Abstract
Background
Validation by data truncation is a common practice in genetic evaluations because of the interest in predicting the genetic merit of a set of young selection candidates. Two of the most used validation methods in genetic evaluations use a single data partition: predictivity or predictive ability (correlation between pre-adjusted phenotypes and estimated breeding values (EBV) divided by the square root of the heritability) and the linear regression (LR) method (comparison of “early” and “late” EBV). Both methods compare predictions with the whole dataset and a partial dataset that is obtained by removing the information related to a set of validation individuals. EBV obtained with the partial dataset are compared against adjusted phenotypes for the predictivity or EBV obtained with the whole dataset in the LR method. Confidence intervals for predictivity and the LR method can be obtained by replicating the validation for different samples (or folds), or bootstrapping. Analytical confidence intervals would be beneficial to avoid running several validations and to test the quality of the bootstrap intervals. However, analytical confidence intervals are unavailable for predictivity and the LR method.
Results
We derived standard errors and Wald confidence intervals for the predictivity and statistics included in the LR method (bias, dispersion, ratio of accuracies, and reliability). The confidence intervals for the bias, dispersion, and reliability depend on the relationships and prediction error variances and covariances across the individuals in the validation set. We developed approximations for large datasets that only need the reliabilities of the individuals in the validation set. The confidence intervals for the ratio of accuracies and predictivity were obtained through the Fisher transformation. We show the adequacy of both the analytical and approximated analytical confidence intervals and compare them versus bootstrap confidence intervals using two simulated examples. The analytical confidence intervals were closer to the simulated ones for both examples. Bootstrap confidence intervals tend to be narrower than the simulated ones. The approximated analytical confidence intervals were similar to those obtained by bootstrapping.
Conclusions
Estimating the sampling variation of predictivity and the statistics in the LR method without replication or bootstrap is possible for any dataset with the formulas presented in this study.
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Background
Validation by data truncation has been proposed to validate models for genetic and genomic predictions [1]. In recent years, its popularity has increased over model-based statistics, such as the Akaike information criterion or likelihood ratio [2]. Widely used statistics for validation by data truncation are those included in the linear regression (LR) method, which compares sets of estimated breeding values (EBV) [3], and predictivity [4], the latter defined as the correlation between EBV and adjusted phenotypes, divided by the square root of the heritability. These validation statistics focus on the performance of the model to predict breeding values. Validation using these methods was done in dairy [5] and beef [6] cattle, pigs [7], chickens [8], sheep [9], goats [10], fish [11], wheat [12], and trees [13], among others. For validation in dairy cattle, using weighted averages and deregressed evaluations could be more robust than the LR method or predictivity [14]. Overall, the validation methods covered in the present study provide measures of bias and accuracy of genomic predictions. Standard errors and confidence intervals of validation statistics can be obtained by k-fold cross-validation [2]. Many studies assessed the variation of the LR method statistics by replicating the validation (e.g., [15, 16]). However, in routine genetic evaluations, k-fold validation is not useful because of population structure [1], it does not account for the reduction in variance in the selected population [3], and the interest is in predicting the genetic merit of young individuals [3]. Therefore, validation by data truncation is a common practice for routine genetic evaluations in animal and plant breeding [17,18,19,20,21,22,23].
In an early stage of developing the LR method, Legarra and Reverter [24] proposed calculating confidence intervals for the dispersion of the predictions (slope of the regression of true on estimated breeding value) using classical regression theory (i.e., considering \({\widehat{\mathbf{u}}}_{p}\) as fixed) [25]. However, the random and correlated nature between \({\widehat{\mathbf{u}}}_{p}\) and \({\widehat{\mathbf{u}}}_{w}\) introduces a systematic underestimation of the standard error of the dispersion. Thus, the estimated confidence intervals are narrower than the true ones.
Two methods are currently used to obtain standard errors and confidence intervals for validation by data truncation in genetic and genomic predictions. The first approach to assess the variation of validation statistics is to perform forward validations at several time points [18, 20, 23]. This practice gives an idea of the variation of the validation statistics over time. However, it cannot predict the variation of any statistics for a specific time point, and it is necessary to correct the statistics because some time periods might be more represented than others [18]. In addition, this method is computationally expensive for large datasets and involves complex manipulations of the available dataset. The second approach uses bootstrapping (sampling with replacement of the validation individuals to create pseudo-replicates of the validation dataset [17, 19, 22, 26]). Bootstrapping is attractive since it is computationally inexpensive and only requires running the validation once. To our knowledge, only Mäntysaari and Koivula [17] tested the adequacy of bootstrapping to obtain the variability of validation statistics for genomic selection, showing a good agreement with the first approach; however, this was only shown for one dairy cattle dataset. In addition, non-sampling-based, analytical confidence intervals for the LR method statistics and predictivity have not been reported, although they are of interest on their own and could simplify the process of assessing the quality of validation statistics. Therefore, the objectives of this study were to derive standard errors and analytical confidence intervals for validation by data truncation statistics used in genetic and genomic evaluations, to benchmark against their simulated sampling distributions, and to compare them against confidence intervals obtained by bootstrapping.
Methods
In the following section, we show the general model used to derive the formulas for the confidence intervals of the different validation statistics, and a useful result for the next derivations. Then, we derive the mathematical expression for each validation statistic and suggest approximations when it is not possible to obtain the exact expressions. Finally, we describe two simulations used for testing the adequacy of the presented confidence intervals. The derivation is frequentist in nature and considers the sampling distribution of the statistics of either validation method, considering the sampling variation in the phenotypes. This is the framework used by many methods to derive confidence intervals and also by related methods such as bootstrap [27]. Indeed, Efron [28] showed that cross-validation methods with replicates have frequentists interpretations.
Theory
For the sake of presentation, we assume a single-trait model with an additive genetic effect as the only random effect, although the results extend to other types of models:
where \(\mathbf{y}\) is the vector of phenotypes, \(\mathbf{b}\) is the vector of fixed effects, \(\mathbf{u}\) is the vector of additive genetic effects, \(\mathbf{e}\) is the vector of errors, and \(\mathbf{X}\) and \(\mathbf{Z}\) are incidence matrices.
The validation methods in this study (LR method and predictivity) consist of splitting the data into a whole and a partial dataset, denoted with the subscripts w and p, respectively. The whole dataset has all the available phenotypes, whereas in the partial dataset the phenotypes after a given date have been removed. Then, validation methods compare EBV versus either EBV (method LR) obtained from the whole dataset, or pre-corrected phenotypes present in the “whole” but not in the “partial” dataset (predictivity). The comparison is usually for a set of individuals, named “focal”; this can be e.g. bulls acquiring progeny records in the “whole” (but not in the partial dataset) or individual pigs acquiring, say, growth records in the “whole” (but not in the partial dataset).
