In the article by Sudhir Kumar and Sudhindra R. Gadagkar (Genetics **158:** 1321–1327) entitled “Disparity Index: A Simple Statistic to Measure and Test the Homogeneity of Substitution Patterns Between Molecular Sequences,” the analytical proof on pages 1321–1322 showing that *E*(*D*_{C}) = *E*(*N*_{d}) is now rewritten (Equations 1–10) as follows. This correction does not affect any of the results or conclusions of the article. In fact, the new proof now extends the equality in Equation 10 even when there is *among-site* (i) heterogeneity in substitution rates (see also Figure 1 in the original article), (ii) correlation in evolutionary rates/patterns, and (iii) variation in substitution patterns.

Let *X* and *Y* be two aligned DNA sequences of length *L* each. Let *x _{i}* be the count of the

*i*th type of nucleotide (

*i*= A, C, G, T) in sequence

*X*and

*y*be the corresponding count in sequence

_{i}*Y.*The composition distance (

*D*

_{C}) between

*X*and

*Y*is

*D*

_{C}is

*k*in sequence

*X*contains nucleotide

*i*, and 0 otherwise.

Equation 3 can be expressed as
*i*th nucleotide at site *k* in sequence *X*, and *i* at sites *k* and *k*′ in sequence *X.*

Similarly,
*k* in sequence *Y* contains nucleotide *i*, and 0 otherwise. Also, *i*th nucleotide at site *k* in sequence *Y*, and *i* at sites *k* and *k*′ in sequence *Y.*

Furthermore,

Because *L*, the right-hand side in (9) becomes the expected number of differences (*N*_{d}). That is,
*e.g.*, 20 in amino acid sequences). Also note that the probability of identity is specified individually for each site in (9) and summed over all sites. Therefore, we do not need to assume that the substitution pattern is the same among sites or that the evolutionary rate is equal among sites. Furthermore, we do not need to assume site independence because the joint probabilities in Equations 4 and 5 are not required to be expressed as multiples of the individual probabilities to get Equation 9. Therefore, Equation 10 holds true under a variety of biologically realistic conditions and requires only that the evolutionary substitution pattern be homogeneous at individual sites between lineages.

- Copyright © 2001 by the Genetics Society of America