Last updated: 2018-10-11

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We illustrate the methods in SER model. It’s straight forward to implement in the general susie model.

\[ \begin{align*} \mathbf{y}_c &= X_c \boldsymbol{\beta} + \boldsymbol{\epsilon} \quad \boldsymbol{\epsilon}\sim N_{n}(0, \frac{1}{\tau}I) \\ \boldsymbol{\beta} &= \boldsymbol{\gamma} \beta \\ \boldsymbol{\gamma} &\sim Multinomial(1,\boldsymbol{\pi}) \\ \beta &\sim N(0, \frac{1}{\tau_{0}}) \end{align*} \]

where \(\mathbf{y}_c\) and \(X_c\) are centered. By default we set \(\frac{1}{\tau} = var(\mathbf{y})\), \(\frac{1}{\tau_{0}} = \sigma_{\beta}^{2} var(\mathbf{y})\).

We use \(\mathbf{y}\) and \(X\) to denote the original data. \(\mathbf{y}_c\) and \(X_c\) denote the centered data. \(\mathbf{y}_{cs}\) and \(X_{cs}\) denote the centered and scaled data.

Data:

library(susieR)
set.seed(1)
n = 800
p = 1000
beta = rep(0,p)
beta[1] = 1
beta[50] = 1
beta[300] = 1
beta[400] = 1
X = matrix(rnorm(n*p),nrow=n,ncol=p)
y = c(X %*% beta + rnorm(n))

X.cs = susieR:::safe_colScale(X, center=TRUE, scale=TRUE)
X.c = susieR:::safe_colScale(X, center=TRUE, scale=FALSE)
X.s = susieR:::safe_colScale(X, center=FALSE, scale=TRUE)

The summary statistics \(\hat{\beta}_{j}\) comes from the \(X_{cs}\)

Know \(\hat{\beta}_{j}\), \(X_c^{T}X_c\), \(var(\mathbf{y})\), n

The \(\hat{\beta}_{j}\) is from \[ \mathbf{y} = \beta_{0j} + \frac{\mathbf{x}_{j}}{\alpha_{j}}\beta_{j} + \epsilon_{j} \] \(\alpha_{j}\) is the scaling parameter for \(\mathbf{x}_{j}\) (\(\alpha_{j} = sd(\mathbf{x}_{j}\)).

\[ \hat{\beta}_{j} = \frac{\alpha_{j}^{-1} (\mathbf{x}_{j}-\bar{\mathbf{x}}_{j})^{T}(\mathbf{y}-\bar{\mathbf{y}})}{\alpha_{j}^{-2}(\mathbf{x}_{j}-\bar{\mathbf{x}}_{j})^{T}(\mathbf{x}_{j}-\bar{\mathbf{x}}_{j})} = \frac{\alpha_{j}^{-1} (\mathbf{x}_{j}-\bar{\mathbf{x}}_{j})^{T}(\mathbf{y}-\bar{\mathbf{y}})}{n-1} \]

We know the correlation matrix R from the \(X_{c}^{T}X_{c}\). The \(X_{cs}^{T}X_{cs} = (n-1) R\). \(X_{cs}\mathbf{y}_{c} = (n-1) \hat{\boldsymbol{\beta}}\). Using \(X_{cs}^{T}X_{cs}\) and \(X_{cs}\mathbf{y}_{c}\) in susie_ss is equivalent to fit susie model with individual level data centered and scaled. \[ \begin{align*} \mathbf{y}_c &= X_{cs} \boldsymbol{\beta} + \boldsymbol{\epsilon} \quad \boldsymbol{\epsilon}\sim N_{n}(0, \frac{1}{\tau}I) \\ \boldsymbol{\beta} &= \boldsymbol{\gamma} \beta \\ \boldsymbol{\gamma} &\sim Multinomial(1,\boldsymbol{\pi}) \\ \beta &\sim N(0, \frac{1}{\tau_{0}}) \end{align*} \]

res0 = susieR::susie(X,y, estimate_residual_variance = FALSE, estimate_prior_variance = FALSE, max_iter = 3)
# summary stat
bhat.cs = numeric(p)
se.cs = numeric(p)
sigmahat.cs = numeric(p)
for(j in 1:p){
  m = summary(lm(y~X.s[,j]))
  bhat.cs[j] = m$coefficients[2,1]
  se.cs[j] = m$coefficients[2,2]
  sigmahat.cs[j] = m$sigma
}

