Henshall et al. Genetics Selection Evolution 2010, 42:34 http://www.gsejournal.org/content/42/1/34 studs, with one outside ram and four home-bred rams used by each stud each year. The intention behind this relatively complicated structure was to simulate popula- tions in which cohorts of ewes were related, with varia- tions in the frequencies of the various Agouti alleles between studs. Parents and progeny of the last genera- tion were assumed to be available for assay. Using the known simulated copy number count, an assay value was simulated using the means and standard deviations described above. The whole simulation was repeated 20 times, producing 1,000 half-sib families (20 replicates × 10 studs × 5 rams) for analysis. The distributions of allele frequencies in the final generations of the simu- lated populations are displayed in Table 4. Relationships between assayed individuals In our simulation study, we performed the assay on half-sib groups of animals and their parents, ignoring relationships between parents. This is equivalent to “cut- ting” inbreeding loops in a relatively naive but systema- tic way, rather than the more sophisticated approaches such as in [25]. We chose this simple approach for a number of reasons, but primarily because we performed many analyses on replicated datasets and therefore needed a fast execution time. However, we believe that the design is justified. In commercial and stud sheep flocks, large half-sib families are usual and although dee- per pedigree data is usually available (especially on the male side), it may not be without errors. In half-sib families, most of the power in the method comes from the confidence with which the sire’s genotype can be estimated, which then adds confidence to estimates for progeny. Software and analysis Genotype probabilities were estimated using a restricted implementation of the Elston-Stewart algorithm [22], restricted in that it operated only on half-sib families. This restriction allowed a very fast execution time enabling the investigation of a wide range of scenarios. The software was validated by comparing the estimated probabilities for test half-sib families to probabilities estimated using the software package Mendel [29]. The allele frequencies assumed for the parents are those Page 4 of 8 shown in Table 2B, that is, in the analysis we used the frequencies that were used in the simulation. Each data- set was evaluated six times, with one of the three for- mulas for the standard deviation of the assay (CV = 3%, 6%, 9%) used to simulate the assay values, and either the correct CV or one half of the correct CV used for the analysis. From the final generation of the simulated populations, families of 1, 2, 4, 8, 16 or 32 half-sib pro- geny were chosen. Assay results were made available for either progeny only, progeny and sire, or progeny, sire and dams. Genotype probabilities were estimated and compared to the simulated copy number alleles. Results Table 5 contains the results for a lamb analyzed without pedigree data. This is not quite the same as relying on assay results alone to estimate the genotype, as popula- tion allele frequencies are still used in the analysis. Without these there is no power at all to declare an ani- mal non-carrier. In all cases, when a lamb was declared to be a carrier with greater than 95% certainty it had the genotype Ac1/Ac2 and in all cases when a lamb had the genotype Ac1/Ac2 it was declared to be a carrier with over 99.9% certainty. The effect of jointly analyzing half-sib families of vary- ing size is displayed in Table 6. The analyses correspond to a situation where the sire and his half-sib progeny are assayed, and the coefficient of variation (CV) of the assay output is 3%. The most important application of a test in an industrial genetic improvement program is the identification of sheep that are non-carriers. We found that with a family size of one, only 46% of non- carriers were identified as non-carriers with a 99% prob- ability. However, this value is almost twice that achieved without genotyping the sire, as in Table 5. Increasing the family size improved the power, and with family sizes of 16, 75% of non-carriers can be declared clear with a 99% probability. Increasing the family size beyond 16 has only a small effect on power, unless the goal is to achieve 99.9% probabilities of being clear. False positives (declaring non-carriers to be carriers) are absent and false negatives (declaring carriers to be non-carriers) are less frequent as family size increases. This is an important result since in industrial Table 4 Frequencies of alleles in the final generation of the simulated populations Frequency Allele Ac1 Ac2 Ac3 Ac4 Ac5 Ac6 0.0:0.1 0.1:0.2 0.2:0.3 0.3:0.4 0.4:0.5 0.5:0.6 0.6:0.7 0.7:0.8 0.8:0.9 0.79 0.07 0.04 0.89 1.00 0.96 0.05 0.14 0.13 0.04 0.00 0.01 0.08 0.14 0.14 0.03 0.00 0.02 0.08 0.11 0.10 0.03 0.00 0.01 0.01 0.18 0.19 0.00 0.00 0.00 0.00 0.16 0.16 0.00 0.00 0.00 0.00 0.07 0.07 0.00 0.00 0.00 0.00 0.08 0.11 0.00 0.00 0.00 0.00 0.04 0.06 0.00 0.00 0.00