Last updated: 2018-10-12

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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)

# z scores
# summary stat
bhat = numeric(p)
se = numeric(p)
sigmahat = numeric(p)
zhat = numeric(p)
for(j in 1:p){
  m = summary(lm(y~X[,j]))
  bhat[j] = m$coefficients[2,1]
  se[j] = m$coefficients[2,2]
  sigmahat[j] = m$sigma
  zhat[j] = abs(qnorm(m$coefficients[2,4]/2)) * sign(m$coefficients[2,1])
}
R = cor(X)

Know \(z\), \(R\)

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

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

\(\mathbf{x}_{.j}\) could be \(\mathbf{x}_{cj}\) or \(\mathbf{x}_{csj}\)

As \(n \rightarrow \infty\) \[ \hat{\sigma}_{j}^{2} \rightarrow \sigma_{j}^{2} \quad \hat{z}_{j}\rightarrow z_{j} = \frac{\mathbf{x}_{.j}^{T}\mathbf{y}_{c}}{\sigma_{j}\sqrt{\mathbf{x}_{.j}^{T}\mathbf{x}_{.j}}} \]

We know \(z_{j}\) for \(j = 1,\cdots,p\). If we set \(\sigma_{j}^{2} = c\), \(\forall j\), we are conservative when making inference of effect.

Assume \[ z_{j} = \frac{\mathbf{x}_{.j}^{T}\mathbf{y}_{c}}{\sqrt{c}\sqrt{\mathbf{x}_{.j}^{T}\mathbf{x}_{.j}}} \]

Let \(XtX = (n-1)R\). Since we know R, \(\mathbf{x}_{.j}\) is \(\mathbf{x}_{csj}\)

\[ X_{cs}^{T}\mathbf{y}_{c} = \sqrt{c} \sqrt{diag(\mathbf{x}_{.j}^{T}\mathbf{x}_{csj})}\mathbf{z} \\ \frac{1}{\sqrt{n-1}}X_{cs}^{T}\frac{1}{\sqrt{c}}\mathbf{y}_{c} = \mathbf{z} \]

This is equivalent to \[ \begin{align*} \frac{1}{\sqrt{c}}\mathbf{y}_{c} &= \frac{1}{\sqrt{n-1}}X_{cs} \boldsymbol{\beta} + \boldsymbol{\epsilon} \quad \boldsymbol{\epsilon}\sim N_{n}(0, I) \\ \boldsymbol{\beta} &= \boldsymbol{\gamma} \beta \quad \boldsymbol{\gamma} \sim Multinomial(1,\boldsymbol{\pi}) \quad \beta \sim N(0, (n-1)\sigma_{\beta}^2) \end{align*} \] We have no information about n, so we estimate the prior variance in susie.

Fit susie model using individual level data:

res0 = susieR::susie(X.cs/sqrt(n-1), (y-mean(y))/sd(y), 
                     estimate_residual_variance = FALSE, 
                     estimate_prior_variance = TRUE, 
                     intercept = FALSE, standardize = TRUE, 
                     max_iter = 2)

Fit susie model using summary statistics with z and R:

res1 = susie_z(z = zhat, R = crossprod(X.c),
               max_iter = 2)

Compare the fitted results:

all.equal(res0$alpha, res1$alpha)
[1] "Mean relative difference: 5.479774e-06"
all.equal(res0$pip, res1$pip)
[1] "Mean relative difference: 5.479774e-06"
{plot(coef(res0), coef(res1))
abline(0,1)}

Expand here to see past versions of unnamed-chunk-4-1.png:
Version Author Date
ca79ff3 zouyuxin 2018-10-10
a990c1f zouyuxin 2018-10-08

Fit susie model using summary statistics with z and covariance matrix:

res2 = susie_z(z = zhat, R = cov(X),
               max_iter = 2)

The output is same as the one with R.

all.equal(res1, res2)
[1] TRUE

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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