Leave-one-study-out influence analysis ($\hat{\mu }-{\hat{\mu }}_{(-i)}$)

Refit a meta-analysis after deleting each study, comparing fitted effects, uncertainty and heterogeneity to the full fit. For fixed weights $w_i>0$, total weight $W$ and weighted effect $\hat{\mu }$, direct subtraction of the two weighted means gives $\hat{\mu }-{\hat{\mu }}_{(-i)}=w_i(y_i-\hat{\mu })/(W-w_i)$. Re-estimating heterogeneity additionally measures influence through the weights. This is a sensitivity diagnostic, not an automatic rule for removing discordant studies.