Direction of Selection (DoS)

MKado reports the Direction of Selection (DoS) statistic from Stoletzki & Eyre-Walker (2011).

Background

The Neutrality Index (NI) has limitations as a measure of selection:

Asymmetric: NI ranges from 0 to infinity, making interpretation difficult
Unbounded: Values can become very large with small counts
Undefined: Cannot be calculated when Dn=0, Ds=0, or Ps=0

DoS addresses these issues by providing a symmetric, bounded measure of selection.

The DoS Formula

\[DoS = \frac{D_n}{D_n + D_s} - \frac{P_n}{P_n + P_s}\]

Where:

D_n = Non-synonymous divergence (fixed differences)
D_s = Synonymous divergence (fixed differences)
P_n = Non-synonymous polymorphisms
P_s = Synonymous polymorphisms

Interpretation

DoS measures the difference between the proportion of non-synonymous changes among divergent sites versus polymorphic sites:

DoS = 0: Neutral evolution (equal proportions)
DoS > 0: Positive selection (excess adaptive substitutions)
DoS < 0: Slightly deleterious polymorphisms (excess non-synonymous polymorphism)

Advantages over NI

Property	NI	DoS
Range	0 to ∞	-1 to +1
Neutral expectation	1	0
Symmetry	Asymmetric	Symmetric around 0
Behavior with small counts	Can be extreme	Well-behaved
Interpretability	Harder	Easier (direct comparison of proportions)

Output

DoS is automatically included in all MK test outputs:

Pretty format:

MK Test Results:
  Divergence:    Dn=6, Ds=8
  Polymorphism:  Pn=1, Ps=8
  Fisher's exact p-value: 0.176
  Neutrality Index (NI):  0.1667
  Alpha (α):              0.8333
  DoS:                    0.3175

JSON format:

{
  "dn": 6,
  "ds": 8,
  "pn": 1,
  "ps": 8,
  "p_value": 0.176,
  "ni": 0.1667,
  "alpha": 0.8333,
  "dos": 0.3175
}

TSV format:

Dn   Ds      Pn      Ps      p_value NI      alpha   DoS
6    8       1       8       0.176   0.1667  0.8333  0.3175

Example Calculation

Using the classic Kreitman Adh data (Dn=6, Ds=8, Pn=1, Ps=8):

\[DoS = \frac{6}{6+8} - \frac{1}{1+8} = \frac{6}{14} - \frac{1}{9} = 0.429 - 0.111 = 0.318\]

This positive value indicates an excess of non-synonymous divergence relative to polymorphism, consistent with positive selection.

Edge Cases

DoS handles edge cases gracefully:

Zero divergence (Dn+Ds=0): The divergence ratio is treated as 0
Zero polymorphism (Pn+Ps=0): The polymorphism ratio is treated as 0
All zeros: Returns None (no data to analyze)

This behavior makes DoS more robust than NI when counts are sparse.

Reference

Stoletzki N, Eyre-Walker A (2011) Estimation of the Neutrality Index. Molecular Biology and Evolution 28(1):63-70. https://doi.org/10.1093/molbev/msq249