💻 bioinformatics

Statistical tests for bivariate spatial association across multi-omics data with disjoint coordinates

This paper introduces the R-package `sbivar`, which provides a suite of modified statistical tests and variance estimators to rigorously assess bivariate spatial associations between multi-omics modalities with disjoint coordinates while properly accounting for spatial autocorrelation and high-dimensional computational challenges.

Original authors: Hawinkel, S., Hu, W., Velten, B., Maere, S.

Published 2026-07-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Hawinkel, S., Hu, W., Velten, B., Maere, S.

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine you are a detective trying to solve a mystery inside a living city. This city is a piece of biological tissue, and the "citizens" are molecules like genes and metabolites. In the past, scientists could only take a blurry photo of the whole city or a tiny, high-definition photo of just one street corner. But now, we have super-powered microscopes that can map out exactly where different types of molecules are living within the tissue. The big question is: Do certain molecules like to hang out together? If a specific gene and a specific chemical are found in the same neighborhood, they might be working together on a secret project, like building a cell wall or fighting an infection. Finding these "buddy pairs" helps us understand how life works, how diseases start, and how to fix them.

However, there's a catch. The different microscopes used to map these molecules often have different resolutions. One might take a picture with tiny, pixel-perfect dots, while the other takes a picture with slightly larger, fuzzier dots. Even if you line up the two pictures perfectly, the dots from one picture rarely land exactly on top of the dots from the other. It's like trying to match a grid of tiny LEGO bricks to a grid of slightly larger Jenga blocks; they just don't line up one-to-one. Furthermore, molecules in biology aren't scattered randomly like marbles on a floor; they tend to clump together in patterns, a phenomenon scientists call "spatial autocorrelation." If you ignore these clumps and the mismatched grids, your math can get very confused, leading you to think two molecules are best friends when they are actually just strangers who happened to be in the same general area.

This is where the new paper steps in with a fresh set of tools. The authors, Stijn Hawinkel and his team, realized that many existing methods for finding these molecular buddy pairs were using the wrong math. They were treating the data as if the molecules were scattered randomly and perfectly aligned, which led to a lot of false alarms—finding "friendships" that didn't actually exist. The team argues that you need a method that can handle mismatched grids (disjoint coordinates) and still respect the natural clumping of molecules.

To solve this, they developed a new software package called sbivar (which you can find on GitHub). They didn't just invent one new trick; they adapted three different statistical approaches to work on these messy, mismatched maps:

The "Neighborhood Watch" (Bivariate Moran's I): Imagine you are checking if two types of people are hanging out in the same neighborhood. Instead of needing them to stand on the exact same spot, this method looks at how close they are to each other. The authors tweaked this old method to calculate the odds of them being together while accounting for the fact that people naturally clump together. They created a new way to measure the "noise" in the data so they don't get fooled by the clumping.
The "Smooth Map" (Generalized Additive Models or GAMs): This method is like drawing a smooth, wavy blanket over the data points to see the big picture. It creates a smooth surface for each molecule type based on where they are, even if the dots are on different grids. Then, it compares the two smooth blankets to see if they rise and fall in the same places. The team figured out how to measure the uncertainty of these blankets so they know if the match is real or just a fluke.
The "Distance Detective" (Bivariate Gaussian Processes): This is a more complex mathematical model that treats the tissue like a continuous field where the relationship between molecules changes based on distance. It's very powerful but can be slow to run on a computer.

The team tested these new tools using computer simulations. They created fake tissue maps with known patterns—some where molecules were truly friends, and some where they were just random neighbors. They found that the old methods often screamed "Friendship!" when there was none, especially when the molecules had natural clumps. In contrast, their new methods (especially the Neighborhood Watch and the Smooth Map) were much better at keeping quiet when there was no friendship and shouting "Friendship!" only when it was real. They also showed that these new tools work well even when the maps are slightly misaligned, which happens often in real experiments.

