The effect of linkage on sample size determination for multiple trait selection

Sufficient sample sizes are needed in breeding programs to be confident, with a specified probability α, of obtaining a specified number of plants of a desired genotype in segregating populations. We develop a method of determining the minimum sample size needed to produce, with specified probability α, at least m individuals of a desired genotype. This method takes into consideration factors affecting differential selection of gametes, segregation at a single locus, and linkage among the loci of interest. We first consider the effects in the gametophyte (haploid level) of fitness and linkage on the frequencies of alleles at two linked loci, then at three or more linked loci. The probability of obtaining at least m successes, or occurrences of the desired allele, among n gametes is given by a formula based on the binomial distribution. This probability is affected by fitness and linkage through their impact on the probability that a single randomly chosen gamete is of the desired type. Using an extension of this approach, we examine the effects of the altered allelic frequencies on the likelihood of obtaining the desired genotype from a randomly chosen pair of gametes in the sporophyte (diploid level). A table and a figure show the sample size required to produce, with probability 0.95, m individuals of the desired g enotype or phenotype, as a function of m and the probability that a randomly selected individual is of the desired type.