# The effect of the **H**^{−1} scaling factors *τ* and *ω* on the structure of **H** in the single-step procedure

## Abstract

### Background

The single-step covariance matrix **H** combines the pedigree-based relationship matrix \({\mathbf {A}}\) with the more accurate information on realized relatedness of genotyped individuals represented by the genomic relationship matrix \({\mathbf {G}}\). In particular, to improve convergence behavior of iterative approaches and to reduce inflation, two weights \(\tau\) and \(\omega\) have been introduced in the definition of \({\mathbf {H}}^{-1}\), which blend the inverse of a part of \({\mathbf {A}}\) with the inverse of \({\mathbf {G}}\). Since the definition of this blending is based on the equation describing \({\mathbf {H}}^{-1}\), its impact on the structure of \({\mathbf {H}}\) is not obvious. In a joint discussion, we considered the question of the shape of \({\mathbf {H}}\) for non-trivial \(\tau\) and \(\omega\).

### Results

Here, we present the general matrix \({\mathbf {H}}\) as a function of these parameters and discuss its structure and properties. Moreover, we screen for optimal values of \(\tau\) and \(\omega\) with respect to predictive ability, inflation and iterations up to convergence on a well investigated, publicly available wheat data set.

### Conclusion

Our results may help the reader to develop a better understanding for the effects of changes of \(\tau\) and \(\omega\) on the covariance model. In particular, we give theoretical arguments that as a general tendency, inflation will be reduced by increasing \(\tau\) or by decreasing \(\omega\).

## Background

Equation (3) is defined on the level of \({\mathbf {H}_{\tau ,\omega }^{-1}}\), but the effect of the introduction of \(\omega\) and \(\tau\) on the shape of \({\mathbf {H}}\) is not obvious. In particular, breeders aiming at implementing the single-step method in breeding programs raised the question of how these parameters affect the relationship model \({\mathbf {H}}_{\tau ,\omega }\). Here, we present \({\mathbf {H}}_{\tau ,\omega }\) in a general form, as a matrix dependent on \(\tau\) and \(\omega\) and discuss some of its properties. Moreover, we provide arguments for a reduction in inflation of predicted breeding values being expected when \(\tau\) increases or when \(\omega\) decreases. Finally, to set a contrast to the widely used cattle data [12, 13, 16, 17], we screened for optimal values of \(\tau\) and \(\omega\) with respect to predictive ability, inflation and iterations to convergence on a well investigated, publicly available wheat data set [18]. Our results may help to develop an understanding for the effects on the covariance model when these parameters are changed. In particular, this may be of interest for people who aim at implementing the single-step method with non-trivial parameters \(\tau\) and \(\omega\) in practical breeding programs.

## \({\mathbf {H}}_{\tau ,\omega }\) and some particular choices of \(\tau\) and \(\omega\)

We will first describe \({\mathbf {H}}_{\tau ,\omega }\) and discuss some special cases. Mathematical arguments for the presented statements are provided in the “Appendix”. If an inverse of a matrix is used, the implicit assumption on invertibility is made (also if not mentioned explicitly). In particular, \({\mathbf {A}}\) is considered invertible on account of its construction from the pedigree (granted clones are absent) [19].

**Central statement**

The structure of Eq. (4) is identical to that of Eq. (1), but with \({\mathbf {G}}\) substituted by Eq. (5). Considering \({\mathbf {H}}_{22}\), we see that the parameterization of the weights \(\omega\) and \(\tau\) is “reverse” in the sense that \(\tau\) and \(\omega\) appear with opposite signs in front of them. In particular, this implies that \({\mathbf {H}}_{\tau ,\omega }\) is not necessarily positive semi-definite when \(\omega > 1\) since this leads to a negative factor for \({\mathbf {A}}_{22}^{-1}\) and thus has to be compensated by \(\tau {\mathbf {G}}^{-1}\) to give a positive semi-definite matrix. However, positive semi-definiteness of \({\mathbf {H}}_{\tau ,\omega }\) is guaranteed, if \({\mathbf {G}}\) and \({\mathbf {A}}\) are positive definite and \(\tau \ge 0\) and \(\omega \le 1\), but not both at their boundary, that is not \(\tau =0\) and \(\omega =1\) at the same time.

**Lemma 1**

*Let* \({\mathbf {A}}\) *and* \({\mathbf {G}}\) *be positive definite and let* \(\tau \ge 0\) *and* \(\omega \le 1\), *but not* \(\tau =0=1-\omega\). *Then* \({\mathbf {H}}_{\tau ,\omega }\) *is positive semi-definite*.

