Random Covariance Research Articles

Deterministic equivalent and error universality of deep random features learning*

Abstract This manuscript considers the problem of learning a random Gaussian network function using a fully connected network with frozen intermediate layers and trainable readout layer. This problem can be seen as a natural generalization of the widely studied random features model to deeper architectures. First, we prove Gaussian universality of the test error in a ridge regression setting where the learner and target networks share the same intermediate layers, and provide a sharp asymptotic formula for it. Establishing this result requires proving a deterministic equivalent for traces of the deep random features sample covariance matrices which can be of independent interest. Second, we conjecture the asymptotic Gaussian universality of the test error in the more general setting of arbitrary convex losses and generic learner/target architectures. We provide extensive numerical evidence for this conjecture. In light of our results, we investigate the interplay between architecture design and implicit regularization.

Investigating Functional Connectivity in Adolescent Depression and Suicide Attempt during Neurofeedback Sessions: A Multivariate Random Covariance Model Approach

Background A recent neurofeedback functional magnetic resonance imaging (NF, fMRI) study on depressed vs. healthy adolescents elicited differential functional connectivity (FC) amongst brain regions of interest (ROIs). Previous results employed univariate methods and included only two seed areas of FC (amygdala and hippocampus). In this study, we propose a new multivariate analysis for whole-network FC estimation. Methods Primary analyses concerned a pre-identified network of 17 salient ROIs reflecting key regions in self-processing and emotion regulation. A random covariance model (RCM) was applied to jointly estimate participant- and group-specific connectivity, where FC was measured by partial correlation conditioned on or adjusted for rest-of-network connectivity patterns. Secondary analyses concerned participant-specific network association with mental functioning changes and the AAL3 whole-brain atlas. Results New findings suggested that depressed adolescents with a suicide attempt expressed significantly higher positive FC between the left temporal gyrus and the left amygdala during NF, compared to negative FC in non-attempting depressed youth, while healthy controls displayed negative FC between the insula, inferior frontal gyrus to inferior parietal lobe connection, compared to mild negative connectivity in depressed adolescents. Previous cross-hemispheric findings in depressed vs. healthy adolescents were corroborated. Conclusion A multivariate RCM uncovered key ROI-pairwise connections differentiating FC patterns between depressed youth vs. healthy controls and among depressed youth, with and without a suicide attempt. Findings were strengthened by enhanced inference vs. univariate methods, and corroboration of previous NF secondary analyses demonstrated future utility for participant-specific study in association with clinical outcomes and/or whole-brain analyses with larger sample sizes.

Some asymptotic results of a kNN conditional mode estimator for functional stationary ergodic data

. This article investigates the conditional mode estimation of a univariate response variable given a functional random covariate (i.e., valued in some infinite-dimensional space) whenever a stationary ergodic data are considered. We construct a new estimator of the conditional mode which is constructed by a kNN approach. Under less restrictive conditions, we show the strong consistency of the proposed estimator. To assess the efficiency of the developed estimator, empirical analysis as well as real data analyses are performed.

Dialysate Sodium Levels, Ambulatory Aortic Blood Pressure and Arterial Stiffness in Hemodialysis Patients.

Increased aortic blood-pressure(BP) and arterial stiffness are associated with higher cardiovascular risk in hemodialysis. Previous works showed that lower dialysate sodium is associated with lower brachial-BP; data on aortic-BP and arterial stiffness are limited. This study aimed to investigate the effects of different dialysate sodium concentrations on 72-h aortic-BP and arterial stiffness parameters in hemodialysis patients. This analysis is part of a prospective, non-randomized interventional study. Twenty-five hemodialysis patients underwent a set of three periods of different dialysate sodium concentrations; six sessions with dialysate sodium of 137meq/L, followed consecutively by six sessions with 139meq/L and, finally, six sessions with 141meq/L. At the start of the sixth hemodialysis session on each sodium concentration, 72-h ABPM (Mobil-O-Graph) was performed to evaluate aortic-BP and arterial stiffness indices during the overall 72-h, different 24-h, day-time and night-time periods. Mean 72-h aortic SBP/DBP were higher with increasing dialysate sodium concentrations (137meq/L: 114.2±15.3/77.0±11.8mmHg vs 139meq/L: 115.4±17.3/77.9±14.0mmHg vs 141meq/L: 120.5±18.4/80.5±14.7mmHg, p=0.002/p=0.057, respectively). Wave-reflections parameters (AIx, AIx(75), AP) were not significantly different between the three dialysate sodium concentrations. Ambulatory PWV was significantly higher with increasing dialysate sodium concentrations (137meq/L: 8.5±1.7m/s vs 139meq/L: 8.6±1.6m/s vs 141meq/L: 8.8±1.6m/s, p<0.001). In generalized linear-mixed-models including 72-h brachial SBP as random covariate, the adjusted marginal-means of 72-h PWV were not significantly different between-groups. This study showed that lower dialysate sodium concentrations are associated with significant decreases in ambulatory aortic BP and PWV. This study showed that higher dialysate sodium concentrations are associated with significant increases in ambulatory aortic BP and PWV. These findings further support the need for modification of dialysate sodium concentration in hemodialysis.

