Tarone-Greenland Alpha (α_TG)

MKado implements the weighted α_TG estimator from Stoletzki & Eyre-Walker (2011), which provides an unbiased estimate of the proportion of adaptive substitutions when analyzing multiple genes.

Background

When analyzing many genes, a common approach is to calculate alpha (α = 1 - NI) for each gene and take the mean. However, this simple average is heavily biased by genes with small sample sizes, where alpha can take extreme values (e.g., -30 or +5) due to sampling noise.

Stoletzki & Eyre-Walker (2011) showed that averaging across genes produces biased estimates even with large sample sizes, and introduced a weighted estimator that corrects this problem.

The NI_TG Formula

The weighted neutrality index is calculated as:

\[NI_{TG} = \frac{\sum_i (D_{si} \times P_{ni}) / (P_{si} + D_{si})}{\sum_i (D_{ni} \times P_{si}) / (P_{si} + D_{si})}\]

Where for each gene i:

  • Dni = nonsynonymous divergence (fixed differences)

  • Dsi = synonymous divergence

  • Pni = nonsynonymous polymorphism

  • Psi = synonymous polymorphism

The weighting by 1/(Psi + Dsi) downweights genes with small denominators, where estimates are unreliable.

Alpha is then: α_TG = 1 - NI_TG

Usage

Use the --alpha-tg flag with batch processing:

# Basic usage
mkado batch alignments/ -i ingroup -o outgroup --alpha-tg

# With more bootstrap replicates for tighter CIs
mkado batch alignments/ -i ingroup -o outgroup --alpha-tg --bootstrap 1000

Example with the included Anopheles data:

mkado batch examples/anopheles_batch/ -i gamb -o afun --alpha-tg

Frequency-Threshold Correction (FWW)

To reduce the bias from low-frequency slightly deleterious polymorphisms, α_TG can be combined with a derived allele frequency cutoff (Fay, Wyckoff & Wu 2001) by passing --min-freq:

# FWW-corrected weighted alpha: drop polymorphisms with derived AF < 0.15
mkado batch alignments/ -i ingroup -o outgroup --alpha-tg --min-freq 0.15

The --min-freq filter is applied per gene before α_TG is computed, so the weighted estimator sees only the high-frequency polymorphisms. --no-singletons is the convenience equivalent of --min-freq 1/n.

Output

The output includes:

  • alpha_TG: Proportion of adaptive substitutions (1 - NI_TG)

  • NI_TG: The weighted neutrality index

  • CI_low, CI_high: 95% bootstrap confidence interval on alpha_TG

  • num_genes: Number of genes analyzed

  • Dn, Ds, Pn, Ps: Total counts across all genes

  • Ln, Ls: Nei-Gojobori non-synonymous and synonymous site totals

  • omega: dN/dS ratio (Dn/Ds) * (Ls/Ln)

  • omega_a, omega_na: Adaptive and non-adaptive substitution rates (Gossmann, Keightley & Eyre-Walker 2012; applied to MK counts by Coronado-Zamora et al. 2019)

  • omega_CI_low/high, omega_a_CI_low/high, omega_na_CI_low/high: 95% bootstrap CIs. Because the gene-resampling bootstrap varies Dn, Ds, Ln, and Ls per replicate, omega itself has a bootstrap distribution here (unlike in the asymptotic test where Ln/Ls are constants). See Omega Decomposition (ω, ω_a, ω_na) for the rationale.

  • ci_method: always "bootstrap" for α_TG. The weighted estimator has no parametric Monte Carlo analog, so the global --ci-method flag has no effect when --alpha-tg is set.

Example output (TSV format, abbreviated):

Dn      Ds      Pn    Ps      alpha_TG  NI_TG     CI_low    CI_high   num_genes  ...  omega    omega_a  omega_na  omega_CI_low  omega_CI_high  ...
18828   49857   7843  25083   0.022781  0.977219  -0.053529 0.088672  400        ...  0.1117   0.0025   0.1092    0.1075        0.1158         ...

Comparison with Other Methods

Different methods for estimating alpha correct for different biases:

Method

Corrects for

Best used when

Simple mean α

Nothing

Never recommended for multi-gene analyses

Imputed MK

Weakly deleterious mutations (by imputation)

Gene-level analyses; maximizing power with limited data

α_TG

Sample size heterogeneity

Comparing species with little slightly deleterious load

Asymptotic α

Slightly deleterious mutations

Most genome-wide analyses

Example comparison (Anopheles gambiae vs. A. funestus, 400 genes):

Method

Alpha estimate

95% CI

Simple mean

-1.19

—

α_TG (weighted)

+0.02

-0.05 to +0.09

Asymptotic α

+0.57

+0.49 to +0.66

The large gap between α_TG and asymptotic α suggests substantial slightly deleterious polymorphism — a common finding. The asymptotic method extrapolates to high frequencies where deleterious variants have been purged, revealing adaptive substitutions masked by segregating deleterious mutations.

When to Use α_TG

Use α_TG when:

  • You want an unbiased multi-gene estimate without frequency spectrum modeling

  • Your species pair has minimal slightly deleterious load

  • You want to compare with published NI_TG values

Use asymptotic α (-a) when:

  • Slightly deleterious mutations are a concern (most cases)

  • You have sufficient polymorphism data for frequency binning

  • You want the most accurate estimate of adaptive substitution rate

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

Gossmann TI, Keightley PD, Eyre-Walker A (2012) The effect of variation in the effective population size on the rate of adaptive molecular evolution in eukaryotes. Genome Biology and Evolution 4(5):658-667. https://doi.org/10.1093/gbe/evs027

Coronado-Zamora M, Salvador-Martínez I, Castellano D, Barbadilla A, Salazar-Ciudad I (2019) Adaptation and conservation throughout the Drosophila melanogaster life-cycle. Genome Biology and Evolution 11(5):1463-1482. https://doi.org/10.1093/gbe/evz046