Predicting \(\mathbf{u}\) for the validation or testing set based on the whole data \(\left({\widehat{\mathbf{u}}}_{w}\right)\) requires solving the model in Eq. (1). The prediction of \(\mathbf{u}\) for the validation set based on the partial data \(\left({\widehat{\mathbf{u}}}_{p}\right)\) is obtained by removing the phenotypes of the individuals in the validation set before solving the model in Eq. (1). As shown in Appendix I, if \(\mathbf{y}\) is assumed to follow a multivariate normal distribution and the predictions are obtained by best linear unbiased prediction in absence of selection (i.e., under random mating and random culling) [29], the joint distribution of \({\widehat{\mathbf{u}}}_{w}\) and \({\widehat{\mathbf{u}}}_{p}\) is:
where \(\mathbf{G}={\text{Var}}\left(\mathbf{u}\right)\), \({\mathbf{C}}_{w}^{22}\) is the prediction error variance of \({\widehat{\mathbf{u}}}_{w}\), and \({\mathbf{C}}_{p}^{22}\) is the prediction error variance of \({\widehat{\mathbf{u}}}_{p}\). If the predictions are obtained from mixed model equations (MME), \({\mathbf{C}}_{w}^{22}\) and \({\mathbf{C}}_{p}^{22}\) are obtained as blocks of the inverse of the MME for the animal effect. Absence of selection is assumed for simplicity and because the variances in Eq. (2) become complicated (and basically impossible in practice, as selection is not easily described algebraically) to obtain (see Appendix I), and this is a standard simplifying assumption in animal breeding applications – for instance, reliabilities are obtained from Eq. (2) or an approximation. As shown in the Appendix I, the conditional distribution of \({\widehat{\mathbf{u}}}_{w}\) given \({\widehat{\mathbf{u}}}_{p}\) is:
Note that Eqs. (2) and (3) also hold for a subvector of \({\widehat{\mathbf{u}}}_{w}\) and \({\widehat{\mathbf{u}}}_{p}\). Thus, the following derivations hold for the entire vectors \({\widehat{\mathbf{u}}}_{w}\) and \({\widehat{\mathbf{u}}}_{p}\) (i.e., the population) as well as for a subvector of \({\widehat{\mathbf{u}}}_{w}\) and \({\widehat{\mathbf{u}}}_{p}\) (i.e., the estimated breeding values of a subset of the population).
Bias
Legarra and Reverter [3] derived the estimate of the bias of predictions (\({\mu }_{wp}\)) as the difference between the averages of \({\widehat{\mathbf{u}}}_{p}\) and \({\widehat{\mathbf{u}}}_{w}\). In matrix notation:
where \(n\) is the number of individuals in the testing set and 1 is a vector of ones. Because of the joint multivariate normality of \({\widehat{\mathbf{u}}}_{p}\) and \({\widehat{\mathbf{u}}}_{w}\), \({\mu }_{wp}\) is normally distributed (see p 92 in [25]). Therefore, a Wald confidence interval [30] for \({\mu }_{wp}\) can be constructed if its standard error is known. Taking the variance of Eq. (4):
The above equation is simply the difference of the averages of the prediction error variances of the predictions. Then, a confidence interval for \({\mu }_{wp}\) is:
where \({z}_{1-\frac{\alpha }{2}}\) is the value of the standard normal distribution quantile function for the confidence level \(1-\frac{\alpha }{2}\). For large datasets, it is computationally unfeasible to obtain \({\mathbf{C}}_{p}^{22}\) and \({\mathbf{C}}_{w}^{22}\). In that situation, we can simplify Eq. (6), assuming that animals are non-inbred and mostly unrelated such that the off-diagonal elements of \(\mathbf{G}\), \({\mathbf{C}}_{w}^{22}\), and \({\mathbf{C}}_{p}^{22}\) can be safely ignored. Thus, \({\mathbf{C}}_{p}^{22}-{\mathbf{C}}_{w}^{22}\approx {\mathbf{R}}_{w}-{\mathbf{R}}_{p}\), where \({\mathbf{R}}_{w}\) and \({\mathbf{R}}_{p}\) are diagonal matrices of genomic (G)EBV’s reliabilities in the whole and partial datasets, respectively. Letting \({\sigma }_{g}^{2}\) be the genetic variance, \({\text{Var}}\left({\mu }_{wp}\right)\approx \frac{{\sigma }_{g}^{2}}{n}\left({\overline{rel} }_{w}-{\overline{rel} }_{p}\right)\) and an approximate confidence interval for \({\mu }_{wp}\) is:
Dispersion
The regression coefficient of \({\widehat{\mathbf{u}}}_{w}\) on \({\widehat{\mathbf{u}}}_{p}\) \(({b}_{wp})\) quantifies the dispersion of the predictions with partial data. If there is no under/over dispersion, the expected value of \({b}_{wp}\) is equal to 1. The mathematical expression for \({b}_{wp}\) is:
where \(cov\) and \(var\) are the sample covariance and variance, respectively, and \(\mathbf{S}=\mathbf{I}-{n}^{-1}{\mathbf{11}}^{\prime}\). A Wald confidence interval for \({b}_{wp}\) can be constructed because \({b}_{wp}\) is asymptotically normal when the number of focal individuals in the validation set increases (see p. 249 in [25]). By the law of the total variance (see p. 167 in [31]):
For the first term in the right-hand side, we have:
Using a first-order Taylor approximation:
By the expectation of quadratic forms [32] and the zero expectation of \({\widehat{\mathbf{u}}}_{p}\) and \({\widehat{\mathbf{u}}}_{w}\), the numerator of the right-hand side in Eq. (11) is \({\text{E}}\left[{\widehat{\mathbf{u}}}_{p}^{\prime}\mathbf{S}\left({\mathbf{C}}_{p}^{22}-{\mathbf{C}}_{w}^{22}\right)\mathbf{S} {\widehat{\mathbf{u}}}_{p}\right]={\text{tr}}\left(\mathbf{S}\left({\mathbf{C}}_{p}^{22}-{\mathbf{C}}_{w}^{22}\right)\mathbf{S}\left(\mathbf{G}-{\mathbf{C}}_{p}^{22}\right)\right)\). For the denominator, we have \({\text{E}}\left[{\left({\widehat{\mathbf{u}}}_{p}^{\prime}\mathbf{S} {\widehat{\mathbf{u}}}_{p}\right)}^{2}\right]={\text{Var}}\left({\widehat{\mathbf{u}}}_{p}^{\prime}\mathbf{S} {\widehat{\mathbf{u}}}_{p}\right)+{\text{E}}{\left[{\widehat{\mathbf{u}}}_{p}^{\prime}\mathbf{S} {\widehat{\mathbf{u}}}_{p}\right]}^{2}=2\,{\text{tr}}\left(\mathbf{S}\left(\mathbf{G}-{\mathbf{C}}_{p}^{22}\right)\mathbf{S}\left(\mathbf{G}-{\mathbf{C}}_{p}^{22}\right)\right)+{\text{tr}}{\left(\mathbf{S}\left(\mathbf{G}-{\mathbf{C}}_{p}^{22}\right)\right)}^{2}\). Thus:
For the second term in the right-hand side of Eq. (9):
Therefore, the variance of \({b}_{wp}\) is:
and the Wald confidence interval for \({b}_{wp}\) is:
