Tajima's D

Under the neutral theory model, for a population at constant size at equilibrium:

${\displaystyle E[\pi ]=\theta =E\left[{\frac {S}{\sum _{i=1}^{n-1}{\frac {1}{i}}}}\right]=4N\mu }$

for diploid DNA, and

${\displaystyle E[\pi ]=\theta =E\left[{\frac {S}{\sum _{i=1}^{n-1}{\frac {1}{i}}}}\right]=2N\mu }$

for haploid.

In the above formulas, S is the number of segregating sites, n is the number of samples, N is the effective population size, ${\displaystyle \mu }$ is the mutation rate at the examined genomic locus, and i is the index of summation. But selection, demographic fluctuations and other violations of the neutral model (including rate heterogeneity and introgression) will change the expected values of ${\displaystyle S}$ and ${\displaystyle \pi }$, so that they are no longer expected to be equal. The difference in the expectations for these two variables (which can be positive or negative) is the crux of Tajima's D test statistic.

${\displaystyle D\,}$ is calculated by taking the difference between the two estimates of the population genetics parameter ${\displaystyle \theta \,}$. This difference is called ${\displaystyle d\,}$, and D is calculated by dividing ${\displaystyle d\,}$ by the square root of its variance ${\displaystyle {\sqrt {{\hat {V}}(d)}}}$ (its standard deviation, by definition).

${\displaystyle D={\frac {d}{\sqrt {{\hat {V}}(d)}}}}$

Fumio Tajima demonstrated by computer simulation that the ${\displaystyle D\,}$ statistic described above could be modeled using a beta distribution. If the ${\displaystyle D\,}$ value for a sample of sequences is outside the confidence interval then one can reject the null hypothesis of neutral mutation for the sequence in question. However, in real world uses, one must be careful as past population changes (for instance, a population bottleneck) can bias the value of the ${\displaystyle D\,}$ statistic.[2]

${\displaystyle D={\frac {d}{\sqrt {{\hat {V}}(d)}}}={\frac {{\hat {k}}-{\frac {S}{a_{1}}}}{\sqrt {[e_{1}S+e_{2}S(S-1)]}}}}$

where

${\displaystyle e_{1}={\frac {c_{1}}{a_{1}}}}$	${\displaystyle e_{2}={\frac {c_{2}}{a_{1}^{2}+a_{2}}}}$
${\displaystyle c_{1}=b_{1}-{\frac {1}{a_{1}}}}$	${\displaystyle c_{2}=b_{2}-{\frac {n+2}{a_{1}n}}+{\frac {a_{2}}{a_{1}^{2}}}}$
${\displaystyle b_{1}={\frac {n+1}{3(n-1)}}}$	${\displaystyle b_{2}={\frac {2(n^{2}+n+3)}{9n(n-1)}}}$
${\displaystyle a_{1}=\sum _{i=1}^{n-1}{\frac {1}{i}}}$	${\displaystyle a_{2}=\sum _{i=1}^{n-1}{\frac {1}{i^{2}}}}$

${\displaystyle {\hat {k}}\,}$ and ${\displaystyle {\frac {S}{a_{1}}}}$ are two estimates of the expected number of single nucleotide polymorphisms (SNPs) between two DNA sequences under the neutral mutation model in a sample size ${\displaystyle n\,}$ from an effective population size ${\displaystyle N}$.

The first estimate is the average number of SNPs found in (n choose 2) pairwise comparisons of sequences ${\displaystyle (i,j)}$ in the sample,

${\displaystyle {\hat {k}}={\frac {\sum \sum _{i<j}k_{ij}}{\binom {n}{2}}}.}$

The second estimate is derived from the expected value of ${\displaystyle S}$, the total number of polymorphisms in the sample

${\displaystyle E(S)=a_{1}M.}$

Tajima defines ${\displaystyle M=4N\mu }$, whereas Hartl & Clark use a different symbol to define the same parameter ${\displaystyle \theta =4N\mu }$.

Suppose you are a geneticist studying an unknown gene. As part of your research you get DNA samples from four random people (plus yourself). For simplicity, you label your sequence as a string of zeroes, and for the other four people you put a zero when their DNA is the same as yours and a one when it is different. (For this example, the specific type of difference is not important.)

                    1           2
Position  12345 67890 12345 67890
Person Y  00000 00000 00000 00000
Person A  00100 00000 00100 00010
Person B  00000 00000 00100 00010
Person C  00000 01000 00000 00010
Person D  00000 01000 00100 00010

Notice the four polymorphic sites (positions where someone differs from you, at 3, 7, 13 and 19 above). Now compare each pair of sequences and get the average number of polymorphisms between two sequences. There are "five choose two" (ten) comparisons that need to be done.

