Foss et al. (1993) offered a simple, one-parameter model for interference which effectively described data from Drosophila and Neurospora. The model supposed that recombinational interactions (C's) were distributed without interference. Any single interaction could lead to conversion with crossing over (Cx) or to conversion without crossing over (Co). These two kinds of outcomes ("resolutions") were effected according to the rule that a fixed number (m) of Co's falls between each pair of adjacent Cx's.

The fit of the model to interference data was especially impressive because the single adjustable parameter, m, was not selected to give a best fit to the published data on interference. Instead, it was estimated from the entirely independent observation of the fraction of all gene conversions that were accompanied by crossing over (Cx/C). The best reported values for Neurospora and Drosophila were 1/3 and 1/5, respectively, implying m = 2 for Neurospora and 4 for Drosophila. These values of m were then seen to give optimal fits to interference data (Foss et al. 1993; Lande and Stahl 1993; Stahl and Lande 1995; Zhao et al. 1995).

Because the model is a simple variation on the Poisson distribution, it is formalized by simple algebra. The mathematical attractions of a model that postulated Poisson-distributed "attempts" and then alternated "successes" with runs of "failures" of fixed number ("counting models") had been noted earlier (for relatively recent treatments, see Cobbs 1978 and Stam 1979). Despite the fact that Cobbs (1978) demonstrated the superior ability of the model to describe Neurospora and Drosophila data, the models have not been widely adopted. Foss et al. (1993) and Lande and Stahl (1993) added to the value of such models by simplifying the algebra and by showing more fully that it was an accurate describer of existing interference data. Subsequently, analysis of extant models by Speed's group (Zhao et al. 1995; McPeek and Speed 1995) demonstrated that the counting model provides a superior description of interference data from Drosophila.

One of the models examined by Speed's group was that of Barratt et al. (1954), which has been widely employed by fungal geneticists for estimating interference. The Barratt model offered an interference parameter (k), which, unlike the coefficient of coincidence, was independent of the lengths of the intervals that happened to be involved in the cross. This was an important conceptual advance because it offered the promise of liberating the estimator of interference (k) from the particulars of the cross used to measure it, as does m in the model of Foss et al. (1993). Snow (1979) provided a maximum likelihood method for the estimation of k from linkage data.

As pointed out by the Speed group, however, the model of Barratt et al. is unrealistic in that it fails to predict the observed dependence of the coefficient of coincidence on the distance between the two intervals used to measure the coefficient. In conflict with abundant data, the model predicts that the coefficient of coincidence is independent of the distance between the intervals concerned, depending only on the length of those two intervals. This basic failure of the model of Barratt et al. may not have been widely appreciated.

In order to facilitate the replacement of the Barratt model by the counting model (e.g., Foss et al.1993), we offer here a computer program for the estimation of m and of its statistical confidence limits. Because most investigations that attempt to reveal the mechanism of interference are currently being conducted in fungi, and for the additional reasons laid out by Stahl and Lande (1995), our program addresses the estimation of m (and of map distance) from two-factor tetrad data that measure the frequencies of nonparental ditype and tetratype tetrads in a tetrad sample of known total size. In so doing, it assumes the absence of chromatid interference, an assessment of which requires crosses involving three linked factors. The computerized procedure offered here is based entirely on the analysis by Stahl and Lande (1995).

The usefulness of the model of Foss et al. (1993) as a description of interference in Drosophila, which is well established, is independent of any assumptions regarding its biological basis, i.e., the distribution of crossovers along the chromosome is well described whether or not that distribution is determined by imposing, between each of two neighboring crossovers, a fixed number of recombination intermediates resolved without crossing over. Just how useful the counting model will prove to be for other organisms remains to be seen. Nevertheless, the model has advantages over current methods (like "NPD ratios") for quantifying interference, especially for comparisons of interference between crosses in which the genetic map lengths of the test interval(s) are different from each other.

On the NPD Ratio and the "Better Way"
Papazian's equation tests for interference by using only a fraction of the available data (the observed frequency of TTs). The resulting inefficiency, which is minimal at small values ofƒN_obs, increases with increasing ƒN_obs. The "Better Way" is a straight-forward, exact method which is free of this shortcoming. Because the Papazian-based method ascribes all of the deviation to two of the three classes, a Chi-Square statistic based on the sum of the squares of those deviations, is exaggerated. Consequently the p values for the Better Way will be greater than those for the Papazian-based method, implying that a Chi-Square test based on the Papazian method will more often falsely claim interference.

LITERATURE CITED

Barratt, R. W., D. Newmeyer, D. D. Perkins, and L. Garnjobst, 1954 Map construction in Neurospora crassa. Advances Genet. 6: 1-93.

Cobbs, G., 1978 Renewal process approach to the theory of genetic linkage: case of no chromatid interference. Genetics 89: 563-581.

Foss, E. and F. W. Stahl, 1995 A test of the counting model for chiasma interference. Genetics 139: 1201-1209.

Foss, E., R. Lande, F. W. Stahl and C. M. Steinberg, 1993 Chiasma interference as a function of genetic distance. Genetics 133: 681-691.

Lande, R., and F. W. Stahl, 1993 Chiasma interference and the distribution of exchanges in Drosophila melanogaster. Cold Spring Harbor Symp. Quant. Biol. 58: 543-552.

McPeek, M. S., and T. P. Speed, 1995 Modeling interference in genetic recombination. Genetics 139: 1031-1044.

Snow, R., 1979 Maximum likelihood estimation of linkage and interference from tetrad data. Genetics 92: 231-245.

Stahl, F. W., and R. Lande, 1995 Estimating interference and linkage map distance from two-factor tetrad data. Genetics 139: 1449-1454.

Stam, P., 1979 Interference in genetic crossing over and chromosome mapping. Genetics 92: 573-594.

Zhao, H., T. P. Speed and M. S. McPeek, 1995 Statistical analysis of crossover interference using the chi-square model. Genetics 139: 1045-1056.