Note that due to the “reverse parameterization” in form of weights \((1-\omega )\) and \(\tau\) in Eq. (5), the sets of parameter values, which guarantee positive semi-definiteness of the single-step matrix \({\mathbf {H}}_{\tau ,\omega }\), are distinct. If both \(\tau\) and \((1-\omega )\) are positive, then positive semi-definiteness of \({\mathbf {H}}_{\tau ,\omega }\) is guaranteed. In particular, this also means that a negative \(\omega\) gives a valid covariance model. Thus, a grid to test combinations would be rather within \((\tau , \omega ) \in [0,2] \times [-1,1]\) than \((\tau , \omega ) \in [0,2] \times [0,1]\), which has often been the frame for the choice of parameters [13, 16, 17].

- (i)
If \(\tau = \omega = 1\), we are dealing with the original single-step method of Eq. (2).

- (ii)
If \(\tau = \omega = 0\), then \({\mathbf {H}}_{22}= {\mathbf {A}}_{22}\) and thus \({\mathbf {H}}={\mathbf {A}}\).

- (iii)
If \(\omega = \tau = \lambda > 0\), then \({\mathbf {H}}_{22}=\left( \lambda {\mathbf {G}}^{-1} + (1-\lambda ){\mathbf {A}}_{22}^{-1}\right) ^{-1}.\)

- (iv)
If \(\omega =1\), then \({\mathbf {H}}_{22}= \tau ^{-1} {\mathbf {G}}\).

- (v)
If \(\tau =1\), then \({\mathbf {H}}_{22}= \left( {\mathbf {G}}^{-1} + (1-\omega ){\mathbf {A}}_{22}^{-1}\right) ^{-1}.\)

Case (i) is already obvious on the level of \({\mathbf {H}}^{-1}\), but it can also be seen on the level of \({\mathbf {H}}_{\tau ,\omega }\) that Eq. (4) coincides in this case with Eq. (1), since \({\mathbf {H}}_{22}= {\mathbf {G}}\). If instead \(\tau =\omega =0\) as for case (ii) then \({\mathbf {H}}_{0,0} = {\mathbf {A}}\) and the single-step BLUP becomes the traditional pedigree-BLUP. Also note that case (iii), for which \(\tau\) and \(\omega\) are equal, has already been addressed in [3] and results in a weighted harmonic mean of \({\mathbf {G}}\) and \({\mathbf {A}}_{22}\).

In case (iv) in which \(\omega\) is equal to 1, \({\mathbf {H}}_{22}=\tau ^{-1}{\mathbf {G}}\). With increasing \(\tau\), the entries of \({\mathbf {H}}_{12}\), \({\mathbf {H}}_{21}\), \({\mathbf {H}}_{22}\) will shrink towards \({\mathbf {0}}\) and \({\mathbf {H}}_{11}\) to the *Schur complement* \({\mathbf {A}}/{\mathbf {A}}_{22}:={\mathbf {A}}_{11}-{\mathbf {A}}_{12} {\mathbf {A}}_{22}^{-1} {\mathbf {A}}_{21}\).

## The effect of \(\tau\) and \(\omega\) on inflation

A main purpose of the introduction of these parameters is the reduction of inflation of the predicted breeding values [13, 16] which is manifested and diagnosed by a slope \(b < 1\) in a regression of observed values (y-axis) on predictions (x-axis). Please recall here that the regression of observed values on predictions should be preferred to a regression of predictions on observed values for model evaluation [20]. We will argue why—as a general tendency—increasing \(\tau\) or decreasing \(\omega\) may lead to a reduced inflation.

**Lemma 2**

*Let*\({\mathbf {A}}\)

*and*\({\mathbf {G}}\)

*be positive definite and*\({\mathbf {H}}_{\tau ,\omega }\)

*as introduced*.

