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Revista mexicana de ciencias pecuarias
On-line version ISSN 2448-6698Print version ISSN 2007-1124
Rev. mex. de cienc. pecuarias vol.14 n.1 Mérida Jan./Mar. 2023 Epub Mar 24, 2023
https://doi.org/10.22319/rmcp.v14i1.6182
Articles
Quantile regression for prediction of complex traits in Braunvieh cattle using SNP markers and pedigree
aUniversidad Autónoma Chapingo. Posgrado en Producción Animal. Carretera Federal México-Texcoco Km 38.5, 56227, Texcoco, Estado de México, México.
b Colegio de Postgraduados. Socio Economía Estadística e Informática. Carretera Federal México-Texcoco Km 36.5, 56230, Texcoco, Estado de México.
Genomic prediction models generally assume that errors are distributed as normal, independent, and identically distributed random variables with zero mean and equal variance. This is not always true, in addition there may be phenotypes distant from the population mean, which are usually removed when making the prediction. Quantile regression (QR) deals with statistical aspects such as high dimensionality, multicollinearity and phenotypic distribution different from the normal one. The objective of this work was to compare QR using marker and pedigree information with alternative methods such as genomic best linear unbiased prediction (GBLUP) and single-step genomic best linear unbiased prediction (ssGBLUP) to analyze the birth (BW), weaning (WW) and yearling (YW) weights of Braunvieh cattle and simulated data with different degrees of asymmetry and proportion of outliers. The predictive capacity of the models was assessed by cross-validation. The predictive performance of QR both with marker information alone and with information of markers plus pedigree, with the actual dataset, was better than the GBLUP and ssGBLUP methodologies for WW and YW. For BW, GBLUP and ssGBLUP were better, however, only quantiles 0.25, 0.50 and 0.75 were used, and the BW distribution was not asymmetric. In the simulated data experiment, correlations between “true” marker effects and estimated effects, as well as “true” and estimated signal correlations were higher when QR was used compared to GBLUP. The advantages of QR were more noticeable with asymmetric distribution of phenotypes and with a higher proportion of outliers, as was the case with the simulated dataset.
Key words Quantile regression; GBLUP; ssGBLUP
Los modelos de predicción genómica generalmente suponen que los errores se distribuyen como variables aleatorias normales, independientes e idénticamente distribuidas con media cero e igual varianza. Esto no siempre se cumple, además puede haber fenotipos distantes de la media poblacional, los que usualmente se eliminan al realizar la predicción. La regresión cuantil (QR) afronta aspectos estadísticos como alta dimensionalidad, multicolinealidad y distribución fenotípica diferente de la normal. El objetivo de este trabajo fue comparar QR utilizando información de marcadores y pedigrí con los métodos alternativos tales como mejor predicción lineal insesgada genómica (GBLUP) y mejor predicción lineal insesgada genómica en un solo paso (ssGBLUP) para analizar los pesos al nacimiento (PN), destete (PD) y al año (PA) de bovinos Suizo Europeo y datos simulados con diferente grado de asimetría y proporción de datos atípicos. La capacidad predictiva de los modelos se evaluó mediante validación cruzada. El desempeño predictivo de QR tanto sólo con información de marcadores como con marcadores más pedigrí, con el conjunto de datos reales, fue mejor que las metodologías GBLUP y ssGBLUP para PD y PA. Para PN GBLUP y ssGBLUP fueron mejores, sin embargo, solo se utilizaron los cuantiles 0.25, 0.50 y 0.75, y la distribución de PN no fue asimétrica. En el experimento de datos simulados, las correlaciones entre efectos de marcador “verdadero” y efectos estimados, así como las correlaciones de señales “verdaderas” y estimadas fueron más altas cuando se usó QR comparado con GBLUP. Las ventajas de QR fueron más notorias con distribución asimétrica de los fenotipos y con mayor proporción de datos atípicos, como fue el caso del conjunto de datos simulados.
Palabras clave Regresión cuantil; GBLUP; ssGBLUP
Introduction
The main motivation of the quantile regression (QR) method is that most models for genetic evaluation assume normality, which is not always true. Another problem is that sometimes phenotypic records very far from the population average are considered as recording errors or outliers and therefore removed from the analyses, seen from the genomic point of view, valuable information of markers associated with certain regions of DNA with strong influence on characteristics of interest is being lost.
