# Information Content

The standard definition of information content of a term in a taxonomy, as proposed by Resnik in 1999 [1], is quantified as the negative log likelihood of a term. The concept expressed by the formula conveys the idea that as the frequency of a term increases, informativeness decreases. The probability of any concept in the taxonomy depends on the sum of the probabilities of the concepts it subsumes and, consequently, the nearer the term is to the root the more frequent and the less informative the concept is (see Fig. 1-ii). This is translated in the following formula:

where *IC* is the Information Content and *p(c)* is the relative frequency of all of the
descendants of the GO term c that are extracted from the GOA database:

Depending on the GOA database release used, the *IC* value changes as a
consequence of the continuous enhancements and updates of both the protein
annotations and the GO graph structure.

The information content range is from 0 (the root of the graph that is uninformative) to 1 (the most distant leaf of the graph carrying the best specific information).

We suggest leaving the default value unless one needs to discard the less informative GO annotations.

# Internal Confidence (InC) and Total Score (TS)

## Weighting the nodes

The scoring method is based on the estimation of some parameters which take into account the global cumulative weight distribution, which is the sum of the considered node weight plus the weights of its children, and local not cumulative weight distribution, which considers only the weight of the term under analysis.
Each node is weighted according to the provided score (for BLAST and HMMER
the logarithms of the e-values are considered). The corrected weights
are reconstructed starting from the leaves of the graph up to the root
following a non-redundant strategy (see Fig. 1-ii).
During the exhaustive reconstruction of the multiple paths leading to
the root of the graph, some parent nodes may be visited many times:
non-redundant means that the score of a child node is added only once to the
weights of the parent nodes during this process. The (cumulative) weight *W _{Gi}* of the node

*G*is calculated as follows:

_{i}that is the absolute value of the sum of the child nodes' scores, and *S _{i}* is the score of the node given in input. As a result of this additive strategy,
only the most generic annotation terms, those near the root, show the highest
scores at the expense of carrying less information. To tackle this behavior a
different measure based on the Information Content (IC) was used to obtain an
acceptable trade-off between detailed knowledge and statistical
significance.

## InC score

The *InC* is a normalized score whose values lie in the 0-1 interval; it has
been specifically designed to assess the statistical significance of the
retrieved hits.

*G*as

_{i}where *W _{Gi}* is the sum of the weights of child nodes of

*G*.

_{i}On the other hand, the non cumulative Internal Confidence for the node

*G*(

_{i}*InC*) is calculated as

_{nci}where *W _{ncGi}* is the not cumulative weight of node

*G*, that is the input score.

_{i}We suggest the user to rely on TS score (see below) and leave the default
value for *InC* to 0. *InC* can help in discriminating GO hits whose Total Score (TS) is
around 2.0. In this case, if *InC* is high and near 1.0, the GO hit can be
considered a reasonable good prediction. If not, the annotation can be
potentially uncertain.

## Z-score and Group Score of the nodes

In order to eliminate the terms that are not functionally related to the initial query, the Z-score and the Group Score have been defined. The Z-score is calculated for node *G _{i}* as follows:

where *W* is the score of the root node divided by the total number of the nodes that compose the trimmed graph, *W _{Gi}* is the weight of node

*i*and

*σ*is the standard deviation assuming a Gaussian distribution of the graph (see Fig. 1-ii). The Group Score is calculated as the sum of the Internal Confidence of the nodes belonging to it. These two measures are used as a first filter for a preliminary skew of the false positive terms.

## Total Score (TS)

For the GO terms which have passed the previous selections based on the Z-score and the Group score, the Total Score for each node *G _{i}* is calculated according to the following formula

where *IC _{i}* is the Information Content,

*W*is the value given in input to Argot

_{ncGi}^{2.5}for then node

*G*and

_{i}*GS*is the non-cumulative Group Score, which in turn is calculated as the sum of the non-cumulative Internal Confidence of the nodes belonging to the group; the TS score is not normalized as it starts from 0 and has no upper bound.

_{nci}We suggest the user a TS score ≥ 200.0 to have a “good” prediction based on our benchmark test conditions. InC (see above) can be evaluated if TS is low.

## Grouping by means of semantic similarity

Once the “final trimmed GO graph” has been obtained, the remaining GO nodes are grouped according to their semantic similarity (see Fig. 1-iii,1-iv). This approach has already been applied in natural language processing and taxonomies providing a manageable metric with which to link different terms carrying similar information. There are many different methods for calculating the shared information between two terms, but we obtained the best results using Lin’s formula [2]. This formula has the advantage of reporting a normalized value between 0 and 1 and has already been proven to outperform other algorithms even though it may be affected by shallow annotations. Lin’s formula is defined as:

where

*S(c _{1}, c_{2})* are the common subsumers of

*c*and

_{1}*c*terms and

_{2}*sim*is the one with the highest Information Content

_{res}(c_{1}, c_{2})*IC*. In other words, the algorithm finds the nearest parent node that is shared by the two terms whose semantic distance is to be calculated.

