Compare two contingency tables with the same structure

Question

I want to compare two contingency tables A and B with the same structure (same columns and rows). My first idea was to use chi squared distance $\sum_{i=1}^{p}{\frac{(O_i-E_i)^2}{E_i}}$. But, some cells $E_i$ are equal to zero.

Have you some ideas to deal with that problem or another distance metrics without this issue ?

---

More precisely, let's consider the contingency table A:

Atoms	Carbone	Hydrogen	Nitrogen
Carbone	2339	2582	305
Hydrogen	2582	0	144
Nitrogen	305	144	29

Each cell $A_{ij}$ of table A shows how many times the two atoms (corresponding to the row i and the column j) are linked together to form a molecule in our data (in molecule table of the Biodegradability database).

I have another list of molecules who is represented by the contingency table B: Given another contingency table B:

Atoms	Carbone	Hydrogen	Nitrogen
Carbone	1293	2664	107
Hydrogen	2664	1310	114
Nitrogen	107	114	0

My objective is to measure how close the table B is from table A by computing the chi-squared distance. As we can see, if we directly compute this distance, we get infinite value because of the null value in the cell B(Nitrogen, Nitrogen) -- Nitrogen atoms are never linked together in table B.

Do you have any idea on how to deal with this issue or have any other distance metric (without this issue) to propose?

Answer 1

You are tabulating edges. Your tables are, more or less, sums of adjacency matrices.

The data can be expressed (without redundancy) as a $2\times 6$ table explicitly indicating the possible edges:

Bond:    CC   CH   CN   HH   HN   NN   Total
-----  ---- ---- ---- ---- ---- ----   -----
    A  2339 2582  305    0  144   29 |  5399
    B  1293 2664  107 1310  114    0 |  5488
Total  3632 5246  412 1310  258   20 | 10887

(Never mind that this is not chemically possible.)

Any column with a total of 0 (which is what I would expect a realistic HH column to be) can be dropped. There is no problem with any remaining cells with zeros, because they will have non-zero expected values. (Some care is needed in computing a p-value when many expected values are small, but that's a separate issue and it does not apply in this example where the smallest expected value is 14.)

As usual, the "expected value" $E_{rc}$ in row $r$ and column $c$ is the product of the row total and column total divided by the grand total and the chi-squared statistic is the sum given in the question. Here, its value is $1739.6.$ You can use that, if you wish, as a measure of how "close" the datasets $A$ and $B$ are.

The usual chi-squared p-value is likely meaningless. It is computed by assuming the cell counts are independent. That's not the case with molecules. Consider, for instance, ethane, $C_2H_6.$ Because there is one CC bond, there must be six CH bonds: that's a lack of independence. (If you are counting double bonds and triple bonds as if they were a single link, then the presence of a single CC link in a hydrocarbon implies there are 2, 4, or 6 CH links -- but that still exhibits non-independence.)

It would be presumptuous to propose a metric or alternative method of comparing the data without more information about how the comparison is intended to be used.