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Phenotypic measurements of a trait can be informative for the genetic values of other traits, if the traits are genetically correlated with one another14,15,32. Recent studies have shown that prediction accuracy of common complex disease can be improved by estimating SNP effects for multiple traits jointly within a multivariate mixed-effects model16,17.
|
study
| 99.94 |
If k traits are measured on different individuals, with Nk observations for trait k, the elements of Eq. (4) become: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\bf y}^ \prime} = [{\bf y}\prime_{\bf 1}...{{\bf y}^ \prime_{k}}], {\bf{W}} = \left[ {\begin{array}{*{20}{c}} {{{\bf W}}_1} & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & {{\bf{W}}_k} \end{array}} \right]$$\end{document}y′=[y′1...yk′],W=W1000⋱000Wk, and R = diag[Rk] = diag\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[ {{\bf{I}}_{N_k}\sigma _{\epsilon _k}^2} \right]$$\end{document}INkσϵk2, a diagonal matrix of length \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N = \mathop {\sum }\limits_k N_k$$\end{document}N= ∑kNk. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bf{B}} = {\bf{\Sigma }}_b \otimes {\bf{I}}_M$$\end{document}B=Σb⊗IM, where Σb is a k × k matrix, with diagonal elements \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{b_k}^2$$\end{document}σbk2 and off-diagonal elements the covariances of SNP effects between traits k and l, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{b_{k,l}}$$\end{document}σbk,l. For Kronecker products, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bf{B}}^{ - 1} = {\bf{\Sigma }}_b^{ - 1} \otimes {\bf{I}}_M$$\end{document}B-1=Σb-1⊗IM and substituting these expressions directly into Eq. (6) means that multi-trait BLUP solutions for b can be obtained in Eq. (7) as:9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{MT}} - {\mathrm{BLUP}}} = \left[ {{{{\bf W}{\prime}{\bf W}}} + {\bf{\Sigma }}_\epsilon {\bf{\Sigma }}_b^{ - 1} \otimes {\bf{I}}_M} \right]^{ - 1}{{{\bf W}{\prime}{\bf y}}}$$\end{document}b^MT-BLUP=W′W+ΣϵΣb-1⊗IM-1W′ywith \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bf{\Sigma }}_\epsilon = {\mathrm{diag}}\left[ {\sigma _{\epsilon _k}^2} \right]$$\end{document}Σϵ=diagσϵk2, a diagonal k × k matrix. For a two-trait example, Eq. (9) expands to:10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{*{20}{l}} {\widehat {\bf{b}}_{\mathrm{MT-BLUP}}} \hfill & = \hfill & {\left[ {\left[ {\begin{array}{*{20}{c}} {{\bf{W}}_1\prime {\bf{W}}_1} & 0 \\ 0 & {{\bf{W}}_2\prime {\bf{W}}_2} \end{array}} \right]} \right.} \hfill \\ {} \hfill & + \hfill & {\left. {\left[ {\begin{array}{*{20}{c}} {{\bf{I}}_M\sigma _{\epsilon _1}^2} & 0 \\ 0 & {{\bf{I}}_M\sigma _{\epsilon _2}^2} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{\bf{I}}_M\sigma _{b_1}^2} & {{\bf{I}}_M\sigma _{b_{1,2}}} \\ {{\bf{I}}_M\sigma _{b_{2,1}}} & {{\bf{I}}_M\sigma _{b_2}^2} \end{array}} \right]^{ - 1}} \right]^{ - 1}} \hfill \\ {} \hfill & {} \hfill & {\left[ {\begin{array}{*{20}{c}} {{\bf{W}}_1} & 0 \\ 0 & {{\bf{W}}_2} \end{array}} \right]\prime \left[ {\begin{array}{*{20}{c}} {{\bf{y}}_1} \\ {{\bf{y}}_2} \end{array}} \right]} \hfill \end{array}$$\end{document}b^MT-BLUP=W1′W100W2′W2+IMσϵ1200IMσϵ22IMσb12IMσb1,2IMσb2,1IMσb22-1-1W100W2′y1y2
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other
| 65.5 |
Estimating SNP effects for multiple traits jointly in Eq. (9) requires individual-level genotype and phenotype data across a range of common complex diseases and quantitative phenotypes, which are not readily available in human medical genetics due to privacy concerns and data sharing restrictions. Additionally, Eq. (9) requires a series of computationally intensive M × k equations to be solved. However, these issues can be overcome by approximating Eq. (9) using publically available GWAS summary statistic data and an independent genomic reference sample.
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study
| 99.94 |
Single-trait approximate BLUP SNP effects can be obtained from GWAS summary statistics (SBLUP: summary statistic approximate BLUP) by replacing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bf{W}}_k\prime {\bf{W}}_k$$\end{document}Wk′Wk and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bf{W}}_k\prime {\bf{y}}_k$$\end{document}Wk′yk of Eq. (8) by their expectation, which are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Bbb E}\left[ {{\bf{W}}_k\prime {\bf{W}}_k} \right] = N_k{\bf{L}}$$\end{document}EWk′Wk=NkL and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Bbb E}\left[ {{\bf{W}}_k\prime {\bf{y}}_k} \right] = N_k\widehat {\bf{b}}_{{\mathrm{OLS}}_k}$$\end{document}EWk′yk=Nkb^OLSk, respectively, where L is an M × M scaled SNP LD correlation matrix estimated from a reference SNP data set and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{OLS}}_k}$$\end{document}b^OLSk are obtained from publically available GWAS summary statistics20. GWAS summary statistics report effect estimates of SNPs on an unstandardised scale and not \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{OLS}}}$$\end{document}b^OLS as it is defined here. To obtain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{OLS}}}$$\end{document}b^OLS from GWAS summary statistics, the effect of each SNP must be multiplied by the standard deviation of each SNP: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{OLS}}_{\mathrm{j}}}$$\end{document}b^OLSj = \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{OLS}} - {\mathrm{UNSCALED}}_{\mathrm{j}}} \times \sqrt {2p_j\left( {1 - p_j} \right)}$$\end{document}b^OLS-UNSCALEDj×2pj1-pj. Equation (8) can then be written as:11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{*{20}{l}} {\widehat {\bf{b}}_{{\mathrm{SBLUP}}_k}} \hfill & = \hfill & {\left[ {N_k{\bf{L}} + {\bf{I}}_M\lambda _k} \right]^{ - 1}N_k\widehat {\bf{b}}_{{\mathrm{OLS}}_k}} \hfill \\ {} \hfill & = \hfill & {\left[ {{\bf{L}} + {\bf{I}}_M\lambda _k{\mathrm{/}}N_k} \right]^{ - 1}\widehat {\bf{b}}_{{\mathrm{OLS}}_k}} \hfill \end{array}$$\end{document}b^SBLUPk=NkL+IMλk-1Nkb^OLSk=L+IMλk∕Nk-1b^OLSk
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study
| 99.9 |
The shrinkage parameter is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _k = \sigma _{{\it{\epsilon }}_k}^2{\mathrm{/}}\sigma _{b_k}^2$$\end{document}λk=σϵk2∕σbk2 = \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M\sigma _{{\it{\epsilon }}_k}^2{\mathrm{/}}h_{{\rm SNP}_k}^2$$\end{document}Mσϵk2∕hSNPk2 = \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M\left( {1 - h_{{\rm SNP}_k}^2} \right){\mathrm{/}}h_{{\rm SNP}_k}^2$$\end{document}M1-hSNPk2∕hSNPk2, under the assumption of phenotypic variance of 1 that makes the proportion of phenotypic variance of trait k attributable to the SNPs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{{\rm SNP}_k}^2 = M\sigma _{b_k}^2$$\end{document}hSNPk2=Mσbk2.
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other
| 98.25 |
This approach was implemented in ref. 21 and is similar to the LDpred model presented by Vilhjálmsson et al.18 but with a few differences. The first is that it only considers the infinitesimal case, where all SNPs are considered to be causal and their effect sizes follow a normal distribution. This corresponds to the LDpred-Inf model. The second difference is that the shrinkage parameter of Vilhjálmsson et al.18 is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _k = M{\mathrm{/}}h_{{\mathrm{SNP}}_k}^2$$\end{document}λk=M∕hSNPk2 as they assume that the error variance is 1 rather than \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 - h_{{\mathrm{SNP}}_k}^2$$\end{document}1-hSNPk2 in our implementation. The third difference is that in the LDpred-Inf model, Vilhjálmsson et al.18 calculate BLUP effects for blocks of a certain number of SNPs following a tiling window approach giving a block diagonal structure to L, whereas our implementation within the software GCTA (see URLs) follows a sliding window approach giving a banded diagonal to L. Assuming an error variance of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 - h_{{\mathrm{SNP}}_k}^2$$\end{document}1-hSNPk2 is more appropriate because cumulatively the SNP markers explain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{{\mathrm{SNP}}_k}^2$$\end{document}hSNPk2 of the phenotypic variance. In both implementations, a window is used to capture the LD around SNP markers in order to avoid the large computational costs of inverting a dense M dimensional SNP LD matrix, with only little loss of information (see below).
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study
| 100.0 |
For multiple phenotypes, the elements of Eq. (11) become: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{{{\rm OLS}{\prime}}}} = \left[ {\widehat {\bf{b}}_{{\mathrm{OLS}}_1{\prime} } \ldots \widehat {\bf{b}}_{{\mathrm{OLS}}_k{\prime} }} \right]$$\end{document}b^OLS′=b^OLS1′…b^OLSk′ and N = \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[ {\begin{array}{*{20}{c}} {{\bf{N}}_1} & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & {{\bf{N}}_k} \end{array}} \right]$$\end{document}N1000⋱000Nk, meaning that Eq. (11) can be extended as:12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{MT}} - {\mathrm{SBLUP}}} = \left[ {{\bf{I}}_k \otimes {\bf{L}} + {\bf{\Sigma }}_\epsilon {\bf{\Sigma }}_b^{ - 1}{\bf{N}}^{ - 1} \otimes {\bf{I}}_M} \right]^{ - 1}\widehat {\bf{b}}_{{\mathrm{OLS}}}$$\end{document}b^MT-SBLUP=Ik⊗L+ΣϵΣb-1N-1⊗IM-1b^OLSEquation (12) approximates Eq. (9) using only publically available GWAS summary statistic data and an independent genomic reference sample. However, there remains the large computational cost associated with the inversion of the non-diagonal matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[ {{\bf{I}}_k \otimes {\bf{L}} + {\bf{\Sigma }}_\epsilon {\bf{\Sigma }}_b^{ - 1}{\bf{N}}^{ - 1} \otimes {\bf{I}}_M} \right]$$\end{document}Ik⊗L+ΣϵΣb-1N-1⊗IM.
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study
| 98.44 |
An alternative to Eq. (12), is to obtain k \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{MT}} - {\mathrm{SBLUP}}}$$\end{document}b^MT-SBLUP effects by combining together k single-trait \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{SBLUP}}}$$\end{document}b^SBLUP estimates of Eq. (11), using an optimal index weighting for each trait. The index weighting to derive \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{MT}} - {\mathrm{SBLUP}}}$$\end{document}b^MT-SBLUP from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{SBLUP}}}$$\end{document}b^SBLUP estimates is identical to the index weighting to derive \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{MT}} - {\mathrm{BLUP}}}$$\end{document}b^MT-BLUP from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{BLUP}}}$$\end{document}b^BLUP estimates.
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study
| 99.94 |
For SNP j and focal trait f, we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{SBLUP}}}$$\end{document}b^SBLUP values for k traits, and we wish to obtain the index weights, wj,k, for each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{SBLUP}}_{j,k}}$$\end{document}b^SBLUPj,k effect as:13\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{\mathrm{wMT-SBLUP}_{j,f}} = \mathop {\sum} \limits_{k} {\bf{w}}_{{\mathrm{SBLUP}},j,k}\widehat{\bf{b}}_{{\mathrm{SBLUP}}_{j,k}} = {\bf{w}}_{{\mathrm{SBLUP}},j}{\prime}\widehat {\bf{b}}_{{\mathrm{SBLUP}}_j}$$\end{document}b^wMT-SBLUPj,f= ∑kwSBLUP,j,kb^SBLUPj,k=wSBLUP,j′b^SBLUPj
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study
| 99.9 |
In animal and plant breeding, selection indices have been developed, which combine many single-trait BLUP predictors of an individual’s genetic value together in an index weighting to optimise the selection of individuals with the most favourable multi-trait phenotype for breeding programs33–36. Utilising a selection index approach, the solution for wSBLUP of Eq. (13) can be obtained as:14\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bf{w}}_{{\mathrm{SBLUP}}}={\bf{V}}_{{\mathrm{SBLUP}}}^{-1}{\bf{C}}_{{\mathrm{SBLUP}}}$$\end{document}wSBLUP=V SBLUP-1CSBLUPwhere CSBLUP a k × 1 column vector of the covariance of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{SBLUP}}_k}$$\end{document}b^SBLUPk values of the k traits, with the true genetic effects of the SNPs for the focal trait, and VSBLUP a k × k variance–covariance matrix of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{SBLUP}}}$$\end{document}b^SBLUP effects:15\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{*{20}{l}} {{\bf{w}}_{{\mathrm{SBLUP}}}} \hfill & = \hfill & {{\bf{V}}_{{\mathrm{SBLUP}}}^{ - 1}{\bf{C}}_{{\mathrm{SBLUP}}}} \hfill \\ {} \hfill & = \hfill & {\left[ {\begin{array}{*{20}{c}} {{\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{SBLUP}}_1}} \right)} & \cdots & {{\mathrm{cov}}\left( {\widehat {\bf{b}}_{{\mathrm{SBLUP}}_1},\widehat {\bf{b}}_{{\mathrm{SBLUP}}_k}} \right)} \\ \vdots & \ddots & \vdots \\ {{\mathrm{cov}}\left( {\widehat {\bf{b}}_{{\mathrm{SBLUP}}_k},\widehat {\bf{b}}_{{\mathrm{SBLUP}}_1}} \right)} & \cdots & {{\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{SBLUP}}_k}} \right)} \end{array}} \right]^{ - 1}} \hfill \\ {} \hfill & {} \hfill & {\left[ {\begin{array}{*{20}{c}} {{\mathrm{cov}}\left( {{\bf{b}}_{\it{f}},\widehat {\bf{b}}_{{\mathrm{SBLUP}}_1}} \right)} \\ \vdots \\ {cov\left( {{\bf{b}}_{\mathit{f}},\widehat {\bf{b}}_{{\mathrm{SBLUP}}_k}} \right)} \end{array}} \right]} \hfill \end{array}$$\end{document}wSBLUP=V SBLUP-1CSBLUP=varb^SBLUP1⋯covb^SBLUP1,b^SBLUPk⋮⋱⋮covb ^SBLUPk,b^SBLUP1⋯varb^SBLUPk-1covbf,b^SBLUP1⋮covbf ,b^SBLUPk
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study
| 100.0 |
Therefore, if VSBLUP and CSBLUP can be approximated then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{MT}} - {\mathrm{SBLUP}}}$$\end{document}b^MT-SBLUP of Eq. (12) can be obtained from k single-trait \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{SBLUP}}}$$\end{document}b^SBLUP estimates from Eq. (11).
