**7.2.4 Malmquist Effects**

There has been a great deal of discussion in the literature during the past decade concerning the significance of selection effects in the determination of extragalactic distances, particularly for those obtained via TF relations. Since the TF relations are applicable to spiral and irregular galaxies, they have until recently been the only quantitative distance indicator which had been applied to a sufficient sample of galaxies such that the effects of Malmquist ``bias'' on distance determinations can be adequately tested.

The samples in question are typically selected on the basis of
apparent magnitude. In such samples, the resulting luminosity function is
progressively biased toward intrinsically brighter systems with
increasing mean
distance. This bias is usually referred to as a ``Malmquist effect'',
although this is not quite the situation
Malmquist
(1920) described.
Tammann (1987) and
Sandage (1988),
for example, argue that the distances are
biased, leading to an apparently small dispersion of the TF relations.
The fact that the *luminosity function* of an apparent magnitude limited
sample is progressively more
biased for ever increasing distance is without doubt (e.g.,
Sandage et al. 1979).
At issue is whether the *estimated distances*
for such a sample are biased.

Additional confusion has arisen when analytical solutions to similar bias problems (e.g., Teerikorpi 1984, 1987; Bottinelli et al. 1986; Lynden-Bell et al. 1988) are applied to the TF samples. However, those solutions may suffer from simplifying assumptions, or they may not be general solutions. In fact, the problem described by Malmquist (1920) is only a special case of the overall problem of estimating astronomical distances.

For example, errors in predicted distance modulus can arise from both
random and systematic errors. Purely random errors obviously do not
result in biased distance estimates, unless they propagate into both
the sample selection criteria and the estimated distance moduli. An
example of this would be the use of cataloged apparent magnitudes to
*both select the sample and estimate individual distance moduli*.
Note, however, that this coupling can be broken if new photometry is
obtained for a sample selected on the basis of cataloged photometry.

An example of a systematic error with distance is that described
originally by
Malmquist (1920).
In this case, an observable (i.e.,
spectral type) is used to predict the *mean* absolute magnitude of
a particular class of stars. This predicted absolute magnitude is then
used to estimate distance moduli, even though the sample selection
criteria have already biased the distribution in absolute magnitude,
such that the mean of the parent population is not the mean of the
selected sample. However, this need not be case for all apparent
magnitude selected samples.

Simulations of the TF relations demonstrate that for the case in which
the dispersion in inclination corrected line-width dominates the
dispersion in apparent magnitude (e.g.,
Bothun and Mould
1987),
either due to an intrinsic scatter or due to observational errors in
*W*_{20}
and inclination, then there is essentially no bias introduced into the
TF distances due to selecting the sample on the basis of apparent
magnitude. The fact that the true luminosities of the sample members
are biased becomes irrelevant, since the unbiased predictor of
luminosity (i.e., *W _{R}^{i}*) will produce an