# Connectedness

The use of BLUP is not a panacea for genetic evaluation. For an across-herd evaluation to be robust, animals in the different herds must either be tied together directly (through relatives) or indirectly through contemporaries. Such is particularly so when herds differ in their mean genetic merit, which is to be anticipated. Contemporaries are usually progeny of sires used widely within a breed. The closer these ties or connectedness, the more reliable the genetic evaluation.

The accuracy of an Expected Progeny Difference (EPD) itself is one item that affects genetic response. However, the level of precision (or error) when comparing EPD of two animals in different herds is equally important and often receives less attention. Because such between herd comparisons often determine which animals are actually chosen, accurate comparisons leading to the right choices have clear impact on genetic progress.

Connectedness affects the quality of selection decisions in two ways: the accuracy of the EPD itself and the precision of the comparison of EPD. As connectedness improves, error in estimating and comparing progeny differences falls. The variance, or the mean squared error, of prediction of the difference between EPD of different animals has therefore been proposed as an appropriate measure of connectedness. Several connectedness statistics consequently have been based on the premise of prediction error variance.

## Prediction error variance and connectedness

The prediction error variance (PEV) of an individual EPD can be obtained from the diagonal elements of the inverse of the coefficient matrix of the mixed-model equations (MME) (in the so called "R" form). Several measures of genetic connectedness have been constructed based on PEV. Kennedy and Trus (1993) proposed using the PEV of the difference between EPD of a pair of animals, $d_{ij}$. The $d_{ij}$ for any two animals, i and j, can also be obtained from functions of i and j's PEV, and their prediction error covariance obtained from the intersecting off-diagonal of the inverse of the MME, $d_{ij}=PEV(\hat{u}_{i}-\hat{u}_{j})=PEV(\hat{u}_{i})+PEV(\hat{u}_{j})-2PEC(\hat{u}_{i},\hat{u}_{j})$

where PEC is the prediction error covariance of the $\hat{u}$ (2 times the EPD). If $d_{ij}$ is small, individuals are said to be connected. Such is the case when EPD are accurately estimated since their PEV are then small, and when individuals in different management units (e.g., herds) are related as their PEC are then positive.

Laloë (1993) proposed the Coefficient of Determination of the difference between EPD of a pair of animals as an alternative measure of connectedness. It differs from $d_{ij}$ by dividing it by a function of the diagonal and off-diagonal elements of the relationship matrix and subtracting that quantity from one. This statistic accounts for reduced variability in true breeding values due to relationships. This value ranges from 0 to 1, with larger values indicating increased connectedness.

Recognizing that the PEC between two animals’ EPDs would be zero if they were not connected, Lewis et al. (1999) proposed the correlation of breeding value prediction errors as a further connectedness statistic. This correlation is $r_{ij}=\frac{PEC(\hat{u}_{i},\hat{u}_{j})}{\sqrt{PEV(\hat{u}_{i})PEV(\hat{u}_{j})}}$

which is also bounded between 0 and 1, with larger values indicating increased connectedness. Where animals are in different management units, values of $r_{ij}$ near zero indicate those units are largely disconnected with comparisons of EPD across management units considered risky.

These statistics have been used to evaluate connectedness in various simulated (e.g., Kennedy and Trus, 1993; Hanocq et al., 1996; Kuehn et al., 2008) and field (e.g., Kuehn et al., 2009; Eikje and Lewis, 2015) data where relationships were based on pedigree. Conclusions were consistent. As relationships between management units strengthened, connectedness of animals within them also improved. Yu et al. (2017), using simulation, demonstrated that genomic relatedness across management units also strengthened connectedness. The coefficient of determination statistic of Laloë (1993), however, behaved most consistently as genomic relatedness increased.

## Sufficiency

As noted by Kennedy and Trus (1993), connectedness itself is not the major issue per se but instead how it contributes to PEV. Sufficient rather than maximal connectedness that ensures comparisons across management units are robust should be the priority.

Lewis et al. (2005) and Kuehn et al. (2007, 2008) proposed a measure of sufficient connectedness based on the contrast among true and expected progeny differences. The average squared contrast measures the mean square error, which is the sum of PEV and squared bias.

With the limited use of sires in common, adequate links were established such that the mean square error quickly decreased to an asymptotic value. This decrease was due to a reduction in bias when comparing EPD across management units. When the average $r_{ij}$ between animals in different units reached 0.05 and 0.10, approximately 20 and 10%, respectively, of the initial bias remained. These values have been suggested as benchmark levels for superior (0.05) and good (0.10) connectedness for reducing the risk of comparing animal EPD among units (Kuehn et al., 2008).

## Summary

Genetic connectedness affects the quality of selection decisions. As connectedness improves, error in estimating and comparing progeny differences across herds falls, particularly where the genetic mean of those herds differ. However, maximizing connectedness is not a goal in its own right. With limited yet systematic exchange of sires among herds, adequate genetic links can be established. Such connections are most easily and efficiently established through wide use of artificial insemination. Monitoring of connectedness in across-herd genetic evaluations may be warranted to confirm that connectedness is sufficiently strong to reduce risk in across-herd selection decisions.