Asymptotic normality of local linear functional regression estimator based upon censored data

This article is concerned with the local linear non parametric regression estimator of a right censored scalar response random variable, given a functional random covariate. First, we investigate its asymptotic normality under the concentration properties of small ball probability functions. Then, we apply the results to present the asymptotic (1−α) confidence interval of the regression function for 0 < α < 1 . To lend further support to our theoretical results, we carry out a simulation study to illustrate the performance and the asymptotic Gaussian behavior of our estimator. Finally, an application to real data is provided.

#1555 The effects of different interdialytic intervals on cardiorespiratory fitness assessed with CPET in hemodialysis patients

Abstract Background and Aims Long interdialytic interval in thrice-weekly hemodialysis is associated with excess cardiovascular and all-cause mortality risk. Impaired cardiorespiratory fitness is a strong predictor of mortality in hemodialysis. This study investigated differences in cardiorespiratory fitness assessed with cardiopulmonary exercise testing (CPET) between the end of the 2-day and the 3-day interdialytic interval. Method A total of 28 hemodialysis patients, randomized in two different sequences of evaluation underwent CPET and spirometry examination at the end of the 2-day and the 3-day intervals. The primary outcome was the difference in oxygen uptake at peak exercise [VO2peak (ml/kg/min)] assessed with CPET. Volume status was assessed with interdialytic-weight-gain (IDWG), lung ultrasound, bioimpedance spectroscopy (BIS), and inferior vena cava measurements. A total of 14 age- and sex-matched controls were also evaluated. Results Hemodialysis patients at the end of both 2-day and 3-day intervals presented lower values in all major CPET parameters than controls. VO2peak (ml/kg/min) was significantly higher at the end of the 2-day than the 3-day interval (15.24 ± 4.18 vs 13.64 ± 2.76; p &amp;lt; 0.001); similar were the results for VO2peak (ml/min) (1188 ± 257 vs 1074 ± 224; p &amp;lt; 0.001) and VO2peak (%predicted) (58.9 ± 9.2 vs 52.3 ± 8.6; p &amp;lt; 0.001). Numerical but no statistically significant differences were detected in VO2-anaerobic-threshold(ml/kg/min) and VO2-anaerobic-threshold(ml/min) between the two time-points. Maximal work load (90.1 ± 23.2 vs 79.3 ± 25.1; p &amp;lt; 0.001), exercise duration, HR at peak exercise and oxygen pulse also showed lower values at the end of the 3-day interval. FEV1 levels were similar between the two evaluations. Generalized-linear-mixed-model analysis including IDWG as random covariate attenuated the observed differences in VO2peak (ml/kg/min). Changes in BIS-derived overhydration indexes were moderately correlated with changes of VO2peak (ml/kg/min). Conclusion The 3-day interval was associated with further impairment of VO2 at peak exercise. This effect was predominantly driven by excess fluid accumulation during the extra interdialytic day.