Finally, they took their new tools to real-world data from human lung and breast cancer samples, and mouse brain samples. They found some exciting new pairs. For instance, in a lung cancer sample, they discovered that a group of genes called Kallikreins were hanging out with a specific fat molecule called PI 34:1. This makes sense biologically because both are involved in a major signaling pathway that cells use to communicate. In the mouse brain, they found a gene (CSRP1) and a fat molecule (palmitoyl-L-carnitine) that were neighbors in the brain's striatum. This pairing hints at how brain cells and their support cells (astrocytes) might be swapping energy sources.

The paper doesn't claim to have solved every mystery in biology, and the authors admit that some of their tools are slower than others. However, they have provided a much more reliable way to find these molecular connections without getting tricked by the messy nature of real-world data. By fixing the math, they hope scientists can now trust their findings about which molecules are truly working together, leading to better insights into how our bodies function and how diseases like cancer might be stopped.

Technical Summary: Statistical Tests for Bivariate Spatial Association Across Multi-Omics Data with Disjoint Coordinates

Problem Statement

The emergence of multimodal spatial omics has enabled the measurement of multiple high-dimensional data types (e.g., transcriptomics and metabolomics) on consecutive tissue slices or co-assayed on the same slice. A primary goal is to detect statistical associations between features of different modalities to infer biological processes. However, current analytical frameworks face three critical challenges:

Disjoint Coordinate Sets: Due to differences in spatial resolution and measurement areas between technologies (e.g., 10x Genomics Visium vs. MALDI-MSI), even after image alignment, there is no one-to-one mapping between measurement spots. Consequently, standard statistical quantities like joint distributions and Pearson correlations are undefined.
Information Loss in Current Practices: The prevailing solution involves aggregating high-resolution spots to match the lower resolution of the other modality. This "granulation" discards spatial information and fails to exploit the full resolution of the data.
Invalid Null Hypotheses and Assumptions: Many existing bivariate association tests (e.g., standard Pearson correlation, naive permutation tests, or certain versions of bivariate Moran's I) implicitly assume independent and identically distributed (i.i.d.) observations or exchangeability. These assumptions are violated in spatial omics data due to inherent Spatial Autocorrelation (SAC). Consequently, these methods often test a "randomisation null hypothesis" (absence of SAC in any variable) rather than the "independence null hypothesis" (absence of association between variables given SAC). This leads to inflated Type I error rates and misleading conclusions where univariate spatial patterns are mistaken for bivariate associations.

Methodology

The authors propose a suite of scalable statistical tests designed to operate directly on disjoint coordinate sets while explicitly accounting for SAC in both variables. The methods are implemented in the R-package sbivar.

1. Bivariate Moran's I under the Independence Null

Concept: A generalization of bivariate Moran's I ( $I_{xy} = x^T W y$ ) to disjoint coordinates using a weight matrix $W$ where elements $w_{ij}$ decay with the distance between spot $i$ of modality X and spot $j$ of modality Y.
Variance Estimation: Unlike previous approaches that assume i.i.d. observations, the authors derive an analytical variance estimator that incorporates SAC. The variance is calculated as $\text{Var}(I_{xy}) = \text{tr}(W^T \hat{\Sigma}_x W \hat{\Sigma}_y)$ , where $\hat{\Sigma}_x$ and $\hat{\Sigma}_y$ are estimated autocorrelation matrices derived from variogram models (e.g., Matheron's estimator).
Testing: The test statistic is assumed to be asymptotically standard normal. To detect various spatial patterns, multiple weight matrices (with different decay parameters) are tested, and their p-values are combined using the Cauchy combination test for dependent p-values.

2. Bivariate Gaussian Processes (GPs)

Concept: Models the concatenated vector of observations from both modalities ( $z$ $z$ ) using a Multivariate Normal distribution. The covariance matrix $\Sigma$ $Σ$ is decomposed into:
- Independent noise.
- Univariate SAC within each modality ( $\Sigma_{SAC}$ ).
- Bivariate spatial covariance ( $\Sigma_{biv}$ ) representing the association of interest.
Testing: A score test is used to test the null hypothesis that the variance component of the bivariate association ( $\tau$ ) is zero. To handle the boundary of the parameter space, a score test statistic is preferred over Wald or likelihood ratio tests. The method tests for both positive and negative associations and combines p-values across a range of length scales ( $l_{xy}$ ) using the Cauchy combination test.

