performances. The results are aligned with our analytic results. Indeed with hold-out data we verified that in the first analysis, with $K=10$ state-level datasets (high heterogeneity; the remaining 40 are used as validation datasets), NDR produced generalist predictions with better accuracy than DR. In contrast, with SPFs developed with county-specific datasets (low cross-study hetherogeneity) DR staking predictions are more accurate than with NDR stacking. These comparisons were confirmed by iterated analyses with random sets of $K=10$ states and $K=10$ counties.

Generalist and Specialist Predictions
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Notation
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We observe $K$ studies $k = 1, \ldots, K$, with sample sizes $n_k$. For individual $i$ in study $k$ we have a vector of features $x_{i,k} \in \mathcal X$ and the individual outcome $y_{i,k} \in \mathbb R$. We use $\mathcal S = \{(x_{i,k}, y_{i,k});i =1, \ldots, n_k, k = 1, \ldots, K\}$ to indicate the collection of all $K$ datasets. Based on these, we define a library of training sets (LTS), denoted as $\mathcal D$, which includes $T$ members $D_1,\ldots ,D_T$. Each $D_t$ is a set of $(i, k)$ indices, where $i \in \{1, 2, \ldots, n_k\}$ is the sample index within a study $k$. The set $D_t$ can include indices with different 