With the QR method, robust results and a broad vision of the explanatory variables on the dependent ones are obtained1. The data generated from omics experiments are often complex and large, so there is a statistical challenge to extract relevant biological information from the large volume of data2,3. Using a robust approach such as QR makes inference less biased and less subject to false positives2. Recent studies using QR describe various applications such as etic association studies4, population genetics5, gene expression6,7, and genomic selection8-10.
One of the first studies where QR was used to predict individual genetic merit was presented by Nascimento et al11, who used simulated data, finding advantages when using QR compared to conventional methodologies. In the same year12, results using QR to adjust growth curves with data from pigs and molecular markers were published; not only did they successfully adjust the growth curves, but they identified important markers associated with the studied characteristic. Another similar work by the same team of researchers was presented by Nascimento et al13, but with bean data. Recently, Pérez-Rodríguez et al10 extended the quantile regression model to include pedigree information through the use of the additive genetic relationship matrix, further improving the predictive ability of the models and at the same time identifying the proportions of the variances attributed to markers, relationships between individuals and the residual, which allows a precise partitioning of the phenotypic variance to be obtained.
The objective of the present study was to study the predictive power of the quantile regression model using simulated data and actual data (birth, weaning and yearling weights) from Braunvieh cattle and the following models were considered: 1) QR with information of SNP molecular markers (QRM), 2) QR simultaneously including molecular marker information and genealogical information derived from pedigree (QRH); 3) GBLUP which, like QRM, only included molecular marker information, and 4) single-step genomic evaluation (ssGBLUP) which included marker and pedigree information.
Material and methods
Genotypes
The information used was from 300 animals (236 females, 64 males) born from 2001 to 2016 in eight herds located in Eastern, Central and Western Mexico. Hair samples were collected for genotyping by the company GeneSeek (Lincoln, https://www.neogen.com/, NE, USA), using the GeneSeek® Genomic Profiler Bovine LDv.4 panel, with 30,000 and 50,000 SNP markers, 150 animals with each Chip. Genotyping was performed on two separate occasions, initially 150 individuals with the 30K Chip and later another group of 150 individuals with the 50K Chip since the 30K Chip was not available at the time. The SNPs in common between the 30K and 50K chips (12,835 SNPs) were used. The proportions of missing values were calculated for each marker and for each individual. The average of missing values per individual was 2.09 % with a standard deviation of 7.50 %. The average call rate (not missing proportion for each marker) was 97.90 % with a standard deviation of 4.66 %. Markers with a call rate of less than 95 % were removed. The genotypes were recoded as AA= 0, AB= 1 and BB= 2, from which a matrix with 300 rows (individuals) and 12,835 columns (markers) was obtained, whose cells take values in the set
Phenotypes
The phenotypic and pedigree information of the Braunvieh cattle population was obtained from the database of the Mexican Association of Breeders of Registered Swiss Cattle. Records of birth (BW), weaning (WW) and yearling (YW) weights were used for analysis. Phenotype editing was similar for BW, WW and YW, records of animals not genetically related to those genotyped or with missing information for herd, dam’s age and management were discarded. Contemporary groups (CG) were defined by removing animals in CG of 2 individuals or with variance equal to zero. For BW, the CGs were defined by combining the effects of the herd (8 herds), year (1998 to 2016) and season of birth; the seasons of birth were defined considering the Julian calendar, from 80 to 171d, spring; from 172 to 264 d, summer; from 265 to 354 d, autumn; from 355 to 366 d and from 1 to 79 d, winter. After editing data, for BW, 330 records were obtained. For WW and YW, the CGs were defined by combining the effects of the herd (6 herds), year (from 1998 to 2016), season of birth (same as BW) and management. In the case of WW, the management groups were defined in three ways: calves fed their mother’s milk; their mother’s milk plus balanced feed; and milk from their mother and nurse plus a balanced diet. For YW, the management groups were defined in three ways: grazing animals; grazing animals with feed concentrate; and housed animals with a balanced diet. The edition of WW and YW data ended with 267 and 232 records for further analyses. Table 1 shows a summary of the number of animals genotyped, and phenotyped for BW, WW and YW. Figure 1 shows the violin plots for BW, WW and YW, the sample mean is represented by the red dot and the sample median by the horizontal line within the box, from the plot, it is clear that the response variables have an asymmetric distribution and the circles with solid filling in it suggest the presence of outliers.
