Silhouette scores

To assess the extent to which the data are grouped based on the batch effects as opposed to biological signals, we computed batch and biology Silhouette scores. Given a partitioning of all cells into groups, if ai denote the average Euclidean distance of the protein expression between the cell i and all other cells in the group to which cell i is assigned, and bi is the minimum of the average distance between the cell i and any cells in other groups not containing cell i, then the silhouette coefficient of cell i is calculated as

The average of the silhouette values across cells using a particular grouping is called the silhouette score for that grouping. Silhouette score ranges from −1 to +one where positive values (b_i is high and a_i is low) indicate that cells are well matched to their own group. In this way, we computed the silhouette score s_batch based on the batches as groups and the silhouette score s_biology based on the grouping of the cells by subpopulation (i.e. clusters).

s_biology is used to quantify the cell-to-cell variation within cell subpopulations compared to other subpopulations. Negative values mean that the data might be mis-clustered as it is more similar to a neighbouring cluster. For example, if two different biological relevant clusters would be merged into a single cluster, s_biology will reflect this merging with a lower s_biology value.