This quantifies the extent to which two concepts are related according to their position, which is highly dependent on the graph connections and density. This measure is much more effective and meaningful than absolute edge distance as it accounts for the real relationships among concepts. For example, the two GO nodes GO:0043734 “pigmentation” and GO:0016032 “viral reproduction” have the root node GO:0008150 “biological process” as parent. These GO terms may be erroneously considered neighbors as their edge distance is 2 but their meaning is completely different. This difference appears immediately evident from a semantic similarity perspective, as the IC of the parent node is 0, being the root of the ontology, and consequently the semantic similarity is 0, as expected. The same edge distance exists between two similar GO terms: GO:0004602 “glutathione peroxidase activity”, which occurs 69 times over 168,919 gene products (as of March 2008 GO release), and GO:0047066 “phospholipid-hydroperoxide glutathione peroxidase activity”, which occurs 5 times. They both descend from a common ancestor GO:0004601 “peroxidase activity” with 409 occurrences, subsuming all the occurrences of its child nodes. In this case, semantic similarity calculated with Lin’s formula is 0.66 as follows:

This reveals that the GO terms are strict neighbors. Comparing the two examples it is clear that edge distance is unsuitable and insufficiently discriminative for finding related concepts.

Semantic similarity is used to group GO terms that belong to the “final
trimmed GO graph” containing the most probable starting paths and allows the
number of similar GO terms in highly populated regions of the graph to be
further reduced (see Fig. 1-iii,-iv). The algorithm attempts to group
recursively the GO terms that have been retrieved by the BLAST hits *R* and
belong to the “final trimmed GO graph” with strict semantic similarity of
over 0.7. Those GO nodes that are part of the different paths leading to the
root of the graph but that have not been found in the annotation of the BLAST
hits, are not considered in this calculation. A matrix distance is computed by
performing an all vs. all comparison of the *IC* scores of the GO terms retrieved
from *R* using the Lin’s formula and if two terms satisfy the cut-off they are
merged into the same group (see Fig. 1-iii):

*g _{Gi}* is the initiator node of the group

*G*. The initiator

_{i}*g*, is another GO node and n is the total number of GO terms extracted from

_{j}*R*and present in the “final trimmed GO graph”. At the beginning of the clustering process any GO term is a potential initiator node and is considered the founder of a forming cluster: the GO terms sharing a semantic similarity of 0.7 are added. After the first clustering step a second and less stringent threshold of 0.6 is applied to further merge acquired groups that have at least one GO term in common. Only the initiator nodes or founders

*g*are involved in this process (see Fig. 1-iv). The grouping strategy has the effect of gathering together semantically similar GO terms thus actually reducing the search space as only one or few representatives per group are chosen according to their scores, described in the next section, and IC.

_{Gi}1)Resnik, P. (1999) *Semantic similarity in a taxonomy: An information-based
measure and its application to problems of ambiguity in natural language.*
Journal of Artificial Intelligence Research, 11, 95-130.

2)Lin D (1998) *An Information-Theoretic Definition of Similarity.*
Proceedings of the Fifteenth International Conference on Machine Learning
(ICML'98): Morgan Kaufmann Publishers Inc. pp. 296-304

**Figure 1:** Schematic representation of the Argot^{2.5}
algorithm. The first step i)
“Trimming the GO graph” involves the trimming of the GO graph to
obtain a slim containing only the putative GO hits extracted in order to
annotate the query protein (black circles). In the second step ii)
“Weighting the GO nodes” and “GO Information content”, the
algorithm calculates the Information Content (IC) (gray bars) of the nodes in
the graph and their cumulative weights (colored bars) derived from the BLAST
scores. In the third step iii) “Grouping by means of semantic
similarity” and “Choosing the most probable paths in the
“initial trimmed GO graph”, GO nodes are clustered into groups
having a given semantic similarity using the Lin formula; GO terms which
populate isolated branches of the graph are discarded on the basis of their
Z-score (red and green triangles) applied to the node weights. In the last step
iv) “Grouping by means of semantic similarity” Argot^{2.5}
tries to merge
similar clusters on the basis of semantic distance calculated among the
groups’ founders (azure circles) using less stringent cut-offs. GO terms with
the highest IC (green big circles) are chosen as representatives of the
clusters obtained.