Restricted maximum-likelihood method for learning latent variance components in gene expression data with known and unknown confounders

Random effects models are popular statistical models for detecting and correcting spurious sample correlations due to hidden confounders in genome-wide gene expression data. In applications where some confounding factors are known, estimating simultaneously the contribution of known and latent variance components in random effects models is a challenge that has so far relied on numerical gradient-based optimizers to maximize the likelihood function. This is unsatisfactory because the resulting solution is poorly characterized and the efficiency of the method may be suboptimal. Here we prove analytically that maximum-likelihood latent variables can always be chosen orthogonal to the known confounding factors, in other words, that maximum-likelihood latent variables explain sample covariances not already explained by known factors. Based on this result we propose a restricted maximum-likelihood method which estimates the latent variables by maximizing the likelihood on the restricted subspace orthogonal to the known confounding factors, and show that this reduces to probabilistic PCA on that subspace. The method then estimates the variance-covariance parameters by maximizing the remaining terms in the likelihood function given the latent variables, using a newly derived analytic solution for this problem. Compared to gradient-based optimizers, our method attains greater or equal likelihood values, can be computed using standard matrix operations, results in latent factors that don’t overlap with any known factors, and has a runtime reduced by several orders of magnitude. Hence the restricted maximum-likelihood method facilitates the application of random effects modelling strategies for learning latent variance components to much larger gene expression datasets than possible with current methods.

Systematic factors and individual variation affecting litter size of Boer and Jawarandu goat cross

We studied the systematic factors and individual variation affecting litter size in the crossbreds between Boer and Jawarandu goat. The data were obtained from the records of litter size of Boerja goats from 2012 to 2015. The systematic factors consisted of season and year of birth, doe breeds and the kid’s sex; along with individual data including pedigree, date of birth, and parental breeds. The data consisted of 107 Boer does, 687 Jawarandu does, and 495 Boerja does with a total of 3804 kids. A linear model was developed to account the effect of systematic factors on litter size of Boerja goats. Later, a mixed model was solved with Restricted Maximum Likelihood (REML) method to estimate the individual variations on litter size. The results showed that litter size trait in goat was influenced by doe breed (P<0.05). Individual variation of this trait was also high (46%). Based on this research, it can be concluded that litter size of Boer goats and their crosses were affected by the doe’s breed with high individual variation. Doe’s selection is potential to improve liter size in goat crossbred population in the future.

GENETIC PARAMETERS OF GIROLANDO CROSSBRED COWS IN DAIRY HERDS IN THE STATE OF ACRE, BRAZIL

This study aimed at estimating genetic parameters for milk production and conformation characteristics in Girolando crossbred dairy cows reared in the High and Low Acre region using the restricted maximum likelihood methodology, under an animal model. We estimated the variance components and genetic parameters using the REML/BLUP procedure (Restricted Maximum Likelihood Methodology/Best Linear Unbiased Prediction). The estimated average for milk production for 305 days of lactation (P305) was of 1523.25 ± 481.11 kg, with a heritability of 0.38 for this characteristic. The conformation characteristics showed no significant correlation with milk production. The phenotypical correlations between the linear characteristics of type were, in general, positive and moderate. The P305 obtained in this study can be considered low and indicates that there is a possibility of increasing milk production through selection in herds along with the use of tested and proven bulls. The heritability estimate found (0.38) indicates that there is genetic variability for milk production, demonstrating that selection for this characteristic would result in genetic progress.

PENDUGAAN PARAMETER GENETIK PERFORMA PRASAPIH DOMBA GARUT DI UPTD-BPPTDK MARGAWATI GARUT

Pengembangan sumber daya genetik ternak lokal penting dilakukan untuk memenuhi permintaan dagingdomba yang tinggi di Indonesia khususnya Jawa Barat melalui kegiatan seleksi bibit unggul Domba garut.Penelitian ini bertujuan untuk mengetahui nilai parameter genetik sifat-sifat prasapih Domba garut. Sifat-sifat yang dianalisis pada penelitian ini yaitu bobot lahir (B0), bobot 30, 60, 90 hari (B30, B60, B90), dan bobot sapih pada 100 hari (B100) Domba garut di UPTD-BPPTDK Margawati Garut yang berasal dari 104 ekor pejantan, 1.809 ekor induk, 2.921 ekor anak domba jantan, dan 2.632 ekor anak domba betina. Total catatan yang dianalisis yaitu 27.019 catatan bobot badan yang terdiri atas 6.559 catatan B0, 5.702 catatan B30, 5.248 catatan B60, 4.843 catatan B90 hari, dan 4.667 catatan B100 Tahun 2012-2019. Analisis data menggunakan Restricted Maximum Likelihood (REML) untuk menduga heritabilitas dengan software Variance Components Estimation (VCE) 6.0, menggunakan model maternal genetic effect (m2) dan lingkungan bersama (c2). Efek tetap yang dimasukkan ke dalam analisis yaitu jenis kelamin dan tipe kelahiran. Hasil penelitian menunjukkan bahwa nilai heritabilitas B0, B30, B60, B90, dan B100 menggunakan model maternal genetic effect dan lingkungan bersama yaitu sebesar 0,133±0,04, 0,108±0,03, 0,099±0,03, 0,122±0,03, 0,123±0,03, artinya nilai-nilai heritabilitas tersebut masuk dalam kategori rendah. Nilai maternal genetic effect dan lingkungan bersama B0, B30, B60, B90, dan B100 Domba garut 0,095±0,03, 0,163±0,03, 0,137±0,03, 0,113± 0,02, 0,115±0,02 dan 0,455±0,16, 0,268±0,13, 0,274±0,13 0,269±0,13 0,278±0,12. Hal ini menunjukkan bahwa pendugaan parameter genetik lebih akurat jika melibatkan maternal genetic effect dan lingkungan bersama.

