Letter to the Editor

The Test Distribution of Modularity Statistics: A Correction and a Clarification

Corresponding author: Günter P. Wagner, Department of Ecology and Evolutionary Biology, Yale University, POB 208106, New Haven, CT 06520-8106., gunter.wagner{at}yale.edu (E-mail)

In a recent article (MEZEY et al. 2000 ) three of us (J. G. Mezey, J. M. Cheverud, and G. P. Wagner) proposed two statistics for testing the hypothesis that the genotype-phenotype map is modular. Specifically, the hypothesis tested is that the effects of mutations are more prevalent among characters that serve a common function while less frequent among characters that belong to different functional groups. We proposed a randomization technique to estimate the test distribution for the modularity statistics. E. Housworth discovered that in describing the technique we made one factual error and failed to fully explain the rationale behind the randomization strategy. In this letter we wish to correct the error and clarify the randomization technique proposed.

One of the statistics, M_P, a measure for the parcellation of a set of characters relative to the rest of the phenotype, is calculated from a 2 by n frequency table. In this table n is the number of quantitative trait loci (QTL) influencing at least two characters and the two columns stand for the set of characters to be tested, T₁ and its complement T₂. The entries in this table are the number of characters a QTL affects in each of the two sets of characters. From this table M_P is calculated exactly like a chi-square statistic. The idea is to test for an association between QTL effects and character sets. We claimed that the test distribution for this statistic is approximately chi-square with 2n degrees of freedom. The latter statement is wrong. The correct degrees of freedom would be n - 1 and we apologize for this error. The previous statement about approximate chi-square distribution is misleading since we subsequently do not use a randomization scheme that is known to approximate a chi-square distribution.

Rationale for the randomization strategy:
The aim of the proposed statistic is to test for a specific feature of the genotype-phenotype map, modularity. We thus seek to use a randomization technique that keeps other aspects of the genetic architecture constant and randomizes only the association between QTL and character sets. The aspects of the QTL data that we did not want to randomize are the distribution of pleiotropy among genes and the polygeny among characters. In other words, we want to keep the number of characters affected by each QTL constant (pleiotropy distribution) as well as the number of QTL that affect each character (polygeny distribution). To honor these constraints we randomized the columns of an n by N incidence matrix in which rows represent the n QTL and the columns the N characters. The entries in the matrix are 0's and 1's, indicating whether a QTL affects a particular character or not. The resulting test distribution is less likely to reject the null hypothesis of no modularity than a chi-square distribution. The chi-square test randomizes the entries in each row of the incidence matrix and thus does not honor the constraints of constant pleiotropy and polygeny.

LITERATURE CITED

MEZEY, J. G., J. M. CHEVERUD, and G. P. WAGNER, 2000  Is the genotype-phenotype map modular? A statistical approach using mouse quantitative trait loci data. Genetics 156:305-311[Abstract/Full Text].