X.ctX.c = t(X.c) %*% X.c
#-------------------------------#
R = cov2cor(X.ctX.c)
X.cstX.cs = (n-1)*R
X.csty.c = (n-1)* bhat.cs
res1 = susieR::susie_ss(X.cstX.cs, X.csty.c, var_y = var(y), n = n,
                        estimate_prior_variance = FALSE, max_iter = 3)

all.equal(res0$alpha, res1$alpha)
[1] TRUE

OR

\(\alpha_{j}^{2} = \frac{\mathbf{x}_{cj}^{T}\mathbf{x}_{cj}}{n-1}\)

\[ X_{c}^{T}\mathbf{y}_{c} = (n-1) \boldsymbol{\alpha}\hat{\boldsymbol{\beta}} = \sqrt{(n-1)\mathbf{x}_{cj}^{T}\mathbf{x}_{cj}} \hat{\boldsymbol{\beta}} \] Using \(X_{c}^{T}X_{c}\) and \(X_{c}\mathbf{y}_{c}\) in susie_ss is equivalent to fit susie model with centered individual level data centered.

\[ \begin{align*} \mathbf{y}_c &= X_{c} \boldsymbol{\beta} + \boldsymbol{\epsilon} \quad \boldsymbol{\epsilon}\sim N_{n}(0, \frac{1}{\tau}I) \\ \boldsymbol{\beta} &= \boldsymbol{\gamma} \beta \\ \boldsymbol{\gamma} &\sim Multinomial(1,\boldsymbol{\pi}) \\ \beta &\sim N(0, \frac{1}{\tau_{0}}) \end{align*} \]

res2 = susie(X, y, standardize = FALSE, estimate_residual_variance = FALSE, max_iter = 3)
X.cty.c = sqrt((n-1)*diag(X.ctX.c)) * bhat.cs

res3 = susie_ss(X.ctX.c, X.cty.c, var_y = var(y), n= n, max_iter = 3, standardize = FALSE)
all.equal(res2$alpha, res3$alpha)
[1] TRUE

Know \(\hat{\beta}_{j}\), \((n-1)R = X_{cs}^{T}X_{cs}\), \(var(\mathbf{y})\), n

Similarly as above to have centered and scaled individual level data.

The summary statistics \(\hat{\beta}_{j}\) comes from the unscaled \(X_{c}\)

Know \(\hat{\beta}_{j}\), \(X_c^{T}X_c\), \(var(\mathbf{y})\), n

The \(\hat{\beta}_{j}\) is from \[ \mathbf{y} = \beta_{0j} + \mathbf{x}_{j}\beta_{j} + \epsilon_{j} \] \(\alpha_{j}\) is the scaling parameter for \(\mathbf{x}_{j}\) (\(\alpha_{j} = sd(\mathbf{x}_{j}\)).

\[ \hat{\beta}_{j} = \frac{ (\mathbf{x}_{j}-\bar{\mathbf{x}}_{j})^{T}(\mathbf{y}-\bar{\mathbf{y}})}{(\mathbf{x}_{j}-\bar{\mathbf{x}}_{j})^{T}(\mathbf{x}_{j}-\bar{\mathbf{x}}_{j})} \]

Using the diagonal element of \(X_c^{T}X_{c}\), we know \(X_{c}^{T}\mathbf{y}_{c}\). Using \(X_{c}^{T}X_{c}\) and \(X_{c}\mathbf{y}_{c}\) in susie_ss is equivalent to fit susie model with centered individual level data. \[ \begin{align*} \mathbf{y}_c &= X_{c} \boldsymbol{\beta} + \boldsymbol{\epsilon} \quad \boldsymbol{\epsilon}\sim N_{n}(0, \frac{1}{\tau}I) \\ \boldsymbol{\beta} &= \boldsymbol{\gamma} \beta \\ \boldsymbol{\gamma} &\sim Multinomial(1,\boldsymbol{\pi}) \\ \beta &\sim N(0, \frac{1}{\tau_{0}}) \end{align*} \]