- (a)
*Let*\(\tau \le \tau '\)*and*\(\omega \ge \omega '\)*be given such that*\(\left( {\mathbf {H}}_{\tau ,\omega } \right) _{22 } \succeq 0 \preceq \left( {\mathbf {H}}_{\tau ',\omega '} \right) _{22 }\) .*Then*$$\begin{aligned} \left( {\mathbf {H}}_{\tau ,\omega } \right) _{22 } \succeq {\left( {\mathbf {H}}_{\tau ',\omega } \right) _{22 }} \succeq \left( {\mathbf {H}}_{\tau ',\omega '} \right) _{22 } \text{ and } \left( {\mathbf {H}}_{\tau ,\omega } \right) _{22 } \succeq {\left( {\mathbf {H}}_{\tau ,\omega '} \right) _{22 }} \succeq \left( {\mathbf {H}}_{\tau ',\omega '} \right) _{22 } \end{aligned}.$$ - (b)
*Moreover*,$$\begin{aligned} \left( {\mathbf {H}}_{\tau ,\omega } \right) _{22 } \succeq \left( {\mathbf {H}}_{\tau ',\omega '} \right) _{22 } \Longleftrightarrow {\mathbf {H}}_{\tau ,\omega } \succeq {\mathbf {H}}_{\tau ',\omega '} \end{aligned}.$$ - (c)
*For two matrices of the shape of the BLUP solution of Eq*. (7)$$\begin{aligned} {\mathbf {K}}_1:=\left( {\mathbf {I}}+ \lambda {\mathbf {H}}_{\tau ,\omega }^{-1} \right) ^{-1} \; \text{ and } \; {\mathbf {K}}_2:=\left( {\mathbf {I}}+ \lambda {\mathbf {H}}_{\tau ',\omega '}^{-1} \right) ^{-1} \end{aligned},$$*with a*\(\lambda > 0\),*we have*$$\begin{aligned} {\mathbf {H}}_{\tau ,\omega } \succeq {\mathbf {H}}_{\tau ',\omega '} \Longleftrightarrow {\mathbf {K}}_1 \succeq {\mathbf {K}}_2 \end{aligned}.$$

Lemma 2(a) illustrates that if we keep \(\tau\) constant and decrease \(\omega\) to \(\omega '\), the resulting matrix \(\left( {\mathbf {H}}_{\tau ,\omega '} \right) _{22}\) will be “smaller” with respect to the Löwner order. The same is true if we keep \(\omega\) constant and increase \(\tau\) to \(\tau '\). Part (b) transfers this observation to the level of \({\mathbf {H}}_{\tau ,\omega }\). Finally, part (c) connects \({\mathbf {H}}_{\tau ,\omega }\) with the BLUP of model (6).

We now illustrate how this reduction with respect to the Löwner order, transfers to the variance of breeding value estimates \({\hat{\mathbf {g}}}\) in this simple model of \({\hat{\mathbf {g}}}:=\left( {\mathbf {I}}+ \lambda {\mathbf {H}}_{\tau ,\omega }^{-1} \right) ^{-1} {\mathbf {y}}\).

**Proposition 1**

*Let*\({\mathbf {K}}_1\succeq {\mathbf {K}}_2\), \({\mathbf {K}}_1 {\mathbf {K}}_1\succeq {\mathbf {K}}_2 {\mathbf {K}}_2\),

*and let*\({\hat{\mathbf {g}}}_i := {\mathbf {K}}_i {\mathbf {y}}\)

*be the corresponding estimate of the breeding values. Moreover, let the empirical mean of both estimates be the same*\({\mathbf {E}}({\hat{\mathbf {g}}}_1)={\mathbf {E}}({\hat{\mathbf {g}}}_2)\)

*and let*\({\text {Var}}({\hat{\mathbf {g}}}_i)\)

*denote the empirical variance of the vector*\({\hat{\mathbf {g}}}_i\),

*defined by*

Proposition 1 illustrates that an important effect of using an \(\omega\) smaller than 1, or a \(\tau\) larger than 1 may be the reduction of the variance of the predicted genetic values. To see this, recall that Lemma 2(a) and (b) stated that reducing \(\omega\) to \(\omega '\) and keeping \(\tau\) fixed implies \({\mathbf {H}}_{\tau ,\omega } \succeq {\mathbf {H}}_{\tau ,\omega '}\). The same is true for increasing \(\tau\) to \(\tau '\) with fixed \(\omega\). Lemma 2(c) then implies that \({\mathbf {K}}_1 \succeq {\mathbf {K}}_2\). Thus, provided that all preconditions are given, Proposition 1 states that the variance of the estimated breeding values is reduced.

The critical assumption is \({\mathbf {K}}_1{\mathbf {K}}_1\succeq {\mathbf {K}}_2{\mathbf {K}}_2\), since this is not implied by \({\mathbf {K}}_1 \succeq {\mathbf {K}}_2\) (for a counter example see [24]). Thus, this will not be totally satisfied in practice. Instead, because we are dealing with a partial order, often neither \({\mathbf {K}}_1{\mathbf {K}}_1\succeq {\mathbf {K}}_2{\mathbf {K}}_2\) nor \({\mathbf {K}}_2{\mathbf {K}}_2\succeq {\mathbf {K}}_1{\mathbf {K}}_1\) will hold, but the difference of the two products may result in an indefinite matrix (i.e. one with both positive and negative eigenvalues). However, if only a few eigenvalues of the difference \({\mathbf {K}}_1{\mathbf {K}}_1 - {\mathbf {K}}_2{\mathbf {K}}_2\) are smaller than zero, this assumption will be correct to a good approximation. Moreover, also the assumption of \(\mathbf {E}({\hat{\mathbf {g}}}_1)=\mathbf {E}({\hat{\mathbf {g}}}_2)\) will only approximately hold in practice. Finally, recall that the variance components are usually estimated and an adapted estimate can compensate the effects of changes of the parameters \(\tau\) and \(\omega\).