By making similar assumptions as for the estimator of the bias, \(\mathbf{G}-{\mathbf{C}}_{p}^{22}\approx {\mathbf{R}}_{p}\). Then, \({\text{tr}}\left(\mathbf{S}\left({\mathbf{C}}_{p}^{22}-{\mathbf{C}}_{w}^{22}\right)\mathbf{S}\left(\mathbf{G}-{\mathbf{C}}_{p}^{22}\right)\right)\approx {\sigma }_{g}^{4}{\sum }_{i=1}^{n}\left(re{l}_{{w}_{i}}-re{l}_{{p}_{i}}\right)re{l}_{{p}_{i}}\), \({\text{tr}}\left(\mathbf{S}\left(\mathbf{G}-{\mathbf{C}}_{p}^{22}\right)\mathbf{S}\left(\mathbf{G}-{\mathbf{C}}_{p}^{22}\right)\right)\approx {\sigma }_{g}^{4}{\sum }_{i=1}^{n}re{l}_{{p}_{i}}^{2}\), and \({\text{tr}}{\left(\mathbf{S}\left(\mathbf{G}-{\mathbf{C}}_{p}^{22}\right)\right)}^{2}\approx {\sigma }_{g}^{4}{\left({\sum }_{i=1}^{n}re{l}_{{p}_{i}}\right)}^{2}\), which results in:
from which an approximate confidence interval for \({b}_{wp}\) can be constructed. Assuming that the increase in reliability from the partial to the whole dataset is constant among the validation animals, \(\frac{re{l}_{{w}_{i}}}{re{l}_{{p}_{i}}}=c\) (which is always higher than 1). Then, an approximate confidence interval for \({b}_{wp}\) is:
Ratio of accuracies
The Pearson correlation coefficient between \({\widehat{\mathbf{u}}}_{p}\) and \({\widehat{\mathbf{u}}}_{w}\) \(\left({\rho }_{wp}\right)\) has an expected value equal to the ratio of accuracies obtained with the partial and the whole dataset. The formula for \({\rho }_{wp}\) is:
In principle, a confidence interval can be obtained that involves explicitly elements in Eq. (2); however, this yielded inelegant expressions that were unusable in practice (see Appendix II). We propose to use a confidence interval for \({\rho }_{wp}\) using the Fisher transformation [25] and (see p. 261 in [29]). The inverse hyperbolic tangent of a correlation coefficient \(r\) \(\left({{\text{tanh}}}^{-1}\left(r\right)=\frac{1}{2}{\text{log}}\left(\frac{1+r}{1-r}\right)\right)\) follows approximately a normal distribution with a standard error equal to \(\frac{1}{\sqrt{n-3}}\) assuming that the samples are identically and independently distributed (which is not the case for genetic evaluations). Thus:
To obtain a confidence interval for \({\rho }_{wp}\), we apply the hyperbolic tangent and get:
where \({\text{tanh}}\left(x\right)=\frac{{e}^{2x}-1}{{e}^{2x}+1}\).
This can be computed for any dataset size but note that this confidence interval is not symmetric around \({\rho }_{wp}\).
Reliability
The reliability of the EBV is defined as the square of the correlation between the true and estimated breeding values. Legarra and Reverter [3] and Macedo et al. [29] proposed that the reliability be estimated as the ratio between the sample covariance of \({\widehat{\mathbf{u}}}_{p}\) and \({\widehat{\mathbf{u}}}_{w}\) and the genetic variance of the validation set \(\left({\sigma }_{{g}_{i}}^{2}\right)\). This variance must account for selection and can be approximated using averages of additive relationships among validation animals, which accounts for e.g. few families of large sibships, or calculated with the method of Sorensen et al. [33] which correctly accounts for selection. We will assume this variance as known. The estimator of the reliability has the following expression:
Although \({\rho }_{co{v}_{wp}}^{2}\) is not normally distributed, we assume that, for large sample sizes, its distribution is approximately normal. Taking the variance of \({\rho }_{co{v}_{wp}}^{2}\) gives:
As in Eq. (9), we apply the law of total variance for the numerator of the right-hand side in Eq. (22):
Following similar arguments as for Eqs. (10) and (11), the first term is equal to \({\text{tr}}\left(\mathbf{S}\left({\mathbf{C}}_{p}^{22}-{\mathbf{C}}_{w}^{22}\right)\mathbf{S}\left(\mathbf{G}-{\mathbf{C}}_{p}^{22}\right)\right)\), whereas the second term is equal to \(2\,{\text{tr}}\left(\mathbf{S}\left(\mathbf{G}-{\mathbf{C}}_{p}^{22}\right)\mathbf{S}\left(\mathbf{G}-{\mathbf{C}}_{p}^{22}\right)\right)\). Therefore:
Finally, a confidence interval for \({\rho }_{co{v}_{wp}}^{2}\) is constructed as:
Following the same assumptions as for the bias and dispersion parameter leads to \({\text{Var}}\left({\rho }_{co{v}_{wp}}^{2}\right)\approx \frac{{\upsigma }_{{\text{g}}}^{4}}{{\left(n {\sigma }_{{g}_{i}}^{2}\right)}^{2}}{\sum }_{i=1}^{n}\left(re{l}_{{w}_{i}}+re{l}_{{p}_{i}}\right)re{l}_{{p}_{i}}\). Assuming that the increase in reliability from the partial to the whole dataset is constant among the validation animals, that is, \(\frac{re{l}_{{w}_{i}}}{re{l}_{{p}_{i}}}=c\), gives \({\text{Var}}\left({\rho }_{co{v}_{wp}}^{2}\right)\approx \frac{\left(1+c\right){\upsigma }_{{\text{g}}}^{4}}{n{\sigma }_{{g}_{i}}^{4}}\left(Var\left(re{l}_{p}\right)+{\overline{rel} }_{p}^{2}\right)\). Thus, an approximate confidence interval for \({\rho }_{co{v}_{wp}}^{2}\) is:
Predictivity
The ratio between the correlation of \({\widehat{\mathbf{u}}}_{p}\) and the phenotypes of the validation set adjusted for fixed effects \(\left({\mathbf{y}}^{*}\right)\) and the square root of the heritability \(\left(h\right)\) is an estimate of the correlation between estimated and true breeding values [4]. This statistic is sometimes called predictivity \(\left({\rho }_{{\mathbf{y}}^{*},{\widehat{\mathbf{u}}}_{p}}\right)\) and has the following mathematical expression:
As with the ratio of accuracies, the Fisher transformation can be used to obtain the following confidence interval for \({\rho }_{{\mathbf{y}}^{*},{\widehat{\mathbf{u}}}_{p}}\):
This can be computed for any dataset size.
Simulations
We tested the adequacy of our analytical [(Eqs. (6), (15), (20), (25), and (28)] and approximated analytical [Eqs. (7), (17), and (26)] confidence intervals using two simulated examples. In both, we obtained the empirical distribution of the validation statistics by replicating the simulation. Then, we compared the standard error and 95% confidence interval of that sampling distribution (i.e., True) versus confidence intervals obtained with the formulas presented in the previous section (i.e., Analytical or Approximated), and by bootstrapping. The confidence intervals with bootstrap were obtained by sampling with replacement of the validation set, replicated 10,000 times.
