## Statistical approach for dealing with uncertain paternity

##### Abstract

A new method of implementing mixed model equations without constructing the inverse of the relationship matrix (WO-A) was developed and compared to the classical best linear unbiased prediction implementation where the inverse of the relationship matrix was constructed (W-A). Both methods were compared in a univariate and multiple-trait situation. For the univariate case, the Pearson correlation between estimates from both methods was 1.00 for fixed and random effects. Similar to the univariate case, parameter estimates were exactly the same using W-A and WO-A for the three traits. These results indicate that W-A could be implemented without the need to construct or to invert the relationship matrix. The second objective was to investigate the implementation of a genetic evaluation when the additive relationship matrix is not completely known due to the presence of uncertain paternity in the pedigree. The average probability of the true sire being identified as such (PSA) and the percentage difference (PD) between PSA and an equal prior probability assigned to each candidate sire were computed. As expected, PSA increased with increasing heritability using one trait. The PD was increased by 50 to 386% when repeated records were used compared with using just one record per animal across varying heritabilities. Using three correlated traits increased PD by 77 to 1,021% when compared with using one record of a trait with varying heritability. The results suggested that the probability of identifying the true sire was the highest when three correlated traits were used to compute PSA and the lowest when only one record was used. The effectiveness of paternity assignments based on 15 markers prior to and after inclusion of phenotypic information was evaluated using simulated data. The molecular information, on average, resolved 94% of the uncertain paternity cases. For the remaining 6%, more than one candidate sires was not excluded based on the marker information. The standardized probability of paternity for each non-excluded candidate sire was used as prior information together with the phenotypic information to discriminate between the non-excluded sires. The prior probability based on marker information was 0.58 and 0.42 for true and non-true candidate sires, respectively. When combined with the phenotypic information, the posterior probability of paternity for the true sires increased to nearly 0.80. Using phenotypic information reduced the posterior probability of paternity of non-true sires by 50% to nearly 0.21. Thus, the true sire was correctly assigned 80% of the time using the phenotypic information when molecular information was not able to unambiguously assign paternity.