Data integration often involves analysing results from different experiments to identify complex patterns. In this context, we consider a scenario where we have a collection of elements $i=1,\dots,n$ (genes, for example) for which the hypotheses : "element $i$ has no effect in condition $q$" have been tested for $Q$ conditions. Each observation $i$, therefore, consists of a vector of $Q$ critical probabilities. The analysis aims to identify the elements that have an effect in all the conditions or a predefined subset of those conditions. The critical probabilities must then be combined flexibly to explore complex hypotheses, called composite hypotheses, while controlling the false positive rate.

We propose a composite hypothesis testing procedure based on a model where the $Q$-uplet of p-value associated with each gene/marker is distributed as a multivariate mixture where each of the $2^Q$ components corresponds to a specific combination of $H_0$ and $H_1$ states. Our method explicitly accounts for the dependence structure across p-value series through a copula function. The inference of this $2^Q$ component mixture model is performed efficiently, allowing its application to cases where the number of markers is $10^5$–$10^6$ and where $Q=20$. The inference procedure consists of two independent steps: first, fitting a non-parametric mixture model to the marginal distribution of each $Q$ series of p-values, then estimating the proportions of the mixture model components and the copula parameters using an EM algorithm. Step (E) is optimised to reduce the memory burden of the procedure from $O(n \times 2^Q)$ to $O(n + 2^Q)$.

Applications on simulated data have been carried out, with conclusive results regarding false positive control and detection power and the method's efficiency (computation time and memory management). The interest in the method is illustrated through two application cases: the first in human genetics, where 14 association studies are jointly analysed to identify new pleiotropic regions associated with psychiatric disorders, and the second in plant genetics, for detecting hotspot regions linked to resistance to multiple viruses in cucumber.