Example 1
The first dataset was created using a publicly available pedigree created by Yutaka Masuda (https://github.com/masuday/data/blob/master/tutorial/rawfiles/rawped). The pedigree had 11 generations without selection (i.e., random mating and random culling) and 4641 individuals. Single-trait models with generation as a fixed effect \(\left(\mathbf{b}\right)\) and additive genetic effect \(\left(\mathbf{u}\right)\) as a random effect were simulated for different heritabilities \(\left({h}^{2}\right)\) and proportions \(\left(prop\right)\) of animals with phenotypes in the population. In total, a grid of 81 scenarios corresponding to \({h}^{2}=\left\{0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9\right\}\) and \(prop=\left\{0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9\right\}\) was evaluated. Each scenario was replicated 50 times by sampling the vector of phenotypes from a multivariate normal distribution with mean \(\mathbf{X}\mathbf{b}\) (b is fixed across replicates) and variance \(\mathbf{Z}\mathbf{A}{\mathbf{Z}}^{\prime}{\sigma }_{a}^{2}+\mathbf{I}{\sigma }_{e}^{2}\), where \(\mathbf{A}\) is the numerator relationship matrix [34], \({\sigma }_{a}^{2}=1\) and \({\sigma }_{e}^{2}=\frac{1}{{h}^{2}}-1\). The validation set was composed of phenotyped animals from the most recent generation. The number of animals in the validation set was constant among heritabilities, and for each \({prop}_{i}\) was equal to 44, 74, 119, 149, 188, 234, 274, 318, and 362, respectively. All the computations were done in Julia [35].
Example 2
For the second example, we replicated the simulation of Vitezica et al. [36], which consists of a dairy cattle selection scheme with single-step genomic best linear unbiased predictor [37,38,39]. In each replicate, the partial dataset was created by removing the phenotypes of the cows in the most recent generation. Two validation sets were created: the cows for which the phenotypes were removed and the sires of those cows. The number of cows in the validation set was equal to 1300, whereas the number of bulls was 200. The simulation was replicated 30 times. Estimated breeding values and exact prediction error variances were obtained with the BLUPF90 + software [40]. Prediction error variances were obtained using sparse inversion techniques, which calculates the elements in the inverse corresponding to the non-zero elements of the original matrix [41, 42]. All the other analyses were done using Julia [35].
Results
Table 1 shows the average squared difference between the estimated and true values of the variance and 95% confidence interval bounds for Example 1. Average squared differences grouped by different \(prop\) and \({h}^{2}\) are reported (see Additional file 1: Tables S1 and S2). The analytical confidence intervals and variances were closer to the true simulated values than those obtained by bootstrapping. The confidence intervals obtained by bootstrapping were very similar to the approximated analytical confidence intervals.
The same patterns can be observed in Figs. 1, 2, 3, 4 and 5, which compare the simulated against estimated standard errors and confidence intervals for a combination of three heritabilities (low, medium, and high) and three proportions of animals with phenotypes (low, medium, and high). For the bias (Fig. 1), the estimation of the confidence intervals was less accurate with low \(prop\). Within each \(prop\), the approximated and bootstrap standard errors and confidence intervals tend to overestimate the simulated ones as the heritability increases.
The situation was the opposite for the dispersion parameter (Fig. 2). In this case, the approximated and bootstrap confidence intervals were too narrow for high \(prop\) with respect to the true confidence intervals obtained from the simulated data. These results suggest that bootstrapping does not consider properly the complex covariance structure between \({\widehat{\mathbf{u}}}_{p}\) and \({\widehat{\mathbf{u}}}_{w}\).
The confidence intervals for the ratio of accuracies were slightly underestimated for the bootstrap method.
The same was observed for the predictivity. However, the variance among replicates was very high for scenarios with low \(prop\) or low \({h}^{2}\). In such cases, the confidence intervals for the predictivity would cover a large portion of its range, making inference based on the predictivity statistic inaccurate.
For reliability (Fig. 5), the analytical confidence intervals were very close to the simulated ones. The approximated analytical and the bootstrap confidence intervals were systematically narrower than the simulated ones. In addition, the simulated confidence intervals were not symmetric around the mean, as the lower bound was closer to the mean than the upper bound. This could indicate that approximate normality is not appropriate.
Results for Example 2 are shown in Fig. 6 for bulls and Fig. 7 for cows. The analytical confidence intervals for reliabilities in cows were closer to the simulated ones than those for bulls. For bulls, the results were overall more variable and showed that the analytical confidence intervals for all the statistics were biased, probably because bulls were highly selected. This violates the assumption of absence of selection and can affect the expressions involving \(\mathbf{G}\). In addition, this issue could have been generated by the sparse inversion [41, 42] implemented in BLUPF90 + , which calculates in an exact manner the elements of \({\mathbf{C}}_{w}^{22}\) and \({\mathbf{C}}_{p}^{22}\) corresponding to the non-zero pattern of the MME and ignores the rest of the elements, which have their values set to zero before sparse inversion. However, these elements, which are not needed for reliabilities or restricted maximum likelihood (REML), are needed to obtain confidence intervals analytically, e.g., in [15]. For instance, the prediction error covariance of two unrelated bulls with daughters in the same herd. Another reason could be that the amount of information removed for the validation bulls was not sufficient, which is shown by a high \({\rho }_{wp}\). Under high \({\rho }_{wp}\), the gain in accuracy from the partial to the whole dataset will be minimal to null, and the standard errors of the validation statistics will tend to zero because \({\mathbf{C}}_{p}^{22}\approx {\mathbf{C}}_{w}^{22}\). Similar to the results from Example 1, the bootstrap confidence intervals were narrower than the simulated ones.
Discussion
The aim of this study was to derive standard errors and analytical confidence intervals for the LR method and predictivity. For the estimators of the bias, dispersion, and reliability from the LR method, we calculated their standard errors and built Wald confidence intervals assuming that the estimators are asymptotically normally distributed. Unlike [24], we used the marginal (unconditional) distribution of the estimators to account for the randomness of \({\widehat{\mathbf{u}}}_{p}\) and the dependence between \({\widehat{\mathbf{u}}}_{p}\) and \({\widehat{\mathbf{u}}}_{w}\). Not accounting for the randomness of \({\widehat{\mathbf{u}}}_{p}\) results in an underestimation of the standard errors of the validation statistics; hence, resulting in narrower confidence intervals. The resulting standard errors and confidence intervals are functions of the relationships between the individuals in the validation set and their prediction error (co)variances in the whole and partial datasets.
For the estimator of the ratio of accuracies from the LR method and the predictivity, we used the Fisher transformation to obtain a confidence interval of those correlation coefficients. Although this method is straightforward, it assumes that all the samples are identically and independently distributed, which is not true when performing validation by data truncation in genetic evaluations. Looking for better formulas that account for heterogeneity in variances and dependence among samples will involve complicated expressions (see Appendix II). In addition, Krishnamoorthy and Xia [43] and Gnambs [44] showed that Fisher’s transformation worked well with a large number of observations regardless of whether its assumptions were violated. Also, unlike the standard errors for bias, dispersion, and reliability, which depend only on the model [see Eqs. (5), (14), and (24)], the variances for the ratio of accuracies and predictivity depend directly on the values of the statistics themselves.