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other
| 59.2 |
To derive the approximations, we first consider the diagonal elements of VSBLUP, which comprise the variance of the SBLUP SNP solutions, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{SBLUP}}_k}} \right)$$\end{document}varb^SBLUPk. These can be approximated from theory under the assumption that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{SBLUP}}_k}$$\end{document}b^SBLUPk have BLUP properties \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathit{E}}\left[ {{\bf{b}}|{\hat{\bf b}}} \right] = {\hat{\bf b}}$$\end{document}Eb∣b^=b^, which in turn implies that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{cov}}\left( {{\bf{b}}_k,\widehat {\bf{b}}_{{\mathrm{SBLUP}}_k}} \right) = {\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{SBLUP}}_k}} \right)$$\end{document}covbk,b^SBLUPk=varb^SBLUPk. Following Daetwyler et al.37 and Wray et al.38, the squared correlation between a phenotype, yk, in an independent sample and a single-trait BLUP predictor of the phenotype, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{g}}_{{\mathrm{BLUP}}_k}$$\end{document}g^BLUPk, is approximately:16\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{{\bf{y}}_k,\widehat {\bf{g}}_{{\mathrm{BLUP}}_k}}^2 = R_k^2 \approx h_k^2{\mathrm{/}}\left( {1 + M_{{\rm eff}}\left( {1 - R_k^2} \right){\mathrm{/}}\left( {N_kh_k^2} \right)} \right)$$\end{document}Ryk,g^BLUPk2=Rk2≈hk2∕1+Meff1-Rk2∕Nkhk2where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{g}}_{{\mathrm{BLUP}}_k} = {\bf{W}}\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}$$\end{document}g^BLUPk=Wb^BLUPk and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_k^2$$\end{document}hk2 is the proportion of phenotypic variance attributable to additive genetic effects for trait k. Note that Meff is the effective number of chromosome segments or the number of independent SNPs, which is a function of effective population size (Ne) and can be empirically obtained as an inverse of the variance of genomic relationships39,40. Here we use an estimate of Meff of 60,000, which is in line both with our estimates from the genomic relationships in our simulation data and with previously reported estimates41. In Eq. (16), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_k^2$$\end{document}Rk2 occurs on both the left- and righ-hand side. Solving for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_k^2$$\end{document}Rk2 gives \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_k^2 = \frac{{\varphi + h^2 - \sqrt {\left( {\varphi + h^2} \right)^2 - 4\varphi h^4} }}{{2\varphi }}$$\end{document}Rk2=φ+h2-φ+h22-4φh42φ, where φ is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{M_{{\mathrm{eff}}}}}{N}$$\end{document}MeffN.
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| 99.8 |
With a phenotypic variance of 1 and individual-level genetic effects gk = Wbk, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_k^2 = \sigma _{g_k}^2 = M\sigma _{b_k}^2$$\end{document}hk2=σgk2=Mσbk2 and the squared correlation between the true, gk, and estimated BLUP effects, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{g}}_{{\mathrm{BLUP}}_k}$$\end{document}g^BLUPk, is:17\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{{\bf{g}}_k,\widehat {\bf{g}}_{{\mathrm{BLUP}}_k}}^2 = R_k^2{\mathrm{/}}h_k^2$$\end{document}Rgk,g^BLUPk2=Rk2∕hk2Rearranging Eq. (17) gives \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_k^2 = h_k^2R_{{\bf{g}}_k,\widehat {\bf{g}}_{{\mathrm{BLUP}}_k}}^2 = h_k^2\frac{{{\mathrm{cov}}\left( {{\bf{g}}_k,\widehat {\bf{g}}_{{\mathrm{BLUP}}_k}} \right)^2}}{{{\mathrm{var}}\left( {{\bf{g}}_k} \right){\mathrm{var}}\left( {\widehat {\bf{g}}_{{\mathrm{BLUP}}_k}} \right)}}$$\end{document}Rk2=hk2Rgk,g^BLUPk2=hk2covgk,g^BLUPk2vargkvarg^BLUPk, which given the BLUP properties \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{cov}}\left( {{\bf{g}}_k,\widehat {\bf{g}}_{{\mathrm{BLUP}}_k}} \right) = {\mathrm{var}}\left( {\widehat {\bf{g}}_{{\mathrm{BLUP}}_k}} \right)$$\end{document}covgk,g^BLUPk=varg^BLUPk and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_k^2 = \sigma _{g_k}^2$$\end{document}hk2=σgk2 with a phenotypic variance of 1, reduces to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_k^2 = {\mathrm{cov}}\left( {{\bf{g}}_k,\widehat {\bf{g}}_{{\mathrm{BLUP}}_k}} \right)$$\end{document}Rk2=covgk,g^BLUPk = \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{var}}\left( {\widehat {\bf{g}}_{{\mathrm{BLUP}}_k}} \right) = M{\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}} \right)$$\end{document}varg^BLUPk=Mvarb^BLUPk. Therefore:18\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}} \right) = \frac{{{\mathrm{var}}\left( {\widehat {\bf{g}}_{{\mathrm{BLUP}}_k}} \right)}}{M} = \frac{{R_k^2}}{M}$$\end{document}varb^BLUPk=varg^BLUPkM=Rk2M
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study
| 99.94 |
Second, we consider the off-diagonal elements of VSBLUP, which are comprised of the covariance of BLUP SNP solutions among the k traits. These can again be approximated from theory given the covariance of genetic effects among traits k and l is cov(bk, bl) = rGhkhl/M, with rG the genetic correlation, and given the squared correlation between the true genetic effects of the SNPs, bk, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}$$\end{document}b^BLUPk which is given by Eq. (17) as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{{\bf{b}}_k,\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}}^2 = \frac{{R_k^2}}{M}{\mathrm{/}}\frac{{h_k^2}}{M} = R_k^2{\mathrm{/}}h_k^2$$\end{document}Rbk,b^BLUPk2=Rk2M∕hk2M=Rk2∕hk2. The covariance of BLUP SNP predictors is then:19\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{cov}}\left( {\widehat {\bf{b}}_{{\rm BLUP}_{k{\prime}}},\widehat {\bf{b}}_{{\rm BLUP}_l}} \right) = \frac{{R_k^2}}{{h_k^2}} \cdot \frac{{R_l^2}}{{h_l^2}}{\mathrm{cov}}\left( {{\bf{b}}_k,{\bf{b}}_{\mathit{l}}} \right) = \frac{{r_{\mathrm{G}}R_k^2R_l^2}}{{h_kh_lM}}$$\end{document}covb^BLUPk′,b^BLUPl=Rk2hk2⋅Rl2hl2covbk,bl=rGRk2Rl2hkhlM
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| 100.0 |
Finally, we can consider the column vector CSBLUP, which is composed of the covariance between the true genetic effects of the SNPs for the focal trait, bf, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{SBLUP}}_k}$$\end{document}b^SBLUPk for all of the k traits. The first element of CSBLUP is covariance between the true genetic effects of the SNPs for the focal trait bf and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{SBLUP}}_f}$$\end{document}b^SBLUPf for the focal trait \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{cov}}\left( {{\bf{b}}_f,\widehat {\bf{b}}_{{\mathrm{BLUP}}_f}} \right) = {\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{BLUP}}_f}} \right) = \frac{{R_f^2}}{M}$$\end{document}covbf,b^BLUPf=varb^BLUPf=Rf2M. The remaining elements of CSBLUP are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{cov}}\left( {{\bf{b}}_f,\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}} \right)$$\end{document}covbf,b^BLUPk, which can be approximated from theory by considering a regression of bf on bk where the regression coefficient \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{f,k} = r_{\mathrm{G}}\sqrt {{\mathrm{var}}\left( {{\bf{b}}_f} \right){\mathrm{/}}{\mathrm{var}}\left( {{\bf{b}}_k} \right)}$$\end{document}βf,k=rGvarbf∕varbk. The covariance of bf and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}$$\end{document}b^BLUPk can then be written as:20\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{cov}}\left( {{\bf{b}}_f,\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}} \right) = {\mathrm{cov}}\left( {\beta _{f,k}{\bf{b}}_k,\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}} \right) = r_{\mathrm{G}}\frac{{R_k^2}}{M} \cdot \frac{{h_f}}{{h_k}}$$\end{document}covbf,b^BLUPk=covβf,kbk,b^BLUPk=rGRk2M⋅hfhkIf we consider a two-trait example where the focal trait that we want to predict is matched to the first of the two traits, Eqs. (18), (19) and (20) combine as:21\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bf{w}}_{{\mathrm{SBLUP}}} = {\bf{V}}_{{\mathrm{SBLUP}}}^{ - 1}{\bf{C}}_{{\mathrm{SBLUP}}} = \left[ {\begin{array}{*{20}{c}} {\frac{{R_1^2}}{M}} & {\frac{{r_{\mathrm{G}}R_1^2R_2^2}}{{h_1h_2M}}} \\ {\frac{{r_{\mathrm{G}}R_1^2R_2^2}}{{h_1h_2M}}} & {\frac{{R_2^2}}{M}} \end{array}} \right]^{ - 1}\left[ {\begin{array}{*{20}{c}} {\frac{{R_1^2}}{M}} \\ {r_{\mathrm{G}}\frac{{R_2^2}}{M} \cdot \frac{{h_1}}{{h_2}}} \end{array}} \right]$$\end{document}wSBLUP=V SBLUP-1CSBLUP=R12MrGR12R22h1h2MrGR12R22h1h2MR22M-1R12MrGR22M⋅h1h2giving the index for the focal trait as: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{SBLUP}}_f} = {\bf{w}}_1\widehat {\bf{b}}_{S{\mathrm{BLUP}}_f} + {\bf{w}}_2\widehat {\bf{b}}_{{\mathrm{SBLUP}}_2}$$\end{document}b^wMT-SBLUPf=w1b^SBLUPf+w2b^SBLUP2 with solutions for the index weights of:22\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{*{20}{l}} {{{w}}_{\mathrm f}} \hfill & = \hfill & {\left( {1 - \frac{{{{r}}_{\mathrm G}^2R_2^2}}{{h_2^2}}} \right){\mathrm{/}}\left( {1 - \frac{{{{r}}_{\mathrm G}^2R_{\mathrm f}^2R_2^2}}{{h_{\mathrm f}^2h_2^2M}}} \right)} \hfill \\ {} \hfill & = \hfill & {\left( {1 - {{r}}_{\mathrm G}^2R_{{\bf{b}}_2,\widehat {\bf{b}}_{{\mathrm{BLUP}}_2}}^2} \right){\mathrm{/}}\left( {1 - {{r}}_{\mathrm G}^2R_{{\bf{b}}_f,\widehat {\bf{b}}_{{\mathrm{BLUP}}_f}}^2R_{{\bf{b}}_2,\widehat {\bf{b}}_{{\mathrm{BLUP}}_2}}^2} \right),{\mathrm{and}}} \hfill \\ {{{w}}_2} \hfill & = \hfill & {\left( {{{r}}_{\mathrm G}/\left( {h_fh_2} \right)\left( {h_f^2 - R_1^2} \right)} \right){\mathrm{/}}\left( {1 - \frac{{{{r}}_{\mathrm G}^2R_f^2R_2^2}}{{h_f^2h_2^2M}}} \right)} \hfill \\ {} \hfill & = \hfill & {{{r}}_{\mathrm G}\left( {h_f{\mathrm{/}}h_2} \right)\left( {1 - R_{{\bf{b}}_f,\widehat {\bf{b}}_{{\mathrm{BLUP}}_f}}^2} \right){\mathrm{/}}\left( {1 - {{r}}_{\mathrm G}^2R_{{\bf{b}}_f,{\hat{\mathbf b}}_{{\mathrm{BLUP}}_f}}^2R_{{\bf{b}}_2,\widehat {\bf{b}}_{{\mathrm{BLUP}}_2}}^2} \right)} \hfill \end{array}$$\end{document}wf=1-rG2R22h22∕1-rG2Rf2R22hf2h22M=1-rG2Rb2,b^BLUP22∕1-rG2Rbf,b^BLUPf2Rb2,b^BLUP22,andw2=rG∕hfh2hf2-R12∕1-rG2Rf2R22hf2h22M=rGhf∕h21-Rbf,b^BLUPf2∕1-rG2Rbf,b^BLUPf2Rb2,b^BLUP22
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| 100.0 |
For traits with low power, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_k^2$$\end{document}Rk2 is usually very small. In that case, VSBLUP can be well approximated by a diagonal matrix with entries \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{R_k^2}}{M}$$\end{document}Rk2M. wf will become 1 and wk for all other traits will be \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{{\mathrm{G}}_{f,k}}\frac{{h_f}}{{h_k}}$$\end{document}rGf,khfhk. It may appear surprising that traits with higher SNP heritability have smaller weights than traits with lower SNP heritability. This can be explained by the fact that the variance of each BLUP predictor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {R_k^2} \right)$$\end{document}Rk2 is approximately proportional to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_k^4N$$\end{document}hk4N if Meff is large, and thus a trait with higher SNP heritability will still have a larger contribution to the multi-trait predictor than a trait with lower SNP heritability.
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study
| 99.94 |
Equation (17) implies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{{\bf{b}}_k,\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}}^2 = R_{{\bf{g}}_k,\widehat {\bf{g}}_{{\mathrm{BLUP}}_k}}^2 = R_k^2{\mathrm{/}}h_k^2$$\end{document}Rbk,b^BLUPk2=Rgk,g^BLUPk2=Rk2∕hk2 and thus the index weights of Eq. (15) can be applied equally to BLUP solutions for the SNP effects or BLUP predictors for individuals of each trait as described in the main text in Eq. (1) through (3). Both \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{{\mathrm{G}}_{k,l}}$$\end{document}rGk,l and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_k^2$$\end{document}hk2 of Eq. (15) can be obtained from summary statistic data using LD score regression22 and therefore \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{MT}} - {\mathrm{BLUP}}}$$\end{document}b^MT-BLUP effects of Eq. (10), which would traditionally require individual-level phenotype–genotype data for all traits, can be approximated accurately in a computationally efficient manner using only publically available GWAS summary statistic data and an independent genomic reference sample.
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| 100.0 |
In the previous section, we have shown how \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{SBLUP}}}$$\end{document}b^SBLUP estimates for multiple traits can be combined to yield more accurate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{SBLUP}}}$$\end{document}b^wMT-SBLUP SNP effects, which can be turned into \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{g}}_{{\mathrm{wMT}} - {\mathrm{SBLUP}}}$$\end{document}g^wMT-SBLUP individual predictors that approach \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{g}}_{{\mathrm{MT}} - {\mathrm{BLUP}}}$$\end{document}g^MT-BLUP accuracy. However, using a similar weighting we can also combine \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{OLS}}}$$\end{document}b^OLS estimates for multiple traits into \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{OLS}}}$$\end{document}b^wMT-OLS.
