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Distance scale and peculiar velocity work have long been plagued by statistical biases. These biases are sufficiently confusing and multifaceted that their effects are often misunderstood or misrepresented. It is worth taking a moment to go over a few of the main issues.

The root problem is that our distance indicators contain scatter: a galaxy with distance d inferred from the DI really lies within some range of distances, approximately (but not exactly) centered on d. This range is characterized by a non-gaussian distribution of characteristic width dDelta, where Delta is the fractional distance error characteristic of the DI. (If sigma is the DI scatter in magnitudes, Delta appeq 0.46 sigma). Thus, the farther away the object is the bigger the distance error. For most DIs, a good approximation is that the distribution of distance errors is log-normal: if the true distance is r, then the distance estimate d has a probability distribution given by

Equation 12 (12)

Two distinct kinds of statistical bias effects can arise when DIs with the above properties are used. Which of the two occurs depends on which of two basic analytic approaches one adopts for treating the DI data. In the first approach, known as Method I, one assumes that the DI-inferred distance d is the best a priori estimate of true distance. Any subsequent averaging or modeling of the data points assumes galaxies with similar values of d to be neighbors in real space as well. The second approach, known as Method II, takes proximity in redshift space as tantamount to real-space proximity; the DI-inferred distances are then treated only in a statistical sense, averaged over objects with similar redshift-space positions. The Method I/Method II terminology originated with Faber & Burstein (1988); a detailed discussion is provided by Strauss & Willick (1995, Section 6.4).

Let us consider this distinction in relation to peculiar velocity or Hubble constant studies. In a Method I approach, one would take objects whose DI-inferred distances are within a narrow range of some value d, and average their redshifts. Subtracting d from the resulting mean redshift yields a peculiar velocity estimate; dividing the mean redshift by d gives an estimate of H0. However, these estimates will be biased, because the distance estimate d itself is biased: It is not the mean true distance of the objects in question. To see this, we reason as follows: if P (d|r) is given by equation 12 above, then the distribution of true distances of our objects is given, according to Bayes' Theorem, by

Equation 13 (13)

where we have taken P(r) propto r2 n(r), where n(r) is the underlying galaxy number density along the line of sight. To obtain the expectation value of the true distance r for a given d, we multiply equation (13) by r and integrate over all r. In general, this integral requires knowledge of the density field n(r) and will have to be done numerically. However, in the simplest case that the density field is constant, the integral can be done analytically. The result is that the expected true distance is de7Delta2/2 (Lynden-Bell et al. 1988; Willick 1991). This effect is called homogeneous Malmquist bias. It tells us that, typically, objects lie further away than their DI-inferred distances. The physical cause is more objects ``scatter in'' from larger true distances (where there is more volume) than ``scatter out'' from smaller ones. In general, however, variations in the number density cannot be neglected. When this is the case, there is inhomogeneous Malmquist bias (IHM). IHM can be computed numerically if one has a model of the density field. Further discussion of this issue may be found in Willick et al. (1997).

The biases which arise in a Method II analysis are quite different. They may be rigorously understood in terms of the probability distribution of the DI-inferred distance d given the redshift cz, P (d|cz) (contrast with equation 13, which underlies Method I). In general, this distribution is quite complicated (cf. Strauss & Willick 1995, Section 8.1.2), and its details are beyond the scope of this Chapter. However, under the assumption of a ``cold'' velocity field - an assumption that appears adequate in ordinary environments - redshifts complemented by a flow model give a good approximation of true distance. Thus, it really is the probability distribution P (d|r) (equation (12), or one similar to it, that counts for a Method II analysis. However, that equation as written does not represent the full story. If severe selection effects such as a magnitude or diameter limit are present, then the log-normal distribution does not apply exactly. Some galaxies are too faint or small to be in the sample; in effect, the large-distance tail of P(d|r) is cut off. It follows that the typical inferred distances are smaller than those expected at a given true distance r. As a result, the peculiar velocity model that allows true distance to be estimated as a function of redshift is tricked into returning shorter distances. This bias goes in the same sense as Malmquist bias, but is fundamentally different. It results not from volume/density effects, but from sample selection effects, and is called selection bias.

Selection bias can be avoided, or at least minimized, by working in the so-called ``inverse direction.'' What that means is most easily illustrated using the TF relation. When viewed in its ``forward'' sense, the TF relation is conceived as a prediction of absolute magnitude given a value of the velocity width parameter, M(eta). However, it is equally valid to view the relation as a prediction of eta given a value of M, i.e., as a function eta0 (M) (the superscript ensures that there is no confusion between the observed width parameter eta and the TF-prediction). When one uses the forward relation, one imagines fitting a line mi = M (etai) + µ by regressing the apparent magnitudes mi on the velocity widths etai; the distance modulus µ is the free parameter solved for. Selection bias then occurs because apparent magnitudes fainter than the magnitude limit are ``missing'' from the sample, so the fitted line is not the same as the true line. However, if one instead fits a line eta0 (mi - µ) by regressing the widths on the magnitudes, the same effect does not occur, provided the sample selection procedure does not exclude large or small velocity widths. In general, this last caveat is more or less valid. Consequently, working in the inverse direction does in fact avoid or at least minimize selection bias.

This fact, first clearly stated by Schechter (1980) and then reiterated in various forms by Aaronson et al. (1982), Tully (1988), Willick (1994), Dekel (1994), and Davis et al. (1996), among others, remains an obscure one, not universally appreciated. It is often heard, for example, that the TF relation applied to relatively distant galaxies will necessarily result in a Hubble constant that is biased high, because the distances are biased low due to selection bias. The clear conclusion of the previous paragraph, however, is that provided the analysis is done using redshift-space information to assign a priori distances - that is, provided that a Method II approach is taken - working in the inverse direction can render selection bias unimportant. It is also the case that a careful analytical methods (Willick 1994) can permit a correction for selection bias even when working in the forward direction. It should be borne in mind, however, that both of these approaches (using the inverse relation or correction for forward selection bias) necessitate a careful characterization of sample selection criteria.

``Method Matrix'' of Distance Indicator Biases Table 1

Another wrinkle in this complicated subject is that the relatively bias-free character of inverse distance indicators does not carry over to a Method I analysis. It is beyond the scope of this Chapter to discuss this issue in full detail; the interested reader is referred to Strauss & Willick (1995, Section 6.5). The main point is that a Method I inverse DI analysis is subject to Malmquist bias in much the same way as a Method I forward analysis; indeed, the inverse Malmquist bias is in some ways considerably more complex, as it depends (unlike forward Malmquist bias) on sample selection criteria. So while it is correct to emphasize the bias-free (or nearly so) nature of working in the inverse direction, it is essential to remember that this property holds only for Method II analyses.

Much of the confusion surrounding the relative bias properties of forward versus inverse DIs stems from neglecting the distinction between Method I and Method II analyses. Recognizing this, Strauss & Willick (1995) summarized the issue with what they called the ``Method Matrix'' (a more memorable term might be the ``magic square'') of peculiar velocity analysis. Their table is reproduced above, in a slightly simpler form (the original alluded to several complications that are unecessary here). Reference to this simple diagram might allay some of the controversies surrounding Malmquist and related biases.

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