Although confidence intervals for predictivity can be obtained with Fisher’s transformation, comparing different models based on those confidence intervals is improper because it does not consider the dependency between the statistics. A bootstrap method to account for this was proposed by [24], but parametric methods exist. In other words, the methods presented in this study explain how to obtain confidence intervals for \({\rho }_{{\mathbf{y}}^{*},{\widehat{\mathbf{u}}}_{p}}\) but they do not assess the null (H0) hypothesis \({\rho }_{{\mathbf{y}}^{*},{\widehat{\mathbf{u}}}_{p}(A)}={\rho }_{{\mathbf{y}}^{*},{\widehat{\mathbf{u}}}_{p}(B)}\), where \(A\) and \(B\) denote different methods or models for prediction. A proper test in this situation is the Williams test [45], which uses the statistic \(T=\left({\rho }_{{\mathbf{y}}^{*},{\widehat{\mathbf{u}}}_{p}(A)}-{\rho }_{{\mathbf{y}}^{*},{\widehat{\mathbf{u}}}_{p}(B)}\right)\sqrt{\frac{\left(n-1\right)\left(1+{\rho }_{{\widehat{\mathbf{u}}}_{p}(A),{\widehat{\mathbf{u}}}_{p}(B)}\right)}{2\left(\frac{n-1}{n-3}\right)\left|\mathbf{R}\right|+{\left(\frac{{\rho }_{{\mathbf{y}}^{*},{\widehat{\mathbf{u}}}_{p}(A)}+{\rho }_{{\mathbf{y}}^{*},{\widehat{\mathbf{u}}}_{p}(B)}}{2}\right)}^{2}{\left(\left(1-{\rho }_{{\widehat{\mathbf{u}}}_{p}(A),{\widehat{\mathbf{u}}}_{p}(B)}\right)\right)}^{3}}}\) to compare correlations \({\rho }_{{\mathbf{y}}^{*},{\widehat{\mathbf{u}}}_{p}(A)}\) and \({\rho }_{{\mathbf{y}}^{*},{\widehat{\mathbf{u}}}_{p}(B)}\), where \({\rho }_{{\widehat{\mathbf{u}}}_{p}(A),{\widehat{\mathbf{u}}}_{p}(B)}\) is the correlation between EBV calculated with methods \(A\) and \(B\), and \(\left|\mathbf{R}\right|=\left(1-{\rho }_{{\mathbf{y}}^{*},{\widehat{\mathbf{u}}}_{p}\left(A\right)}-{\rho }_{{\mathbf{y}}^{*},{\widehat{\mathbf{u}}}_{p}\left(B\right)}-{\rho }_{{\widehat{\mathbf{u}}}_{p}(A),{\widehat{\mathbf{u}}}_{p}(B)}\right)+2\left({\rho }_{{\mathbf{y}}^{*},{\widehat{\mathbf{u}}}_{p}(A)}{\rho }_{{\mathbf{y}}^{*},{\widehat{\mathbf{u}}}_{p}(B)}{\rho }_{{\widehat{\mathbf{u}}}_{p}(A),{\widehat{\mathbf{u}}}_{p}(B)}\right)\). Statistic \(T\) follows approximately a \(t\) distribution with \(n-3\) degrees of freedom. Indeed, this test has already been used but not in the context of the LR method [46].
According to the results of our study, analytical confidence intervals should be preferred over bootstrap confidence intervals. However, the analytical confidence intervals for the bias, dispersion, and reliability are computationally expensive to obtain in large datasets because they need the prediction error variances and covariances for the validation animals in the whole and partial datasets. Alternatives for large-scale genetic evaluations could be to approximate \({\mathbf{C}}_{w}^{22}\) and \({\mathbf{C}}_{p}^{22}\) with Markov chain Monte Carlo methods [47]. In this study, the approximations that we propose assume \(\left({\mathbf{C}}_{p}^{22}-{\mathbf{C}}_{w}^{22}\right)\) and \(\left(\mathbf{G}-{\mathbf{C}}_{p}^{22}\right)\) to be diagonal. In such a case, \({\mathbf{C}}_{w}^{22}\) and \({\mathbf{C}}_{p}^{22}\) can be obtained from the (G)EBV reliabilities reported in the evaluation, which corresponds to, for instance, the information that is used in Interbulls’ tests [48, 49]. The robustness of the diagonal assumption depends on the data. For not-very-related individuals with high reliabilities, the assumption holds. More complex scenarios, for instance, families with half-sibs and low to medium reliabilities will require further inspections due to a block structure of \(\left({\mathbf{C}}_{p}^{22}-{\mathbf{C}}_{w}^{22}\right)\) and \(\left(\mathbf{G}-{\mathbf{C}}_{p}^{22}\right)\). Assuming that the increase in reliability from partial to whole data \(\left(c\right)\) is constant among animals leads to an expression where only \(re{l}_{p}\) or \(re{l}_{w}\) are required. This could be attractive in cases where performing validation by data truncation is not possible (e.g., when phenotypes could not be shared or \(re{l}_{p}\) might not be available) or when adding a source of information or calculating the reliability is not possible (\(re{l}_{w}\) might not be available). The assumption of a constant increase in reliability, using the average increase of the reliability for the calculations, was shown to be robust in this study in spite of the range of \(c\), which ranged up to (1.86-8.91) for some scenarios in Example 1. In our simulations, the approximated analytical confidence intervals were similar to those obtained by bootstrapping.
In many scenarios in both examples, the bootstrap confidence intervals were narrower than the simulated ones. In other words, bootstrapping was “too optimistic” regardless of the variation of the empirical distribution of the validation statistic. The reason could be the correlated data structure shown by populations under artificial selection. Bickel et al. [50] reviewed situations and presented scenarios where classical bootstrap fails. In such a case, they proposed sampling with replacement using fewer observations than the total number. In addition, this could increase the efficiency of bootstrapping. The number of observations to sample would depend on the data and could be calibrated with the analytical confidence intervals in case they are too expensive to obtain for routine evaluations.
The additive genetic variance and the accuracy of the EBV change when selection occurs [51, 52]. To our knowledge, the interaction between predictivity and selection has not been studied. That statistic depends on the square root of the heritability. Thus, if the estimate of the heritability under selection is biased, the predictivity could also be biased. In case the model and genetic parameters are correct, the predictivity could be biased if selected animals are chosen for the validation set. Simulation studies reported that the LR method worked well when the model used to estimate breeding values matches the true data-generating process [14, 29]. According to these results, one could infer that the LR method would estimate the bias, dispersion, and accuracy properly in the presence of selection if it is correctly taken into account in the model with, for instance, the method of Henderson [53, 54]. However, this is rarely used in genetic evaluations, and selection is often ignored in the estimation of breeding values. In such a case, the LR method can estimate the direction of the bias but not its magnitude if the model is incorrect but reasonably robust [29]. The LR method cannot estimate the bias when the model is seriously mis-specified, which in the case of [29] was when a simulated environmental trend was ignored in the model. Macedo et al. [29] found that the dispersion and accuracy were well estimated in all scenarios. However, Himmelbauer et al. [14] found that the LR method performed well for males but not for females in dairy cattle selection schemes. In addition, they reported that the estimator of the reliability depends heavily on how the additive genetic variance for the validation set is calculated. The confidence intervals derived in this study can be affected by selection in two ways: (i) by the bias of the validation statistics and (ii) by the effect of selection on the standard error of the dispersion and reliability estimators. The first way affects the location of the confidence interval, and given a biased estimator, it is not possible to correct. The second way affects the confidence interval length because the additive relationships in the validation group change due to selection [53, 54]. Specifically, the affected term is \(\left(\mathbf{G}-{\mathbf{C}}_{p}^{22}\right)\), which is the variance of \({\widehat{\mathbf{u}}}_{p}\). According to Henderson [53, 54], the variance of \({\widehat{\mathbf{u}}}_{p}\) is reduced under selection. However, the effect on the standard error of the estimators of the dispersion and reliability is hard to assess because the variance of \({\widehat{\mathbf{u}}}_{p}\) is involved in convoluted algebraic operations.