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For SNP j and focal trait f, we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{OLS}}}$$\end{document}b^OLS values for k traits, and we wish to obtain the index weights, wj,k, for each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{OLS}}_{j,k}}$$\end{document}b^OLSj,k effect as:23\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{OLS}}_{j,f}} = \mathop {\sum }\limits_k w_{j,k}{\hat{\bf b}}_{{\mathrm{OLS}}_{j,k}} = {\bf{w}}_j\prime \widehat {\bf{b}}_{{\mathrm{OLS}}_j}$$\end{document}b^wMT-OLSj,f= ∑kwj,kb^OLSj,k=wj′b^OLSj
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study
| 99.9 |
Just like before, the optimal weights can be derived as:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bf{w}}_{{\mathrm{OLS}}} = {\bf{V}}_{{\mathrm{OLS}}}^{ - 1}{\bf{C}}_{{\mathrm{OLS}}}$$\end{document}wOLS=V OLS-1COLS, where COLS is now a k × 1 column vector of the covariances of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{OLS}}_k}$$\end{document}b^OLSk values of the k traits with the true genetic effects of the SNPs for the focal trait, and VOLS is a k × k variance–covariance matrix of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{OLS}}}$$\end{document}b^OLS effects:24\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{*{20}{l}} {{\bf{w}}_{{\mathrm{OLS}}}} \hfill & = \hfill & {{\bf{V}}_{{\mathrm{OLS}}}^{ - 1}{\bf{C}}_{{\mathrm{OLS}}}} \hfill \\ {} \hfill & = \hfill & {\left[ {\begin{array}{*{20}{c}} {{\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{OLS}}_1}} \right)} & \cdots & {{\mathrm{cov}}\left( {\widehat {\bf{b}}_{{\mathrm{OLS}}_1},\widehat {\bf{b}}_{{\mathrm{OLS}}_k}} \right)} \\ \vdots & \ddots & \vdots \\ {{\mathrm{cov}}\left( {\widehat {\bf{b}}_{{\mathrm{OLS}}_k},\widehat {\bf{b}}_{{\mathrm{OLS}}_1}} \right)} & \cdots & {{\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{OLS}}_k}} \right)} \end{array}} \right]^{ - 1}} \hfill \\ {} \hfill & {} \hfill & {\left[ {\begin{array}{*{20}{c}} {{\mathrm{cov}}\left( {{\bf{b}}_{\mathit{f}},\widehat {\bf{b}}_{{\mathrm{OLS}}_1}} \right)} \\ \vdots \\ {{\mathrm{cov}}\left( {{\bf{b}}_{\mathit{f}},\widehat {\bf{b}}_{{\mathrm{OLS}}_k}} \right)} \end{array}} \right]} \hfill \end{array}$$\end{document}wOLS=V OLS-1COLS=varb^OLS1⋯covb^OLS1,b^OLSk⋮⋱⋮covb ^OLSk,b^OLS1⋯varb^OLSk-1covbf ,b^OLS1⋮covbf ,b^OLSkThe diagonal elements of VOLS are:25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{OLS}}_k}} \right) = \frac{{h_k^2}}{M} + \frac{1}{{N_k}}$$\end{document}varb^OLSk=hk2M+1NkThe off-diagonal elements for trait k and l are26\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{cov}}\left( {\widehat {\bf{b}}_{{\mathrm{OLS}}_k},\widehat {\bf{b}}_{{\mathrm{OLS}}_l}} \right) = \frac{{r_{\mathrm{G}}h_kh_l}}{M}$$\end{document}covb^OLSk,b^OLSl=rGhkhlMCOLS now has elements27\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{cov}}\left( {{\bf{b}}_{\mathrm{k}},\widehat {\bf{b}}_{{\mathrm{OLS}}_k}} \right) = \frac{{r_{\mathrm{G}}h_kh_l}}{M}$$\end{document}covbk,b^OLSk=rGhkhlMIf we again consider a two-trait example, Eqs. (25), (26) and (27) combine as:28\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bf{w}}_{{\mathrm{OLS}}} = {\bf{V}}_{{\mathrm{OLS}}}^{ - 1}{\bf{C}}_{{\mathrm{OLS}}} = \left[ {\begin{array}{*{20}{c}} {\frac{{h_1^2}}{M} + \frac{1}{{N_1}}} & {\frac{{r_{\mathrm{G}}h_1h_2}}{M}} \\ {\frac{{r_{\mathrm{G}}h_1h_2}}{M}} & {\frac{{h_2^2}}{M} + \frac{1}{{N_2}}} \end{array}} \right]^{ - 1}\left[ {\begin{array}{*{20}{c}} {\frac{{h_1^2}}{M}} \\ {\frac{{r_{\mathrm{G}}h_1h_2}}{M}} \end{array}} \right]$$\end{document}wOLS=V OLS-1COLS=h12M+1N1rGh1h2MrGh1h2Mh22M+1N2-1h12MrGh1h2M
|
study
| 99.94 |
These weights are considerably different from the BLUP weights, which reflects the different variances of BLUP effects and OLS effects. Here we include this section for completeness but focus our analyses on multi-trait BLUP effects, because they are more accurate in expectation than multi-trait OLS effects.
|
study
| 99.3 |
For phenotypes with a genetic architecture characterised by a few loci of very large effect sizes, this approach may not be ideal. Models that assume a mixture distribution for SNP effects, such as LDpred or BayesR, can yield higher prediction accuracies in traits of non-infinitesimal genetic architecture18,42. As outlined above, Eq. (17) implies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{{\bf{b}}_k,\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}}^2 = R_{{\bf{g}}_k,\widehat {\bf{g}}_{{\mathrm{BLUP}}_k}}^2 = R_k^2{\mathrm{/}}h_k^2$$\end{document}Rbk,b^BLUPk2=Rgk,g^BLUPk2=Rk2∕hk2 and thus the index weights of Eq. (15) can be applied equally to BLUP solutions for the SNP effects or BLUP predictors for individuals of each trait as described in the main text in Eq. (1) through (3). LDpred aims to estimate the posterior mean phenotype that provides best unbiased prediction. Therefore, single-trait individual-level predictors obtained from LDPred can also be weighted together to create an approximate multi-trait predictor.
|
study
| 100.0 |
The prediction accuracy of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}}$$\end{document}b^wMT-BLUP effects obtained from Eq. (15) can be derived by considering the correlation of bf and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}_k}$$\end{document}b^wMT-BLUPk as:29\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{{\bf{b}}_f,\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}_f}} = \frac{{{\mathrm{cov}}\left( {{\bf{b}}_f,\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}_f}} \right)}}{{\sqrt {{\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}_f}} \right){\mathrm{var}}\left( {{\bf{b}}_f} \right)} }}$$\end{document}rbf,b^wMT-BLUPf=covbf,b^wMT-BLUPfvarb^wMT-BLUPfvarbfEquation (13) gives \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}_f} = {{{\bf w}{\prime}}}\widehat {\bf{b}}_{{\mathrm{BLUP}}}$$\end{document}b^wMT-BLUPf=w′b^BLUP and thus the covariance of bf and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}_f}$$\end{document}b^wMT-BLUPf is:30\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{cov}}\left( {{\bf{b}}_f,\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}_f}} \right) = {\mathrm{cov}}\left( {{\bf{b}}_f,{{{\bf w}{\prime}}}\widehat {\bf{b}}_{{\mathrm{BLUP}}}} \right) = {{{\bf w}{\prime}}}{\mathrm{cov}}\left( {{\bf{b}}_f,\widehat {\bf{b}}_{{\mathrm{BLUP}}}} \right) = {{{\bf w}{\prime}\bf C}}$$\end{document}covbf,b^wMT-BLUPf=covbf,w′b^BLUP=w′covbf,b^BLUP=w′CThe variance of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}}$$\end{document}b^wMT-BLUP effects obtained from Eq. (15) is:31\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}}} \right) = {\mathrm{var}}\left( {{{{\bf w}{\prime}}}\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}} \right) = {{{\bf w}{\prime}}}{\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}} \right){\bf{w}} = {{{\bf w}{\prime}\bf Vw}}$$\end{document}varb^wMT-BLUP=varw′b^BLUPk=w′varb^BLUPkw=w′V wAdditionally, w = V−1C and Vw = C, and thus w′C = w′Vw or written another way \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{cov}}\left( {{\bf{b}}_f,\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}_f}} \right) = {\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}}} \right)$$\end{document}covbf,b^wMT-BLUPf=varb^wMT-BLUP following BLUP properties. Substituting into Eq. (19), the correlation of bf and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}_k}$$\end{document}b^wMT-BLUPk can then be written as:32\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{*{20}{l}} {r_{{\bf{b}}_f,\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}_f}}} \hfill & = \hfill & {{\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}}} \right){\mathrm{/}}\sqrt {{\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}}} \right){\mathrm{var}}\left( {{\bf{b}}_f} \right)} } \hfill \\ {} \hfill & = \hfill & {\sqrt {{\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}}} \right){\mathrm{/}}{\mathrm{var}}\left( {{\bf{b}}_f} \right)} } \hfill \end{array},$$\end{document}rbf,b^wMT-BLUPf=varb^wMT-BLUP∕varb^wMT-BLUPvarbf=varb^wMT-BLUP∕varbf,which gives the squared correlation as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{{\bf{b}}_f,\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}_f}}^2$$\end{document}Rbf,b^wMT-BLUPf2 = \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}}} \right){\mathrm{/}}{\mathrm{var}}\left( {{\bf{b}}_f} \right)$$\end{document}varb^wMT-BLUP∕varbf = \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{R_f^2}}{M}{\mathrm{/}}\frac{{h_k^2}}{M} = R_f^2{\mathrm{/}}h_k^2$$\end{document}Rf2M∕hk2M=Rf2∕hk2. Therefore, the squared correlation between a phenotype and a multiple trait index weighted BLUP predictor of the phenotype is approximately:33\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{{\bf{y}}_k,\widehat {\bf{g}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}_k}}^2 = M{\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}}} \right) = M{{{\bf w}{\prime}{\bf Vw}}}$$\end{document}Ryk,g^wMT-BLUPk2=Mvarb^wMT-BLUP=Mw′V wIf we consider a two-trait example then prediction accuracy for a focal trait \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{{\bf{y}}_f,\widehat {\bf{g}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}_k}}^2$$\end{document}Ryf,g^wMT-BLUPk2 can be written as:34\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{{\bf{y}}_f,\widehat {\bf{g}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}_f}}^2 = {\bf{w}}_f^2R_{{\bf{y}}_f,\widehat {\bf{g}}_{{\mathrm{BLUP}}_f}}^2 + {\bf{w}}_2^2R_{{\bf{y}}_2,\widehat {\bf{g}}_{{\mathrm{BLUP}}_2}}^2 + 2{\bf w}_f{\bf w}_2{\bf V}_{1,2}$$\end{document}Ryf,g^wMT-BLUPf2=wf2Ryf,g^BLUPf2+w22Ry2,g^BLUP22+2wfw2V 1,2where V1,2 is the off-diagonal element of the matrix V of Eqs. (15) and (21). The value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{{\bf{y}}_f,\widehat {\bf{g}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}_f}}^2$$\end{document}Ryf,g^wMT-BLUPf2 can then be compared to the prediction accuracy of the single-trait BLUP predictor of Eq. (16) and to the prediction accuracy of a cross-trait predictor43, where a BLUP predictor of the second trait is used to predict the focal trait phenotype, which is given by: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{{\bf{y}}_f,\widehat {\bf{g}}_{{\mathrm{BLUP}}_2}}^2 = R_{{\bf{y}}_2,\widehat {\bf{g}}_{{\mathrm{BLUP}}_2}}^2r_{\mathrm{G}}\sqrt {\left( {h_2{\mathrm{/}}h_f} \right)}$$\end{document}Ryf,g^BLUP22=Ry2,g^BLUP22rGh2∕hf. This comparison is of interest, because we expect the multi-trait predictor to be more accurate than any available single-trait predictor, even if the most accurate single-trait predictor is across two different traits. Cross-trait prediction is equivalent to the proxy-phenotype method, which has been used to predict cognitive performance from educational attainment GWAS data44.
|
study
| 97.2 |
Equations (16), (17), (18), (19), (20), (21), (22), (23), (24), (25), (26), (27), (28), (29), (30), (31), (32), (33) and (34) assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{cov}}\left( {{\bf{b}}_k,\widehat {\bf{b}}_{{\mathrm{SBLUP}}_k}} \right)$$\end{document}covbk,b^SBLUPk = \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{SBLUP}}_k}} \right) = {\mathrm{var}}\left( {\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}} \right)$$\end{document}varb^SBLUPk=varb^BLUPk, or in other words that SBLUP SNP solutions have BLUP properties. The use of an independent LD reference sample to create an approximate single-trait BLUP predictor in Eq. (11) does not affect the covariance between the true SNP effect sizes and the approximate BLUP SNP solution, meaning that the approximate single-trait BLUP predictors have BLUP properties. However, the variance of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{SBLUP}}}$$\end{document}b^SBLUP is likely affected, which may potentially result in a loss of prediction accuracy of a weighted multi-trait BLUP predictor. The variance of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{SBLUP}}}$$\end{document}b^SBLUP is:35\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{*{20}{l}} {\sigma _{\widehat {\bf{b}}_{{\mathrm{SBLUP}}}}^2} \hfill & = \hfill & {\left[ {\left[ {N{\bf{L}} + {\bf{I}}_M\lambda } \right]^{ - 1}{{{\bf W}{\prime}}}} \right]\left[ {{{{\bf W}{\prime}{\bf W}}}\sigma _{\mathit{b}}^2 + {\bf{I}}\sigma _{\mathit{e}}^2} \right]} \hfill \\ {} \hfill & {} \hfill & {\left[ {{\bf{W}}\left[ {N{\bf{L}} + {\bf{I}}_M\lambda } \right]^{ - 1}} \right]} \hfill \\ {} \hfill & = \hfill & {\left[ {N{\bf{L}} + {\bf{I}}_M\lambda } \right]^{ - 1}\left[ {\left( {{{{\bf W}{\prime}{\bf W}}}} \right)\left( {{{{\bf W}{\prime}{\bf W}}}} \right)\sigma _{\mathit{b}}^2} \right.} \hfill \\ {} \hfill & {} \hfill & {\left. { + {{{\bf W}{\prime}{\bf W}}}\sigma _{\mathit{e}}^2} \right]\left[ {N{\bf{L}} + {\bf{I}}_M\lambda } \right]^{ - 1}} \hfill \\ {} \hfill & = \hfill & {\left[ {\left( {\left[ {N{\bf{L}} + {\bf{I}}_M\lambda } \right]^{ - 1}\left( {{{{\bf W}{\prime}{\bf W}}}} \right)\left( {{{{\bf W}{\prime}{\bf W}}}} \right)\left[ {N{\bf{L}} + {\bf{I}}_M\lambda } \right]^{ - 1}} \right.} \right.} \hfill \\ {} \hfill & {} \hfill & {\left. {\left. { + \left[ {N{\bf{L}} + {\bf{I}}_M\lambda } \right]^{ - 1}{{{\bf W}{\prime}{\bf W}}}\lambda _k} \right)\left[ {N{\bf{L}} + {\bf{I}}_M\lambda } \right]^{ - 1}} \right]\sigma _{\mathit{b}}^2} \hfill \end{array}$$\end{document}σb^SBLUP2=NL+IMλ-1W′W′Wσb2+Iσe2WNL+IMλ-1=NL+IMλ-1W′WW′Wσb2+W′Wσe2NL+IMλ-1=NL+IMλ-1W′WW′WNL+IMλ-1+NL+IMλ-1W′WλkNL+IMλ-1σb2The loss of information from using an independent data set as an LD reference to obtain L, rather than directly using the individual-level data to calculate W′W, can be approximated by considering the scenario where SNP makers are unlinked, resulting in diag[L]. The diagonal elements of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{\widehat {\bf{b}}_{{\mathrm{SBLUP}}_{jj}}}^2$$\end{document}σb^SBLUPjj2 for SNP j are then:36\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{\widehat {\bf{b}}_{{\mathrm{SBLUP}}_{jj}}}^2 = \left( {\left[ {N + \lambda } \right]^{ - 2}{\mathrm{diag}}\left[ {\left( {{{{\bf W}{\prime}{\bf W}}}} \right)\left( {{{{\bf W}{\prime}{\bf W}}}} \right)} \right] + N\lambda \left[ {N + \lambda } \right]^{ - 2}} \right)\sigma _{\mathit{b}}^2$$\end{document}σb^SBLUPjj2=N+λ-2diagW′WW′W+NλN+λ-2σb2The diagonal elements of diag[(W′W)(W′W)] can be approximated as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{diag}}\left[ {\left( {{{{\bf W}{\prime}{\bf W}}}} \right)\left( {{{{\bf W}{\prime}{\bf W}}}} \right)} \right]$$\end{document}diagW′WW′W ≈ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N^2\left( {1 + {\Bbb E}\left[ {r^2} \right]M} \right)$$\end{document}N21+Er2M + \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N^2\left( {1 + M{\mathrm{/}}N} \right)$$\end{document}N21+M∕N, where the expectation of the LD correlation of the SNPs, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Bbb E}\left[ {r^2} \right]$$\end{document}Er2, is 1/N as the SNP markers are unlinked. Equation (36) can then be written as:37\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{*{20}{l}} {\sigma _{\widehat {\bf{b}}_{{\mathrm{SBLUP}}_{jj}}}^2} \hfill & = \hfill & {\left( {\left( {N^2 + NM + N\lambda } \right){\mathrm{/}}\left( {N + \lambda } \right)^2} \right)\sigma _{\mathit{b}}^2} \hfill \\ {} \hfill & = \hfill & {\sigma _{\mathit{b}}^2N{\mathrm{/}}(N + \lambda ) + \sigma _{\mathit{b}}^2NM{\mathrm{/}}(N + \lambda )^2} \hfill \end{array}$$\end{document}σb^SBLUPjj2=N2+NM+Nλ∕N+λ2σb2=σb2N∕(N+λ)+σb2NM∕(N+λ)2From Eq. (37), the squared correlation between true SNP effects and SBLUP SNP effects can be written as:38\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{{\bf{b}},\widehat {\bf{b}}_{{\mathrm{SBLUP}}}}^2 = N{\mathrm{/}}\left( {N + M + \lambda } \right) = N{\mathrm{/}}\left( {N + M{\mathrm{/}}h^2} \right)$$\end{document}Rb,b^SBLUP2=N∕N+M+λ=N∕N+M∕h2This can be contrasted to Eq. (17), which gives the squared correlation between the true genetic effects of the SNPs, bk, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}$$\end{document}b^BLUPk as:39\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{*{20}{l}} {R_{{\bf{b}}_k,\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}}^2} \hfill & = \hfill & {\frac{{R_k^2}}{M}{\mathrm{/}}\frac{{h_k^2}}{M} = R_k^2{\mathrm{/}}h_k^2} \hfill \\ {} \hfill & = \hfill & {1{\mathrm{/}}\left( {1 + M\left( {1 - R_k^2} \right){\mathrm{/}}\left( {N_kh_k^2} \right)} \right)} \hfill \\ {} \hfill & = \hfill & {N_k{\mathrm{/}}\left( {N_k + M\left( {1 - R_k^2} \right){\mathrm{/}}h_k^2} \right)} \hfill \end{array}$$\end{document}Rbk,b^BLUPk2=Rk2M∕hk2M=Rk2∕hk2=1∕1+M1-Rk2∕Nkhk2=Nk∕Nk+M1-Rk2∕hk2Equation (39) is similar to Eq. (38) apart from the factor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 - R_k^2$$\end{document}1-Rk2. Therefore, the relative loss of prediction accuracy from using an SBLUP predictor is given as a ratio of Eqs. (39) and (38) as:40\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{R_{{\bf{b}},\widehat {\bf{b}}_{{\mathrm{SBLUP}}}}^2}}{{R_{{\bf{b}}_k,\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}}^2}} = \frac{{Nh^2 + M}}{{Nh^2 + M\left( {1 - R_k^2} \right)}}$$\end{document}Rb,b^SBLUP2Rbk,b^BLUPk2=Nh2+MNh2+M1-Rk2
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| 99.94 |
For a phenotype of SNP heritability 0.5, with effective number of independent markers (independent genomic segments), Meff, of ~ 60,000 and sample size, N, of 500,000, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{{\bf{b}},\widehat {\bf{b}}_{{\mathrm{SBLUP}}}}^2$$\end{document}Rb,b^SBLUP2 from summary statistics in an independent reference sample will be 91% of the value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{{\bf{b}}_k,\widehat {\bf{b}}_{{\mathrm{BLUP}}_k}}^2$$\end{document}Rbk,b^BLUPk2 if individual-level data were available. Likewise, for a two-trait example where both traits have h2 = 0.5 and N = 500,000, the accuracy of the multi-trait SBLUP predictor will also be 91% of the accuracy of the multi-trait BLUP predictor.
