D-statistics, ABBA-BABA, and f4-ratios

The D-statistic, also called the ABBA-BABA test, was introduced in the 2010 Neanderthal genome paper by Green and colleagues. It asks a precise question: given four populations arranged on a known phylogeny, do two of them share a disproportionate fraction of derived alleles, beyond what genetic drift alone would predict?

The four-population setup

Pick four populations: P1 and P2 (sister groups in the modern tree), P3 (the candidate introgressor, such as Neanderthal), and an outgroup O (such as chimpanzee). For every biallelic SNP, classify each population's allele as A (ancestral, matching the outgroup) or B (derived, differing from the outgroup). The pattern across (P1, P2, P3, O) is a four-letter code: AABA, ABBA, BABA, BBBA, and so on.

Under a null model of incomplete lineage sorting and no introgression, the patterns ABBA (P2 and P3 share derived) and BABA (P1 and P3 share derived) should occur with equal frequency. Any imbalance indicates that one of P1 or P2 shares more recent ancestry with P3, beyond the shared phylogeny. That asymmetry is exactly the signature of admixture.

The math

D = (n_ABBA - n_BABA) / (n_ABBA + n_BABA). The statistic ranges from -1 to +1. Values significantly different from zero indicate introgression: positive D suggests P2-P3 admixture, negative D suggests P1-P3 admixture. The 2010 Neanderthal paper reported D ≈ 0.05 with French versus Yoruba versus Vindija, supporting Neanderthal contribution to non-African genomes at roughly 2-4%.

f4-ratios for admixture proportion

The D-statistic flags admixture but does not quantify it. The f4-ratio, introduced by Patterson and colleagues in 2012, extends the same framework to estimate the fraction of ancestry contributed by the source population. Given a five-population setup (a focal admixed population X, a reference admixed population, two source proxies, and an outgroup), the f4-ratio is f4(X, O; P3, A) / f4(B, O; P3, A), where the numerator estimates the absolute drift shared between X and the source and the denominator normalises against a known admixed reference.

When applied to European genomes versus Vindija Neanderthal versus West African and chimpanzee outgroups, the f4-ratio recovers the headline 2% Neanderthal contribution that has become the canonical figure. The same machinery applied to Melanesians versus Denisova produces estimates of 3-6% Denisovan contribution.

Practical limitations

• D-statistics test for asymmetry, not for the source: a significant D means SOMETHING introgressed, but a separate analysis is needed to confirm the source identity.
• The four-population assumption is fragile when the true demographic history involves more populations or pulses than the model captures. Multiple introgression events can produce confusing D values.
• Reference allele ascertainment matters. If the SNP set was ascertained in modern humans, archaic-private variants are systematically underrepresented and D is biased.
• The f4-ratio assumes the admixed reference population truly represents the source, an assumption that is only approximate even for the cleanest reference panels.