A robust, scalable K-statistic for quantifying immune cell clustering in spatial proteomics data

Spatial summary statistics based on point process theory are widely used to quantify the spatial organization of cell populations in single-cell spatial proteomics data. Among these, Ripley's $K$ is a popular metric for assessing whether cells are spatially clustered or are randomly dispersed. However, the key assumption of spatial homogeneity is frequently violated in spatial proteomics data, leading to overestimates of cell clustering and colocalization. To address this, we propose a novel $K$-based method, termed \textit{KAMP} (\textbf{K} adjustment by \textbf{A}nalytical \textbf{M}oments of the \textbf{P}ermutation distribution), for quantifying the spatial organization of cells in spatial proteomics samples. \textit{KAMP} leverages background cells in each sample along with a new closed-form representation of the first and second moments of the permutation distribution of Ripley's $K$ to estimate an empirical null model. Our method is robust to inhomogeneity, computationally efficient even in large datasets, and provides approximate $p$-values for testing spatial clustering and colocalization. Methodological developments are motivated by a spatial proteomics study of 103 women with ovarian cancer, where our analysis using \textit{KAMP} shows a positive association between immune cell clustering and overall patient survival. Notably, we also find evidence that using $K$ without correcting for sample inhomogeneity may bias hazard ratio estimates in downstream analyses. \textit{KAMP} completes this analysis in just 5 minutes, compared to 538 minutes for the only competing method that adequately addresses inhomogeneity.