We will give an example of how a reduced empirical variance may reduce inflation.

*Example 1*

Let \(\mathbf {y}\) be a vector of measured data and \(\mathbf {g}_1:=\mathbf {K}_1 \mathbf {y}\) with \(\mathbf {K}_1\succeq \mathbf {0}\). Moreover, let \(\mathbf {g}_2:=0.5 \mathbf {K}_1 \mathbf {y}\) which means \(\mathbf {K}_2=0.5\mathbf {K}_1\). Then \(\mathbf {K}_1 \succeq \mathbf {K}_2\) and \(\mathbf {K}_1\mathbf {K}_1 \succeq \mathbf {K}_2\mathbf {K}_2\) and \({\text {Var}}(\mathbf {g}_2) = 0.25{\text {Var}}(\mathbf {g}_1)\).

*b*of an ordinary least squares regression of \(\mathbf {y}\) on \(\mathbf {g}\)

Example 1 illustrates that the reduced variance of the predicted genetic values may reduce inflation. It is worth highlighting that the scaling factor used in this example was formulated on the level of \(\mathbf {K}_i\) which does not simply translate to a scaled variance component for \(\mathbf {H}_{\tau ,\omega }\). In the next section, we give a small example with a well investigated wheat data set [18].

## An example with wheat data

We assessed predictive ability, inflation and number of iterations up to convergence with varying parameters \(\tau\) and \(\omega\) on a publicly available wheat data set [18, 25]. The aim was to seek for the optimal combinations of both parameters, which maximize the predictive ability or minimize the inflation or the number of iterations to convergence, respectively. Moreover, we were interested in the general behavior of inflation when \(\tau\) and \(\omega\) are varied.

### Data

The data set which we used consists of 599 CIMMYT wheat lines, genotyped with 1279 Diversity Array Technology markers indicating whether a certain allele is present (1) or not (0) in the respective line. The lines were grown in four different environments and grain yield was recorded for each line and each environment (for more details see [18]). We used only the phenotypic data of environment 1 for our comparisons. To see whether the choice of which lines are considered as (not) genotyped has a significant impact on properties of the single-step procedure, we split the lines into two parts according to the order in the data set and considered two scenarios: In scenario 1 (hereinafter referred to as SC1), lines 1 to 300 were treated as not genotyped and the remaining lines 301 to 599 were used as genotyped group. Thus, the pedigree relationship of lines 301 to 599 represents \(\mathbf {A}_{22}\) and their genomic relationship represents \(\mathbf {G}\). The genomic relationship matrix was calculated according to VanRaden [26]: \(\mathbf {G}= (\mathbf {Z}-\mathbf {P})(\mathbf {Z}-\mathbf {P})^T/ \sum _{j=1}^{p} p_j (1-p_j)\), with \(\mathbf {Z}\) denoting the \(n \times p\) matrix giving the states of the *p* markers of the *n* individuals, and \(\mathbf {P}\) denoting the matrix with identical rows giving the column averages of \(\mathbf {Z}\). The same procedure was repeated in scenario 2 (hereinafter referred to as SC2) but the genotyped group consisted of lines 1 to 300. Note again that the order was used as provided by the data set.

### Parameter grid

To seek for the optimal values for both parameters, 420 combinations of \(\tau\) and \(\omega\) were tested for each scenario. This number of combinations resulted from varying both parameters on a grid defined by 0.10 steps dividing the interval \([-1,1]\) for \(\omega\), or [0.1, 2] for \(\tau\). To evaluate the performance of each parameter combination, we constructed \(\mathbf {H}_{\tau ,\omega }^{-1}\) by Eq. (3) for each combination of the parameters. Consequently, 420 different \(\mathbf {H}_{\tau ,\omega }^{-1}\) matrices were calculated in R [27] and transferred to the blupf90 software [28] to estimate the breeding values using the single-step procedure.

