DECOMPOSE: Stata module to compute decompositions of wage differentials

Given the results from two regressions (one for each of two groups), decompose computes several decompositions of the outcome variable differential. The decompositions shows how much of the gap is due to differing endowments between the two groups, and how much is due to discrimination. Usually this is applied to wage differentials using Mincer type earnings equations.

On ANOVA Decompositions of Kernels and Gaussian Random Field Paths

The FANOVA (or “Sobol’-Hoeffding”) decomposition of multivariate functions has been used for high-dimensional model representation and global sensitivity analysis. When the objective function f has no simple analytic form and is costly to evaluate, computing FANOVA terms may be unaffordable due to numerical integration costs. Several approximate approaches relying on Gaussian random field (GRF) models have been proposed to alleviate these costs, where f is substituted by a (kriging) predictor or by conditional simulations. Here we focus on FANOVA decompositions of GRF sample paths, and we notably introduce an associated kernel decomposition into 4 d 4d terms called KANOVA. An interpretation in terms of tensor product projections is obtained, and it is shown that projected kernels control both the sparsity of GRF sample paths and the dependence structure between FANOVA effects. Applications on simulated data show the relevance of the approach for designing new classes of covariance kernels dedicated to high-dimensional kriging.

How does spatial extent of fMRI datasets affect independent component analysis decomposition?

Spatial independent component analysis (sICA) of functional magnetic resonance imaging (fMRI) time series can generate meaningful activation maps and associated descriptive signals, which are useful to evaluate datasets of the entire brain or selected portions of it. Besides computational implications, variations in the input dataset combined with the multivariate nature of ICA may lead to different spatial or temporal readouts of brain activation phenomena. By reducing and increasing a volume of interest (VOI), we applied sICA to different datasets from real activation experiments with multislice acquisition and single or multiple sensory-motor task-induced blood oxygenation level-dependent (BOLD) signal sources with different spatial and temporal structure. Using receiver operating characteristics (ROC) methodology for accuracy evaluation and multiple regression analysis as benchmark, we compared sICA decompositions of reduced and increased VOI fMRI time-series containing auditory, motor and hemifield visual activation occurring separately or simultaneously in time. Both approaches yielded valid results; however, the results of the increased VOI approach were spatially more accurate compared to the results of the decreased VOI approach. This is consistent with the capability of sICA to take advantage of extended samples of statistical observations and suggests that sICA is more powerful with extended rather than reduced VOI datasets to delineate brain activity.

Sheaf representations of MV-algebras and lattice-ordered abelian groups via duality

We study representations of MV-algebras -- equivalently, unital lattice-ordered abelian groups -- through the lens of Stone-Priestley duality, using canonical extensions as an essential tool. Specifically, the theory of canonical extensions implies that the (Stone-Priestley) dual spaces of MV-algebras carry the structure of topological partial commutative ordered semigroups. We use this structure to obtain two different decompositions of such spaces, one indexed over the prime MV-spectrum, the other over the maximal MV-spectrum. These decompositions yield sheaf representations of MV-algebras, using a new and purely duality-theoretic result that relates certain sheaf representations of distributive lattices to decompositions of their dual spaces. Importantly, the proofs of the MV-algebraic representation theorems that we obtain in this way are distinguished from the existing work on this topic by the following features: (1) we use only basic algebraic facts about MV-algebras; (2) we show that the two aforementioned sheaf representations are special cases of a common result, with potential for generalizations; and (3) we show that these results are strongly related to the structure of the Stone-Priestley duals of MV-algebras. In addition, using our analysis of these decompositions, we prove that MV-algebras with isomorphic underlying lattices have homeomorphic maximal MV-spectra. This result is an MV-algebraic generalization of a classical theorem by Kaplansky stating that two compact Hausdorff spaces are homeomorphic if, and only if, the lattices of continuous [0, 1]-valued functions on the spaces are isomorphic.

SMITHWELCH: Stata module to compute trend decomposition of outcome differentials

smithwelch computes decompositions of differences in mean outcome differentials. Smith and Welch (1989) used such decomposition techniques in their analysis of the change in the black-white wage differential over time. An alternative application would be the decomposition of country differences in the male-female wage gap.