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| 100.0 |
It should be noted that here we assume L to be a a diagonal matrix, which will lead to a conservative estimate of the accuracy of SBLUP relative to the accuracy of BLUP, and that this estimate is in fact equivalent to the expected accuracy of a polygenic risk predictor based on marginal OLS effects28. In practice, approximating L through an external reference data set leads to SBLUP predictors, which are more accurate than predictors based on marginal OLS effects but less accurate than predictors based on BLUP effects.
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Computing weights and combing up to 930,000 SNP effects of 34 traits takes <10 min on an Intel Xeon E7-8837 processor with 2.76 GHz. Memory requirements do not extend much beyond the amount necessary to read in the summary statistics. Calculation of single-trait SBLUP effects is more computationally demanding, so we split the data by chromosome. Runtime for chromosome one was <40 min, with memory usage just under 10 GB.
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other
| 99.3 |
To compare the accuracy of single-trait and multi-trait genetic predictors created from SNP effects obtained from both individual-level and summary statistic data, we conducted a simulation study based on real genotypes from the Kaiser Permanente study (Genetic Epidemiology Research on Adult Health and Aging: GERA cohort) and simulated phenotypes.
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From the GERA cohort, we selected 50,000 individuals of European ancestry (for definitions of European individuals and quality control (QC) of the genotypic data, see ref. 45). SNP QC procedures on the initial data sets consisted of per-SNP missing data rate of <0.01, minor allele frequency >0.01, per-person missing data rate <0.01 and Hardy–Weinberg disequilibrium p-value < 1 × 10−6. For the subsequent imputation, the data were first haplotyped using HAPI-UR. After that, Impute2 was used to impute the haplotypes to the 1000 genomes reference panel (release 1, version 3). Best-guess genotypes at common SNPs included in the HapMap 3 European sample were then extracted and filtered for imputation info score >0.5, missing data rate of <0.01, minor allele frequency >0.01, per-person missing data rate <0.01 and Hardy–Weinberg disequilibrium p-value < 1 × 10−6. Next, we performed principal component analysis and removed individuals with principal eigenvector values that were >7 SD from the mean of the European cluster. Lastly, we identified pairs of individuals with genetic relatedness matrix >0.05 and removed one individual from each of these pairs.
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The Atherosclerosis Risk in Communities study (ARIC data) was used as an independent LD reference when estimating SBLUP SNP effects of Eq. (11). Eight thousand seven hundred and forty-four European individuals were selected and the data were imputed and QC conducted in the same way as described above for the GERA cohort. We then reduced the SNPs used in both the GERA and ARIC cohorts to overlapping HapMap3 SNPs, which gave 557,034 SNPs that were used in the simulation study.
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| 100.0 |
We then randomly assigned 20,000, 20,000 and 10,000 individuals from the GERA cohort to create three data sets: training set one, training set two, and a testing set. We simulated two genetically correlated traits by randomly selecting 2000 causal SNPs. Effect sizes for the causal markers were simulated from a bivariate normal distribution with mean 0, variances of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{h_1^2}}{M}$$\end{document}h12M and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{h_2^2}}{M}$$\end{document}h22M and covariance of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{\mathrm{G}}\sqrt {h_1^2h_2^2}$$\end{document}rGh12h22. These effect sizes were then multiplied with the standardised genotype dosages (mean 0 and variance 1) to create a genetic value for each individual. Normally distributed environmental effects e ~ N(0, 1 − h2) were added to this genetic value for each individual to create phenotypes with mean 0 and variance of 1. To remove any effects of population stratification, the simulated phenotypes were then regressed against the first 20 genetic principal components, and the residuals from this regression were used in all subsequent analyses.
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study
| 100.0 |
In training set 1, we simulated trait 1 and we then estimated: (i) OLS SNP effects using Eq. (5) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{b}}_{{\mathrm{OLS}}}} \right)$$\end{document}b^OLS, (ii) BLUP SNP effects from the individual-level data using Eq. (8) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{b}}_{{\mathrm{BLUP}}}} \right)$$\end{document}b^BLUP, and (iii) approximate SBLUP effects using the OLS SNP effects from Eq. (5) and the ARIC data as a reference \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{b}}_{{\mathrm{SBLUP}}}} \right)$$\end{document}b^SBLUP. In training set 2, we simulated trait 2 and estimated \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{OLS}}}$$\end{document}b^OLS, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{BLUP}}}$$\end{document}b^BLUP and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat {\bf{b}}_{{\mathrm{SBLUP}}}$$\end{document}b^SBLUP in the same manner. We then estimated multi-trait BLUP SNP effects using Eq. (9) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{b}}_{{\mathrm{MT}} - {\mathrm{BLUP}}}} \right)$$\end{document}b^MT-BLUP from individual-level data by combining trait 1 from training set 1 and trait 2 from training set 2.
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study
| 100.0 |
In the testing set, we then used the estimated SNP effects from the training sets to produce genetic predictors for both traits. Single-trait genetic predictors were created for both simulated traits from (i) the OLS SNP effects \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{g}}_{{\mathrm{OLS}}}} \right)$$\end{document}g^OLS, (ii) the BLUP SNP effects \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{g}}_{{\mathrm{BLUP}}}} \right)$$\end{document}g^BLUP and (iii) the SBLUP SNP effects \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{g}}_{{\mathrm{SBLUP}}}} \right)$$\end{document}g^SBLUP. We then created multi-trait predictors where trait 1 was the focal trait from: (i) individual-level multi-trait BLUP predictor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{g}}_{{\mathrm{MT}} - {\mathrm{BLUP}}}} \right)$$\end{document}g^MT-BLUP, (ii) weighted multi-trait SBLUP predictor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{g}}_{{\mathrm{wMT}} - {\mathrm{SBLUP}}}} \right)$$\end{document}g^wMT-SBLUP, (iii) a weighted multi-trait BLUP predictor-based individual-level single-trait BLUP estimates \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{g}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}}} \right)$$\end{document}g^wMT-BLUP, and (iv) a weighted multi-trait GWAS predictor based on GWAS OLS estimates \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{g}}_{{\mathrm{wMT}} - {\mathrm{OLS}}}} \right)$$\end{document}g^wMT-OLS. We simulated phenotypic values for both traits using the same effect sizes as those used to generate the phenotypes in the training sets and normally distributed environmental effects sampled independently for each trait as e ~ N(0, 1 − h2). We also compared estimates obtained using LDPred with the proportion of SNPs in the model of 0.001, 0.003, 0.01, 0.03, 0.1, 0.3, 1 or using the LDPred-Inf option.
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study
| 100.0 |
We created two simulation scenarios. Heritability of the first and second trait and genetic correlations were \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_1^2$$\end{document}h12 = 0.2, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_2^2$$\end{document}h22 = 0.8 and rG = 0.8, respectively, in the first scenario and were \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_1^2$$\end{document}h12 = 0.5, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_2^2$$\end{document}h22 = 0.5, and rG = 0.5, respectively, in the second scenario. In each set-up, six replicates were conducted, each with a different set of randomly selected causal markers. We then repeated all analyses on a permuted data set, where the values of the genotype matrix were permuted across all individuals, for each SNP. This creates a genotype matrix where the allele frequency distribution remains the same, but all LD structure is removed, allowing us to determine the degree to which differences between the simulations results are driven by the LD structure in the real genotype data. Finally, because prediction accuracy is expected to be reduced by the error introduced by using an external LD reference data set and a restricted LD window when implementing Eq. (8) (see above), we examined how changing the LD reference and restricting the LD window size influences to optimal value of shrinkage parameter λ when implementing Eq. (8) (see Supplementary Fig. 3).
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study
| 99.94 |
We then applied our approach to the schizophrenia (SCZ) and bipolar disorder (BIP) samples from both wave 1 and wave 2 data of the PGC (PGC1 and PGC2). A description of the data collection and imputation of the SNP genotype data can be found elsewhere25,26,46.
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study
| 100.0 |
We selected these two disorders because there is a high genetic correlation between them (estimate for rG between schizophrenia and bipolar disorder using ldsc: 0.72, SE: 0.03; estimated using meta-analysis of all PGC2 schizophrenia and bipolar cohorts, excluding cohorts which were used as test set in the initial PGC1 analysis), and it enabled us to draw a direct comparison between the approach described here and a previous study, which estimated multi-trait BLUP SNP effects \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{b}}_{{\mathrm{MT}} - {\mathrm{BLUP}}}} \right)$$\end{document}b^MT-BLUP from individual-level data in an approach equivalent to Eq. (9). The previous study used PGC1 data in the training set and selected four cohorts for schizophrenia and three cohorts for bipolar disorder as test sets. For schizophrenia, the training set comprised 17 cohorts (8826 cases, 6106 controls) and for bipolar disorder the training set comprised 11 cohorts (5867 cases, 3328 controls). The test set of 4 cohorts for schizophrenia contained 4068 cases and 5471 controls, and the test set of 3 cohorts for bipolar disorder contained 2029 cases and 5338 controls. The analyses on the PGC1 data were performed on 745,705 HapMap3 SNPs in common across all data sets. To have a direct comparison to our previous study, we began by re-analysing the same PGC1 training set data to estimate: (i) OLS SNP effects using Eq. (5) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{b}}_{{\mathrm{OLS}}}} \right)$$\end{document}b^OLS, (ii) BLUP SNP effects from the individual-level data using Eq. (8) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{b}}_{{\mathrm{BLUP}}}} \right)$$\end{document}b^BLUP, and (iii) approximate SBLUP effects using the OLS SNP effects from Eq. (5) and the ARIC data as a reference \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{b}}_{{\mathrm{SBLUP}}}} \right)$$\end{document}b^SBLUP using Eq. (11). For the estimation of schizophrenia SBLUP effects, λ was set to 1,100,000, corresponding roughly to 1,000,000 markers and an observed scale SNP heritability estimate of 0.47, and for the estimation of bipolar disorder SBLUP effects, lambda was set to 1,200,000, corresponding roughly to 1,000,000 markers and an observed scale SNP heritability estimate of 0.45. For the four SCZ testing cohorts and the three BIP testing cohorts used in the previous study, we created: (i) weighted multi-trait SBLUP predictors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{g}}_{{\mathrm{wMT}} - {\mathrm{SBLUP}}}} \right)$$\end{document}g^wMT-SBLUP, (ii) weighted multi-trait BLUP predictor-based individual-level single-trait BLUP estimates \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{g}}_{{\mathrm{wMT}} - {\mathrm{BLUP}}}} \right)$$\end{document}g^wMT-BLUP, and (iii) weighted multi-trait GWAS predictor based on GWAS OLS estimates \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{g}}_{{\mathrm{wMT}} - {\mathrm{OLS}}}} \right)$$\end{document}g^wMT-OLS. We then compared the prediction accuracy we obtained using the weighted multi-trait SBLUP predictors to the individual-level multi-trait BLUP predictor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{g}}_{{\mathrm{MT}} - {\mathrm{BLUP}}}} \right)$$\end{document}g^MT-BLUP used in the previous study16.
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study
| 100.0 |
We then extended our analysis to the PGC2 data set. There were 36 cohorts for schizophrenia (26,412 cases and 32,440 controls in total) and 23 cohorts for bipolar disorder (18,865 cases and 30,460 controls in total) available to us. The number of SNPs used in the PGC2 analyses varied between cohorts. Summary statistics for each of the PGC2 cohorts was available to an imputed SNP set of >10,000,000 SNPs. After intersecting this set of SNPs with the HapMap3 SNPs and the ARIC SNPs, 932,344 SNPs remained that were used to create predictors.
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study
| 100.0 |
We applied a cross-validation approach as we observed that prediction accuracy as well as accuracy differences between predictors can be highly dependent on the choice of the test set in the extended PGC2 data set (Supplementary Figs. 5 and 6), which is supported by previous results showing highly variable prediction accuracy across cohorts in the PGC2 data set25. A cross-validation approach allowed us to get a more robust estimate of the increase of prediction accuracy achieved by our multi-trait prediction method compared to a single-trait predictor. We employed a leave-one-out cross-validation approach, where, for each test set cohort, all cohorts of the same disease without any highly related individuals were chosen to be in the training set for the single-trait predictor and all cohorts of both diseases without any highly related individuals were chosen to be in the training set for the multi-trait predictor. To identify pairs of cohorts with highly related individuals, genetic relatedness for all pairs of individuals (across all pairs of cohorts) was calculated based on chromosome 22, and whenever at least one pair of individuals had relatedness >0.8, that pair of cohorts was not simultaneously used in the training set and the test set.
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study
| 100.0 |
The full genotypes from the PGC2 cohorts that were used as test sets underwent stringent QC and only comprised 458,744–860,576 SNPs for schizophrenia and 556,278–859,034 SNPs for bipolar disorder. We refrained from using the intersection between all these cohorts to not reduce the number of SNPs used in prediction by too much. This meant that different iterations in the cross-validations were based on predictions using a different number of SNPs. However, each comparison between a single-trait predictor and a multi-trait predictor is based on the same number of SNPs.
