Many studies in biomedical and health sciences involve small sample sizes due to logistic or financial constraints. Often, identifying weak (but scientifically interesting) associations between a set of predictors and a response necessitates pooling datasets from multiple diverse labs or groups. While there is a rich literature in statistical machine learning to address distributional shifts and inference in multi-site datasets, it is less clear when such pooling is guaranteed to help (and when it does not) – independent of the inference algorithms we use. In this paper, we present a hypothesis test to answer this question, both for classical and high dimensional linear regression. We precisely identify regimes where pooling datasets across multiple sites is sensible, and how such policy decisions can be made via simple checks executable on each site before any data transfer ever happens. With a focus on Alzheimer’s disease studies, we present empirical results showing that in regimes suggested by our analysis, pooling a local dataset with data from an international study improves power.

Consider samples from two different data sources $\{\mathbf{x_s^i}\} \sim P_{\rm source}$ and $\{\mathbf{x_t^i}\} \sim P_{\rm target}$. We only observe their transformed versions $h(\mathbf{x_s^i})$ and $g(\mathbf{x_t^i})$, for some known function class $h(\cdot)$ and $g(\cdot)$. Our goal is to perform a statistical test checking if $P_{\rm source}$ = $P_{\rm target}$ while removing the distortions induced by the transformations. This problem is closely related to concepts underlying numerous domain adaptation algorithms, and in our case, is motivated by the need to combine clinical and imaging based biomarkers from multiple sites and/or batches, where this problem is fairly common and an impediment in the conduct of analyses with much larger sample sizes. We develop a framework that addresses this problem using ideas from hypothesis testing on the transformed measurements, where in the distortions need to be estimated {\it in tandem} with the testing. We derive a simple algorithm and study its convergence and consistency properties in detail, and we also provide lower-bound strategies based on recent work in continuous optimization. On a dataset of individuals at risk for neurological disease, our results are competitive with alternative procedures that are twice as expensive and in some cases operationally infeasible to implement.

BACKGROUND: In systems biology, the task of reverse engineering gene pathways from data has been limited not just by the curse of dimensionality (the interaction space is huge) but also by systematic error in the data. The gene expression barcode reduces spurious association driven by batch effects and probe effects. The binary nature of the resulting expression calls lends itself perfectly to modern regularization approaches that thrive in high-dimensional settings. RESULTS: The Partitioned LASSO-Patternsearch algorithm is proposed to identify patterns of multiple dichotomous risk factors for outcomes of interest in genomic studies. A partitioning scheme is used to identify promising patterns by solving many LASSO-Patternsearch subproblems in parallel. All variables that survive this stage proceed to an aggregation stage where the most significant patterns are identified by solving a reduced LASSO-Patternsearch problem in just these variables. This approach was applied to genetic data sets with expression levels dichotomized by gene expression bar code. Most of the genes and second-order interactions thus selected and are known to be related to the outcomes. CONCLUSIONS: We demonstrate with simulations and data analyses that the proposed method not only selects variables and patterns more accurately, but also provides smaller models with better prediction accuracy, in comparison to several alternative methodologies.

In discrete undirected graphical models, the conditional independence of node labels Y is specified by the graph structure. We study the case where there is another input random vector X (e.g. observed features) such that the distribution P (Y | X) is determined by functions of X that characterize the (higher-order) interactions among the Y ’s. The main contribution of this paper is to learn the graph structure and the functions conditioned on X at the same time. We prove that discrete undirected graphical models with feature X are equivalent to mul- tivariate discrete models. The reparameterization of the potential functions in graphical models by conditional log odds ratios of the latter offers advantages in representation of the conditional independence structure. The functional spaces can be flexibly determined by kernels. Additionally, we impose a Structure Lasso (SLasso) penalty on groups of functions to learn the graph structure. These groups with overlaps are designed to enforce hierarchical function selection. In this way, we are able to shrink higher order interactions to obtain a sparse graph structure.