Selection, covariance, and Malmquist bias

The correction for Malmquist bias is generally thought to have the form α{DELTA}^2^, where {DELTA} is the rms fractional error for a given distance indicator and α is a numerical coefficient which depends on the galaxy distribution and other aspects of the problem. In fact, this form of the correction is valid only when the objects are selected on the same observable that is used to make the distance measurement. In many applications of the Tully-Fisher relation, the galaxies are selected from B catalogs, but the distances are measured using H, I, or R magnitudes. In these cases, the proper form for the mean Malmquist bias is αc_12_, where c_12_ is the covariance between the fractional distance error in the band in which the measurements are made with the fractional error in the band on which the galaxies are selected. The Malmquist bias for the D_n_-σ relation is also proportional to a covariance. Failure to use the proper Malmquist correction may lead to significant errors in estimates of the Hubble parameter and to the introduction of spurious motions in peculiar-velocity studies.