THE EFFECTS OF DIFFERENT INTERDIALYTIC INTERVALS ON CARDIORESPIRATORY FITNESS ASSESSED WITH CARDIOPULMONARY EXERCISE TESTING IN HEMODIALYSIS PATIENTS

Objective: Long interdialytic interval in thrice-weekly hemodialysis is associated with excess cardiovascular and all-cause mortality risk. Impaired cardiorespiratory fitness is a strong predictor of mortality in hemodialysis. This study investigated differences in cardiorespiratory fitness assessed with cardiopulmonary exercise testing (CPET) between the end of the 2-day and the 3-day interdialytic interval. Design and method: A total of 28 hemodialysis patients, randomized in two different sequences of evaluation underwent CPET and spirometry examination at the end of the 2-day and the 3-day intervals. The primary outcome was the difference in oxygen uptake at peak exercise [VO2peak(ml/kg/min)] assessed with CPET. Volume status was assessed with interdialytic-weight-gain (IDWG), lung ultrasound, bioimpedance spectroscopy (BIS), and inferior vena cava measurements. A total of 14 age- and sex-matched controls were also evaluated. Results: Hemodialysis patients at the end of both 2-day and 3-day intervals presented lower values in all major CPET parameters than controls. VO2peak(ml/kg/min) was significantly higher at the end of the 2-day than the 3-day interval (15.24±4.18 vs 13.64±2.76;p&lt;0.001); similar were the results for VO2peak(ml/min) (1188±257 vs 1074±224;p&lt;0.001) and VO2peak(%predicted) (58.9±9.2 vs 52.3±8.6;p&lt;0.001). Numerical but no statistically significant differences were detected in VO2-anaerobic-threshold(ml/kg/min) and VO2-anaerobic-threshold(ml/min) between the two time-points. Maximal work load (90.1±23.2 vs 79.3±25.1;p&lt;0.001), exercise duration, HR at peak exercise and oxygen pulse also showed lower values at the end of the 3-day interval. FEV1 levels were similar between the two evaluations. Generalized-linear-mixed-model analysis including IDWG as random covariate attenuated the observed differences in VO2peak(ml/kg/min). Changes in BIS-derived overhydration indexes were moderately correlated with changes of VO2peak(ml/kg/min). Conclusions: The 3-day interval was associated with further impairment of VO2 at peak exercise. This effect was predominantly driven by excess fluid accumulation during the extra interdialytic day.

A data-adaptive method for investigating effect heterogeneity with high-dimensional covariates in Mendelian randomization

BackgroundMendelian randomization is a popular method for causal inference with observational data that uses genetic variants as instrumental variables. Similarly to a randomized trial, a standard Mendelian randomization analysis estimates the population-averaged effect of an exposure on an outcome. Dividing the population into subgroups can reveal effect heterogeneity to inform who would most benefit from intervention on the exposure. However, as covariates are measured post-“randomization”, naive stratification typically induces collider bias in stratum-specific estimates.MethodWe extend a previously proposed stratification method (the “doubly-ranked method”) to form strata based on a single covariate, and introduce a data-adaptive random forest method to calculate stratum-specific estimates that are robust to collider bias based on a high-dimensional covariate set. We also propose measures based on the Q statistic to assess heterogeneity between stratum-specific estimates (to understand whether estimates are more variable than expected due to chance alone) and variable importance (to identify the key drivers of effect heterogeneity).ResultWe show that the effect of body mass index (BMI) on lung function is heterogeneous, depending most strongly on hip circumference and weight. While for most individuals, the predicted effect of increasing BMI on lung function is negative, it is positive for some individuals and strongly negative for others.ConclusionOur data-adaptive approach allows for the exploration of effect heterogeneity in the relationship between an exposure and an outcome within a Mendelian randomization framework. This can yield valuable insights into disease aetiology and help identify specific groups of individuals who would derive the greatest benefit from targeted interventions on the exposure.

A Novel Method for Analyzing the Spatiotemporal Characteristics of GNSS Time Series: A Case Study in Sichuan Province, China

The motion of a continuously operating reference station is usually dominated by the long-term crustal motions of the tectonic block on which the station is located. Monitoring changes in the coordinates of reference stations located at tectonic plate boundaries allows for the calculation of velocity fields that reflect the spatial and temporal characteristics of the region. This study analyzes the spatiotemporal relationships of regional reference frame points with GNSS data from 25 reference stations in Sichuan, China, from 2015 to 2021. The common mode errors are extracted and eliminated by principal component analysis. A time series function model is developed for the reference stations and their constituent baselines for calculating the velocity field. Subsequently, the spatiotemporal characteristics of the regional reference frame in Sichuan is analyzed by a stochastic model. The results show that the influences of the common mode error on the horizontal and vertical directions of the reference stations is 2.5 mm and 4.3 mm, respectively. Generally, the horizontal motion of the reference stations in the Sichuan region tends to be in the southeast direction and the vertical motion trend is mainly uplifting. The east–west and vertical components of the baseline tend to be shortened, and the random influence among the reference stations is larger in the north–south and east–west directions—0.39 mm and 0.54 mm, respectively. Polynomial functions are more appropriate for constructing the fitted random influence covariance model.

