Selection
BEAR fits a mixture model to the absolute z-values in each dataset. The model allows a Hedges-style selection effect: results with \(|z| < 1.96\) may be observed at a lower rate than results with \(|z| \geq 1.96\) (Hedges 1984, 1992).
The fitted selection parameter is
\[ \omega = \frac{\Pr(\text{observed} \mid |z| < 1.96)} {\Pr(\text{observed} \mid |z| \geq 1.96)}. \]
Values below one indicate lower observation probability below the conventional two-sided significance threshold.
The interpretation of \(\omega\) depends on the unit represented by each z-value. In datasets where each observation is close to a focal estimate from a paper, such as a headline result or main meta-analytic estimate, \(\omega\) is close to a conventional publication-selection parameter. It compares the probability that a non-significant versus significant result enters the published evidence base. In datasets that extract many coefficients from the same paper, however, selection need not operate independently on each coefficient. A paper may be published because of one main result while other coefficients appear because they are reported alongside it, and authors may also choose which outcomes and specifications to report within a published paper. In these cases, \(\omega\) is better read as a reported-result selection parameter: it measures how much less visible non-significant z-values are in BEAR, without by itself distinguishing journal publication bias, within-paper selective reporting, or p-hacking. Comparisons across datasets should therefore take the extraction rule seriously, especially the number of observations per paper and whether the extracted estimates correspond to pre-specified or focal claims.
The solid curve is the fitted observed distribution of absolute z-values; the dashed curve shows the corresponding distribution after removing the fitted selection effect.
Panels are ordered by fitted \(\hat\omega\). Colours denote dataset families: blue for curated datasets, red for meta-analysis datasets, green for article-wide harvests, and gold for original-study replications.