GAMETOPHYTIC SELF‐FERTILIZATION IN HOMOSPOROUS PLANTS: DEVELOPMENT, EVALUATION, AND APPLICATION OF A STATISTICAL METHOD FOR EVALUATING ITS IMPORTANCE

The methods described here make it possible to use data on sporophytic genotype frequencies to estimate the frequency of gametophytic self-fertilization in populations of homosporous plants. Bootstrap bias reduction is effective in reducing or eliminating the bias of the maximum likelihood estimate of the gametophytic selfing rate. The bias-corrected percentile method provides the most reliable confidence intervals for allele frequencies. The percentile method gives the most reliable confidence intervals for the gametophytic selfing rate when selfing is common. The maximum likelihood intervals, the percentile intervals, the bias-corrected percentile intervals, and the bootstrap t intervals are all overly conservative in their construction ofconfidence intervals for the gametophytic selfing rate when self-fertilization is rare. Application of the recommended methods indicates that gametophytic self-fertilization is quite rare in two sexually reproducing populations of Pellaea andromedifolia studied by Gastony and Gottlieb (1985). DATA ON THE GENETIC structure of populations of vascular cryptogams is gradually beginning to accumulate. Levin and Crepet (1973) published electrophoretic data on populations of Lycopodium lucidulum, although they did not determine the genetic basis of the banding patterns observed. Haufler and Soltis (1984) used electrophoretic data from natural populations of Bommeria hispida to confirm the prediction from laboratory experiments that it is an obligate outcrosser. Gastony and Gottlieb (1985) reported the results of an electrophoretic survey of five polymorphic allozyme loci in nine sexually reproducing populations of the facultatively apogamous fern Pellaea andromedifolia. McCauley, Whittier, and Reilly (1985) recently provided estimates of the gametophytic selfing rate in Botrychium dissectum. Soltis and Soltis (1986 and personal communication) have conducted large population surveys of species in the fern genera Botrychiurn, Polystichum, and Dryopteris. Part of the interest of these studies is that 'Received for publication 15 April 1986; revision accepted 5 December 1986. 2 Current address is Department of Ecology and Evolutionary Biology, U-43, University of Connecticut, 75 North Eagleville Road, Storrs, CT 06268. I am indebted to Gerald Gastony for piquing my interest in this topic and encouraging me to pursue it. Marc Feldman, Doug Soltis, and Pam Soltis provided helpful comments on an earlier draft of this paper. A listing (in Pascal) of a program that performs the procedures described here is available upon request. A compiled version is available for IBM PCs and PC-compatibles. This research was supported in part by NIH grants GM28016 and GM 10452. they allow us to determine how frequently selffertilization of gametophytes occurs in natural populations of homosporous plants. In particular, it may be possible to evaluate one aspect of the hypothesis that gametophytes of homosporous ferns frequently self-fertilize in nature and that they compensate for the consequent complete homozygosity by storing and releasing heterozygosity via duplicated loci on homoeologous polyploid chromosome sets (Klekowski and Baker, 1966; Klekowski, 1973; Chapman, Klekowski, and Selander, 1979). [Genetic evidence already indicates that many presumed polyploids among the homosporous ferns are genetically diploid (Haufler and Soltis, 1986).] Since gametophytic self-fertilization will result in complete homozygosity at all loci, Gastony and Gottlieb (1985) were able to demonstrate that a minimum of 81.3% of the sporophytes they examined were the result of outcrossing events. This estimate is clearly a lower bound, however, since outcrossed matings could occur between different individuals that have identical genotypes at the loci they sampled. A variety of statistical techniques for estimating rates of self-fertilization has been developed in seed plants. (See Clegg, 1980, or Ritland, 1983, for recent reviews and references.) These techniques are all based on the analysis of progeny arrays, i.e., the analysis of genotype frequencies within a set of halfor full-sib progeny arrays drawn from different mothers. Ferns and other vascular cryptogams are not susceptible to such an analytical tech-