E-mail: kosman@post.tau.ac.il

# Nei's gene diversity and the index of average differences are identical measures of diversity within populations

Article first published online: 29 OCT 2003

DOI: 10.1046/j.1365-3059.2003.00923.x

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#### How to Cite

Kosman, E. (2003), Nei's gene diversity and the index of average differences are identical measures of diversity within populations. Plant Pathology, 52: 533–535. doi: 10.1046/j.1365-3059.2003.00923.x

#### Publication History

- Issue published online: 29 OCT 2003
- Article first published online: 29 OCT 2003
- Accepted 12 June 2003

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### Keywords:

- diversity indices;
- population genetics

### Abstract

It has been established that Nei's measure of the average genetic diversity per locus, *H*_{S}, and the measure of average differences between isolates with respect to simple mismatch dissimilarity, are identical measures of diversity within populations. The Müller index of diversity can be considered as the correction of Nei's measure for small samples.

Various indices are used for analyses and comparisons of diversity within plant pathogen populations. These include Nei's measure of the average gene diversity per locus, *H*_{S} (Nei, 1973); the measure of average differences (dissimilarities) within populations, *ADW* (McCain *et al*., 1992); and the Müller index, *Mu* (Müller *et al*., 1996). The Nei index is the most frequently used in applications, although the average dissimilarity between isolates is also employed quite often (Adhikari *et al*., 1999; Kolmer & Liu, 2000; Menzies *et al*., 2003). Sometimes these two indices are used together (Gale *et al*., 2002) to enhance the significance of conclusions. The objective of this short communication is to prove that the three above-mentioned indices are actually the same measure of diversity within populations in the case of binary data. This statement also holds true in the case of multiple states of multiallelic loci. The proof for multiallelic loci is similar to that for binary data, but is rather lengthy – it is not included in this letter, but is available from the author on request.

Consider a sample from population *P*, which consists of *n* individuals tested on *k* differentiating factors and represented by binary patterns. The frequency of appearance 1 at the *s*th differentiating factor for population *P* is denoted by *q*_{s}. For example, if the differentiating factors represent a typical set of differential host lines used in virulence tests, *q*_{s} would be the frequency of virulence in population *P* on the *s*th differential line.

The indices of average differences within populations depend on the measure of dissimilarity between isolates. Consider the simple mismatch coefficient of dissimilarity

- (1)

where *x*_{i} and *x*_{j} are isolates from population *P*(*i*, *j* = 1, … , *n*) and *d*(*x*_{i}, *x*_{j}) is the number of characters for which two isolates *x*_{i} and *x*_{j} respond differently. The index of average differences with respect to the simple mismatch coefficient *m* is defined as:

- (2)

The product of *ADW*_{m} by the number of differentiating characters *k* is also a usable index (Table 1 in Gale *et al*., 2002):

- (3)

This is the average number of characters for which two arbitrary isolates from population *P* respond differently.

Müller's mean dissimilarity index *Mu* (Müller *et al*. (1996) was defined as the mean number of virulence loci differences between all pairs of different isolates in a sample. Its normalized version (dividing by the total number of loci) has the form:

- (4)

where 1 ≤ *i* < *j* ≤ *n*, that is, only one of the pairs of isolates (*x*_{i}, *x*_{j}) and (*x*_{j}, *x*_{i}) is considered, and [*n*(*n* − 1)]/2 is the total number of such pairs.

The Müller diversity *Mu* within population *P* can be expressed by the index of average differences within population *ADW*_{m} with respect to the simple mismatch dissimilarity:

- (5)

Nei's measure of the average gene diversity per locus *H*_{S} (Nei, 1973) is determined by the formula:

- (6)

where *k* is the total number of loci (differentiating factors), *H*_{Ss} = 1 − − (1 − *q*_{s})^{2}, and *q*_{s} is the frequency of one of the two alleles at the *s*th diallelic locus (or virulence frequency, or band frequency, or frequency of appearance 1 at the *s*th differentiating factor).