# summary stat
bhat.c = numeric(p)
se.c = numeric(p)
sigmahat.c = numeric(p)
for(j in 1:p){
  m = summary(lm(y~X[,j]))
  bhat.c[j] = m$coefficients[2,1]
  se.c[j] = m$coefficients[2,2]
  sigmahat.c[j] = m$sigma
}

X.ctX.c = t(X.c) %*% X.c
#-------------------------------#
X.cty.c = diag(X.ctX.c)* bhat.c
res3 = susieR::susie_ss(X.ctX.c, X.cty.c, var_y = var(y), n = n,
                        estimate_prior_variance = FALSE, max_iter = 3, standardize = FALSE)

all.equal(res2$alpha, res3$alpha)
[1] TRUE

Or,

\(\alpha_{j}^2 = \frac{\mathbf{x}_{cj}^{T}\mathbf{x}_{cj}}{n-1}\), \(\Lambda = diag(\sqrt{\mathbf{x}_{cj}^{T}\mathbf{x}_{cj}})\) \[ X_{cs}^{T}\mathbf{y}_{c} = diag(\alpha^{-1})\Lambda^2\hat{\boldsymbol{\beta}} = \sqrt{n-1}\Lambda\hat{\boldsymbol{\beta}} \] We know the correlation matrix R from the \(X_{c}^{T}X_{c}\). The \(X_{cs}^{T}X_{cs} = (n-1) R\). Using \(X_{cs}^{T}X_{cs}\) and \(X_{cs}\mathbf{y}_{c}\) in susie_ss is equivalent to fit susie model with centered and scaled individual level data. \[ \begin{align*} \mathbf{y}_c &= X_{cs} \boldsymbol{\beta} + \boldsymbol{\epsilon} \quad \boldsymbol{\epsilon}\sim N_{n}(0, \frac{1}{\tau}I) \\ \boldsymbol{\beta} &= \boldsymbol{\gamma} \beta \\ \boldsymbol{\gamma} &\sim Multinomial(1,\boldsymbol{\pi}) \\ \beta &\sim N(0, \frac{1}{\tau_{0}}) \end{align*} \]

X.csty.c = sqrt(n-1) * sqrt(diag(X.ctX.c)) * bhat.c
res4 = susie_ss(X.cstX.cs, X.csty.c, var_y = var(y), n = n, max_iter = 3)
all.equal(res0$alpha, res4$alpha)
[1] TRUE

Know \(\hat{\beta}_{j}\), the standard errors of effect size (\(\hat{s}_{j}\)), \((n-1)R = X_{cs}^{T}X_{cs}\), \(var(\mathbf{y})\), n

\[ \hat{\beta}_{j} = \frac{ \mathbf{x}_{cj}^{T}\mathbf{y}_{c}}{\mathbf{x}_{cj}^{T}\mathbf{x}_{cj}} \] \[ \hat{s}_{j}^2 = \frac{(\mathbf{y}_{c} - \mathbf{x}_{cj}\hat{\beta}_{j})^{T}(\mathbf{y}_{c} - \mathbf{x}_{cj}\hat{\beta}_{j})}{(n-2)\mathbf{x}_{cj}^{T}\mathbf{x}_{cj}} = \frac{(n-1)var(\mathbf{y}) - (\mathbf{x}_{cj}^{T}\mathbf{x}_{cj})^{-1}(\mathbf{x}_{cj}^{T}\mathbf{y}_{c})^2}{(n-2)\mathbf{x}_{cj}^{T}\mathbf{x}_{cj}} \]

We can compute \(\Lambda = diag(\sqrt{\mathbf{x}_{cj}^{T}\mathbf{x}_{cj}})\).