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study
| 100.0 |
In each iteration of the cross-validation, a different cohort acts as the test set and a different set of cohorts comprises the training set. To create a predictor from a particular set of cohorts, we first had to obtain effect size estimates from this particular set of cohorts. This is achieved by performing a meta-analysis of the summary statistics of the cohorts that comprise the training set. The meta-analysed beta values bMETA are calculated as:41\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_{{\mathrm{META}}} = \frac{{\mathop {\sum }\nolimits_s \frac{{b_s}}{{{\mathrm{SE}}_s^2}}}}{{\mathop {\sum }\nolimits_s \frac{1}{{{\mathrm{SE}}_s^2}}}}$$\end{document}bMETA=∑sbsSEs2∑s1SEs2where bs is the effect size in cohort s and SEs is the standard error in cohort s. Conversion between beta values and odds ratios (OR) simply follows the equality b = log(OR). The weights derived for each trait make assumptions about the variance of SNP effects. We found that, in the summary statistics we used, the observed variance across SNP effects often departed from the expected value. To correct for that, we scaled the SNP effect estimates for each trait to have a variance of one and multiplied the weights for the unscaled SNP effects by the expected standard deviation across all SNPs.
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study
| 100.0 |
We created approximate SBLUP effects \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{b}}_{{\mathrm{SBLUP}}}} \right)$$\end{document}b^SBLUP using the OLS SNP effects from Eq. (5) and the ARIC data as an LD reference using Eq. (11) and set the shrinkage parameter, λ, to 1,300,000 for schizophrenia and to 2,000,000 for bipolar disorder, corresponding to observed scale SNP heritability estimates of 0.43 and 0.33 for schizophrenia and bipolar disorder, respectively. We then used the PLINK “--score” function to turn SNP effects \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{b}}_{{\mathrm{SBLUP}}},\widehat {\bf{b}}_{{\mathrm{GWAS}}}} \right)$$\end{document}b^SBLUP,b^GWAS into individual predictors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{g}}_{{\mathrm{SBLUP}}},\widehat {\bf{g}}_{{\mathrm{GWAS}}}} \right)$$\end{document}g^SBLUP,g^GWAS for each meta-analysed schizophrenia or bipolar disorder cross-validation set. For the multi-trait weighting, we estimated the heritability of schizophrenia and bipolar disorder and their genetic correlation using LD score regression from publicly available PGC2 schizophrenia summary statistics and the PGC1 bipolar disorder summary statistics. These estimates were then used to calculate the index weights of Eq. (15) for the weighted multi-trait SBLUP predictors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{g}}_{{\mathrm{wMT}} - {\mathrm{SBLUP}}},\widehat {\bf{g}}_{{\mathrm{wMT}} - {\mathrm{GWAS}}}} \right)$$\end{document}g^wMT-SBLUP,g^wMT-GWAS of SCZ and BIP, and these were not altered between different cross-validation sets.
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study
| 100.0 |
To test the degree to which the choice of weights affects the accuracy of the multi-trait predictor, we compared the accuracy of multi-trait predictors based on a spectrum of other weights (Supplementary Figs. 5 and 6). For this, we took advantage of two things: First, when individual predictors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{g}}_{{\mathrm{SBLUP}}},\widehat {\bf{g}}_{{\mathrm{GWAS}}}} \right)$$\end{document}g^SBLUP,g^GWAS are weighted rather than SNP effects \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{b}}_{{\mathrm{SBLUP}}},\widehat {\bf{b}}_{{\mathrm{GWAS}}}} \right)$$\end{document}b^SBLUP,b^GWAS, the conversion from SNP effects to individual effects does not have to be repeated for different weights. Second, the scaling of a predictor does not influence its accuracy in terms of correlation between prediction and outcome. Therefore, rather than testing each combination of weights of schizophrenia and bipolar disorder, it is sufficient to vary the relative weight of schizophrenia to bipolar disorder to explore the whole range of possible multi-trait predictors for these two traits. For each test cohort, this enabled us to test whether the weights of our multi-trait predictor derived from theory deviate from the weights that would result in the highest prediction accuracy for that data set.
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study
| 99.94 |
We applied our approach to a large range of phenotypes for which GWAS summary statistics are publicly available. We started with GWAS summary statistics for 46 phenotypes. However, in some circumstance the same studies (i.e., based on the same individuals) had generated summary statistics for multiple similar phenotypes, so we chose only one phenotype per study, which left us with 34 phenotypes. For example, out of “Cigarettes per day” and “Smoking Ever” we only selected the latter to have only one trait for smoking. We used 112,338 unrelated individuals of European descent in the UK biobank data as the testing set. We paired 6 phenotypes out of the 34 summary statistic phenotypes to phenotypes in the UK Biobank: Height, BMI, fluid intelligence score, depression, angina, and diabetes. The first three are quantitative traits and the latter three are disease traits for which we could identify at least 1000 cases in the UK Biobank data. For details, see Supplementary Table 1.
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study
| 100.0 |
For the disease traits, we used the self-reported diagnoses rather than ICD10 diagnoses, as they tend to have larger sample sizes. For depression, we used a more refined definition of cases and controls, where individuals were not counted as cases if they had any history of psychiatric symptoms or diagnoses other than depression or if they were prescribed drugs that are indicative of such diagnoses. Individuals were selected as controls only when there was an absence of any psychiatric symptoms or diagnoses and only when they were not prescribed any drugs that could be indicative of such diagnoses. All 6 traits in the UK Biobank were corrected for age, sex and the first 10 principal components by regressing the phenotype on these covariates and using the residuals from that regression for further analysis. For each trait, the SNPs that went into the analysis were based on the overlap between the GWAS summary statistics, the HapMap3 SNPs, the GERA data set, which was used as an LD reference in the SBLUP analysis, and the imputed SNPs from the UK Biobank. (For details on the QC process and imputation, see URLs.) Depending on the trait, the total number of SNPs ranged from around 660,000 to around 930,000.
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| 100.0 |
We created single-trait \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{g}}_{{\mathrm{SBLUP}}}} \right)$$\end{document}g^SBLUP as well as multi-trait \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{g}}_{{\mathrm{wMT}} - {\mathrm{SBLUP}}}} \right)$$\end{document}g^wMT-SBLUP predictors for the six paired phenotypes. To create SBLUP SNP effects \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{b}}_{{\mathrm{SBLUP}}}} \right)$$\end{document}b^SBLUP from summary statistic trait, we used a λ value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M\left( {1 - h_{{\mathrm{SNP}}_k}^2} \right){\mathrm{/}}h_{{\mathrm{SNP}}_k}^2$$\end{document}M1-hSNPk2∕hSNPk2 for each trait k, where M is assumed to be 1,000,000. As LD reference set, we used a random subset of 10,000 people of European descent from the GERA data set, and we set the LD window size to 2000 kb. We then used the PLINK “–score” function to turn SNP effects \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{b}}_{{\mathrm{SBLUP}}}} \right)$$\end{document}b^SBLUP into individual predictors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{g}}_{{\mathrm{SBLUP}}}} \right)$$\end{document}g^SBLUP for each trait. For the multi-trait weighting, we used LD score regression to calculate SNP heritability and genetic correlation between all pairs of cohorts. For dichotomous disease traits, SNP heritability was calculated on the observed scale. For each phenotype for which a multi-trait predictor was created, we selected all phenotypes that had a genetic correlation estimate significantly different from 0 at p = 0.05 with the focal trait, as well as the focal trait itself. The summary statistics based single-trait SBLUP predictors of the selected phenotypes were then combined into multi-trait SBLUP \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{g}}_{{\mathrm{wMT}} - {\mathrm{SBLUP}}}} \right)$$\end{document}g^wMT-SBLUP predictors. The weights for each phenotype were calculated according to Eq. (15). These weights require the single-trait predictors to have exactly the right variance. Since the summary statistics data slightly diverged from this expectation, we scaled each single-trait SBLUP predictor to have mean 0 and variance 1 and then multiplied it with its expected standard deviation, to ensure everything is on exactly the correct scale. We followed the same approach when using single-trait LDPred predictors.
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| 99.94 |
We compared the performance of the multi-trait predictors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{g}}_{{\mathrm{wMT}} - {\mathrm{SBLUP}}}} \right)$$\end{document}g^wMT-SBLUP not only to the performance of the single-trait predictor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\widehat {\bf{g}}_{{\mathrm{SBLUP}}}} \right)$$\end{document}g^SBLUP for the same trait but also to the performance of all other (cross-trait) single-trait predictors for the traits that exhibited significant rG with the focal trait (Fig. 4). This is appropriate because in some traits the single-trait predictor from the same trait is not the most accurate single-trait predictor.
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| 100.0 |
Motor mimicry supports the decoding of perceived emotions by the healthy brain1,2. Viewing emotional facial expressions rapidly and involuntarily engages the facial muscles of neurologically normal observers3,4. Emotional mimesis may have evolved as a specialized ‘exaptation’ of action observation, and by promoting emotional contagion and affective valuation may have facilitated the development of advanced human social behaviour and theory of mind2,5,6. In line with this interpretation, motor recoding of observed emotion correlates with empathy and emotion identification ability7 and predicts authenticity judgments on facial expressions8; while conversely, facial paralysis induced by botulinum toxin attenuates emotional reactivity9. The linkage between emotion observation, recognition and mimesis is precise: viewing of universal facial emotional expressions10 produces signature profiles of electromyographic (EMG) activity in the facial muscles conveying each expression3,11. This phenomenon is mediated by distributed, cortico-subcortical brain regions that may together instantiate a hierarchically organised neural substrate for inferring the intentions and subjective states of others12–15: primary visual representations of emotions would comprise the lowest level of the hierarchy, ascending through sensorimotor representations of emotional movement kinematics, prediction of movement goals and affective states, and encoding of intentions, including affective mentalising.
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review
| 97.6 |
On clinical, pathophysiological and neuroanatomical grounds, altered motor recoding might be anticipated to underlie impaired emotional and social signal processing in the frontotemporal dementias (FTD). This diverse group of neurodegenerative diseases manifests as three canonical clinico-anatomical syndromes16; behavioural variant (bvFTD), semantic variant primary progressive aphasia (svPPA) and nonfluent variant primary progressive aphasia (nfvPPA). These broad syndromic groupings encompass various sub-syndromes: in particular, within the heterogeneous bvFTD syndrome at least two major variants can be defined, based on the relative selectivity of right temporal lobe atrophy17,18. Deficits in emotion recognition, empathy and social understanding and behaviour are defining features of bvFTD but integral to all FTD syndromes19–24 and collectively engender substantial distress and care burden25. Impaired facial emotion recognition in bvFTD, svPPA and nfvPPA has been linked to atrophy of an overlapping network of cerebral regions including orbitofrontal cortex, posterior insula and antero-medial temporal lobe23,26, implicated in evaluation of facial emotional expressions and integration with bodily signals27–29. Moreover, various abnormalities of physiological reactivity have been documented in FTD, including changes in resting skin conductance and heart rate variability in bvFTD and altered homeostatic and affective autonomic responses in bvFTD, svPPA and nfvPPA30–36. Patients with bvFTD have been noted to have reduced facial expressivity37 and indeed, deficient volitional imitation of emotional faces38. However, whereas impaired facial EMG reactivity to facial expressions has been linked to emotion processing deficits in Parkinson’s disease39,40, Huntington’s disease41 and schizophrenia42, the motor physiology of emotional reactivity has not been addressed in the FTD spectrum.
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review
| 99.9 |
In this study, we investigated facial motor responses to viewing facial emotional expressions in a cohort of patients representing all major phenotypes of FTD (bvFTD, svPPA and nfvPPA) relative to healthy older individuals. In addition to the canonical syndromic FTD variants, we identified a subset of patients presenting with behavioural decline and selective right temporal lobe atrophy (right temporal variant, rtvFTD): this entity has been proposed previously to account for much of the heterogeneity of the broader bvFTD syndromic spectrum and is associated with particularly severe disturbances of facial empathy18,38,43,44. We compared facial EMG response profiles with emotion identification accuracy on a stimulus set comprising video recordings of dynamic, natural facial expressions: such expressions are more faithful exemplars of the emotions actually encountered in daily life and are anticipated to engage mechanisms of motor imitation more potently than the static images conventionally used in neuropsychological studies45,46. Neuroanatomical associations of facial expression identification and EMG reactivity in the patient cohort were assessed using voxel-based morphometry (VBM). Based on previous clinical and physiological evidence3,4,30,31,33,34,36,37,43,47, we hypothesised that healthy older individuals would show rapid and characteristic patterns of facial muscle responses to perceived emotional expressions coupled with efficient emotion identification. In contrast, we hypothesised that all FTD syndromes would be associated with impaired emotion identification but would exhibit separable profiles of facial muscle reactivity. In particular, we predicted that bvFTD and rtvFTD would be associated with reduced EMG responses while svPPA would be associated with aberrant coupling of muscle reactivity to emotion identification and nfvPPA with a more selective, emotion-specific reactivity profile. Based on previous neuroimaging studies both in the healthy brain and in FTD14,23,26,45,48–50, we further hypothesised that facial emotion identification and EMG reactivity would have partly overlapping neuroanatomical correlates within the extensive cortical circuitry previously implicated in the decoding of visual emotional signals, supplementary motor and insular cortices mediating the integration of somatic representations and antero-medial temporal and prefrontal circuitry involved in the evaluation of emotion. Within these distributed networks (given the known neuroanatomical heterogeneity of the target syndromes) we predicted a differential emphasis of grey matter correlates, with more marked involvement of inferior frontal, anterior cingulate and insular cortices in bvFTD and nfvPPA and more extensive, lateralised temporal lobe involvement in svPPA and rtvFTD16–18.