Conclusions
We derived analytical standard errors and confidence intervals for predictivity and the LR method statistics of bias, dispersion, ratio of accuracies, and reliability. Based on the examples shown in this study, the analytical confidence intervals were more accurate than the confidence intervals obtained by bootstrapping. We also developed approximated analytical confidence intervals for situations where the analytical ones are not feasible due to computational limitations. This study provided a framework for proper validation by data truncation statistical inference applied to genetic evaluation when replication is not possible.
Availability of data and materials
Data and material are available upon request.
References
Thompson R. Statistical validation of genetic models. Livest Prod Sci. 2001;72:129–34.
Gianola D, Schön CC. Cross-validation without doing cross-validation in genome-enabled prediction. G3 (Bethesda). 2016;6:3107–28.
Legarra A, Reverter A. Semi-parametric estimates of population accuracy and bias of predictions of breeding values and future phenotypes using the LR method. Genet Sel Evol. 2018;50:53.
Legarra A, Robert-Granié C, Manfredi E, Elsen JM. Performance of genomic selection in mice. Genetics. 2008;180:611–8.
Alkhoder H, Liu Z, Segelke D, Reents R. Comparison of a single-step with a multistep single nucleotide polymorphism best linear unbiased predictor model for genomic evaluation of conformation traits in German Holsteins. J Dairy Sci. 2022;105:3306–22.
Cardoso FF, Matika O, Djikeng A, Mapholi N, Burrow HM, Yokoo MJI, et al. Multiple country and breed genomic prediction of tick resistance in beef cattle. Front Immunol. 2021;12: 620847.
Aliakbari A, Zemb O, Cauquil L, Barilly C, Billon Y, Gilbert H. Microbiability and microbiome-wide association analyses of feed efficiency and performance traits in pigs. Genet Sel Evol. 2022;54:29.
Bermann M, Legarra A, Hollifield MK, Masuda Y, Lourenco D, Misztal I. Validation of single-step GBLUP genomic predictions from threshold models using the linear regression method: an application in chicken mortality. J Anim Breed Genet. 2021;138:4–13.
Macedo FL, Astruc JM, Meuwissen THE, Legarra A. Removing data and using metafounders alleviates biases for all traits in Lacaune dairy sheep predictions. J Dairy Sci. 2022;105:2439–52.
Massender E, Brito LF, Maignel L, Oliveira HR, Jafarikia M, Baes CF, et al. Single-step genomic evaluation of milk production traits in Canadian Alpine and Saanen dairy goats. J Dairy Sci. 2022;105:2393–407.
Silva RMO, Evenhuis JP, Vallejo RL, Gao G, Martin KE, Leeds TD, et al. Whole-genome mapping of quantitative trait loci and accuracy of genomic predictions for resistance to columnaris disease in two rainbow trout breeding populations. Genet Sel Evol. 2019;51:42.
Raffo MA, Sarup P, Andersen JR, Orabi J, Jahoor A, Jensen J. Integrating a growth degree-days based reaction norm methodology and multi-trait modeling for genomic prediction in wheat. Front Plant Sci. 2022;13: 939448.
Callister AN, Bermann M, Elms S, Bradshaw BP, Lourenco D, Brawner JT. Accounting for population structure in genomic predictions of Eucalyptus globulus. G3 (Bethesda). 2022;12:jkac180.
Himmelbauer J, Schwarzenbacher H, Fuerst C, Fuerst-Waltl B. Comparison of different validation methods for single-step genomic evaluations based on a simulated cattle population. J Dairy Sci. 2023;2023(106):9026–43.
Duenk P, Calus MPL, Wientjes YCJ, Breen VP, Henshall JM, Hawken R, et al. Validation of genomic predictions for body weight in broilers using crossbred information and considering breed-of-origin of alleles. Genet Sel Evol. 2019;51:38.
Pravia MI, Navajas EA, Aguilar I, Ravagnolo O. Prediction ability of an alternative multi-trait genomic evaluation for residual feed intake. J Anim Breed Genet. 2023;140:508–18.
Mäntysaari EA, Koivula M. GEBV validation test revisited. Interbull Bull. 2012;45:1–5.
Macedo FL, Christensen OF, Astruc JM, Aguilar I, Masuda Y, Legarra A. Bias and accuracy of dairy sheep evaluations using BLUP and SSGBLUP with metafounders and unknown parent groups. Genet Sel Evol. 2020;52:47.
Junqueira VS, Lopes PS, Lourenco D, Silva FFE, Cardoso FF. Applying the metafounders approach for genomic evaluation in a multibreed beef cattle population. Front Genet. 2020;11: 556399.
Alexandre PA, Li Y, Hine BC, Duff CJ, Ingham AB, Porto-Neto LR, et al. Bias, dispersion, and accuracy of genomic predictions for feedlot and carcase traits in Australian Angus steers. Genet Sel Evol. 2021;53:77.
Bonifazi R, Calus MPL, Ten Napel J, Veerkamp RF, Michenet A, Savoia S, et al. International single-step SNPBLUP beef cattle evaluations for Limousin weaning weight. Genet Sel Evol. 2022;54:57.
Raffo MA, Sarup P, Guo X, Liu H, Andersen JR, Orabi J, et al. Improvement of genomic prediction in advanced wheat breeding lines by including additive-by-additive epistasis. Theor Appl Genet. 2022;135:965–78.
Wicki M, Raoul J, Legarra A. Effect of subdivision of the Lacaune dairy sheep breed on the accuracy of genomic prediction. J Dairy Sci. 2023;106:5570–81.
Legarra A, Reverter A. Can we frame and understand cross-validation results in animal breeding? Proc Assoc Advmt Anim Breed Genet. 2017;22:73–80.
Rencher A, Schaalje B. Linear models in statistics. Hokoben: Wiley; 2008.
Legarra A, Baloche G, Barillet F, Astruc JM, Soulas C, Aguerre X, et al. Within- and across-breed genomic predictions and genomic relationships for Western Pyrenees dairy sheep breeds Latxa, Manech, and Basco-Béarnaise. J Dairy Sci. 2014;97:3200–12.
Efron B, Tibshirani RJ. An introduction to the bootstrap. New York: Chapman and Hall; 1993.
Efron B. The estimation of prediction error: covariance penalties and cross-validation. J Am Stat Assoc. 2004;99:619–32.
Macedo FL, Reverter A, Legarra A. Behavior of the linear regression method to estimate bias and accuracies with correct and incorrect genetic evaluation models. J Dairy Sci. 2020;103:529–44.
Wald A. Tests of statistical hypotheses concerning several parameters when the number of observations is large. Trans Am Math Soc. 1943;54:426–82.
Casella G, Berger R. Statistical inference. Pacific Grove: Duxbury; 2022.
Searle S. Linear models. New York: Wiley; 1971.
Sorensen D, Fernando R, Gianola D. Inferring the trajectory of genetic variance in the course of artificial selection. Genet Res. 2001;77:83–94.
Emik LO, Terrill CE. Systematic procedures for calculating inbreeding coefficients. J Hered. 1949;40:51–5.
Bezanson J, Edelman A, Karpinski S, Shah V. Julia: a fresh approach to numerical computing. SIAM Rev. 2017;59:65–98.
Vitezica ZG, Aguilar I, Misztal I, Legarra A. Bias in genomic predictions for populations under selection. Genet Res (Camb). 2011;93:357–66.
Legarra A, Aguilar I, Misztal I. A relationship matrix including full pedigree and genomic information. J Dairy Sci. 2009;92:4656–63.