\[ \mathbf{x}_{cj}^{T}\mathbf{x}_{cj} = \frac{n*var(\mathbf{y})}{\hat{s}_{j}^2(n-2)+\hat{\beta}_{j}^2} \quad \mathbf{x}_{cj}^{T}\mathbf{y} = \hat{\beta}_{j}\mathbf{x}_{cj}^{T}\mathbf{x}_{cj} \] Using \(X_{c}^{T}X_{c}\) and \(X_{c}\mathbf{y}_{c}\) in susie_ss is equivalent to fit susie model with centered individual level data. \[ \begin{align*} \mathbf{y}_c &= X_{c} \boldsymbol{\beta} + \boldsymbol{\epsilon} \quad \boldsymbol{\epsilon}\sim N_{n}(0, \frac{1}{\tau}I) \\ \boldsymbol{\beta} &= \boldsymbol{\gamma} \beta \\ \boldsymbol{\gamma} &\sim Multinomial(1,\boldsymbol{\pi}) \\ \beta &\sim N(0, \frac{1}{\tau_{0}}) \end{align*} \]

XtXbetase = (n-1)*var(y)/(se.c^2 * (n-2) + bhat.c^2)
Xtybetase = bhat.c * XtXbetase
XtXbetase = t(R * sqrt(XtXbetase)) * sqrt(XtXbetase)

res4 = susieR::susie_ss(XtXbetase,Xtybetase,var_y = var(y), n = n, max_iter = 3, standardize = FALSE)
all.equal(res2$alpha, res4$alpha)
[1] TRUE

To fit the model with scaled X, we can transfer \(\hat{\beta}_{j}\), \(\hat{s}_{j}\) into z scores.

Know z scores

With z scores, we don’t care whether the summary statisitcs are from the scaled X or unscaled X.

\[ \hat{z}_{j} = \frac{\hat{\beta}_{j}}{\hat{s}_{j}} = \frac{\mathbf{x}_{cj}^{T}\mathbf{y}_{c}}{\hat{\sigma}_{j}\sqrt{\mathbf{x}_{cj}^{T}\mathbf{x}_{cj}}} = \frac{\frac{1}{\alpha_{j}}\mathbf{x}_{cj}^{T}\mathbf{y}_{c}}{\hat{\sigma}_{j}\sqrt{\frac{\mathbf{x}_{cj}}{\alpha_{j}}^{T}\frac{\mathbf{x}_{cj}}{\alpha_{j}}}} \]

\(\hat{\Sigma} = diag(\hat{\sigma}_{1}^2, \cdots, \hat{\sigma}_{p}^2)\)

\[ R_{j}^2 = \frac{\hat{z}_{j}^2}{\hat{z}_{j}^2+n-2} \]

Then \[ 1-R_{j}^{2} = \frac{RSS}{SST} = \frac{\hat{\sigma}_{j}^2(n-2)}{\mathbf{y}_{c}^{T}\mathbf{y}_{c}} \quad \hat{\sigma}_{j}^2 = \frac{\mathbf{y}_{c}^{T}\mathbf{y}_{c}(1-R_{j}^2)}{n-2} \]

Know \(X_{c}^{T}X_{c}\)

\(\Lambda = diag(\sqrt{\mathbf{x}_{cj}^{T}\mathbf{x}_{cj}}) = \sqrt{diag(X_{c}^{T}X_{c})}\) \[ X_{c}^{T}\mathbf{y}_{c} = \hat{\Sigma}^{1/2}\Lambda\hat{\mathbf{z}} \] Using \(X_{c}^{T}X_{c}\) and \(X_{c}\mathbf{y}_{c}\) in susie_ss is equivalent to fit susie model with centered individual level data. \[ \begin{align*} \mathbf{y}_c &= X_{c} \boldsymbol{\beta} + \boldsymbol{\epsilon} \quad \boldsymbol{\epsilon}\sim N_{n}(0, \frac{1}{\tau}I) \\ \boldsymbol{\beta} &= \boldsymbol{\gamma} \beta \\ \boldsymbol{\gamma} &\sim Multinomial(1,\boldsymbol{\pi}) \\ \beta &\sim N(0, \frac{1}{\tau_{0}}) \end{align*} \]

