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348 13. MULTILEVEL MODELSabsolute error of estimate0.00 0.10 0.205 10 1025 25351 15 30 45 60tankFIGURE 13.2. Error of no-pooling and varying effects estimates, for the simulatedtadpoles. e horizontal axis displays pond number, j. e verticalaxis measures the absolute error in the predicted proportion of survivors,compared to the true value used in the simulation. e higher the point,the worse the estimate. No-pooling empirical estimates of proportion survivingin each tank are shown by the filled blue points. Varying estimatesare shown by the black circles. e dashed blue and black lines show the averageerror for each kind of estimate, across each initial density of tadpoles(sample size). Smaller tanks produce more error, but the varying estimatesare also always better, on average, especially in smaller tanks.estimates always have the advantage, at least on average. ird, and this is the observationthat will help you grasp shrinkage, the distance between the blue dashed line and the blackdashed line grows, the smaller the pond gets. So while both kinds of estimates suffer fromreduced sample size, the varying effect estimates suffer less.Okay, so what are we to make of all of this? Remember, back in FIGURE 13.1 (page 342),the smaller tanks demonstrated more shrinkage towards the mean. Here, the ponds withthe smallest sample size show the greatest improvement over the naive no-pooling estimates.is is no coincidence. Shrinkage towards the mean results from trying to negotiate the underfittingand overfitting risks of the grand mean on one end and the individual means ofeach pond on the other. e smaller tanks/ponds contain less information, and so their varyingestimates are influenced more by the pooled information from the other ponds. In otherwords, small ponds are prone to overfitting (high variance), and so they receive a bigger doesof the underfit grand mean (bias), to move them towards the optimal tradeoff of underfittingand overfitting. Likewise, the larger ponds shrink much less, because they contain more informationand are prone to less overfitting. erefore they need less correcting to negotiate