Group | Birth weight | Weaning weight | Yearling weight |
---|---|---|---|
Genotyped | 300 | 300 | 300 |
Genotyped and phenotyped | 232 | 218 | 191 |
Phenotyped in QRM and GBLUP | 232 | 218 | 191 |
Phenotyped in QRH and ssGBLUP | 330 | 267 | 232 |
QRM=Quantile regression using marker information, QRH=Quantile regression using marker and pedigree information, GBLUP=Genomic best linear unbiased predictor, ssGBLUP=Single-step genomic evaluation.
Models
Quantile regression model with markers (QRM)
The model for quantile regression is:
where
where
where
where
The QR model can be extended to include other terms, in particular for growth characteristics, the following model is used:
where
GBLUP
The model is given by:
where
Single-step quantile regression (QRH) model
This method is considered an extension of the quantile model for a relationship matrix constructed using matrices of relationships for genotyped and non-genotyped animals and of which a pedigree is available. The resulting matrix is known in the literature as matrix H16,17, this matrix is given by:
where, A
gg
is a submatrix of A for genotyped animals, G
a
= β
G + α;
The QRH model is given by:
where
Single-step GBLUP regression (ssGBLUP) model
The ssGBLUP model is equivalent to the GBLUP model described above with the difference that the genomic relationship matrix G is replaced with the extended genetic relationship matrix H, it is assumed that
Cross-validation
The predictive capacity of the models was evaluated by cross-validation, which was performed as follows. The dataset was divided into five groups of the same size
Simulation
In order to evaluate the predictive power of the QR model against GBLUP, an asymmetric data simulation with the presence of outliers was also carried out; the simulation of the present work is analogous to that used by Pérez-Rodríguez et al10. The main idea is to highlight that the quantile regression model works adequately in the presence of atypical observations, inhomogeneous variances and response variables with responses with asymmetric distribution. In the context of selection, for example, it is not unusual to have asymmetric distributions for phenotypes due to the process itself, since, if one selects for some characteristic Y, and if there is in addition to this another characteristic of interest O, then the conditional distribution of Y |O>o19 is asymmetric. In the context of genomic selection, it is also common to find subsets of observations that differ significantly from the rest and these observations could be considered atypical. Montesinos-López et al20 proposed a model with Laplace errors and showed that it predicts adequately even in the presence of outliers, the proposed model is a special case of the quantile regression model that is studied in the present work. The 9,628 SNPs resulting from the quality control described above for 300 animals were considered, the simulation of the data was carried out considering the linear model:
where
Software and model fitting