z = bhat.c/se.c
R2 = z^2/(z^2 + n-2)
sigma2 = var(y)*(n-1)*(1-R2)/(n-2)
Xtyz = sigma2^(0.5) * sqrt(diag(X.ctX.c)) * z

res5 = susieR::susie_ss(X.ctX.c,Xtyz,var_y = var(y), n = n, max_iter = 3, standardize = FALSE)
all.equal(res2$alpha, res5$alpha)
[1] TRUE

Or change \(X_{c}^{T}X_{c}\) into R, fit model with centered and scaled individual level data.

Know \((n-1)R = X_{cs}^{T}X_{cs}\)

\(\Lambda = diag(\sqrt{\mathbf{x}_{cj}^{T}\mathbf{x}_{cj}}) = diag(\sqrt{n-1})\) \[ X_{cs}^{T}\mathbf{y}_{c} = \hat{\Sigma}^{1/2}\Lambda\hat{\mathbf{z}} \] Using \(X_{cs}^{T}X_{cs}\) and \(X_{cs}\mathbf{y}_{c}\) in susie_ss is equivalent to fit susie model with centered and scaled individual level data. \[ \begin{align*} \mathbf{y}_c &= X_{cs} \boldsymbol{\beta} + \boldsymbol{\epsilon} \quad \boldsymbol{\epsilon}\sim N_{n}(0, \frac{1}{\tau}I) \\ \boldsymbol{\beta} &= \boldsymbol{\gamma} \beta \\ \boldsymbol{\gamma} &\sim Multinomial(1,\boldsymbol{\pi}) \\ \beta &\sim N(0, \frac{1}{\tau_{0}}) \end{align*} \]

z = bhat.c/se.c
R2 = z^2/(z^2 + n-2)
sigma2 = var(y)*(n-1)*(1-R2)/(n-2)
Xtyz = sigma2^(0.5) * sqrt(diag(X.cstX.cs)) * z

res6 = susieR::susie_ss(X.cstX.cs,Xtyz,var_y = var(y), n = n, max_iter = 3, standardize = FALSE)
all.equal(res0$alpha, res6$alpha)
[1] TRUE

Don’t know \(var(y)\)

\[ 1-R_{j}^{2} = \frac{RSS}{SST} = \frac{\hat{\sigma}_{j}^2(n-2)}{\mathbf{y}_{c}^{T}\mathbf{y}_{c}} \quad \hat{\sigma}_{j}^2 = \frac{c(n-1)(1-R_{j}^2)}{n-2} \] where \(\mathbf{y}_{c}^{T}\mathbf{y}_{c} = (n-1)c\).

Define \[ \tilde{\sigma}_{j}^2 = \frac{(n-1)(1-R_{j}^2)}{n-2} = \frac{1}{c} \hat{\sigma}_{j}^2 \] \(\tilde{\Sigma} = diag(\tilde{\sigma}_{1}^2, \cdots, \tilde{\sigma}_{p}^2) = \frac{1}{c}\hat{\Sigma}\)

\(X_{.}\) could be \(X_{c}\) or \(X_{cs}\) \[ X_{.}^{T}\mathbf{y} = \hat{\Sigma}^{1/2} \Lambda \hat{\mathbf{z}} = \sqrt{c} \Lambda\tilde{\Sigma}^{1/2} \hat{\mathbf{z}} \\ X_{.}^{T}\frac{1}{\sqrt{c}}\mathbf{y} = \Lambda\tilde{\Sigma}^{1/2} \hat{\mathbf{z}} \]