It was mentioned by Manisterski *et al*. (2000) that the Müller index is a function of virulence frequencies, and the corresponding formula was presented. It will be demonstrated here that the measure of average differences with respect to the simple mismatch coefficient *ADW*_{m} equals the Nei gene diversity parameter *H*_{S}, and the Müller index *Mu* could be considered as the correction of Nei's measure *H*_{S} for small samples.

Dissimilarity *d*_{s} between two isolates *x*_{i} and *x*_{j} with regard to any character *s* can be measured as follows: *d*_{s}(*x*_{i}, *x*_{j}) = 0 if *x*_{i} and *x*_{j} respond identically on *s*, and *d*_{s}(*x*_{i}, *x*_{j}) = 1 if *x*_{i} and *x*_{j} respond differently on *s*. The dissimilarity *d* between these isolates equals to the following sum:

- (7)

Therefore:

- (8)

There are *nq*_{s} and *n*(*1 − q*_{s}) isolates with positive (1) and negative (0) responses, respectively, on the differential *s*. Then (*nq*_{s})^{2} pairs of isolates respond positively on *s*, [*n*(*1 − q*_{s})]^{2} pairs of isolates respond negatively on *s*, and *n*^{2} *–* (*nq*_{s})^{2} *–*[*n*(1 *– q*_{s})]^{2} = 2*n*^{2}*q*_{s}(1 − *q*_{s}) pairs of isolates respond differently on *s*, where *n*^{2} is the total number of pairs. Thus:

- (9)

and the following equalities are fulfilled:

- (10)

This means that Nei's measure of the average gene diversity per locus, *H*_{S}, and the index of average differences with respect to the simple mismatch coefficient, are identical measures of diversity within populations. In the case of diallelic loci (binary data) the maximum value of the *H*_{S} and *ADW*_{m} indices equals 0·5. The Müller index *Mu* could be considered as the correction of Nei's measure *H*_{S} for small samples because:

- (11)

A more accurate unbiased estimate of *H*_{S} for a small sample size is given by the formula:

- (12)

(Nei, 1978).

### Example

Let population *P* consist of six isolates (*n* = 6) tested on eight differentials (*k* = 8):

s | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

x_{1} | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |

x_{2} | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 |

x_{3} | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 |

x_{4} | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |

x_{5} | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |

x_{6} | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 |

q_{s} | 2/3 | 1 | 5/6 | 2/3 | 1/2 | 1/3 | 1/6 | 1/6 |

H_{Ss} | 4/9 | 0 | 5/18 | 4/9 | 1/2 | 4/9 | 5/18 | 5/18 |

The Nei index scores the following value of within-population diversity:

- (13)

The matrix of dissimilarities between isolates according to the simple mismatch coefficient *m* has the following form:

m | x_{1} | x_{2} | x_{3} | x_{4} | x_{5} | x_{6} |

x_{1} | 0 | 1/2 | 1/8 | 3/8 | 1/8 | 3/8 |

x_{2} | 1/2 | 0 | 3/8 | 1/8 | 5/8 | 7/8 |

x_{3} | 1/8 | 3/8 | 0 | 1/4 | 1/4 | 1/2 |

x_{4} | 3/8 | 1/8 | 1/4 | 0 | 1/2 | 3/4 |

x_{5} | 1/8 | 5/8 | 1/4 | 1/2 | 0 | 1/4 |

x_{6} | 3/8 | 7/8 | 1/2 | 3/4 | 1/4 | 0 |

The index of average differences with respect to the simple mismatch coefficient equals the sum of all entries of the dissimilarity matrix divided by the total number of entries:

- (14)

and the Müller index equals the sum of all entries above the diagonal of the dissimilarity matrix divided by the number of entries above the diagonal:

- (15)

Thus the results obtained by Nei's measure of the average gene diversity per locus, *H*_{S}, the measure of average differences within population, *ADW*_{m}, and the Müller index are absolutely comparable. The simultaneous application of these measures is unnecessary.

### Acknowledgements

This study was partially supported by the Leiberman–Okinow foundation.

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