The quantile regression models were fitted using a computational strategy similar to that described by Pérez-Rodríguez et al10. Adaptations of algorithms to include fixed and random effects do not present great computational difficulty. The codes for the fitting of the models were developed in the programming languages R25 and C. The codes for the fitting of the models were organized in such a way that they can be easily run from the statistical software R and are available by requesting them to the first author of the present study. In all cases, three quantiles were selected,
Results
Real data
Tables 2, 3, and 4 show the results of the experiment conducted with BW, WW, and YW data from a Braunvieh cattle population, evaluated under two scenarios 1) with marker information only, and 2) marker and pedigree information. In general, the highest correlations between observed and predicted values were obtained with QR, except for BW, where the correlations of GBLUP and ssGBLUP were higher than those obtained with QRM and QRH, however, the correlations of QRM
Model | Cor( |
MSE |
|
DIC |
---|---|---|---|---|
QRM |
0.7521 | 3.9973 | 2.7260 | 513.5014 |
(0.0753) | (1.6108) | (1.9762) | (531.5701) | |
QRM |
0.5619 | 7.3249 | 8.6297 | 970.7680 |
(0.1501) | (0.4561) | (0.2660) | (6.9791) | |
QRM |
0.7902 | 3.6535 | 2.4268 | 716.4237 |
(0.0766) | (0.0943) | (0.4829) | (35.7161) | |
GBLUP | 0.7924 | 2.3269 | 3.0035 | 803.0675 |
(0.0874) | (0.2063) | (0.5578) | (31.9814) | |
QRH |
0.6713 | 3.5026 | 2.3645 | 872.3949 |
(0.1329) | (1.2848) | (1.9670) | (432.0737) | |
QRH |
0.6816 | 2.9988 | 2.7372 | 659.1450 |
(0.1253) | (0.7769) | (1.8239) | (1079.8674) | |
QRH |
0.6981 | 4.1405 | 2.8610 | 1077.2027 |
(0.1140) | (0.6187) | (0.8666) | (60.6781) | |
ssGBLUP | 0.7055 | 2.4463 | 3.2641 | 1189.4282 |
(0.1191) | (0.2204) | (0.4244) | (26.5023) |
Cor(
Model | Cor( |
MSE |
|
DIC |
---|---|---|---|---|
QRM |
0.5661 | 476.5293 | 419.4138 | 1550.5339 |
(0.2212) | (17.4612) | (23.3216) | (13.9644) | |
QRM |
0.5695 | 357.7328 | 396.8138 | 1576.8871 |
(0.2307) | (8.9681) | (47.7433) | (21.5826) | |
QRM |
0.5493 | 175.1298 | 67.9660 | 737.2216 |
(0.2196) | (47.6181) | (82.0807) | (1150.7340) | |
GBLUP | 0.5677 | 294.5807 | 376.7794 | 1583.2355 |
(0.2377) | (36.6279) | (24.1379) | (16.2187) | |
QRH |
0.4816 | 644.1278 | 551.5150 | 1962.1296 |
(0.0672) | (50.8464) | (64.8091) | (20.9916) | |
QRH |
0.4797 | 366.5940 | 356.9005 | 1537.7760 |
(0.0274) | (56.8604) | (238.5303) | (903.3492) | |
QRH |
0.3918 | 216.1753 | 5.9471 | -706.1573 |
(0.0544) | (53.2417) | (11.7834) | (2034.7757) | |
ssGBLUP | 0.4712 | 303.0404 | 421.8316 | 1982.3314 |
(0.0502) | (37.6933) | (55.2774) | (21.9229) |
Cor(
Model | Cor( |
MSE |
|
DIC |
---|---|---|---|---|
QRM |
0.5421 | 1037.6529 | 953.6807 | 1487.1104 |
(0.1350) | (175.2648) | (261.8652) | (35.8873) | |
QRM |
0.5341 | 868.3651 | 964.4477 | 1524.0511 |
(0.1355) | (34.0429) | (113.1832) | (12.4648) | |
QRM |
0.5115 | 938.8244 | 700.7849 | 1284.0829 |
(0.1290) | (241.2205) | (465.2109) | (402.9787) | |
GBLUP | 0.5330 | 725.7579 | 924.8388 | 1526.7596 |
(0.1389) | (71.3999) | (90.0089) | (11.6346) | |
QRH |
0.5306 | 1277.9493 | 1172.2877 | 1850.7122 |
(0.1411) | (44.0948) | (108.7991) | (17.2025) | |
QRH |
0.5098 | 894.4148 | 1061.3157 | 1883.6773 |
(0.1700) | (35.3996) | (129.4702) | (15.4422) | |
QRH |
0.5027 | 915.1871 | 666.8830 | 1706.4933 |
(0.1748) | (162.7629) | (413.5046) | (209.8455) | |
ssGBLUP | 0.4712 | 778.6416 | 1071.3096 | 1891.9029 |