Person Y is you!

You vs A: 3 polymorphisms

Person Y     00000 00000 00000 00000
Person A     00100 00000 00100 00010

You vs B: 2 polymorphisms

Person Y     00000 00000 00000 00000
Person B     00000 00000 00100 00010

You vs C: 2 polymorphisms

Person Y     00000 00000 00000 00000
Person C     00000 01000 00000 00010

You vs D: 3 polymorphisms

Person Y     00000 00000 00000 00000
Person D     00000 01000 00100 00010

A vs B: 1 polymorphism

Person A     00100 00000 00100 00010
Person B     00000 00000 00100 00010

A vs C: 3 polymorphisms

Person A     00100 00000 00100 00010
Person C     00000 01000 00000 00010

A vs D: 2 polymorphisms

Person A     00100 00000 00100 00010
Person D     00000 01000 00100 00010

B vs C: 2 polymorphisms

Person B     00000 00000 00100 00010
Person C     00000 01000 00000 00010

B vs D: 1 polymorphism

Person B     00000 00000 00100 00010
Person D     00000 01000 00100 00010

C vs D: 1 polymorphism

Person C     00000 01000 00000 00010
Person D     00000 01000 00100 00010

The average number of polymorphisms is ${\displaystyle {3+2+2+3+1+3+2+2+1+1 \over 10}=2}$.

The second estimate of the equilibrium is M=S/a1

Since there were n=5 individuals and S=4 segregating sites

a1=1/1+1/2+1/3+1/4=2.08

M=4/2.08=1.92

The lower-case d described above is the difference between these two numbers—the average number of polymorphisms found in pairwise comparison (2) and M. Thus ${\displaystyle d=2-1.92=.08}$.

Since this is a statistical test, you need to assess the significance of this value. A discussion of how to do this is provided below.

A negative Tajima's D signifies an excess of low frequency polymorphisms relative to expectation, indicating population size expansion (e.g., after a bottleneck or a selective sweep). A positive Tajima's D signifies low levels of both low and high frequency polymorphisms, indicating a decrease in population size and/or balancing selection. However, calculating a conventional "p-value" associated with any Tajima's D value that is obtained from a sample is impossible. Briefly, this is because there is no way to describe the distribution of the statistic that is independent of the true, and unknown, theta parameter (no pivot quantity exists). To circumvent this issue, several options have been proposed.

However, this interpretation should be made only if the D-value is deemed statistically significant.

When performing a statistical test such as Tajima's D, the critical question is whether the value calculated for the statistic is unexpected under a null process. For Tajima's D, the magnitude of the statistic is expected to increase the more the data deviates from a pattern expected under a population evolving according to the standard coalescent model.

Tajima (1989) found an empirical similarity between the distribution of the test statistic and a beta distribution with mean zero and variance one. He estimated theta by taking Watterson's estimator and dividing it by the number of samples. Simulations have shown this distribution to be conservative,[3] and now that the computing power is more readily available this approximation is not frequently used.

A more nuanced approach was presented in a paper by Simonsen et al.[4] These authors advocated constructing a confidence interval for the true theta value, and then performing a grid search over this interval to obtain the critical values at which the statistic is significant below a particular alpha value. An alternative approach is for the investigator to perform the grid search over the values of theta which they believe to be plausible based on their knowledge of the organism under study. Bayesian approaches are a natural extension of this method.

A very rough rule of thumb to significance is that values greater than +2 or less than -2 are likely to be significant. This rule is based on an appeal to asymptotic properties of some statistics, and thus +/- 2 does not actually represent a critical value for a significance test.

Finally, genome wide scans of Tajima's D in sliding windows along a chromosomal segment are often performed. With this approach, those regions that have a value of D that greatly deviates from the bulk of the empirical distribution of all such windows are reported as significant. This method does not assess significance in the traditional statistical sense, but is quite powerful given a large genomic region, and is unlikely to falsely identify interesting regions of a chromosome if only the greatest outliers are reported.

• Fay and Wu's H

• Hartl, Daniel L.; Clark, Andrew G. (2007). Principles of Population Genetics (4th ed.). Sinauer Associates. ISBN 978-0878933082.

Computational tools:

• DNAsp (Windows)
• Variscan (Mac OS X, Linux, Windows)
• Arlequin (Windows)
• Online view of Tajima's D values in human genome
• Python3 package for computation of Tajima's D
• MEGA4 or MEGA5
• Bio::PopGen::Statistics in BioPerl

• A video explanation of Tajima's D, and its application to DNA sequences, is available online.