Almost sure asymptotic representation for the conditional copula estimator

. This note establishes the almost sure asymptotic representation for the conditional copula estimator based on multivariate observations and a random covariate. Some auxiliary results on almost sure convergence are of independent interest.

Multivariate nearest‐neighbors Gaussian processes with random covariance matrices

AbstractWe propose a non‐stationary spatial model based on a normal‐inverse‐Wishart framework, conditioning on a set of nearest‐neighbors. The model, called nearest‐neighbor Gaussian process with random covariance matrices is developed for both univariate and multivariate spatial settings and allows for fully flexible covariance structures that impose no stationarity or isotropic restrictions. In addition, the model can handle duplicate observations and missing data. We consider an approach based on integrating out the spatial random effects that allows fast inference for the model parameters. We also consider a full hierarchical approach that leverages the sparse structures induced by the model to perform fast Monte Carlo computations. Strong computational efficiency is achieved by leveraging the adaptive localized structure of the model that allows for a high level of parallelization. We illustrate the performance of the model with univariate and bivariate simulations, as well as with observations from two stationary satellites consisting of albedo measurements.

Moment inequalities for multinomial choice with fixed effects

This paper proposes a new approach to identification of the semiparametric multinomial choice model with fixed effects. The framework employed is the semiparametric version of the traditional multinomial logit with the fixed‐effects model (Chamberlain (1980)). This semiparametric multinomial choice model places no restrictions on either the joint distribution of the random utility disturbances across choices or their within group (or across time) correlations. We show that a novel within‐group comparison leads to a set of conditional moment inequalities. Our main finding shows that the derived conditional moment inequalities yield the sharp identified set for the random utility covariate index, while avoiding the incidental parameter problem. Specializing this result to the binary choice case shows that Manski (1987)'s conditional moment inequalities still lead to sharp bounds without restrictions on covariates.

Enhancing second-order blind identification for underdetermined operational modal analysis through bandlimited source separation

Blind source separation (BSS) has been demonstrated to be a promising method for operational modal analysis (OMA). However, the conventional BSS-based modal identification methods can only handle the determined or overdetermined identification where the number of sensors must equal or exceed that of active modes, limiting their wider application. Furthermore, the existing methods for underdetermined BSS require sparsity assumption of sources, optimization of model parameters, or structural prior information, undermining the non-parametric and “blind” properties of BSS. This paper proposes a novel method for underdetermined OMA. Firstly, the bandlimited source separation using sinc-dictionaries is performed to efficiently transform the underdetermined problem into several determined or overdetermined ones. An improved second-order blind identification (SOBI) involving Hilbert transform and random time-lagged covariances is then applied for each bandlimited signal to recover modal responses and mode shapes. To attenuate noise, an adaptive singular spectrum analysis (SSA) can be optionally applied to the recovered modal responses for the highly noisy systems. Finally, the natural frequencies and damping ratios are estimated by the single-degree-of-freedom (SDOF) fitting technique. Several numerical examples, an experimental frame case, and a field test of a pedestrian bridge are studied to verify the effectiveness of the proposed strategy.

Calibrating Sales Forecasts in a Pandemic Using Competitive Online Nonparametric Regression