(0.0502) | (84.9871) | (128.2878) | (17.5592) |
Cor(
Simulated data
The results of the simulation exercise where QR is compared with GBLUP under different degrees of asymmetry and proportions of outliers are shown in Table 5. Column 2 records the correlations between the “true” marker effects and the estimated marker effects, the correlations obtained with QR were higher than those obtained with GBLUP. Column 3 shows the correlations between the “true signals” and the estimated ones, the highest correlations were obtained with QR. Column 4 records the estimation of the variance components associated with the error and column 5 the DIC values, the lowest values in both columns were obtained with QR
Model | Cor( |
Cor( |
|
DIC |
---|---|---|---|---|
| ||||
QR |
0.0784 | 0.4963 | 0.6821 | 620.5455 |
(0.0034) | (0.0336) | (0.1806) | (49.3305) | |
QR |
0.0766 | 0.4643 | 0.6644 | 665.8219 |
(0.0042) | (0.0493) | (0.0703) | (16.3032) | |
QR |
0.0606 | 0.4269 | 0.1438 | 290.6870 |
(0.0132) | (0.0386) | (0.1421) | (148.9695) | |
GBLUP | 0.0722 | 0.4910 | 0.7375 | 691.6503 |
(0.0064) | (0.0398) | (0.0723) | (19.9391) | |
| ||||
QR |
0.0614 | 0.4369 | 0.4683 | 407.6496 |
(0.0183) | (0.0329) | (0.4030) | (330.6304) | |
QR |
0.0728 | 0.4579 | 0.7947 | 706.7931 |
(0.0045) | (0.0420) | (0.1063) | (20.5797) | |
QR |
0.0574 | 0.4061 | 0.4482 | 381.4644 |
(0.0092) | (0.0399) | (0.3225) | (474.7138) | |
GBLUP | 0.0654 | 0.4556 | 0.8717 | 731.9104 |
(0.0057) | (0.0314) | (0.0890) | (21.8563) | |
| ||||
QR |
0.0773 | 0.4835 | 0.5578 | 582.4254 |
(0.0087) | (0.0562) | (0.2523) | (83.0548) | |
QR |
0.0771 | 0.4689 | 0.6369 | 662.0337 |
(0.0074) | (0.0515) | (0.0868) | (23.8018) | |
QR |
0.0598 | 0.4169 | 0.2398 | 219.1691 |
(0.0128) | (0.0450) | (0.2033) | (444.5060) | |
GBLUP | 0.0703 | 0.4804 | 0.7316 | 692.6392 |
(0.0056) | (0.0333) | (0.0831) | (24.0645) | |
| ||||
QR |
0.0731 | 0.4386 | 0.8739 | 677.0858 |
(0.0081) | (0.0789) | (0.1077) | (23.5472) | |
QR |
0.0734 | 0.4529 | 0.8154 | 711.2935 |
(0.0078) | (0.0615) | (0.0845) | (14.9809) | |
QR |
0.0541 | 0.3945 | 0.3628 | 385.6030 |
(0.0056) | (0.0583) | (0.2572) | (393.1935) | |
GBLUP | 0.0640 | 0.4491 | 0.8913 | 736.7880 |
(0.0077) | (0.0517) | (0.0654) | (14.8343) | |
| ||||
QR |
0.0615 | 0.5286 | 0.1535 | 205.6973 |
(0.0144) | (0.0271) | (0.1657) | 277.5807 | |
QR |
0.0741 | 0.5514 | 0.4860 | 614.2282 |
(0.0037) | (0.0167) | (0.0663) | 15.7647 | |
QR |
0.0467 | 0.4855 | 0.0166 | -271.4761 |
(0.0112) | (0.0150) | (0.0192) | 288.4509 | |
GBLUP | 0.0737 | 0.5428 | 0.5305 | 625.9703 |
(0.0030) | (0.0121) | (0.0353) | 11.3632 | |
| ||||
QR |
0.0768 | 0.4807 | 0.7817 | 650.8593 |
(0.0080) | (0.0687) | (0.0888) | 22.8417 | |
QR |
0.0696 | 0.4630 | 0.6154 | 511.5287 |
(0.0148) | (0.0600) | (0.3369) | 412.6645 | |
QR |
0.0507 | 0.3967 | 0.0204 | -160.1660 |
(0.0031) | (0.0505) | (0.0127) | 213.0462 | |
GBLUP | 0.0659 | 0.4649 | 0.7876 | 709.7240 |
(0.0065) | (0.0418) | (0.0528) | 14.8566 |
Cor(
Discussion
In this study, QR analysis methodologies were compared with GBLUP and ssGBLUP. This comparison was made with simulated phenotypes with different degrees of asymmetry and proportions of outliers and actual data for birth, weaning and yearling weights.
