In this supplementary, we give a brief introduction of following coefficients that measure the similarity between two gene sets based on gene overlaps: Jaccard coefficient, Dice coefficient, overlap coefficient and kappa coefficient.
Denote two sets of genes as \(A\) and \(B\), the Jaccard coeffcient between the two sets is calculated as:
\[{|A \cap B|}\over{|A \cup B|}\]
The Dice coeffcient is calculated as:
\[\frac{|A \cap B|}{(|A| + |B|)/2}\]
Jaccard coefficient has a relationship with Dice coefficient in the form of:
\[Jaccard = \frac{Dice}{2 - Dice}\]
The overlap coeffcient is calculated as:
\[\frac{| A \cap B | }{\min(|A|,|B|)}\]
The symbol \(|A|\) means the size of set \(A\) (number of elements).
The definition of kappa coeffient is a little bit complex. First let’s format the two sets into a contigency table:
| In set B | |||
| Yes | No | ||
| In set A | Yes | a | b |
| No | c | d | |
where \(a\), \(b\), \(c\), \(d\) are the numbers of genes that fall in each category.
Let’s calculate \(p_o\) and \(p_e\) as:
\[p_o = \frac{a+d}{a+b+c+d}\]
\[p_{Yes} = \frac{a+b}{a+b+c+d} \cdot \frac{a+c}{a+b+c+d}\]
\[p_{No} = \frac{c+d}{a+b+c+d} \cdot \frac{b+d}{a+b+c+d}\]
\[p_e = p_{Yes} + p_{No}\]
where \(p_o\) is the probability of a gene in both gene sets or neither in the two sets, \(p_{Yes}\) is the probability of a gene in both gene sets by random (by assuming the events of a gene in set \(A\) and set \(B\) are independent), \(p_{No}\) is the probability of a gene not in the two sets by random, and \(p_e\) is the probability of a gene either both in the two sets or not in the two sets by random.
and the kappa coeffcient is calculated as:
\[\frac{p_o - p_e}{1 - p_e}\]