Genomic selection and Threshold Validation

Important measures in genetic evaluations

• Accuracy
  
• Bias
  
• dispersion

Accuracy

This is the correlation between the estimated breeding value \((EBV)\) and the true breeding value \((TBV)\). values near 1 indicate strong associations while values near zero indicate weak associations. Weak associations imply that the animals are poorly ranked and selection decisions will be undesirable.

Bias

Bias is the difference between the average prediction and the average true value. It is desirable that the bias is zero. Biased estimates lead to incorrect comparisons between animals of different generations and inaccurate estimates of genetic trends.

Dispersion

This is the slope of the regression of true breeding value and the estimated breeding value. It can also be refered to as the predicted bias and it has an expected value of one. Lower than one means that the mean prediction of the test animals is biased downwards and greater than one means that the mean is biased upwards.

Validation of genetic and genomic models using the LR method

Let \(u\) and \(\hat{u}\) be the vectors of \(TBV\) and \(EBVs\) respectively, the the (true) bias is defined as \(\bar{u}-\bar{\hat{u}}\).
The (true) dispersion, interpreted as the slope of the regression of \(u\) on \(\hat{u}\) is equal to: \[ \frac{cov(\mathbf{u,\hat u})} {var(\mathbf{\hat u})} \] , where cov and var denote the sample covariance and variance respectively.
The accuracy is defined as the sample Pearson correlation coefficient between \(u\) and \(\hat u\), which is equal to: \[ \frac{cov(\mathbf{u,\hat{u}})} {\sqrt{var(\mathbf{u})var(\mathbf{\hat{u}})}} \] While using the LR method, the dataset is split into two i.e. the whole data set containing all the phenotyopes and the partial data set that contains phenotypes upto a certain date. The individuals of focus in this case will be the young ones that might be selected at a given time given early information(partial).
The partial data set is interpreted as the evaluation at the time of selection and the whole data set the posterior confirmation of the goodness of these selections decisions.
The LR method suugestes the following statistics: (a) \[\mu_{wp} = \mathbf{\hat{u_p}-\hat{u_w}}\] for the bias with the value of zero if the evaluation is unbiased. (b)\[b_{w,p}=\frac{cov(\mathbf{\hat u_w,\hat u_p})} {var(\mathbf{\hat u_p})}\] for the slope of the regression of the \(EBVs\) computed with the whole data set on the \(EBVs\) estimated using the partial data set. The value of one indicating neither underdispersion nor overdispersion. (c)\[\rho^2_{cov(w,p)}=\frac{cov(\mathbf{\hat u_w,\hat u_p})} {(1+\bar{F}-2\bar f)\sigma^2_{u,\infty}}\] for the reliability (square of accuracy), where \(\bar F\) is the average inbreeding coefficient, \(2\bar f\) is the average relationship between individuals, and \(\sigma^2_{u,\infty}\) is the genetic variance of the validation individuals.
Additionally:
\(\rho_{w,p}=cor(\mathbf{\hat u}_w,\mathbf{\hat u}_p)\) estimates the ration between the accuracies obtained with partial and whole data sets i.e \(\frac{acc_p}{acc_w}\).
\(\rho_{A,G}=corr(\mathbf{\hat u}_A ,\mathbf{\hat u}_G)\), this estimates the ration between the accuracies obtained by pedigree-based and genomic-based evaluations in the partial data sets i.e \(\frac{acc_A}{acc_G}\).
The former can be expressed as the relative increase in accuracy by adding phenotypic information \((inc_G=\rho^{-1}_{A,G} -1)\).
The latter can be expressed as the increase in accuracy by adding genomic information \((inc_{Phen}=\rho^{-1}_{w,p} -1)\).