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| 99.94 |
Thirty-seven consecutive patients fulfilling consensus criteria for a syndrome of FTD51,52 (19 with bvFTD, nine with svPPA, nine with nfvPPA) and 21 healthy older individuals with no history of neurological or psychiatric illness participated. General characteristics of the participant groups are summarised in Table 1. No participant had a history of facial palsy or clinically significant visual loss after appropriate correction. There was clinical evidence of orofacial apraxia in seven patients in the nfvPPA group, but none in any of the other participant groups. General neuropsychological assessment (see Table 1) and brain MRI corroborated the syndromic diagnosis in all patients; no participant had radiological evidence of significant cerebrovascular damage. Based on visual inspection of individual brain MR images, six patients with a behavioural syndrome and relatively selective right temporal lobe atrophy were re-categorised as a rtvFTD subgroup (throughout this paper, we use ‘bvFTD’ to refer to those patients with a behavioural presentation not re-classified as rtvFTD). Between group differences in demographic and neuropsychological variables were analysed using ANOVAs with post hoc T-tests when main effects were found, except for categorical variables, for which a chi-squared test was used.Table 1Demographic, clinical and neuropsychological characteristics of participant groups.CharacteristicControlsbvFTDrtvFTDsvPPAnfvPPADemographic/clinicalNo. (male:female)9:1210:36:0a7:24:5Age (years)69.1 (5.3)66.2 (6.3)63.8 (6.4)66.1 (6.5)69.6 (6.5)Handedness (R:L)20:112:16:08:17:2Education (years)15.7 (3.5)13.2c (2.5)18.0 (3.1)14.9 (2.8)15.0 (2.1)MMSE (/30)29.6 (0.6)24.5a (4.6)25.3a (4.3)21.8a (6.9)23.7a (6.0)Symptom duration (years)N/A7.7 (6.0)6.5 (3.5)5.6 (3.0)4.7 (2.2)Neuropsychological General intellect WASI Verbal IQ125 (6.7)89a (21.9)87a (22.2)77a (19.7)80a (17.3)WASI Performance IQ125 (10.2)104a (20.3)107 (24.6)108 (23.5)99a (21.5) Episodic memory RMT words (/50)49.0 (1.4)37.4a (7.9)37.2a (9.3)30.0a,c (6.3)41.4a (9.5)RMT faces (/50)44.7 (3.5)33.5a (6.9)34.8a (7.9)32.8a (6.9)39.5 (6.6)Camden PAL (/24)20.4 (3.3)10.8a (8.1)12.5a (6.2)2.2a,b,c,e (3.7)16.3 (7.8) Executive skills WASI Block Design (/71)44.8 (10.5)32.5 (16.7)37.2 (22.1)39.1 (21.7)25.1a (19.7)WASI Matrices (/32)26.6 (3.9)19.3a (9.4)19.0a (9.8)19.8a (10.6)17.4a (9.0)WMS-R DS forward (max)7.1 (1.1)6.6 (1.2)6.8 (1.2)6.7 (1.2)4.8a,b,c,d (0.8)WMS-R DS reverse (max)5.6 (1.2)4.0a (1.5)4.7 (1.4)5.3 (1.8)3.0a, d (0.7)D-KEFS Stroop:color (s)33.4 (7.2)48.0 (20.5)48.8 (21.4)53.2a (28.2)87.0a,b,c,d (6.7)word (s)23.9 (5.6)32.5 (19.0)38.7 (26.1)36.0 (24.0)85.4a,b,c,d (10.3)interference (s)57.6 (16.7)99.6a (47.5)98.3 (45.1)90.1 (56.1)165a,b,c,d (30.8)Fluency:letter (F total)18.1 (5.6)7.8a (4.6)9.0a (4.7)8.9a (7.1)3.5a (1.7)category (animals total)24.4 (6.0)13.8a (7.5)10.3a (2.3)5.7a,b (5.1)8.8a (3.5)Trails A (s)33.7 (7.3)56.5 (32.3)59.8 (32.9)49.7 (20.1)81.7a (48.4)Trails B (s)67.3 (21.5)171.7a (88.2)186.7a (100.4)134.9 (101.7)211.1a (94.6) Language skills WASI Vocabulary72.3 (3.2)42.4a (21.5)47.0a (19.1)33.6a (22.0)31.7a (13.9)BPVS148.6 (1.1)120.8 (38.7)141.8 (7.2)85.8a,b,c,e (53.8)142.6 (10.1)GNT26.1a (2.7)12.2a (10.2)12.5 (10.1)1.6a,b,c,e (4.7)15.5a (6.6) Other skills GDA (/24)15.8 (5.3)7.8a (6.6)7.5a (6.3)11.9 (8.6)5.4a (1.9)VOSP (/20)19.0 (1.5)15.9a (3.4)16.7 (2.3)15.8 (4.5)15.3a (4.7)Mean (standard deviation) scores are shown unless otherwise indicated; maximum scores are shown after tests (in parentheses). asignificantly different from healthy controls, bsignificantly different from bvFTD, csignificantly different from rtvFTD, dsignificantly different from svPPA, esignificantly different from nfvPPA (all p < 0.05). BPVS, British Picture Vocabulary Scale (Dunn LM et al., 1982); bvFTD, patient group with behavioural variant frontotemporal dementia (excluding right temporal cases); Category fluency totals for animal category and letter fluency for the letter F in one minute (Gladsjo et al., 1999); Controls, healthy control group; D-KEFS, Delis Kaplan Executive System (Delis et al., 2001); DS, digit span; GDA, Graded Difficulty Arithmetic (Jackson M and Warrington, 1986); GNT, Graded Naming Test (McKenna and Warrington, 1983); MMSE, Mini-Mental State Examination score (Folstein et al., 1975); N/A, not assessed; NART, National Adult Reading Test (Nelson, 1982); nfvPPA, patient group with nonfluent variant primary progressive aphasia; PAL, Paired Associate Learning test (Warrington, 1996); RMT, Recognition Memory Test (Warrington, 1984); rtvFTD, patient subgroup with right temporal variant frontotemporal dementia; svPPA, patient group with semantic variant primary progressive aphasia; Synonyms, Single Word Comprehension: A Concrete and Abstract Word Synonyms Test (Warrington et al., 1998); Trails-making task based on maximum time achievable 2.5 minutes on task A, 5 minutes on task B (Lezak et al., 2004); VOSP, Visual Object and Spatial Perception Battery – Object Decision test (Warrington and James, 1991); WAIS-R, Wechsler Adult Intelligence Scale‐-Revised (Wechsler, 1981); WASI, Wechsler Abbreviated Scale of Intelligence (Wechsler, 1997); WMS, Wechsler Memory Scale (Wechsler, 1987).
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| 99.94 |
Mean (standard deviation) scores are shown unless otherwise indicated; maximum scores are shown after tests (in parentheses). asignificantly different from healthy controls, bsignificantly different from bvFTD, csignificantly different from rtvFTD, dsignificantly different from svPPA, esignificantly different from nfvPPA (all p < 0.05). BPVS, British Picture Vocabulary Scale (Dunn LM et al., 1982); bvFTD, patient group with behavioural variant frontotemporal dementia (excluding right temporal cases); Category fluency totals for animal category and letter fluency for the letter F in one minute (Gladsjo et al., 1999); Controls, healthy control group; D-KEFS, Delis Kaplan Executive System (Delis et al., 2001); DS, digit span; GDA, Graded Difficulty Arithmetic (Jackson M and Warrington, 1986); GNT, Graded Naming Test (McKenna and Warrington, 1983); MMSE, Mini-Mental State Examination score (Folstein et al., 1975); N/A, not assessed; NART, National Adult Reading Test (Nelson, 1982); nfvPPA, patient group with nonfluent variant primary progressive aphasia; PAL, Paired Associate Learning test (Warrington, 1996); RMT, Recognition Memory Test (Warrington, 1984); rtvFTD, patient subgroup with right temporal variant frontotemporal dementia; svPPA, patient group with semantic variant primary progressive aphasia; Synonyms, Single Word Comprehension: A Concrete and Abstract Word Synonyms Test (Warrington et al., 1998); Trails-making task based on maximum time achievable 2.5 minutes on task A, 5 minutes on task B (Lezak et al., 2004); VOSP, Visual Object and Spatial Perception Battery – Object Decision test (Warrington and James, 1991); WAIS-R, Wechsler Adult Intelligence Scale‐-Revised (Wechsler, 1981); WASI, Wechsler Abbreviated Scale of Intelligence (Wechsler, 1997); WMS, Wechsler Memory Scale (Wechsler, 1987).
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| 99.94 |
This study was approved by the University College London institutional ethics committee and all methods were performed in accordance with the relevant guidelines and regulations. All participants gave informed consent in accordance with the Declaration of Helsinki.
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other
| 99.94 |
Videos of emotional facial expressions were obtained from the Face and Gesture Recognition Research Network (FG-NET) database53. This database comprises silent recordings of healthy young adults viewing emotional scenarios: the scenarios were designed to elicit spontaneous, canonical facial expressions, but were presented without any instruction to pose or inhibit particular expressions (further details in Supplementary Material). In order to sample the spectrum of facial expressions, for each of the canonical emotions of anger, fear, happiness, surprise and disgust10 we selected 10 videos (50 stimuli in total; see Table S1) that clearly conveyed the relevant expression (the canonical emotion of sadness was omitted because its more diffuse time course sets it apart from other emotional expressions). Each video stimulus lasted between four and eight seconds (mean 4.9 seconds), commencing as a neutral facial expression and evolving into an emotional expression (further information in Supplementary Material). The video frame in which an emotional expression first began to develop unambiguously from the neutral baseline (previously determined by independent normal raters and provided with the FG-NET database) was used to align data traces across trials.
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| 100.0 |
Stimuli were presented in randomised order via the monitor of a notebook computer running the Cogent toolbox of Matlab R2012b. The participant’s task on each trial was to identify from among the five alternatives (verbally or by pointing to the appropriate written name) which emotion was displayed; participant responses were recorded for offline analysis. Participants were first familiarised with the stimuli and task to ensure they understood and were able to comply with the protocol. During the test, no feedback was given and no time limits were imposed on responses. Emotion identification scores were compared among groups using ANOVAs, with Bonferroni-corrected post hoc T-tests when main effects were found.
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| 100.0 |
While participants viewed the video stimuli, facial EMG was recorded continuously from left corrugator supercilii, levator labii and zygomaticus major muscles with bipolar surface electrodes, according to published guidelines for the use of EMG in research54. These facial muscles were selected as the key drivers of the canonical expressions represented by the video stimuli3,11. Expressions of anger and fear engage corrugator supercilii (which knits the brow) and inhibit zygomaticus major (which raises the corner of the mouth); expressions of happiness and surprise are associated with the reverse muscle activity profile, while disgust engages both corrugator supercilii and levator labii (which curls the top lip). EMG data were sampled at 2048Hz with a 0.16–100Hz band-pass filter and the EMG signal was rectified, high-pass filtered to correct for baseline shifts and smoothed with a 100 data point sliding filter using MATLAB R2012b; trials with signal amplitude >3 standard deviations from the mean (attributable to large artifacts, e.g., blinks) were removed prior to analysis. For each trial, the mean change in EMG activity from baseline (mean activity during a 500 ms period prior to trial onset) was analysed for each muscle in 500 ms epochs, starting 1s before the onset of expression change in the video stimuli; the EMG response for each muscle was calculated as the area under the curve of EMG signal change from baseline.
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study
| 100.0 |
We first assessed the presence of automatic imitation (any EMG change from baseline) and emotion-specific muscle activation (any interaction of muscle EMG response with emotion) for the healthy control group, using a repeated measures ANOVA (mean EMG activity for five emotions in eight 500 ms time bins for the three muscles). To determine if there was an overall effect of participant group on the degree of emotion-specific muscle activation, EMG responses were compared across all participants using a restricted maximum likelihood mixed effects model incorporating interactions between emotion, muscle and participant group, with participant identity as a level variable and time bin as a covariate of no interest. After assessing the overall effect of participant group in the omnibus test, we proceeded to establish the basis for any group differences by examining particular emotion-specific muscle contrasts. Emotion-specific EMG response profiles were quantified for each trial by combining individual muscle responses pairwise as follows: for anger and fear, (corrugator response minus zygomaticus response); for happiness and surprise, (zygomaticus response minus corrugator response); for disgust, (corrugator response plus levator response). These pairwise muscle contrasts have been shown to improve reliability and internal consistency of facial EMG analysis55. Muscle contrast EMG reactivity for each trial was then analysed as a dependent variable in an ANOVA incorporating participant group and emotion as fixed factors. Significant main effects in the ANOVA were explored with post hoc T-tests, using Bonferroni correction for multiple comparisons.
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| 100.0 |
To test the hypothesis that emotional imitation supports identification, we assessed any relationship between overall EMG reactivity and emotion identification score using Spearman’s rank correlation across the participant cohort. In addition, we compared EMG responses on trials with correct versus incorrect emotion identification and assessed any interaction with participant group membership using an ANOVA.
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| 100.0 |
To generate an overall measure of reactivity for each participant for use in the voxel based morphometry analysis, EMG reactivity was averaged over all trials for that participant and then normalised as the square root of the absolute value of the change in muscle activity from baseline (subzero values corresponding to muscle activity changes in the reverse direction to that expected were restored).
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study
| 100.0 |
For both emotion recognition and EMG reactivity, we assessed correlations with neuropsychological instruments indexing general nonverbal intellectual ability (nonverbal executive performance on the WASI Matrices task) and semantic knowledge (performance on the British Picture Vocabulary Scale), to examine the extent to which the experimental parameters of interest were influenced by disease severity and background semantic deficits.
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study
| 100.0 |
Each patient had a sagittal 3-D magnetization-prepared rapid-gradient-echo T1-weighted volumetric brain MR sequence (echo time/repetition time/inversion time 2.9/2200/900 msec, dimensions 256 256 208, voxel size 1.1 1.1 1.1 mm), acquired on a Siemens Trio 3T MRI scanner using a 32-channel phased-array head-coil. Pre-processing of brain images was performed using the New Segment56 and DARTEL57 toolboxes of SPM8 (www.fil.ion.ucl.ac.uk/spm), following an optimised protocol58. Normalisation, segmentation and modulation of grey and white matter images were performed using default parameter settings and grey matter images were smoothed using a 6 mm full width-at-half-maximum Gaussian kernel. A study-specific template mean brain image was created by warping all bias-corrected native space brain images to the final DARTEL template and calculating the average of the warped brain images. Total intracranial volume was calculated for each patient by summing grey matter, white matter and cerebrospinal fluid volumes after segmentation of tissue classes.
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study
| 100.0 |
Processed brain MR images were entered into a VBM analysis of the patient cohort. Separate regression models were used to assess associations of regional grey matter volume (indexed as voxel intensity) with mean overall emotion identification score and EMG reactivity, for each syndromic group. Age, total intracranial volume and WASI Matrices score (a measure of nonverbal executive function and index of disease severity) were incorporated as covariates of no interest in all models. Statistical parametric maps of regional grey matter associations were assessed at threshold p < 0.05 after family-wise error (FWE) correction for multiple voxel-wise comparisons within pre-specified regional volumes of interest. For the emotion identification contrast, these regions were informed by previous studies of emotion processing in FTD and in the healthy brain, comprising insula, anteromedial temporal lobe (including amygdala, fusiform gyrus and temporal pole), inferior frontal cortex, anterior cingulate and supplementary motor cortices23,26,48. For the EMG reactivity contrast, regions of interest were based on previous functional imaging studies of facial mimicry and dynamic facial stimuli14,45,49,50, comprising visual (V1, MT/V5, parahippocampal and fusiform gyri) and primary and supplementary motor cortices.
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study
| 100.0 |
General clinical characteristics of the participant groups are presented in Table 1. There was a significant gender difference between participant groups (chi24 = 10.31, p = 0.036), but no significant age difference. The patient groups did not differ in mean symptom duration or level of overall cognitive impairment (as indexed using WASI Matrices score; ANOVAs and post hoc T-tests all p > 0.4).
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study
| 100.0 |
Group data for facial emotion identification are summarised in Table 2.Table 2Summary of emotion identification and EMG reactivity findings for participant groupsResponse parameterControlsbvFTDrtvFTDsvPPAnfvPPAEmotion identificationAnger4.6 (2.2)1.8 (1.4)a2.5 (1.6)1.1 (0.9)a3.4 (1.7)Disgust8.1 (1.0)5.3 (3.3)a3.5 (3.9)a3.8 (3.3)a5.4 (3.3)Fear5.4 (2.1)2.6 (2.0)a2.0 (1.7)a3.9 (2.0)4.4 (2.4)Happiness9.2 (0.8)8.0 (3.2)8.3 (1.9)7.0 (3.2)7.8 (1.6)Surprise8.4 (1.0)4.9 (2.8)a3.7 (2.8)a4.1 (3.2)a5.8 (3.0)Overall (/50)35.7 (4.6)22.7 (9.5)a20.0 (9.7)a20.2 (7.9)a26.9 (9.3)aFacial EMG reactivityAnger1.3 (3.3)0.5 (1.5)0.2 (1.0)1.2 (5.1)0.3 (4.0)Disgust2.6 (8.9)−0.9 (9.0)a0.5 (1.7)1.4 (6.2)0.9 (3.7)Fear0.7 (2.9)0.3 (1.3)−0.1 (1.9)0.8 (4.4)−0.9 (3.5)a,b,cHappiness1.3 (2.3)0.5 (1.3)d0.2 (1.6)d1.8 (8.2)2.3 (4.9)Surprise1.0 (2.5)0.01 (3.1)c,d0.3 (1.8)1.7 (5.3)1.7 (3.8) Overall 1.4 (4.7)0.09 (4.4)a,c,d0.2 (1.6)a,c1.4 (6.0)0.9 (4.2)Mean (standard deviation) scores on the emotion identification task and mean facial EMG reactivity (as defined in Fig. 1) to viewed emotional expressions are shown for each emotion, in each participant group. asignificantly less than healthy controls, bsignificantly less than bvFTD, csignificantly less than svPPA, dsignificantly less than nfvPPA (all pbonf < 0.05). bvFTD, patient group with behavioural variant frontotemporal dementia (excluding right temporal cases); Controls, healthy control group; nfvPPA, patient group with nonfluent variant primary progressive aphasia; rtvFTD, patient subgroup with right temporal variant frontotemporal dementia; svPPA, patient group with semantic variant primary progressive aphasia.