Motivated by our collaboration with Anheuser-Busch InBev (AB InBev), a consumer packaged goods (CPG) company, we consider the problem of forecasting sales under the coronavirus disease 2019 (COVID-19) pandemic. Our approach combines nonparametric regression, game theory, and pandemic modeling to develop a data-driven competitive online nonparametric regression method. Specifically, the method takes the future COVID-19 case estimates, which can be simulated via the susceptible-infectious-removed (SIR) epidemic model as an input, and outputs the level of calibration for the baseline sales forecast generated by AB InBev. In generating the calibration level, we focus on an online learning setting where our algorithm sequentially predicts the label (i.e., the level of calibration) of a random covariate (i.e., the current number of active cases) given past observations and the generative process (i.e., the SIR epidemic model) of future covariates. To provide robust performance guarantee, we derive our algorithm by minimizing regret, which is the difference between the squared [Formula: see text]-norm associated with labels generated by the algorithm and labels generated by an adversary and the squared [Formula: see text]-norm associated with labels generated by the best isotonic (nondecreasing) function in hindsight and the adversarial labels. We develop a computationally efficient algorithm that attains the minimax-optimal regret over all possible choices of the labels (possibly non-i.i.d. and even adversarial). We demonstrate the performances of our algorithm on both synthetic and AB InBev’s data sets of three different markets (each corresponds to a country) from March 2020 to March 2021. The AB InBev’s numerical experiments show that our method is capable of reducing the forecast error in terms of weighted mean absolute percentage error (WMAPE) and mean squared error (MSE) by more than 37% for the company. This paper was accepted by J. George Shanthikumar, data science. Funding: This work was partially supported by the Massachusetts Institute of Technology Data Science Lab and AB-InBev Corporation. Supplemental Material: The e-companion and data are available at https://doi.org/10.1287/mnsc.2023.4969 .

Estimation of zero-inflated bivariate Poisson regression with missing covariates

Zero-inflated bivariate Poisson regression (ZIBP) models are most often applied to correlated bivariate count data that contain a large proportion of zeros. These models have been applied in various fields, such as medical research, health economics, insurance, sports, etc. Because, in practice, some of the covariates involved in ZIBP regression modeling often have missing values, we propose methods for estimating the parameters of the ZIBP regression model with missing at random covariates. Assuming that the selection probability is unknown and estimated non parametrically (by a kernel estimator), we propose inverse probability weighting (IPW) and multiple imputation (MI) methods for estimating the parameters of the ZIBP regression model with missing at random covariates. Asymptotic properties of the proposed estimators are studied under certain regularity conditions. A simulation study is conducted to evaluate the performance of the proposed methods. Finally, the practical application of the proposed methodology is illustrated with data from the health-care field.

Regression-based rectangular multivariate normal prediction regions for setting reference regions

Reference intervals are among the most widely used decision-making tools in the medical field, and are invaluable in the interpretation of laboratory test results. Moreover, when several biochemical analytes are needed to diagnose the same condition, a multivariate reference region is necessary to take the correlations among the analytes into account. Traditionally, multivariate reference regions have been constructed as ellipsoidal in shape. Since such regions are difficult to interpret and cannot detect the outlyingness of a specific analyte, many authors prefer rectangular regions. Reference intervals and regions may also depend on covariates such as age and sex. The covariate values are often random quantities since they are typically not controlled in reference interval determination studies. In this study, we propose procedures to construct rectangular multivariate reference regions that incorporate random covariate information. The reference regions are computed in a multivariate normal setting, using a prediction region criterion. A parametric bootstrap approach is employed to compute the prediction factor. Numerical results show that the parametric bootstrap approach is quite accurate, with coverage probabilities close to the desired nominal value. Finally, we apply the proposed methods to real-life data from a study to compute covariate-dependent reference regions for insulin-like growth factors.

Effect of reduced versus usual lipid emulsion dosing on bilirubin neurotoxicity and neurodevelopmental impairment in extremely preterm infants: study protocol for a randomized controlled trial

