shows a path diagram that could be used to compare measurements from 3 different methods, for example, 3 different photometric methods. The path diagrams can be directly translated into the corresponding set of structural equations. Full information maximum likelihood estimation can be used to determine model parameters. Common mistakes made when designing methods comparison studies are discussed.