I’m guessing that most readers of this blog will be familiar with the concept of shrinkage estimation. But if not, here’s an example to give you the idea. Imagine you’re trying to estimate the true winning percentage of each team in a professional soccer league–the percentage of games each team would win if, hypothetically, it played each of the other teams many, many times. But it’s early in the season, and so each team has only played each of the other teams once. You could take each team’s observed winning percentage as an estimate of its unknown true winning percentage. But those estimates come from small samples of games, and the better team doesn’t necessarily win every game because chance events play a role. So observed winning percentages after just a few games are imprecise estimates of those unknown true winning percentages. Meaning that the variance among teams in their observed winning percentages surely overestimates the variance among teams in their unknown true winning percentages. In all likelihood, the team with the highest observed winning percentage so far is not only good, it’s also gotten lucky. It’s good, but likely not as good as its observed winning percentage suggests. And in all likelihood, the team with the lowest observed winning percentage so far is not only bad, it’s also gotten unlucky. It’s bad, but likely not as bad as its observed winning percentage suggests. Put another way, as the teams play more games, they’re likely to regress to the mean. So in the aggregate, you can improve your estimates of the teams’ true winning percentages if you shrink the observed winning percentages towards 50% (the average winning percentage). You make the bias-variance trade-off work in your favor by biasing all of your estimates towards the mean, in order to reduce their variance. There are ways to work out exactly how much shrinkage is optimal.
I think we need shrinkage estimation for mean effect sizes in ecological meta-analyses. That is, I think many ecological meta-analyses provide very imprecise estimates of the unknown “true” mean effect size. So that, in aggregate, those estimated mean effect sizes would be improved if they were shrunk towards the mean. Here, see for yourself:Continue reading