BackgroundBilirubin neurotoxicity (BN) occurs in premature infants at lower total serum bilirubin levels than term infants and causes neurodevelopmental impairment. Usual dose lipid infusions in preterm infants may increase free fatty acids sufficiently to cause bilirubin displacement from albumin, increasing passage of unbound bilirubin (UB) into the brain leading to BN and neurodevelopmental impairment not reliably identifiable in infancy. These risks may be influenced by whether cycled or continuous phototherapy is used to control bilirubin levels.ObjectiveTo assess differences in wave V latency measured by brainstem auditory evoked responses (BAER) at 34–36 weeks gestational age in infants born ≤ 750 g or < 27 weeks’ gestational age randomized to receive usual or reduced dose lipid emulsion (half of the usual dose) irrespective of whether cycled or continuous phototherapy is administered.MethodsPilot factorial randomized controlled trial (RCT) of lipid dosing (usual and reduced) with treatment groups balanced between cycled or continuous phototherapy assignment. Eligible infants are born at ≤ 750 g or < 27 weeks’ gestational age enrolled in the NICHD Neonatal Research Network RCT of cycled or continuous phototherapy. Infants will randomize 1:1 to reduced or usual dose lipid assignment during the first 2 weeks after birth and stratified by phototherapy assignment. Free fatty acids and UB will be measured daily using a novel probe. BAER testing will be performed at 34–36 weeks postmenstrual age or prior to discharge. Blinded neurodevelopmental assessments will be performed at 22–26 months. Intention-to-treat analyses will be performed with generalized linear mixed models with lipid dose and phototherapy assignments as random effects covariates, and assessment for interactions. Bayesian analyses will be performed as a secondary analysis.DiscussionPragmatic trials are needed to evaluate whether lipid emulsion dosing modifies the effect of phototherapy on BN. This factorial design presents a unique opportunity to evaluate both therapies and their interaction. This study aims to address basic controversial questions about the relationships between lipid administration, free fatty acids, UB, and BN. Findings suggesting a reduced lipid dose can diminish the risk of BN would support the need for a large multicenter RCT of reduced versus usual lipid dosing.Trial registrationClinical Trials.gov, NCT04584983, Registered 14 October 2020, https://clinicaltrials.gov/ct2/show/NCT04584983Protocol version: Version 3.2 (10/5/2022).

Blinded sample size recalculation in multiple composite population designs with normal data and baseline adjustments.

The increasing interest in subpopulation analysis has led to the development of various new trial designs and analysis methods in the fields of personalized medicine and targeted therapies. In this paper, subpopulations are defined in terms of an accumulation of disjoint population subsets and will therefore be called composite populations. The proposed trial design is applicable to any set of composite populations, considering normally distributed endpoints and random baseline covariates. Treatment effects for composite populations are tested by combining p-values, calculated on the subset levels, using the inverse normal combination function to generate test statistics for those composite populations while the closed testing procedure accounts for multiple testing. Critical boundaries for intersection hypothesis tests are derived using multivariate normal distributions, reflecting the joint distribution of composite population test statistics given no treatment effect exists. For sample size calculation and sample size, recalculation multivariate normal distributions are derived which describe the joint distribution of composite population test statistics under an assumed alternative hypothesis. Simulations demonstrate the absence of any practical relevant inflation of the type I error rate. The target power after sample size recalculation is typically met or close to beingmet.

Bayesian Detectors for Wideband Radar Target With Range Walking in Compound Gaussian Clutter

In this paper, the problem of the wideband radar target detection with the range walking in compound Gaussian clutter without the secondary data is addressed. The wideband radar target return with the range walking is represented as a canonical linear model in the multi-rank subspace, and the clutter is modeled as a two-dimensional random process with separable complex inverse Wishart distributed random covariance matrices in the pulse and range-frequency domains. The generalized likelihood ratio test (GLRT), the Wald test and the Rao test based on the above target and clutter models are designed under the Bayesian framework. The Laplace approximation is adopted to compute the second-kind confluent hypergeometric function of matrix argument (SKCHFMA) in the proposed detectors. Finally, the performance of the detectors is validated by the simulations.

Random regression models to estimate genetic parameters for milk yield, fat, and protein contents in Tunisian Holsteins.

This study aimed to find the parsimonious random regression model (RRM) to evaluate the genetic potential for milk yield (MY), fat content (FC), and protein content (PC) in Tunisian Holstein cows. For this purpose, 551,139; 331,654; and 302,396 test day records for MY, FC, and PC were analysed using various RRMs with different Legendre polynomials (LP) orders on additive genetic (AG) and permanent environmental (PE) effects, and different types of residual variances (RV). The statistical analysis was performed in a Bayesian framework with Gibbs sampling, and the model performances were assessed, mainly, on the predictive ability criteria. The study found that the optimal model for evaluating these traits was an RRM with a third LP order and nine classes of heterogeneous RV. In addition, the study found that heritability estimates for MY, FC, and PC ranged from 0.11 to 0.22, 0.11 to 0.17, and 0.12 to 0.18, respectively, indicating that genetic improvement should be accompanied by improvements in the production environment. The study also suggested that new selection rules could be used to modify lactation curves by exploiting the canonical transformation of the random coefficient covariance (RC) matrix or by using the combination of slopes of individual lactation curves and expected daily breeding values.