Publisher Correction: Robustification of GWAS to explore effective SNPs addressing the challenges of hidden population stratification and polygenic effects

Correction to: Scientific Reports 10.1038/s41598-021-90774-7, published online 22 June 2021

The original version of this Article contained typographical errors in the Equations.

In Table 1, fourth column, the Equation “Polygenic effect variation” was incorrect:

$$\text{var}\left(\sum _{k-m_1+1}^{{\text{m}}_2}{\text{b}}_{\text{k}}{\text{Z}}_{\text{k}}\right)$$

now reads:

$$\text{var}\left(\sum _{k=m_1+1}^{{\text{m}}_2}{\text{b}}_{\text{k}}{\text{Z}}_{\text{k}}\right)$$

In the Materials and methods section, under the subheading “Robustification of LMM based GWAS by using the outlier modification rule (proposed)”,

“b is the vector of random polygenic effects which follows N(0,σ_g²K), where σ_g² is the polygenic variance component, and K = (k_ijt) is the m × m genomic relationship matrix.”

now reads:

“b is the vector of random polygenic effects which follows N(0,σ_g²K), where σ_g² is the polygenic variance component, and K = (k_jt) is the m × m genomic relationship matrix.”

And,

“When the maximum likelihood (ML) or restricted maximum likelihood (REML) variance component $\widehat{\mathbf{V}}={\widehat{\sigma }}_g^2\mathbf{K}+{\widehat{\sigma }}_{\epsilon }^2\mathbf{I}$ is estimated, the classical F-statistic for testing the null hypothesis Ma = 0 for an arbitrary full-rank p × q matrix M^13,60.”

now reads:

“When the maximum likelihood (ML) or restricted maximum likelihood (REML) variance compontent $\widehat{\mathbf{V}}={\widehat{\sigma }}_g^2{\mathbf{Z}\mathbf{K}\mathbf{Z}}^{{}^{\prime }}+{\widehat{\sigma }}_{\epsilon }^2\mathbf{I}$ is estimated, the classical F-statistic for testing the null hypothesis Ma = 0 for an arbitrary full-rank p × q matrix M^13,60.”

And,

“(ii) Divide the phenotypic data into m groups corresponding to the m genotypic labels of the selected most significant SNP. For example, let

$$\mathit{y}={(y_1,y_2,...,y_n)}^{\prime }={(y_{11},..,y_{{}_1{n}_1},...,y_{m1},..,y_{m{n}_m})}^{\prime }$$

be the partition of phenotypic observations corresponding to the selected SNP, where, n = n₁ + n₂ + ……..+ n_k.

(iii) Detect the outlying observations from the l^th (l = 1,2,…,k) group using the β-weight function defined by

$$W_{\beta }(y_{\mathit{li}}|{\widehat{\theta }}_l)=exp\{-\frac{\beta }{2{\sigma }_{\mathit{li}}^2}(y_{\mathit{li}}-{\widehat{\mu }}_l{)}^2\}$$ 4

where i =  1, 2, …, n”

now reads:

“(ii) Divide the phenotypic data into m groups corresponding to the m genotypic labels of the selected most significant SNP. For example, let

$$\mathit{y}={(y_1,y_2,...,y_n)}^{\prime }={(y_{11},..,y_{{}_1{n}_1},...,y_{m1},..,y_{m{n}_m})}^{\prime }$$

be the partition of phenotypic observations corresponding to the selected SNP, where, n = n₁ + n₂ + …….. + n_m.

(iii) Detect the outlying observations from the l^th (l = 1,2,….,k) group using the β-weight function defined by

$$W_{\beta }(y_{\mathit{li}}|{\widehat{\theta }}_l)=exp\{-\frac{\beta }{2{\sigma }_{\mathit{li}}^2}(y_{\mathit{li}}-{\widehat{\mu }}_l{)}^2\}$$ 4

where i = 1, 2,  …, n_l”

Additionally, Equation 7 was incorrect:

“An outlying phenotypic observation y_lk in the l^th group is defined based on the β-weight function mentioned below:

$$W_{\beta }(x_{\mathit{jk}}|{\widehat{\theta }}_{j,\beta })=\left\{\begin{array}{c}>{\tau }_j,\text{if}{x}_{\mathit{jk}}\text{is}\text{not}\text{an}\text{outlier}\hfill \\ \le {\tau }_j,\text{if}{x}_{\mathit{jk}}\text{is}\text{an}\text{outlier}\hfill \end{array}\right)$$ 7

where the threshold value τ_l is the p^th quantile value of the empirical distribution of $W_{\beta }(x_{\mathit{li}}|{\widehat{\theta }}_{l,\beta })$.”

now reads:

“An outlying phenotypic observation y_li in the l^th group is defined based on the β-weight function mentioned below:

$$W_{\beta }(y_{\mathit{li}}|{\widehat{\theta }}_{j,\beta })=\left\{\begin{array}{c}>{\tau }_j,\text{if}{y}_{\mathit{li}}\text{is}\text{not}\text{an}\text{outlier}\hfill \\ \le {\tau }_j,\text{if}{y}_{\mathit{li}}\text{is}\text{an}\text{outlier}\hfill \end{array}\right)$$ 7

where the threshold value τ_l is the p^th quantile value of the empirical distribution of $W_{\beta }(y_{\mathit{li}}|{\widehat{\theta }}_{l,\beta }$).”

