Conditional Contact Odds

Answers the question: Which group pairs remain associated after the complete composition of each cell’s neighbourhood is taken into account?

The tool is based on a softmax model that relates each cell’s group identity to the composition of its local neighbourhood. The probability that cell $i$ belongs to group $A$ is modeled as:

Neighbourhoods are defined by an undirected $k$-nearest-neighbour graph, and $N(i,B)$ is the number of group-$B$ neighbours connected to cell $i$. The model jointly estimates the group-specific offsets $h(A)$ and the coefficients $J(A,B)$ from the observed group labels. While $h(A)$ accounts for the baseline tendency of a cell to belong to group $A$, the main quantities of interest are the coefficients $J(A,B)$. They describe the association between groups $A$ and $B$ after the complete neighbourhood composition has been taken into account.

More precisely, while all other neighbour counts remain fixed, one additional $B$-neighbour changes the log-odds of group $A$ versus group $C$ by $J(A,B)-J(C,B)$.

Map. Using the fitted softmax model above, each cell is colored by $\log P(z(i)=\text{observed group}|N(i))$, the log probability assigned to its observed group, given its neighbourhood. Values closer to 0 indicate that the model assigns a high probability to the observed group, while more negative values indicate that the cell’s identity is unusual for its neighbourhood.

Matrix. Both axes list all groups, and each square shows the corresponding fitted coefficient $J(A,B)$. The coefficients are centered to have a mean of zero, $J\prime (A,B)=J(A,B)-\mathrm{mean}(J)$. Positive values indicate stronger-than-average conditional association; negative values indicate weaker association or relative avoidance. The diagonal shows conditional self-association.

Table. Each row represents one group pair. Conditional log-odds is the centered $J(A,B)$ value from the matrix. Marginal log2 obs/exp is the Cell-Type Co-occurrence score calculated on the same neighbour graph, without accounting for other groups. Contacts is the number of observed connections between the groups. Rank shift shows how the pair’s position changes when pairs are ranked by conditional association instead of ordinary co-occurrence. A positive value means that the pair ranks higher by conditional association, while a negative value means that it ranks lower.