Real data
The observed and predicted phenotype correlations obtained from cross-validation with actual data were higher when using QRM and QRH in the WW and YW characteristics. For BW, the highest correlations were obtained with GBLUP and ssGBLUP; however, in this study, only three quantiles 0.25, 0.50 and 0.75 were tested, there is evidence in other studies where QR is better than GBLUP, as in the case of the work of Nascimento et al4, who compared QR with models such as BLASSO, BayesB and RR-BLUP. These authors found a 15.15 % gain in the predictive capacity of QR compared to RR-BLUP, it should be noted that, mathematically, RR-is equivalent to GBLUP, in addition to the fact that the datasets used in this experiment presented asymmetry.
The values of the mean squared error (MSE) measure the average of the squared error, that is, the difference between the estimator and what is estimated, so low values are preferred; the MSE averages of QRM and QRH were lower than those obtained with GBLUP and ssGBLUP only for WW. The residual variance estimator is an indication that how well or poorly the model fits the observed data, low values are preferred; the smallest variance components of the error were obtained with QRM and QRH for the three characteristics analyzed. Finally, the DIC value is used to select candidate models and, like MSE and error variance components, low values are preferred. The lowest DIC values were obtained with QRM
In the analysis of real data, a limitation of the present study is the sample size, which can impact the variability of the parameters estimated with the models and consequently the variability of the predictions, however, all the models were fitted using the same information and therefore the comparison of the predictive capacity of the models is considered reasonable, the ideal would be to have large sample sizes, but, due to economic limitations, this is not always possible. On the other hand, it is currently very common to use prediction models in which the number of phenotypic records is smaller than the number of predictors (SNPS), that is
Simulated data
In the simulated data experiment, the correlations between “true” marker effects and estimated effects as well as correlations of “true” and estimated signals were higher when QR was used compared to GBLUP. These results are similar to those obtained by other researchers10, who simulated data with three different coefficients of asymmetry 0.75, 0.95, 0.999 with 5 % and 10 % of outliers and found that the correlations obtained with QR were higher than those obtained with Bayesian ridge regression (BRR), in addition, this pattern was more evident with a greater asymmetry and proportion of outliers. In this study, simulations with asymmetry coefficients of 0.950, 0.975, 0.999 were carried out and the quantiles with which higher correlations were obtained varied between 0.25 and 0.50; the advantage of QR is that different quantiles can be tested, obtaining better results depending on the quantile used, this advantage in the ability to predict the effects of markers and signals has been taken advantage of by other researchers4 , who found no trait association using the traditional GWAS model of single SNP, but, when using QR with extreme quantiles such as 0.1, the model was able to find up to 7 SNPs associated with the characteristics studied.
The coefficients of variance of the error indicate how well the proposed model fits the studied data, the smaller this value, the better the fit, the DIC is another value that is used to compare candidate models. Models with a smaller DIC are preferred to models with a larger DIC24. The lowest residual variance estimators and DIC values were obtained with QR
QR performed equally or better than GBLUP and ssGBLUP to predict growth characteristics BW, WW and YW, the advantages of this method are more noticeable when the data are more biased and present a higher proportion of outliers, as in the case of the simulation experiment.
Conclusions and implications
The predictive performance of QR both with marker information alone and with information of markers plus pedigree, with the actual dataset, was better than the GBLUP and ssGBLUP methodologies for WW and YW. For BW, GBLUP and ssGBLUP were better; however, only quantiles 0.25, 0.50 and 0.75 were used, and the BW distribution was not asymmetric. In the simulated data experiment, correlations between “true” marker effects and estimated effects, as well as correlations of “true” and estimated signals were higher when QR was used compared to GBLUP. The advantages of QR were more noticeable with asymmetric distribution of phenotypes and with a higher proportion of outliers, as was the case with the simulated dataset.
Acknowledgements
To the National Council of Science and Technology, Mexico, for the financial support for the first author during his doctoral studies. The authors also thank the Mexican Association of Breeders of Registered Swiss Cattle for allowing the use of their databases, and the cooperating breeders for their kind cooperation in this study.
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Received: March 30, 2022; Accepted: August 04, 2022