In Table 3, fourth column, the Equation “Polygenic effect variation” was incorrect:

$$\text{var}\left(\sum _{j=1}^{{\text{m}}_2}{\text{b}}_{\text{k}}{\text{x}}_{\text{k}}\right)$$

now reads:

$$\text{var}\left(\sum _{k=m_1+1}^{{\text{m}}_2}{\text{b}}_{\text{k}}{\text{z}}_{\text{k}}\right)$$

Under the subheading “Genotype simulation”,

“For this purpose, m = 2000 SNPs were generated for n = 1000 individuals, and these individuals were taken from k = 3 distinct population by considering different minor allele frequencies (MAFs). To do this, first, a set of mlatent vectors {v₁, v₂, …..v_m} was generated from a multivariate normal distribution with mean zero and variance–covariance matrix Cov(v_j,v_k) = ρ^|j-k|64,65. In our simulation, we considered ρ = 0.5 to avoid the linkage disequilibrium (LD) between the SNPs. Finally, two cutoff values s₁ and s₂ were used to convert the design matrix V = [v₁, v₂, …, v_m ] = [v_ij] of latent vectors to the genotypic score matrix $x_{\mathit{ij}}(i=1,2,\dots ,n,j=1,2,\dots ,m_1)\text{and}{z}_{\mathit{ij}}(i=m_1,1,\dots ,n,j=1,2,\dots ,m_2)$ as follows:

$$x_{\mathit{ij}},z_{\mathit{ij}}=\left\{\begin{array}{cc}-1,\hfill & v_{\mathit{ij}}<s_1\\ 0,\hfill & s_1\le v_{\mathit{ij}}\le s_2\\ 2,\hfill & v_{\mathit{ij}}>s_2\end{array}\right)$$

where s₁ and s₂ determine the minor allele frequency.”

now reads:

“For this purpose, m* = 2000 SNPs were generated for n = 1000 individuals, and these individuals were taken from k = 3 distinct population by considering different minor allele frequencies (MAFs). To do this, first, a set of latent vectors {v₁, v₂, …..v_m*} was generated from a multivariate normal distribution with mean zero and variance–covariance matrix Cov(v_j,v_k) = ρ^|j-k|64,65. In our simulation, we considered ρ = 0.5 to avoid the linkage disequilibrium (LD) between the SNPs. Finally, two cutoff values s₁ and s₂ were used to convert the design matrix V = [v₁, v₂, …, v_m* ] = [v_ij] of latent vectors to the genotypic score matrix $x_{\mathit{ij}}(i=1,2,\dots ,n,j=1,2,\dots ,m_1)\text{and}{z}_{\mathit{ij}}(i=1,2,\dots ,n;j=1,2,\dots ,m_2)$ as follows:

$$x_{\mathit{ij}},z_{\mathit{ij}}=\left\{\begin{array}{cc}0,\hfill & v_{\mathit{ij}}<s_1\\ 1,\hfill & s_1\le v_{\mathit{ij}}\le s_2\\ 2,\hfill & v_{\mathit{ij}}>s_2\end{array}\right)$$

where s₁ and s₂ determine the minor allele frequency.”

Furthermore, in the section “Phenotype simulation,”

“In every situation, m₁ = 4 SNPs were considered as causal variants and the remaining m₂ = m-m₁ = 1996 SNPs were allocated as polygenic variants (effects).”

now reads:

“In every situation, m₁ = 4 SNPs were considered as causal variants and the remaining m₂ = m* - m₁ = 1996 SNPs were allocated as polygenic variants (effects).”

Lastly, in the section “Consequence of phenotypic outliers on the partition of total variations”,

$$h=\frac{\text{var}\left(\sum _{k=1}^{m_1},a_k,x_k\right)}{\text{var}\left(y\right)}>\frac{\text{var}\left(\sum _{k=1}^{m_1},a_k,x_k\right)}{\text{var}\left(y^{\ast }\right)}=h^{2\ast }$$

now reads:

$$h^2=\frac{\text{var}\left(\sum _{k=1}^{m_1},a_k,x_k\right)}{\text{var}\left(y\right)}>\frac{\text{var}\left(\sum _{k=1}^{m_1},a_k,x_k\right)}{\text{var}\left(y^{\ast }\right)}=h^{2\ast }$$

The original Article has been corrected.