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| 100.0 |
Mean (standard deviation) scores on the emotion identification task and mean facial EMG reactivity (as defined in Fig. 1) to viewed emotional expressions are shown for each emotion, in each participant group. asignificantly less than healthy controls, bsignificantly less than bvFTD, csignificantly less than svPPA, dsignificantly less than nfvPPA (all pbonf < 0.05). bvFTD, patient group with behavioural variant frontotemporal dementia (excluding right temporal cases); Controls, healthy control group; nfvPPA, patient group with nonfluent variant primary progressive aphasia; rtvFTD, patient subgroup with right temporal variant frontotemporal dementia; svPPA, patient group with semantic variant primary progressive aphasia.
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Overall accuracy of facial emotion identification showed a main effect of participant group (F4 = 10.89, p < 0.001), and was reduced in all syndromic groups relative to controls (all pbonf < 0.012) (Table 2). There was no significant relationship between emotion identification accuracy and age but a significant effect of gender (p = 0.04), with higher identification scores overall in female participants. The main effect of participant group persisted after covarying for gender (F4 = 13.852, p < 0.001). Emotion identification accuracy in the patient cohort correlated with standard measures of nonverbal executive function (WASI Matrices score, an index of disease severity; rho = 0.547, p < 0.001) and semantic competence (British Picture Vocabulary Scale; rho = 0.676, p < 0.001).
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Mean time courses of EMG responses for each facial muscle and emotion are shown for all participant groups in Fig. 1. Group data for EMG reactivity are summarised in Table 2 and Fig. 2.Figure 1Patterns of EMG reactivity for each muscle in each participant group. For each participant group, the plots show the time course of average EMG reactivity (in microvolts) for key facial muscles while participants watched videos of emotional facial expressions. EMG reactivity, here indexed in arbitrary units as mean EMG change from baseline, is shown on the y-axis (after rectifying, high-pass filtering and removing artefacts as described in Methods). Onset of the viewed facial expression (as determined in a prior independent analysis of the video stimuli) is at time 0 (dotted line) in each panel. In healthy controls, corrugator supercilii (CS) was activated during viewing of anger, fear and disgust, but inhibited during viewing of happiness and surprise; zygomaticus major (ZM) was activated during viewing of happiness and surprise, but inhibited during viewing of anger and fear; and levator labii (LL) was inhibited during viewing of anger and fear, and maximally activated during viewing of disgust. Note that in healthy controls muscle responses consistently preceded the unambiguous onset of viewed emotional expressions. bvFTD, patient group with behavioural variant frontotemporal dementia (excluding right temporal cases); Control, healthy control group; nfvPPA, patient group with nonfluent variant primary progressive aphasia; rtvFTD, patient subgroup with right temporal variant frontotemporal dementia; svPPA, patient group with semantic variant primary progressive aphasia.Figure 2EMG reactivity in each participant group, and the relationship with identification accuracy. For each participant group, the histograms show mean overall facial muscle EMG reactivity (top) and EMG reactivity separately (below) for those trials on which viewed emotional expressions were identified correctly (corr) versus incorrectly (incorr); error bars indicate standard error of the mean (see also Table 2). bvFTD, patient group with behavioural variant frontotemporal dementia; Control, healthy control group; nfvPPA, patient group with nonfluent variant primary progressive aphasia; rtvFTD, patient subgroup with right temporal variant frontotemporal dementia; svPPA, patient group with semantic variant primary progressive aphasia.
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| 100.0 |
Patterns of EMG reactivity for each muscle in each participant group. For each participant group, the plots show the time course of average EMG reactivity (in microvolts) for key facial muscles while participants watched videos of emotional facial expressions. EMG reactivity, here indexed in arbitrary units as mean EMG change from baseline, is shown on the y-axis (after rectifying, high-pass filtering and removing artefacts as described in Methods). Onset of the viewed facial expression (as determined in a prior independent analysis of the video stimuli) is at time 0 (dotted line) in each panel. In healthy controls, corrugator supercilii (CS) was activated during viewing of anger, fear and disgust, but inhibited during viewing of happiness and surprise; zygomaticus major (ZM) was activated during viewing of happiness and surprise, but inhibited during viewing of anger and fear; and levator labii (LL) was inhibited during viewing of anger and fear, and maximally activated during viewing of disgust. Note that in healthy controls muscle responses consistently preceded the unambiguous onset of viewed emotional expressions. bvFTD, patient group with behavioural variant frontotemporal dementia (excluding right temporal cases); Control, healthy control group; nfvPPA, patient group with nonfluent variant primary progressive aphasia; rtvFTD, patient subgroup with right temporal variant frontotemporal dementia; svPPA, patient group with semantic variant primary progressive aphasia.
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| 100.0 |
EMG reactivity in each participant group, and the relationship with identification accuracy. For each participant group, the histograms show mean overall facial muscle EMG reactivity (top) and EMG reactivity separately (below) for those trials on which viewed emotional expressions were identified correctly (corr) versus incorrectly (incorr); error bars indicate standard error of the mean (see also Table 2). bvFTD, patient group with behavioural variant frontotemporal dementia; Control, healthy control group; nfvPPA, patient group with nonfluent variant primary progressive aphasia; rtvFTD, patient subgroup with right temporal variant frontotemporal dementia; svPPA, patient group with semantic variant primary progressive aphasia.
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| 100.0 |
Healthy older participants showed the anticipated profiles of facial muscle activity in response to viewing facial expressions (Fig. 1): corrugator supercilii was activated by anger, fear and disgust, and inhibited by happiness and surprise; zygomaticus major was activated by happiness and surprise, and inhibited by anger and fear; and levator labii activity was maximal for disgust. Due to the proximity of levator labii and zygomaticus major, and the limited spatial specificity of surface electrodes54, there was substantial electrical leakage between these two muscles. However, zygomaticus major was maximally activated by happiness and surprise, and levator labii by disgust; moreover, these muscles were not combined in any of the pairwise muscle contrasts.
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| 100.0 |
EMG reactivity to viewed facial expressions was modulated in an emotion- and muscle-specific manner in healthy controls (F(2.20,43.94) = 5.03, p = 0.009) and the participant cohort as a whole (chi2(8) = 80.05, p < 0.001). There was further evidence that this interaction between emotion and muscle reactivity varied between participant groups (interaction of group, emotion and muscle: (chi2(32) = 143.91, p < 0.001). After the generation of a muscle contrast reactivity measure for each trial, ANOVA revealed significant main effects of participant group (F(4) = 10.84, p < 0.001), emotion (F(4) = 3.40, p = 0.009) and the interaction of group and emotion (F(16) = 2.79, p < 0.001; Table 2). In post hoc T-tests comparing participant groups (with Bonferroni correction), overall EMG reactivity across the five emotions was significantly reduced in the bvFTD group relative to the healthy control group (pbonf < 0.001), the svPPA group (pbonf < 0.001) and the nfvPPA group (pbonf = 0.042); and significantly reduced in the rtvFTD group relative to the healthy control group (pbonf = 0.001) and the svPPA group (pbonf = 0.005).
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| 100.0 |
There was no significant relationship between EMG reactivity and age (p = 0.1), gender (p = 0.42), or WASI Matrices score (used here as a measure of disease severity; p = 0.63) in the patient cohort, nor with a standard measure of semantic knowledge (British Picture Vocabulary Scale; p = 0.5).
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| 100.0 |
Across the participant cohort, overall EMG reactivity was significantly correlated with emotion identification accuracy (rho = 0.331, p = 0.011) and mean trial EMG reactivity was significantly higher for trials on which the emotion was correctly identified (n = 1586) than on error trials (n = 1314; p = 0.002). This differential effect of correct versus incorrect trials showed a significant interaction with participant group (F(4) = 4.18, p = 0.002; see Fig. 2). Among healthy controls, there was a strong trend towards greater reactivity predicting correct identification (p = 0.087). Comparing trial types within patient groups, EMG reactivity was significantly higher on correct identification trials than error trials in the bvFTD group (p = 0.009) and the nfvPPA group (p = 0.01) but not the rtvFTD group (p = 0.76) or the svPPA group (p = 0.06, here signifying a trend towards greater EMG reactivity on incorrect trials).
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| 100.0 |
Significant grey matter associations of emotion identification and EMG reactivity for the patient cohort are summarised in Table 3 (all thresholded at pFWE < 0.05 within pre-specified anatomical regions of interest); statistical parametric maps are presented in Fig. 3.Table 3Neuroanatomical correlates of emotion identification and reactivity in patient groups.GroupRegionSideClusterPeak (mm)T scorePFWE(voxels)xyz Emotion identification bvFTDAnterior cingulateL196−844125.590.003Anterior insulaL123−302704.070.047Supplementary motor areaL5−104503.810.044Opercular IFGL32−5712185.140.003Anteromedial temporal:Temporal poleL2133−328−385.110.010Amygdala−242−384.940.015Fusiform gyrus−30−9−384.820.019rtvFTDSupplementary motor areaL34−3−10574.150.022Temporo-occipital junctionR1866−50−84.060.038svPPASTG/STSL536−58−30147.210.005Supplementary motor areaL19−4−2504.230.019Opercular IFGL25−5712185.050.003Anterior cingulateL24−24434.110.042Fusiform gyrusR4422−4−444.430.042nfvPPASupplementary motor areaL37−4−2504.140.023Opercular IFGL9−528183.960.033 Facial EMG reactivity bvFTDSupplementary motor areaL12−8−9563.990.030Temporo-occipital junctionL25−54−45−44.290.064rtvFTDTemporo-occipital junctionR862−6223.960.046svPPAParahippocampal gyrusL59−20−28−244.250.028Parahippocampal gyrusR7218−33−185.250.003nfvPPAPrimary visual cortexR29112−8035.920.001Primary motor cortexR521568275.430.007Supplementary motor areaR1888684.420.012The table presents regional grey matter correlates of mean overall emotion identification score and facial EMG reactivity (as defined in Fig. 1) during viewing of facial expressions in the four patient groups, based on voxel-based morphometry. Coordinates of local maxima are in standard MNI space. P values are all significant after family-wise error (FWE) correction for multiple voxel-wise comparisons within pre-specified anatomical regions of interest (see text). bvFTD, patient group with behavioural variant frontotemporal dementia (excluding right temporal cases); IFG, inferior frontal gyrus; nfvPPA, patient group with nonfluent variant primary progressive aphasia; rtvFTD, patient subgroup with right temporal variant frontotemporal dementia; STG/S, superior temporal gyrus/sulcus; svPPA, patient group with semantic variant primary progressive aphasia.Figure 3Neuroanatomical correlates of emotion identification and EMG reactivity for each syndromic group. Statistical parametric maps (SPMs) show regional grey matter volume positively associated with overall emotion identification accuracy and facial EMG reactivity during viewing of emotional facial expressions, based on voxel-based morphometry of patients’ brain MR images (see also Table 3); T-scores are coded on the colour bar. SPMs are overlaid on sections of the normalised study-specific T1-weighted mean brain MR image; the MNI coordinate (mm) of the plane of each section is indicated (coronal and axial sections show the left hemisphere on the left). Panels code syndromic profiles of emotion identification (ID) or EMG reactivity (EMG). Note that the correlates of emotion identification and EMG reactivity in different syndromes overlapped in particular brain regions, including supplementary motor cortex and temporo-occipital junction (see Table 3). SPMs are thresholded for display purposes at p < 0.001 uncorrected over the whole brain, however local maxima of areas shown were each significant at p < 0.05 after family-wise error correction for multiple voxel-wise comparisons within pre-specified anatomical regions of interest (see Table 3). bvFTD, patient group with behavioural variant FTD; nfvPPA, patient group with nonfluent variant primary progressive aphasia; rtvFTD, patient subgroup with right temporal variant frontotemporal dementia; svPPA, patient group with semantic variant primary progressive aphasia.
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study
| 100.0 |
The table presents regional grey matter correlates of mean overall emotion identification score and facial EMG reactivity (as defined in Fig. 1) during viewing of facial expressions in the four patient groups, based on voxel-based morphometry. Coordinates of local maxima are in standard MNI space. P values are all significant after family-wise error (FWE) correction for multiple voxel-wise comparisons within pre-specified anatomical regions of interest (see text). bvFTD, patient group with behavioural variant frontotemporal dementia (excluding right temporal cases); IFG, inferior frontal gyrus; nfvPPA, patient group with nonfluent variant primary progressive aphasia; rtvFTD, patient subgroup with right temporal variant frontotemporal dementia; STG/S, superior temporal gyrus/sulcus; svPPA, patient group with semantic variant primary progressive aphasia.
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study
| 100.0 |
Neuroanatomical correlates of emotion identification and EMG reactivity for each syndromic group. Statistical parametric maps (SPMs) show regional grey matter volume positively associated with overall emotion identification accuracy and facial EMG reactivity during viewing of emotional facial expressions, based on voxel-based morphometry of patients’ brain MR images (see also Table 3); T-scores are coded on the colour bar. SPMs are overlaid on sections of the normalised study-specific T1-weighted mean brain MR image; the MNI coordinate (mm) of the plane of each section is indicated (coronal and axial sections show the left hemisphere on the left). Panels code syndromic profiles of emotion identification (ID) or EMG reactivity (EMG). Note that the correlates of emotion identification and EMG reactivity in different syndromes overlapped in particular brain regions, including supplementary motor cortex and temporo-occipital junction (see Table 3). SPMs are thresholded for display purposes at p < 0.001 uncorrected over the whole brain, however local maxima of areas shown were each significant at p < 0.05 after family-wise error correction for multiple voxel-wise comparisons within pre-specified anatomical regions of interest (see Table 3). bvFTD, patient group with behavioural variant FTD; nfvPPA, patient group with nonfluent variant primary progressive aphasia; rtvFTD, patient subgroup with right temporal variant frontotemporal dementia; svPPA, patient group with semantic variant primary progressive aphasia.
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| 100.0 |
Accuracy identifying dynamic emotional expressions was correlated with regional grey matter volume in left supplementary motor cortex in all syndromic groups. Additional regional grey matter correlates of emotion identification were delineated for particular syndromic groups. The bvFTD, svPPA and nfvPPA groups showed syndromic grey matter correlates within a bi-hemispheric (predominantly left-lateralised) frontotemporal network including opercular inferior frontal gyrus, anterior cingulate, anterior insula and antero-inferior temporal lobe; while the svPPA group showed a further correlate in left posterior superior temporal cortex and the rtvFTD group showed a correlate in right temporo-occipital junctional cortex in the vicinity of MT/V5 complex59.
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| 100.0 |
Across the patient cohort, overall mean EMG reactivity was correlated with regional grey matter in an overlapping but more posteriorly directed and right-lateralised network, with variable emphasis in particular syndromic groups. The bvFTD and nfvPPA groups showed grey matter correlates of EMG reactivity in supplementary and primary motor cortices, while all syndromic groups showed grey matter associations in cortical areas implicated in the analysis of visual signals, comprising primary visual cortex in the nfvPPA group; temporo-occipital. junction (MT/V5 complex) in the bvFTD and rtvFTD groups; and parahippocampal gyrus in the svPPA group.
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| 100.0 |
Here we have demonstrated facial motor signatures of emotional reactivity in the FTD spectrum. As anticipated, healthy older individuals showed characteristic profiles of facial muscle engagement by observed facial emotions; moreover, facial muscle reactivity predicted correct trial-by-trial identification of facial emotions. These findings provide further evidence that (in the healthy brain) facial mimesis is an automatic, involuntary mechanism supporting stimulus decoding and evaluation, rather than simply an accompaniment of conscious emotion recognition. In contrast, overall facial muscle reactivity and the normal coupling of muscle reactivity to facial emotion identification were altered differentially in the patient groups representing major FTD syndromes. As predicted, identification of facial expressions was impaired across the patient cohort: however, whereas the bvFTD group showed globally reduced facial muscle reactivity to observed emotional expressions, the svPPA group had preserved overall muscle reactivity but loss of the linkage between muscle response and correct expression identification. Among those patients with syndromes dominated by behavioural decline, the profile of facial muscle reactivity stratified cases with rtvFTD from other cases of bvFTD: the subgroup with rtvFTD had a particularly severe phenotype, exhibiting both globally reduced facial reactivity and also aberrant coupling of muscle reactivity to facial expression identification.