3. Generalized Additive Models (GAMs)

Concept: Models the spatial patterns of each modality as smooth functions of space using 2D thin-plate smoothing splines ( $s_x(c_i)$ and $s_y(e_j)$ ).
Handling Disjointness: The fitted spline surfaces are evaluated on a common, evenly spaced grid within the concave hull of the data. The covariance and correlation between these evaluated surfaces are calculated.
Variance Estimation: Because spline evaluations are dependent, standard variance formulas are invalid. The authors use the delta method to propagate the uncertainty of the spline coefficients to the correlation estimate, allowing for a valid z-test.
Extensions: The framework supports different outcome distributions (e.g., negative binomial for transcriptomics, gamma for metabolomics) and can be extended to Generalized Additive Mixed Models (GAMMs) to account for residual SAC.

4. Multi-Image Analysis

For replicated experiments (multiple tissue sections), the authors propose a pragmatic approach: calculate an association measure (e.g., Moran's I or spline correlation) for each image individually, then feed these measures into a linear model that accounts for the experimental design. This approach quantifies variability across replicates rather than within a single image, naturally absorbing SAC-induced variability.

Key Contributions

Theoretical Advancement: The paper redefines the null hypothesis for bivariate spatial association, distinguishing between the "randomisation null" (no SAC) and the "independence null" (no association given SAC). It provides analytical variance estimators for bivariate Moran's I and GAMs that correctly account for SAC in disjoint coordinate sets.
Methodological Innovation: It introduces three scalable methods (Bivariate Moran's I, Bivariate GPs, GAMs) that do not require spot matching or data aggregation, thereby preserving spatial resolution.
Software Implementation: The development of the sbivar R-package, which provides a unified interface for these tests, including support for single-section and multi-section analyses.

Results

Simulation Studies:

Type I Error Control: Nonparametric and parametric simulations demonstrate that many existing methods (including standard Pearson correlation, bivariate Geary's C, and naive permutation tests) suffer from inflated Type I errors when SAC is present in one or both variables. In contrast, the proposed methods (Bivariate Moran's I with independence null, Bivariate GPs, and GAMs) successfully control Type I error rates.
Power: The new methods generally exhibit good power to detect various bivariate patterns (e.g., "streaks," "zones," and Gaussian process dependencies). While Bivariate GPs and GAMMs are computationally intensive, Bivariate Moran's I and GAMs offer a favorable balance of speed and power.
Robustness: The methods maintain Type I error control even in the presence of alignment errors, though power decreases as distortion increases.

Real Data Applications:

Godfrey Data (Human Lung/Breast Cancer): Applied to co-assayed transcriptomics and metabolomics. The new methods identified significant gene-metabolite pairs that were missed by the original correlation-based analysis (which used a fixed threshold). Notably, a positive association was found between Kallikrein transcripts (KLK10-13) and the metabolite PI 34:1 in lung cancer, linking them to the PI3K/AKT signaling pathway.
Vicari Data (Mouse Brain): Applied to Parkinson's disease models. The analysis confirmed some previous findings but also revealed new associations, such as a positive link between CSRP1 transcripts and palmitoyl-L-carnitine in the striatum, consistent with known astrocyte-neuron lipid transfer mechanisms.

Significance and Claims

The authors claim that their work addresses a critical gap in spatial multi-omics analysis by providing statistically valid tools for detecting bivariate associations across disjoint coordinate sets. They argue that ignoring SAC leads to false discoveries and that aggregating data to force coordinate matching results in unnecessary information loss.

The paper modestly positions its contributions as a necessary correction to existing practices:

It provides the first suite of tests that explicitly account for SAC while testing the independence null hypothesis on disjoint coordinates.
It offers a practical framework for analyzing replicated spatial experiments without requiring prohibitively complex single-model fitting.
It emphasizes that valid inference requires separating the estimation of univariate spatial structures from the testing of bivariate associations.

The authors conclude that while no single method is perfect (e.g., GAMs may struggle with short-range SAC if residual autocorrelation is not modeled, and GPs are computationally heavy), the combination of Bivariate Moran's I and GAMs, potentially confirmed by GAMMs, offers a robust strategy for uncovering biologically meaningful spatial colocalization.