ESTIMASI RATA-RATA LAMA SEKOLAH TINGKAT KECAMATAN DI KABUPATEN PADANG PARIAMAN DENGAN METODE EMPIRICAL BEST LINEAR UNBIASED PREDICTOR

Rata-rata lama sekolah penduduk umur 25 tahun ke atas merupakan salah satu indikator yang menggambarkan tingkat pendidikan penduduk secara keseluruhan. Dari 19 kabupaten/kota di Sumatera Barat, Kabupaten Padang Pariaman memiliki rata-rata lama sekolah terendah kedua setelah Kabupaten Kepulauan Mentawai. Penanganan rendahnya rata-rata lama sekolah membutuhkan tersedianya data rata-rata lama sekolah yang up to date dan menjangkau level wilayah yang kecil seperti kecamatan dan desa/nagari, agar kebijakan yang diambil pemerintah bisa tepat sasaran. Ketersediaan data tersebut belum mampu diakomodir oleh Badan Pusat Statistik (BPS), karena survei yang dilakukan oleh BPS dirancang untuk pendugaan data area besar, yaitu provinsi dan kabupaten. Salah satu solusi untuk masalah tersebut adalah dengan menggunakan metode estimasi tidak langsung, yaitu Small Area Estimation (SAE). Salah satu estimasi parameter secara tidak langsung berbasiskan model SAE adalah Empirical Best Linear Unbiased Predictor (EBLUP). Tujuan penelitian ini adalah melakukan estimasi rata-rata lama sekolah tingkat kecamatan di Kabupaten Padang Pariaman menggunakan metode EBLUP dengan prosedur maximum likelihood (ML) dan prosedur restricted maximum likelihood (REML). Variabel penyerta yang digunakan dalam penelitian ini yang diduga berpengaruh terhadap variabel respon adalah rasio jumlah SLTA/sederajat per 10.000 penduduk, rata-rata jarak terhadap SLTA/sederajat dan persentase keluarga pertanian. Hasil penelitian menunjukkan SAE metode EBLUP dengan prosedur REML menghasilkan nilai estimasi rata-rata lama sekolah tingkat kecamatan di kabupaten Padang Pariaman memiliki akurasi yang lebih baik dibandingkan dengan hasil estimasi langsung (direct) dan prosedur ML.

Restricted maximum-likelihood method for learning latent variance components in gene expression data with known and unknown confounders

Linear mixed modelling is a popular approach for detecting and correcting spurious sample correlations due to hidden confounders in genome-wide gene expression data. In applications where some confounding factors are known, estimating simultaneously the contribution of known and latent variance components in linear mixed models is a challenge that has so far relied on numerical gradient-based optimizers to maximize the likelihood function. This is unsatisfactory because the resulting solution is poorly characterized and the efficiency of the method may be suboptimal. Here we prove analytically that maximumlikelihood latent variables can always be chosen orthogonal to the known confounding factors, in other words, that maximum-likelihood latent variables explain sample covariances not already explained by known factors. Based on this result we propose a restricted maximum-likelihood method which estimates the latent variables by maximizing the likelihood on the restricted subspace orthogonal to the known confounding factors, and show that this reduces to probabilistic PCA on that subspace. The method then estimates the variance-covariance parameters by maximizing the remaining terms in the likelihood function given the latent variables, using a newly derived analytic solution for this problem. Compared to gradient-based optimizers, our method attains equal or higher likelihood values, can be computed using standard matrix operations, results in latent factors that don’t overlap with any known factors, and has a runtime reduced by several orders of magnitude. We anticipate that the restricted maximum-likelihood method will facilitate the application of linear mixed modelling strategies for learning latent variance components to much larger gene expression datasets than currently possible.

Morphometric Evaluation of Phenotypic Groups of Creole Cattle of Southern Ecuador

The breeding of creole cattle from the southern region of Ecuador, also known as Criollo Lojano, is a source of economic support and work for the communities located in the remote areas of the Andes mountains in this region. These cattle are grouped into four biotypes based on their phenotypic characteristics: Negro Lojano, Encerado, Colorado, and Cajamarca or Pintado. This study analyzes the morphometric variability of these creole cattle using least squares means (LSM) and restricted maximum likelihood Restricted Maximum Likelihood (REML) variance components. The evaluation parameters used to characterize these cattle were live weight plus 15 morphometric characteristics and nine morphometric indexes. The measurements came from 151 adult animals (28 male and 123 females). With the exception of Height at Withers (P = 0.06), the other morphometric characteristics do not show significant difference among these creole biotypes. Sexual dimorphism was found in live weight, thoracic circumference, height at withers, chest width, length of thorax, length of body, depth of thorax, depth of abdomen, length of head, and length of horns (P < 0.05). The adult Creole Lojano has an average live weight of 288 ± 12.9 kg (mean ± standard error), The Cephalic index is 45.6, the Corporal index is 115.9, the Pelvic index is 90.5, the Thoracic index is 58.3, the Proportionality index is 62.6, the Thoracic Capacity index is 2.1, the Lower Leg–Thoracic index is 9.9, the Transverse Pelvic index is 34.7, and the Pelvic Length index is 38.4. This creole bovine breed presents 4 biotypes that are similar; there are differences in the analysis with respect to sex (males are higher in 10 of the 16 characteristics analyzed); and on the basis of the indexes, this animal is small, has a triangular head, is longilinear with a long and narrow hip. It is a dual-purpose milk type with the exception of the Colorado biotype which is a dual purpose meat type.