How to interprest li.mt.bivariate output?

[RunyuXia]

I was hoping to get some clarification regarding the output of li.mt.bivariate. I understand that the layers column includes categories and p-values as explained in the tutorial, but I’m not quite sure what the X layer represents. Could you please clarify?

Thank you!

[j-bac]

I have a related question. I'm confused by the formula given for bivariate metrics:

$$\text{wCosine}_i = \frac{\sum_{j=1}^n w_{ij} x_j y_j}{\sqrt{\sum_{j=1}^n w_{ij} x_j^2} \sqrt{\sum_{j=1}^n w_{ij} y_j^2}}$$

This means the metric multiplies variable x and y within the same spot j. Whereas I would have expected a cell-cell communication metric to correlate receptor x in spot i to ligand y in each of the neighbor spots j. For example this is what is done in SpatialDM's Moran implementation, which you also reimplement.

Is this just a typo in your formula ?

Moreover you mention that Moran's R is bound between -1 and 1 but this is not true AFAIK ?

[dbdimitrov]

Hi @j-bac,

I agree defining j and i is confusing here, if I remember correctly I did it mostly for consistency with the rest of the furmulas. I will think how to represent this a bit better.
This is the implimentation tho:

liana-py/liana/method/sp/_bivariate/_local_functions.py

Line 290 in e0c86b1

def _vectorized_cosine(x_mat, y_mat, weight):

re: local Moran's R is not bound between -1 and 1, but the global version is. See the 5th paragraph in the discussion of the original paper: https://www.nature.com/articles/s41467-023-39608-w#Sec10

Hope this helps.

[dbdimitrov]

Hi @RunyuXia,

Sorry for the delated response. The bivariate scores (X layer) you can think of in the simplest terms as simply weighted summaries of the expression of ligand and receptors from surrounding cells. Essentially, it will tell you which interactions are plausible to happen in the surrounding of each spot. The X layer is then z-scaled for the categories, and we do permutations for the p-values where we shuffle the neighbouring spots essentially.

Hope this helps :)

[j-bac]

I think you mean the opposite, local values are (roughly, after clipping) bound to -1 and 1 ? But I didn't find anything for global values

Quote from SpatialDM:

local Moran’s R value is not normalised to a fixed bound, e.g., (−1, 1), but it is refined to a reasonable range after standardising the expression matrices (i.e., −10 to 10), and we have further clipped the extreme values out this range

From the implementation code you linked I'm not sure if x_mat, y_mat come from the same cells (i.e. just neighbors j as in your formula) or if x_mat would be the center spot i of the neighborhood.

It matters a lot because multiplying ligand and receptor within the same cell is not meaningful as a communication score. It would just measure coexpression of these genes inside a given cell but not between cells

[dbdimitrov]

the multiplication is correct and across neighbors

EDIT: I pointed you to the wrong place, sorry, if spatialdm does not say it explicitly, this one should: https://link.springer.com/article/10.1007/s101090100064

[j-bac]

Yes sorry it says they bound local values between -10 and 10 after clipping not -1 and 1. But once again, there is nothing said about global values ?

[dbdimitrov]

Yes, sorry - see edit :)

[j-bac]

I didn't find the section on the bounds from your linked study. Moran's I wikipedia
"Expected value" section mentions bounds are "minimum and maximum eigenvalues of the weight matrix"

[dbdimitrov]

Yeah, you're actually right - I also coulnd't find it either, my bad.

EDITED: "values near zero imply spatial independence, while positive or negative values reflect spatial co-clustering or spatial cross-dispersion, respectively"

PS. Thanks for noticing.

[j-bac]

that's not exactly right either, the expected value under the null is different from 0, per wikipedia