Modern data sets in various domains often include units that were sampled non-randomly from the population and have a latent correlation structure. Here we investigate a common form of this setting, where every unit is associated with a latent variable, all latent variables are correlated, and the probability of sampling a unit depends on its response. Such settings often arise in case-control studies, where the sampled units are correlated due to spatial proximity, family relations, or other sources of relatedness. Maximum likelihood estimation in such settings is challenging from both a computational and statistical perspective, necessitating approximations that take the sampling scheme into account. We propose a family of approximate likelihood approaches which combine composite likelihood and expectation propagation. We demonstrate the efficacy of our solutions via extensive simulations. We utilize them to investigate the genetic architecture of several complex disorders collected in case-control genetic association studies, where hundreds of thousands of genetic variants are measured for every individual, and the underlying disease liabilities of individuals are correlated due to genetic similarity. Our work is the first to provide a tractable likelihood-based solution for case-control data with complex dependency structures.

A popular approach for testing if two univariate random variables are statistically independent consists of partitioning the sample space into bins, and evaluating a test statistic on the binned data. The partition size matters, and the optimal partition size is data dependent. While for detecting simple relationships coarse partitions may be best, for detecting complex relationships a great gain in power can be achieved by considering finer partitions. We suggest novel consistent distribution-free tests that are based on summation or maximization aggregation of scores over all partitions of a fixed size. We show that our test statistics based on summation can serve as good estimators of the mutual information. Moreover, we suggest regularized tests that aggregate over all partition sizes, and prove those are consistent too. We provide polynomial-time algorithms, which are critical for computing the suggested test statistics efficiently. We show that the power of the regularized tests is excellent compared to existing tests, and almost as powerful as the tests based on the optimal (yet unknown in practice) partition size, in simulations as well as on a real data example.

Deemed “one of the top ten data mining mistakes”, leakage is the introduction of information about the data mining target that should not be legitimately available to mine from. In addition to our own industry experience with real-life projects, controversies around several major public data mining competitions held recently such as the INFORMS 2010 Data Mining Challenge and the IJCNN 2011 Social Network Challenge are evidence that this issue is as relevant today as it has ever been. While acknowledging the importance and prevalence of leakage in both synthetic competitions and real-life data mining projects, existing literature has largely left this idea unexplored. What little has been said turns out not to be broad enough to cover more complex cases of leakage, such as those where the classical independently and identically distributed (i.i.d.) assumption is violated, that have been recently documented. In our new approach, these cases and others are explained by explicitly defining modeling goals and analyzing the broader framework of the data mining problem. The resulting definition enables us to derive general methodology for dealing with the issue. We show that it is possible to avoid leakage with a simple specific approach to data management followed by what we call a learn-predict separation, and present several ways of detecting leakage when the modeler has no control over how the data have been collected. We also offer an alternative point of view on leakage that is based on causal graph modeling concepts.

Deemed "one of the top ten data mining mistakes", leakage is essentially the introduction of information about the data mining target, which should not be legitimately available to mine from. In addition to our own industry experience with real-life projects, controversies around several major public data mining competitions held recently such as the INFORMS 2010 Data Mining Challenge and the IJCNN 2011 Social Network Challenge are evidence that this issue is as relevant today as it has ever been. While acknowledging the importance and prevalence of leakage in both synthetic competitions and real-life data mining projects, existing literature has largely left this idea unexplored. What little has been said turns out not to be broad enough to cover more complex cases of leakage, such as those where the classical i.i.d. assumption is violated, that have been recently documented. In our new approach, these cases and others are explained by explicitly defining modeling goals and analyzing the broader framework of the data mining problem. The resulting definition enables us to derive general methodology for dealing with the issue. We show that it is possible to avoid leakage with a simple specific approach to data management followed by what we call a learn-predict separation, and present several ways of detecting leakage when the modeler has no control over how the data have been collected.