This is equivalent to \[ \begin{align*} \frac{1}{\sqrt{c}}\mathbf{y}_{c} &= X_{.} \boldsymbol{\beta} + \boldsymbol{\epsilon} \quad \boldsymbol{\epsilon}\sim N_{n}(0, I) \\ \boldsymbol{\beta} &= \boldsymbol{\gamma} \beta \quad \boldsymbol{\gamma} \sim Multinomial(1,\boldsymbol{\pi}) \quad \beta \sim N(0, \sigma_{\beta}^2) \end{align*} \]

z = bhat.c/se.c
R2 = z^2/(z^2 + n-2)
sigma2 = (n-1)*(1-R2)/(n-2)
Xtyz = sigma2^(0.5) * sqrt(diag(X.cstX.cs)) * z

res7 = susieR::susie_ss(X.cstX.cs,Xtyz,var_y = 1, n = n, max_iter = 3)
all.equal(res0$alpha, res7$alpha)
[1] TRUE

Simulated sata

dat = readRDS(system.file("datafiles", "N3finemapping.rds", package = "susieR"))

# the 3 “true” signals in the first data-set
b = dat$data$true_coef[,1]
plot(b, pch=16, ylab='effect size')

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Version Author Date
cd773c5 zouyuxin 2018-09-18 

which(b != 0) ## 403 653 773
[1] 403 653 773
z_scores = dat$sumstats[1,,] / dat$sumstats[2,,]
z_scores = z_scores[,1]
susie_plot(z_scores, y = "z", b=b)

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Version Author Date
73cf69a zouyuxin 2018-10-06
cd773c5 zouyuxin 2018-09-18

SuSiE model with individual level data

fitted = susie(dat$data$X, dat$data$Y[,1],
               L=5,
               estimate_residual_variance = FALSE, 
               estimate_prior_variance = FALSE,
               scaled_prior_variance = 0.1, 
               tol=1e-3)
sets = susie_get_CS(fitted)
pip = susieR::susie_get_PIP(fitted, sets$cs_index)
susieR::susie_plot(fitted, y='PIP', b=b, main = paste('SuSiE, ', length(sets$cs), 'CS identified'))

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Version Author Date
73cf69a zouyuxin 2018-10-06
cd773c5 zouyuxin 2018-09-18

SuSiE model using summary stat

n = nrow(dat$data$X)
R = cor(dat$data$X)

R2 = z_scores^2/(z_scores^2 + n-2)
sigma2 = (n-1)*(1-R2)/(n-2)
sXtXz = (n-1)*R
sXtyz = sqrt(n-1) * sqrt(sigma2) * z_scores

fitted_ss = susie_ss(sXtXz, sXtyz,
               L=5, var_y = 1, n = n,
               residual_variance = 1,
               estimate_prior_variance = FALSE,
               scaled_prior_variance = 0.1, 
               tol=1e-3, max_iter = 6)

all.equal(fitted$alpha, fitted_ss$alpha)
[1] TRUE
sets_ss = susie_get_CS(fitted_ss)
pip_ss = susieR::susie_get_PIP(fitted, sets_ss$cs_index)
susieR::susie_plot(fitted_ss, y='PIP', b=b, main = paste('SuSiE, ', length(sets_ss$cs), 'CS identified'))

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Version Author Date
73cf69a zouyuxin 2018-10-06
cd773c5 zouyuxin 2018-09-18

Session information

sessionInfo()
R version 3.5.1 (2018-07-02)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.6

Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] susieR_0.4.30.0332

loaded via a namespace (and not attached):
 [1] workflowr_1.1.1    Rcpp_0.12.19       matrixStats_0.54.0
 [4] lattice_0.20-35    digest_0.6.15      rprojroot_1.3-2   
 [7] R.methodsS3_1.7.1  grid_3.5.1         backports_1.1.2   
[10] git2r_0.23.0       magrittr_1.5       evaluate_0.11     
[13] stringi_1.2.4      whisker_0.3-2      R.oo_1.22.0       
[16] R.utils_2.6.0      Matrix_1.2-14      rmarkdown_1.10    
[19] tools_3.5.1        stringr_1.3.1      yaml_2.2.0        
[22] compiler_3.5.1     htmltools_0.3.6    knitr_1.20

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