TITLE:

Linear Regression for Astronomical Data with Measurement Errors
and Intrinsic Scatter

AUTHORS:  Akritas, M. G., Bershady, M. A.

PUBLISHED: 1996, The Astrophysical Journal, in press

ABSTRACT: 

Two new methods are proposed for linear regression analysis for
data with measurement errors. Both methods are designed to
accommodate intrinsic scatter in addition to measurement
errors. The first method is a direct extension of the ordinary
least squares (OLS) estimator to allow for measurement errors.
It is quite general in that a) it allows for measurement errors
on both variables, b) it allows the measurement errors for the
two variables to be dependent, c) it allows the magnitudes of
the measurement errors to depend on the measurements, and d)
other `symmetric' lines such as the bisector and the orthogonal
regression can be constructed.  We refer to this method as BCES
estimators (for Bivariate Correlated Errors and intrinsic
Scatter). The second method is a weighted least squares (WLS)
estimator, which applies only in the case where the
`independent' variable is measured without error and the
magnitudes of the measurement errors on the 'dependent' variable
are independent from the measurements.

Several applications are made to extragalactic astronomy: The
BCES method, when applied to data describing the
color-luminosity relations for field galaxies, yields
significantly different slopes than OLS and other estimators
used in the literature. Simulations with artificial data sets
are used to evaluate the small sample performance of the
estimators.  Unsurprisingly, the least-biased results are
obtained when color is treated as the dependent variable. The
Tully-Fisher relation is another example where the BCES method
should be used because errors in luminosity and velocity are
correlated due to inclination corrections.  We also find, via
simulations, that the WLS method is by far the best method for
the Tolman surface-brightness test, producing the smallest
variance in slope by an order of magnitude. Moreover, with WLS
it is not necessary to ``reduce'' galaxies to a fiducial
surface-brightness, since this model incorporates intrinsic
scatter.