Aguilar I, Misztal I, Johnson DL, Legarra A, Tsuruta S, Lawlor TJ. Hot topic: a unified approach to utilize phenotypic, full pedigree, and genomic information for genetic evaluation of Holstein final score. J Dairy Sci. 2010;93:743–52.
Christensen OF, Lund MS. Genomic prediction when some animals are not genotyped. Genet Sel Evol. 2010;42:2.
Lourenco D, Tsuruta S, Masuda Y, Bermann M, Legarra A, Misztal I. Recent updates in the BLUPF90 software suite. In: Proceedings of the 12th World Congress on Genetics Applied to Livestock Production: 3–8 July 2022; Rotterdam. 2022.
Takahashi K, Fagan J, Chen MS. Formation of a sparse bus impedance matrix and its application to short circuit study. In: Proceedings of the 8th Power Industry Computer Applications Conference: 3–6 June 1973; Minneapolis. 1973.
Misztal I, Perez-Enciso M. Sparse matrix inversion for restricted maximum likelihood estimation of variance components by expectation-maximization. J Dairy Sci. 1993;76:1479–83.
Krishnamoorthy K, Xia Y. Inferences on correlation coefficients: one-sample, independent and correlated cases. J Stat Plan Inference. 2007;137:2362–79.
Gnambs T. A brief note on the standard error of the Pearson correlation. Collabra Psychol. 2023;9:87615.
Steiger JH. Tests for comparing elements of a correlation matrix. Psychol Bull. 1980;87:245–51.
Xiang T, Nielsen B, Su G, Legarra A, Christensen OF. Application of single-step genomic evaluation for crossbred performance in pig. J Anim Sci. 2016;94:936–48.
Hickey JM, Veerkamp RF, Calus MP, Mulder HA, Thompson R. Estimation of prediction error variances via Monte Carlo sampling methods using different formulations of the prediction error variance. Genet Sel Evol. 2009;41:23.
Boichard D, Bonaiti B, Barbat A, Mattalia S. Three methods to validate the estimation of genetic trend in dairy cattle. J Dairy Sci. 1995;78:431–7.
Mäntysaari EA, Liu Z, VanRaden P. Interbull validation test for genomic evaluations. Interbull Bull. 2010;41:17–22.
Bickel PJ, Götze F, van Zwet WR. Resampling fewer than n observations: gains, losses, and remedies for losses. Stat Sin. 1997;7:1–31.
Sorensen DA, Kennedy BW. Estimation of genetic variances from unselected and selected populations. J Anim Sci. 1984;59:1213–23.
Dekkers JCM. Asymptotic response to selection on best linear unbiased predictors of breeding values. Anim Sci. 1992;54:351–60.
Henderson CR. Best linear unbiased estimation and prediction under a selection model. Biometrics. 1975;31:423–47.
Henderson CR. Best linear unbiased prediction in populations that have undergone selection. In: Proceedings of the World Congress on Sheep and Beef Cattle Breeding: 28 October–13 November 1980; Palmerston North and Christchurch. 1980.
Reverter A, Golden BL, Bourdon RM, Brinks JS. Technical note: detection of bias in genetic predictions. J Anim Sci. 1994;72:34–7.
Henderson CR. Applications of linear models in animal breeding. Guelph: University of Guelph; 1984.
Acknowledgements
The authors thank Yutaka Masuda for making the pedigree for the first example available for this study.
Funding
This study was partially funded by Agriculture and Food Research Initiative Competitive Grant no. 2020–67015-31030 from the US Department of Agriculture's National Institute of Food and Agriculture.
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MB conceived the study and derived the exact formulas. MB and DL designed the comparisons. MB and AAM ran the examples. MB and AL co-wrote the manuscript and derived the approximated formulas. DL and IM edited the manuscript. All the authors read and approved the final manuscript.
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Supplementary Information
Additional file 1: Table S1.
Average squared differences between estimated and true variance, lower bound of the 95% confidence interval (lCI), and upper bound of the 95% confidence interval (uCI) for Example 1 for different heritabilities. Table S2. Average squared differences between estimated and true variance, lower bound of the 95% confidence interval (lCI), and upper bound of the 95% confidence interval (uCI) for Example 1 for different proportions of phenotyped animals.
Appendices
Appendix I
This is essentially a rewriting of properties shown by several authors [53,54,55,56], some of them difficult to find, that we put together here for readers’ convenience. The full model is:
Assuming \(\mathbf{u}\sim {\text{MVN}}({\mathbf{0}},\mathbf{G})\), \(\mathbf{e}\sim {\text{MVN}}({\mathbf{0}},\mathbf{R})\), and \({\text{cov}}\left(\mathbf{u},\mathbf{e}\right)=0\) gives \(\mathbf{y}\sim {\text{MVN}}(\mathbf{X}\mathbf{b},\mathbf{V}=\mathbf{Z}\mathbf{G}{\mathbf{Z}}^{\prime}+\mathbf{R})\). The best linear unbiased prediction of \(\mathbf{u}\) is obtained as \({\widehat{\mathbf{u}}}_{w}={\mathbf{C}}_{w}\mathbf{y}=\left(\mathbf{G}{\mathbf{Z}}^{\prime}{\mathbf{V}}^{-1}-\mathbf{G}{\mathbf{Z}}^{\prime}{\mathbf{V}}^{-1}\mathbf{X}{\left({\mathbf{X}}^{\prime}{\mathbf{V}}^{-1}\mathbf{X}\right)}^{-}{\mathbf{X}}^{\prime}{\mathbf{V}}^{-1}\right)\mathbf{y}\). The partial model (the model without the phenotypes of the validation set) is:
In this case, the best linear unbiased prediction of \(\mathbf{u}\) is obtained as \({\widehat{\mathbf{u}}}_{p}={\mathbf{C}}_{p}\mathbf{y}=\left[\begin{array}{cc}\left(\mathbf{G}{\mathbf{Z}}_{p}^{\prime}{\mathbf{V}}_{p}^{-1}-\mathbf{G}{\mathbf{Z}}_{p}^{\prime}{\mathbf{V}}_{p}^{-1}{\mathbf{X}}_{p}{\left({\mathbf{X}}_{p}^{\prime}{\mathbf{V}}_{p}^{-1}{\mathbf{X}}_{p}\right)}^{-}{\mathbf{X}}_{p}^{\prime}{\mathbf{V}}_{p}^{-1}\right)& {\mathbf{0}}\end{array}\right]\left[\begin{array}{c}{\mathbf{y}}_{p}\\ {\mathbf{y}}_{w-{\varvec{p}}}\end{array}\right]\). Then:
By affine transformation of \(\mathbf{y}\) (see p. 92 in [25]):
To account for selection, [29] showed that \(\mathbf{G}\) should be replaced by \({\mathbf{G}}^{*}=\mathbf{G}-{\mathbf{B}}_{u}{\mathbf{H}}_{0}{\mathbf{B}}_{u}^{\prime}\) [53], where \({\mathbf{B}}_{u}\) represents the selection process and \({\mathbf{H}}_{0}\) represents the decrease in variance under selection. However, information about the selection process to write down matrices \({\mathbf{B}}_{u}\) and \({\mathbf{H}}_{0}\) is unknown, except for idealized settings. For this reason and for simplicity, absence of selection is assumed to result in Eq. (33). Then, by multivariate normality, \({\widehat{\mathbf{u}}}_{w}|{\widehat{\mathbf{u}}}_{p}\) follows a multivariate normality distribution with mean:
and variance:
Thus,
Appendix II
Let \(r=\frac{{\sigma }_{12}}{\sqrt{{\sigma }_{1}^{2}{\sigma }_{2}^{2}}}\) be a Pearson sample correlation coefficient. Using a first-order Taylor approximation, its variance is:
Applying Eq. (37) to the estimator of the ratio of accuracies \(\left({\rho }_{wp}\right)\) (Eq. 18) we have:
Recalling that \({\text{Var}}\left({\widehat{\mathbf{u}}}_{p}^{\prime}\mathbf{S}{\widehat{\mathbf{u}}}_{p}\right)=2\,{\text{tr}}\left(\mathbf{S}\left(\mathbf{G}-{\mathbf{C}}_{p}^{22}\right)\mathbf{S}\left(\mathbf{G}-{\mathbf{C}}_{p}^{22}\right)\right)\), \({\text{Var}}\left({\widehat{\mathbf{u}}}_{w}^{\prime}\mathbf{S}{\widehat{\mathbf{u}}}_{w}\right)=2\,{\text{tr}}\left(\mathbf{S}\left(\mathbf{G}-{\mathbf{C}}_{w}^{22}\right)\mathbf{S}\left(\mathbf{G}-{\mathbf{C}}_{w}^{22}\right)\right)\), and \({\text{Var}}\left({\widehat{\mathbf{u}}}_{p}^{\prime}\mathbf{S}{\widehat{\mathbf{u}}}_{w}\right)={\text{tr}}\left(\mathbf{S}\left({\mathbf{C}}_{p}^{22}-{\mathbf{C}}_{w}^{22}\right)\mathbf{S}\left(\mathbf{G}-{\mathbf{C}}_{p}^{22}\right)\right)+2\,{\text{tr}}\left(\mathbf{S}\left(\mathbf{G}-{\mathbf{C}}_{p}^{22}\right)\mathbf{S}\left(\mathbf{G}-{\mathbf{C}}_{p}^{22}\right)\right)\), it can proved using (see p. 66 [32]) that \({\text{Cov}}\left({\widehat{\mathbf{u}}}_{p}^{\prime}\mathbf{S}{\widehat{\mathbf{u}}}_{p}, {\widehat{\mathbf{u}}}_{w}^{\prime}\mathbf{S}{\widehat{\mathbf{u}}}_{w}\right)={\text{Cov}}\left({\widehat{\mathbf{u}}}_{p}^{\prime}\mathbf{S}{\widehat{\mathbf{u}}}_{p}, {\widehat{\mathbf{u}}}_{p}^{\prime}\mathbf{S}{\widehat{\mathbf{u}}}_{w}\right)={\text{Var}}\left({\widehat{\mathbf{u}}}_{p}^{\prime}\mathbf{S}{\widehat{\mathbf{u}}}_{p}\right)\) and \({\text{Cov}}\left({\widehat{\mathbf{u}}}_{w}^{\prime}\mathbf{S}{\widehat{\mathbf{u}}}_{p}, {\widehat{\mathbf{u}}}_{w}^{\prime}\mathbf{S}{\widehat{\mathbf{u}}}_{w}\right)=2\,{\text{tr}}\left(\mathbf{S}\left(\mathbf{G}-{\mathbf{C}}_{w}^{22}\right)\mathbf{S}\left(\mathbf{G}-{\mathbf{C}}_{p}^{22}\right)\right)\). Then, Eq. (38) results in:
Then, applying Eq. (37) to the predictive ability \({\rho }_{{\mathbf{y}}^{*},{\widehat{\mathbf{u}}}_{p}}=\frac{1}{h}\frac{cov\left({\mathbf{y}}^{*},{\widehat{\mathbf{u}}}_{p}\right)}{\sqrt{var\left({\mathbf{y}}^{*}\right) var\left({\widehat{\mathbf{u}}}_{p}\right)}}\) (Eq. 27) we have:
The joint distribution of \({\widehat{\mathbf{u}}}_{p}\) and \({\mathbf{y}}^{*}\) is [3]:
where \(\mathbf{K}=\mathbf{G}+\mathbf{R}-\mathbf{X}{\mathbf{C}}^{11}\mathbf{X}\), with \(\mathbf{R}={\text{Var}}\left(\mathbf{e}\right)\) and \({\mathbf{C}}^{11}\) the block of the generalized inverse of the mixed model equations pertaining to the fixed effects. After algebra, \({\text{Var}}\left({{\mathbf{y}}^{\mathbf{*}}}^{\mathbf{^{\prime}}}\,\mathbf{S}{\mathbf{y}}^{*}\right)=2\,{\text{tr}}\left(\mathbf{S}\mathbf{K}\mathbf{S}\mathbf{K}\right)\), \({\text{Var}}\left({{\mathbf{y}}^{\mathbf{*}}}^{\mathbf{^{\prime}}}\mathbf{S}\,{\widehat{\mathbf{u}}}_{p}\right)={\text{tr}}\left(\mathbf{S}\mathbf{L}\mathbf{S}\left(\mathbf{G}-{\mathbf{C}}_{p}^{22}\right)\right)+{\text{tr}}\left(\mathbf{S}\left(\mathbf{G}-{\mathbf{C}}_{p}^{22}\right)\mathbf{S}\left(\mathbf{G}-{\mathbf{C}}_{p}^{22}\right)\right)\) where \(\mathbf{L}={\mathbf{C}}_{p}^{22}+\mathbf{R}-\mathbf{X}{\mathbf{C}}^{11}\mathbf{X}\), \({\text{Cov}}\left({\widehat{\mathbf{u}}}_{p}^{\prime}\mathbf{S}{\widehat{\mathbf{u}}}_{p}, {{\mathbf{y}}^{\mathbf{*}}}^{\mathbf{^{\prime}}}\mathbf{S}{\mathbf{y}}^{*}\right)={\text{Cov}}\left({\widehat{\mathbf{u}}}_{p}^{\prime}\mathbf{S}{\widehat{\mathbf{u}}}_{p},{{\mathbf{y}}^{\mathbf{*}}}^{\mathbf{^{\prime}}}\mathbf{S}{\widehat{\mathbf{u}}}_{p}\right)={\text{Var}}\left({\widehat{\mathbf{u}}}_{p}^{\prime}\mathbf{S}{\widehat{\mathbf{u}}}_{p}\right)\), and \({\text{Cov}}\left({{\mathbf{y}}^{\mathbf{*}}}^{\mathbf{^{\prime}}}\mathbf{S}{\widehat{\mathbf{u}}}_{p},{{\mathbf{y}}^{\mathbf{*}}}^{\mathbf{^{\prime}}}\mathbf{S}{\mathbf{y}}^{*}\right)=2\,{\text{tr}}\left(\mathbf{S}\mathbf{K}\mathbf{S}\left(\mathbf{G}-{\mathbf{C}}_{p}^{22}\right)\right)\). Then, Eq. (37) results in:
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Bermann, M., Legarra, A., Munera, A.A. et al. Confidence intervals for validation statistics with data truncation in genomic prediction. Genet Sel Evol 56, 18 (2024). https://doi.org/10.1186/s12711-024-00883-w
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DOI: https://doi.org/10.1186/s12711-024-00883-w