### Evaluation of the prediction

To evaluate the predictions obtained with the different matrices, a cross-validation was run by partitioning the 599 wheat lines into 10 disjoint groups of approximately 60 lines each (regardless of whether their genomic information had been used in the single-step covariance matrix). The partitions used were those provided with the data set, which had been generated randomly [18]. Iteratively, each group was used as a test set and models were fit with the remaining lines. Prediction quality was evaluated for these 60 lines in terms of predictive ability and inflation. The former was measured as Pearson’s correlation between the phenotype and the estimated breeding value (EBV) for the test set. Inflation was calculated as the coefficient of regression of the phenotype on the EBV (for the test set). The optimal combination of parameter values should have a regression coefficient close to 1 (neither inflation nor deflation). The number of iterations to convergence was also recorded.

### Results

*b*were \((\tau ,\omega )=(2,-1)\) in both scenarios, as suggested by our theoretical results. We see the tendency that both, increasing \(\tau\) or decreasing \(\omega\) reduces inflation in the sense of increasing

*b*. However, note that in our example, we are already in a situation of deflation and reducing the variance of \({\hat{\mathbf {g}}}\) increases the predictive bias.

Lastly, the optimal values of the parameters in terms of a minimal number of iterations to convergence were \((\tau ,\omega )=(1.9,0.8)\) for SC1 and \((\tau ,\omega )=(1.0,-\,0.8)\) for SC2. However, for most combinations, the number of iterations was between 26 and 32 which indicates that the influence of \((\tau ,\omega )\) on the number of iterations required is limited for this data set (results not shown).

## Discussion

Here we presented the general form of the single-step relationship matrix \(\mathbf {H}_{\tau ,\omega }\), when blending parameters \(\tau\) and \(\omega\) are defined on the level of its inverse [11, 12]. The matrix obtained (Eq. 4) is similar to the original single-step relationship matrix (Eq. 1) but with the role of \(\mathbf {G}\) replaced by expression Eq. (5). Moreover, we discussed some special choices of these parameters including the case for which \(\tau\) and \(\omega\) are equal, which was also the first adjustment of \(\mathbf {H}\) discussed in the literature [3].

The reduction in inflation was one of the main motivations for using the blending parameters [13, 16]. We illustrated with theoretical considerations that increasing \(\tau\) or decreasing \(\omega\) tends to reduce the empirical variance of \({\hat{\mathbf {g}}}_i\), which again may lead to a reduced inflation. Our theoretical arguments are limited by their assumptions, but should hold to a good approximation. To reinforce these results with an empirical exploration, we gave a small example with a well investigated wheat data set [18]. There, the pattern observed for inflation was largely in accordance with what we expected from our theoretical considerations. With regard to predictive ability, the parameters showed broad optimality and varied strongly across the two scenarios SC1 and SC2. Both observations may be data set specific and the latter a consequence of the small population size.

Finally, note that similar effects on inflation can also be achieved with other methods as for instance by explicitly reducing the additive variance or by accounting for inbreeding [5] (see in this context also Example 1). It may be worth considering the single-step method in more detail from a theoretical perspective to address the causes of inflation. Recent studies reported results in this direction by for instance attributing inflation to inconsistencies between genomic and pedigree relationships and by suggesting that accounting for inbreeding and unknown parent groups in a proper way may reduce this problem [5]. Moreover, it has also been highlighted that selective genotyping and selective imputation may have an impact on the properties of ssBLUP [29].

## Conclusion

We provided theoretical arguments that increasing \(\tau\) or decreasing \(\omega\) may mainly decrease inflation by decreasing the variance of the estimated breeding values \({\hat{\mathbf {g}}}\). Alternative solutions that address the problems of single-step predictions from a more theoretical point of view may be found by investigating the consistency problems of \(\mathbf {A}\) and \(\mathbf {G}\) with respect to scaling and coding further.

## Notes

### Authors' contributions

ECGP posed the research questions and defined the main content; JWRM, MFS and CAGB calculated the theoretical results, performed the analysis of the wheat data set and wrote the manuscript; all authors discussed the structure of \(\mathbf {H}_{\tau ,\omega }\), the special choices of the parameters and the content of the manuscript. All authors read and approved the final manuscript.

## Acknowledgements

JWRM thanks KWS SAAT SE for financial support. Moreover, we thank the DAAD for financial support in the context of the exchange program 57335814 “Genomic selection and measures of kinship”. We also thank two unknown reviewers for their valuable comments. In particular, we would like to credit one of them for the simplified approach to prove the central statement. Finally, we acknowledge support by the Open Access Publication Funds of the University of Göttingen.

## Competing interests

The authors declare that they have no competing interests.

## Ethics approval and consent to participate

Not applicable.

## Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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