  Publication bias, the fact that studies identified for inclusion in a meta
analysis do not represent all studies on the topic of interest, is commonly
recognized as a threat to the validity of the results of a meta analysis. One
way to explicitly model publication bias is via selection models or weighted
probability distributions. We adopt the nonparametric approach initially
introduced by Dear (1992) but impose that the weight function $w$ is monotonely
non-increasing as a function of the $p$-value. Since in meta analysis one
typically only has few studies or "observations", regularization of the
estimation problem seems sensible. In addition, virtually all parametric weight
functions proposed so far in the literature are in fact decreasing. We discuss
how to estimate a decreasing weight function in the above model and illustrate
the new methodology on two well-known examples. The new approach potentially
offers more insight in the selection process than other methods and is more
flexible than parametric approaches. Some basic properties of the
log-likelihood function and computation of a $p$-value quantifying the evidence
against the null hypothesis of a constant weight function are indicated. In
addition, we provide an approximate selection bias adjusted profile likelihood
confidence interval for the treatment effect. The corresponding software and
the datasets used to illustrate it are provided as the R package selectMeta.
This enables full reproducibility of the results in this paper.
