  Scargle (2000) has discussed Rosenthal and Rubin's (1978) "fail-safe number"
(FSN) method for estimating the number of unpublished studies in meta-analysis.
He concluded that this FSN cannot possibly be correct because a central
assumption the authors used conflicts with the very definition of publication
bias. While this point has been made by others before (Elsahoff, 1978;
Darlington, 1980; Thomas, 1985, Iyengar & Greenhouse, 1988), Scargle showed, by
way of a simple 2-parameter model, how far off Rosenthal & Rubin' s estimate
can be in practice. However, his results relied on the assumption that the
decision variable is normally distributed with zero mean. In this case the
ratio of unpublished to published papers is large only in a tiny region of the
parameter plane.
  Building on these results, we now show that (1) Replacing densities with
probability masses greatly simplifies Scargle's derivations and permits an
explicit statement of the relation between the probability alpha of Type I
errors and the step-size beta; (2) This result does not require any
distribution assumptions; (3) The distinction between 1-sided and 2-sided
rejection regions becomes immaterial; (4) This distribution-free approach leads
to an immediate generalization to partitions involving more than two intervals,
and thus covers more general selection functions.