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| 100.0 |
Considered collectively, the motor signatures of emotional reactivity identified in our patient cohort amplify previous clinical, neuropsychological and physiological evidence in particular FTD syndromes. The generalised impairment of emotional mimesis in our bvFTD and rtvFTD groups is consistent with the clinical impression of facial impassivity37,60, impaired intentional imitation38 and blunting of autonomic responsiveness30,31,33,35,36 in these patients. Abnormal coupling of facial mimesis to facial expression identification in our svPPA group is in line with the disordered autonomic signalling of affective valuation previously documented in this syndrome33,35, and suggests a method of dissociating emotional reactivity from the declarative, semantic categorisation of emotions. The present findings suggest that aberrant motor recoding of perceived expressions may constitute a core physiological mechanism for impaired emotion processing in FTD.
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| 100.0 |
This mimetic mechanism may be particularly pertinent to the dynamically shifting and subtle emotions of everyday interpersonal encounters. Our own emotional expressions are normally subject to continual modulation by the expressed emotions of others, including tracking of transient ‘micro-expressions’61; this modulation occurs over short timescales (a few hundred milliseconds) and contributes importantly to the regulation of social interactions, prosociality and empathy28,62–64. If facial mimesis plays a key role in tuning such responses, loss of this modulatory mechanism (most notably in bvFTD and rtvFTD) might underpin not only impaired socio-emotional awareness in FTD but also the ‘poker-faced’ sense of unease these patients commonly provoke in others37.
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| 99.94 |
The neuroanatomical correlates we have identified speak to the coherent nature of dynamic emotion mimesis and identification. In line with previous evidence38, these processes mapped onto a distributed cerebral network within which FTD syndromes showed separable profiles of grey matter atrophy. Involvement of supplementary motor cortex was a feature across syndromes and associated both with emotion identification and motor reactivity, though joint correlation was observed in the bvFTD and nfvPPA groups but not the rtvFTD and svPPA groups (see Table 3). Supplementary motor cortex is a candidate hub for the computation of sensorimotor representations unfolding over time, an integral function of the mirror neuron system: this region generates both facial sensory-evoked potentials and complex facial movements65 and it is activated during facial imitation and empathy66 as well as by dynamic auditory emotional signals48. Furthermore, transcranial magnetic stimulation of the supplementary motor region disrupts facial emotion recognition67. The uncoupling of motor reactivity from emotion identification in the rtvFTD and svPPA groups may reflect disconnection of this key hub from linked mechanisms for affective semantic appraisal12, perhaps accounting for lack of an EMG reactivity correlate in supplementary motor cortex in these syndromic groups. Two further cortical hubs correlating both with emotion identification and mimesis were delineated in our patient cohort. In the svPPA and rtvFTD groups, a joint correlate was identified in the temporo-occipital junction zone, overlapping posterior superior temporal sulcus and MT/V5 visual motion cortices59,68: this region has been implicated in the imitation and decoding of dynamic facial expressions15,49,69,70, integration of dynamic social percepts, action observation and theory of mind71,72. In the svPPA group, infero-medial temporal cortex was linked both to emotion identification and mimesis: this region has previously been shown to respond to dynamic facial stimuli45.
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| 100.0 |
Additional grey matter associations of facial expression identification accuracy were delineated in cingulo-insular, antero-medial temporal and inferior frontal areas previously implicated both in the detection and evaluation of salient affective stimuli and in canonical FTD syndromes15,20,21,23,26,73,74. Additional grey matter associations of facial motor reactivity were identified (for the nfvPPA group) in primary visual and motor cortices: enhanced responses to emotional facial expressions have previously been demonstrated in visual cortex75, while motoric responses to social stimuli have been located in precentral gyrus14. However, it is noteworthy that certain grey matter associations emerging from this analysis - in particular, the ‘hub regions’ of supplementary motor cortex and temporo-occipital junction and (in the nfvPPA group) primary visual and motor cortices - lie beyond the brain regions canonically targeted in particular FTD syndromes or indeed, in previous studies of emotion processing in FTD21. It is likely that the dynamic expression stimuli employed here allowed a more complete picture of the cerebral mechanisms engaged in processing naturalistic emotions. Moreover, involvement of brain regions remote from zones of maximal atrophy may reflect distributed functional network effects (for example, visual cortical activity has been shown to be modulated by amygdala75) in conjunction with disease-related network connectivity changes, which are known to extend beyond the atrophy maps that conventionally define particular FTD syndromes76. Taken together, the present neuroanatomical findings are compatible with the previously proposed, hierarchical organisation of embodied representations supporting emotional decoding and empathy13,48,77,78: whereas early visual and motor areas may support automatic imitation via low-level visual and kinematic representations, higher levels of the processing hierarchy engage the human ‘mirror’ system and substrates for semantic, evaluative and mentalising processes that drive explicit emotion identification.
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| 99.94 |
From a clinical perspective, this work suggests a pathophysiological framework for deconstructing the complex social and emotional symptoms that characterise FTD syndromes. Such symptoms are difficult to measure using conventional neuropsychological tests, and may only be elicited by naturalistic social interactions: dynamic motor physiological surrogates might index both the affective dysfunction of patients’ daily lives and the underlying disintegration of culprit neural networks38. These physiological metrics might facilitate early disease detection and tracking over a wider spectrum of severity than is currently possible and enable socio-emotional assessment in challenging clinical settings (such as aphasia), especially since our results suggest that (in contrast to explicit emotion recognition) automatic motor reactivity may be relatively insensitive to semantic deficits. Our findings further suggest that such metrics are not simply ciphers of reduced cognitive capacity but may help stratify broad disease groupings (such as the heterogeneous bvFTD syndrome) and at the same time, may capture mechanisms that transcend traditional syndromic boundaries. We therefore propose that the paradigm of emotional sensorimotor reactivity may yield a fresh perspective on FTD nosology and candidate novel biomarkers of FTD syndromes. Looking forward, this paradigm suggests a potential strategy for biofeedback-based retraining of emotional responsiveness, perhaps in conjunction with disease-modifying therapies79.
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| 99.5 |
This study establishes a preliminary proof of principle but the findings require further corroboration. There are several clear limitations that suggest caution in interpreting our findings and directions for future work. We have studied a small, intensively phenotyped patient cohort: the most pressing issue will be to replicate the findings in larger clinical populations. Future studies should encompass a wider range of pathologies, in order to determine the general applicability of the paradigm and the specificity of syndromic motor profiles; it would be of interest, for example, to assess the heightened emotional contagion previously documented in Alzheimer’s disease80 in this context. Longitudinal cohorts including presymptomatic mutation carriers will be required in order to assess the diagnostic sensitivity of mimetic indices and their utility as biomarkers; ultimately, histopathological correlation will be necessary to establish any molecular correlates of the syndromic stratification suggested here. It will be relevant to explore the cognitive milieu of emotional motor responses in greater detail: for example, the effects of other sensory modalities (in particular, audition48, micro-expressions61, sincere versus social emotions81 and emotional ‘caricatures’ in FTD82) and the correlation of mimetic markers with measures of social cognition and daily life empathy38. Emotional reciprocity might be modeled using virtual reality techniques to generate model social interactions62. Beyond mimesis, integration of somatic and cognitive mechanisms during social emotional exchanges demands the joint processing of autonomic and neuroendocrine signals under executive control29,83,84: future work should assess other physiological modalities alongside EMG. Functional MRI and magnetoencephalography would amplify the present structural neuroanatomical correlates by capturing disease-related changes in underlying brain network connectivity and dynamics. Multimodal studies of this kind may set motor mimicry in the context of a comprehensive physiology of socio-emotional reactivity in neurodegenerative diseases. The ultimate goal will be to identify practical physiological markers that can be widely translated for the diagnosis and dynamic tracking of these diseases and the evaluation of new therapies.
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review
| 98.9 |
The feeding of preterm newborns, especially those weighing less than 1500 g at birth, has been a source of increasing concern. There has been debate about the ideal type of feeding that would allow an adequate development for these babies after birth, with growth and weight gain rates close to those observed during intrauterine life (1).
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review
| 95.7 |
Human milk is recommended for the enteral feeding of premature and term newborns since it is better tolerated, and is associated with low rates of necrotizing enterocolitis, sepsis, retinopathy of prematurity, and a better neurocognitive development (4 –8). Human milk is known to contain many antioxidant substances that are important for the reduction of oxidative stress (7).
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review
| 99.8 |
Infant milk formulas are used in situations in which the milk of the mother or of other donors is not available. Some neonatal units use these formulas to improve the weight gain of very low birth weight newborns (9,10). The prevention of growth restriction of preterm newborns during hospitalization is of extreme importance since there is evidence suggesting that appropriate postnatal growth is associated with better neurological development and other benefits such as improved immune response and reduced risk of infections (11,12).
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review
| 99.9 |
We selected patients included in the Brazilian Neonatal Network of infants with birth weight ≤1500 g, born from January 2006 to December 2013 at a university hospital. Exclusion criteria were: newborns who died, babies with malformations, babies transferred to other institutions, and babies whose medical charts were incomplete.
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study
| 99.75 |
We calculated differences of Z-scores for weight, length and head circumference measured at hospital discharge and at birth for each patient (Δ= hospital discharge - birth). Simple and multiple linear regression models were adjusted to compare the three groups in terms of Δ Z-score for the anthropometric measurements. For all multiple models, we considered covariates such as bronchopulmonary dysplasia, peri-intraventricular hemorrhage, SNAPPE II (Score for Neonatal Acute Physiology with Perinatal extension-II), weight and gestational age at birth, occurrence of necrotizing enterocolitis and sepsis. Estimates of the differences among group means and their respective 95% confidence intervals were obtained using the SAS 9.3 software (SAS Institute Inc., USA).
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| 100.0 |
The patients included in the study were divided into 3 groups. Group 1 consisted of infants who were exclusively breastfed at discharge, group 2 infants received mixed feeding, i.e., breast milk complemented with infant milk formula, and group 3 consisted of patients exclusively receiving a formula.
|
study
| 99.94 |
In our service, all preterm infants start to be fed milk expressed from their own mother unprocessed or pasteurized breast milk from the milk bank. When a volume of 100 mL·kg-1·day-1 is reached, the milk is enriched with a fortifying agent. When the patient is clinically stable and receiving a full diet (140–160 mL·kg-1·day-1), and if the maintenance of maternal lactation is not possible, a full or partial transition to infant milk formulas is started.
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other
| 99.9 |
According to the nutritional routine of the hospital service, group 1 infants received a greater proportion of human milk, frequently fortified, while group 2 infants received breast milk at least once a day associated with milk formula. Group 3 infants received fortified human milk at the beginning of life, followed by transition to an artificial formula due to the impossibility of maintaining maternal lactation.
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other
| 99.9 |
The mean birth weight was 1338.7 g for group 1, 1104.0 g for group 2 and 1254.7 g for group 3. Mean gestational age at birth was 31.9, 30.0 and 31.2 weeks, respectively, and mean corrected age at discharge was 37.8, 41.7 and 38.2 weeks, respectively. The groups were comparable, with no significant difference between them.
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study
| 99.94 |
Table 1.Characteristics of the mothers and comorbidities of the newborns according to type of feeding at discharge.Group 1Group 2Group 3 Comorbidities BPDYes6 (3.11)128 (66.32)59 (30.57)PIVHGrade 1 and 29 (6.72)66 (49.25)59 (44.03)Grade 3 and 43 (5.77)34 (65.38)15 (28.85)Apgar score≤52 (5.56)23 (63.89)11 (30.56)>564 (10.56)231 (38.12)311 (51.32)SNAPPE<2056 (13.18)140 (32.94)229 (53.88)≥2010 (4.65)117 (54.42)88 (40.93)GenderMale32 (9.28)143 (41.45)170 (49.28)Female33 (10.96)115 (38.21)153 (50.83)Late onset sepsisYes12 (4.41)148 (54.41)112 (41.18)Necrotizing enterocolitisYes2 (3.77)30 (56.60)21 (39.63) Maternal characteristics Age<20 years7 (5.69)56 (45.53)60 (48.78)≥20 years58 (11.09)201 (38.43)264 (50.48)Schooling<10 years15 (6.85)96 (43.84)108 (49.32)>10 years50 (12.38)149 (36.88)205 (50.74)ChorioamnionitisYes3 (3.39)30 (50.85)27 (45.76)Antenatal steroidNo18 (7.03)111 (43.36)127 (49.61)DeliveryVaginal21 (9.86)94 (44.13)98 (46.01)Cesarean45 (10.37)163 (37.56)226 (52.07)Data are reported as numbers (%). Study groups: 1, exclusive breast milk; 2, mixed feeding; 3: exclusive artificial formula. BPD: bronchopulmonary dysplasia; PIVH: periintraventricular hemorrhage; SNAPPE-II (Score for Neonatal Acute Physiology with Perinatal extension-II).
|
study
| 99.94 |
Data are reported as numbers (%). Study groups: 1, exclusive breast milk; 2, mixed feeding; 3: exclusive artificial formula. BPD: bronchopulmonary dysplasia; PIVH: periintraventricular hemorrhage; SNAPPE-II (Score for Neonatal Acute Physiology with Perinatal extension-II).
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study
| 99.94 |
Table 2.Differences in Z-scores for weight, length, and head circumference between discharge and birth according to type of feeding at discharge.Study groupsWeightLengthHead circumference1−0.84 (0.68)−1.10 (1.18)−0.21 (1.23)2−1.02 (0.75)−1.54 (1.37)−0.52 (1.64)3−0.86 (0.71)−0.97 (1.21)−0.08 (1.24)Data are reported as ΔZ-Score [mean differences in Z-scores (SD)] for weight, length and skull perimeter between discharge and birth. Study groups: 1, exclusive breast milk; 2: mixed feeding; 3: exclusive artificial feeding.
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study
| 100.0 |
Table 3.Adjusted linear regression model for comparative analysis of mean differences in Z-scores for weight, length and head circumference between discharge and birth for the different study groupsZ-Score/Multiple comparisonsΔZ-Score (discharge–birth)P value95%CILowerUpperWeight1–2−0.160.05−0.330.0011–3−0.100.19−0.260.052–30.060.22−0.040.16Length1–20.070.73−0.330.471–3−0.270.17−0.650.112–3−0.340.01*−0.58−0.10Head circumference1–2−0.010.97−0.460.441–3−0.210.33−0.630.212–3−0.200.15−0.470.07Study groups: 1, exclusive breast milk; 2, mixed feeding; 3, exclusive artificial feeding. ΔZ-Score: mean difference in Z-score for weight, length, and head circumference between discharge and birth. Linear regression model adjusted for bronchopulmonary dysplasia, periintraventricular hemorrhage, SNAPPE II, weight and gestational age at birth, occurrence of enterocolitis and sepsis. *P<0.05.
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study
| 100.0 |
Study groups: 1, exclusive breast milk; 2, mixed feeding; 3, exclusive artificial feeding. ΔZ-Score: mean difference in Z-score for weight, length, and head circumference between discharge and birth. Linear regression model adjusted for bronchopulmonary dysplasia, periintraventricular hemorrhage, SNAPPE II, weight and gestational age at birth, occurrence of enterocolitis and sepsis. *P<0.05.
|
study
| 100.0 |
No significant difference in weight or head circumference was observed between the infants studied regardless of the type of feeding they were receiving at discharge. Only length [Δ Z-score −0.34 (P-value=0.01; 95%CI=-0.58 to −0.10)] was impaired in group 2 compared to group 3, although without clinical significance.
|
study
| 100.0 |
It is known that insufficient postnatal growth of very low birth weight premature babies may result from complex interactions between genetic and environmental factors and not simply from an inadequate nutritional supply. Growth might also be affected by morbidities, endocrinological abnormalities, and administration of medications that might interfere with nutritional requirements and nutrient metabolism (14).
|
study
| 89.5 |
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