= RNA oligonucleotide association coefficient
{c}
{title2=$S_{AB}=2N_{AB}/(N_A+N_B)$}
The <RNA oligonucleotide association coefficient> measures the overlap of two catalogs produced through <ribosomal RNA oligonucleotide cataloguing>, weighting the retained shared fragments by their nucleotide content. In the 1977 comparison, catalogued sequences had at least six residues, and
$$
S_{AB}=\frac{2N_{AB}}{N_A+N_B},
$$
where $N_A,N_B$ count nucleotide residues represented in the two eligible catalogs and $N_{AB}$ counts those represented in their shared portion. It is a catalog-overlap statistic, not the percentage identity of a complete <sequence alignment>.<Phylogenetic structure of the prokaryotic domain: the primary kingdoms by Carl R. Woese and George E. Fox>[p.5089, Table1 legend: association coefficient, nucleotide counts and hexamer-or-larger cutoff]{ref}
An original example uses distinct fragment labels with specified lengths. CatalogA contains $a_6,b_8,c_{10}$; catalogB contains $a_6,b_8,d_6$. The subscripts denote nucleotide counts, and differently named fragments are different sequences. Thus
$$
N_A=24,\quad N_B=20,\quad N_{AB}=14,\qquad S_{AB}=\frac{28}{44}=\frac7{11}\approx0.636.
$$
Counting only matched labels would instead give two shared labels out of three per catalog, producing $2/3$. The difference exposes the effect of weighting: a shared eight-residue fragment contributes more than a shared six-residue fragment.
For a nonzero denominator, $0\leq N_{AB}\leq\min(N_A,N_B)$ gives $0\leq S_{AB}\leq1$. A useful equivalent form is
$$
S_{AB}=1-\frac{N_A+N_B-2N_{AB}}{N_A+N_B}.
$$
The numerator in the subtracted fraction represents the catalog material outside the overlap under this counting model. If both catalogs are empty, the original formula has denominator zero; assigning a value would require an extra convention.
The coefficient is also sensitive to catalog coverage. Add the same new six-residue fragment to both example catalogs: the totals become30 and26, with20 shared, so the coefficient rises to $40/56=5/7$. Add it only toA instead, and the coefficient is $28/50=0.56$. These constructed changes alter the statistic without specifying any new evolutionary history.
The original <ribosomal RNA oligonucleotide cataloguing> experiments used comparisons of this kind as evidence for <phylogenetics>. Turning a similarity into a particular <phylogenetic tree> or a divergence date needs additional assumptions and evidence. The statistic itself records the specified molecular overlap.<Phylogenetic structure of the prokaryotic domain: the primary kingdoms by Carl R. Woese and George E. Fox>[Abstract and Table1: molecular comparison and phylogenetic interpretation]{ref}
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