Last updated: 2020-11-10

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We further investigate the BF for imputed data here. We fit model with L = 1 and a simple rank 1 prior. We don't estimate prior scalar.

Load data and impute missing data.

library(mmbr)
Loading required package: mashr
Loading required package: ashr
Loading required package: susieR
dat = readRDS('data/ENSG00000140265.12.Multi_Tissues.problem.rds')
Ymean = colMeans(dat$Y, na.rm=T)
Y = t(t(dat$Y) - Ymean)
Y[is.na(Y)] = 0

Model with dense residual varaince

u = rep(1,49)
U = tcrossprod(u)
m = mmbr::msusie(dat$X, Y, prior_variance = U, residual_variance = dat$residual_var, L = 1, compute_objective = T, estimate_prior_variance = FALSE)
susie_plot(m, y='PIP', main=paste0('ELBO=', round(m$elbo[m$niter],2)))

Version Author Date
28d20bb zouyuxin 2020-11-10

Model with diagonal residual variance

m_diag = mmbr::msusie(dat$X, Y, prior_variance = U, residual_variance = diag(diag(dat$residual_var)), L = 1, compute_objective = T, estimate_prior_variance = FALSE)
susie_plot(m_diag, y='PIP', main=paste0('ELBO=', round(m_diag$elbo[m_diag$niter],2)))

Version Author Date
28d20bb zouyuxin 2020-11-10

Check CS details:

m_diag$sets
$cs
$cs$L1
[1] 243 249 253 260 263 264 265 268 270


$purity
   min.abs.corr mean.abs.corr median.abs.corr
L1    0.9458794     0.9857346       0.9962252

$cs_index
[1] 1

$coverage
[1] 0.95

Suppose there is only one SNP with non-zero effect, \[ \mathbf{y}_i \sim N_R( x_i \mathbf{b}, V) \quad \mathbf{b} \sim N_R(0, \mathbf{u}\mathbf{u}^\intercal) \] Let \(z \sim N(0, 1)\), then \(\mathbf{b} = z \mathbf{u}\). The model becomes \[ \mathbf{y}_i \sim N_R( x_i z \mathbf{u}, V) \quad z \sim N(0,1) \] The BF for comparing this model with the null model (z = 0) is \[ BF = (1 + \mathbf{u}^\intercal V^{-1} \mathbf{u} \sum_{i=1}^{N} x_i^2 )^{-1/2} \exp\{\frac{1}{2} \frac{(\mathbf{u}^\intercal V^{-1} \sum_{i=1}^N x_i \mathbf{y}_i)^2}{1 + \mathbf{u}^\intercal V^{-1} \mathbf{u} \sum_{i=1}^{N} x_i^2 } \} \]

The difference in \(V^{-1}\) is

summary(as.vector(solve(dat$residual_var) - diag(1/diag(dat$residual_var))))
    Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
-0.38050 -0.12217 -0.08411 -0.08261 -0.05275  1.99170

In our example, \(u^T V^{-1} u =\) 77.89 when V is dense; \(u^T V^{-1} u =\) 276.23 when V is diagonal.

par(mfrow=c(1,2))
{barplot(colSums(solve(dat$residual_var))/sqrt(sum(colSums(solve(dat$residual_var))^2)), main = 'normalized u^T Vinv, V dense')
barplot((1/diag(dat$residual_var))/sqrt(sum((1/diag(dat$residual_var))^2)), main = 'normalized u^T Vinv, V diagonal')}

Version Author Date
28d20bb zouyuxin 2020-11-10

The difference in results may caused by the difference in the inverse of residual variance matrix.

# # BF
# compute_bf = function(X, Y, u, residual_var){
#   J = ncol(X)
#   Y = t(t(Y) - colMeans(Y))
#   X = scale(X)
#   d = colSums(X^2)
#   resid_inv = solve(residual_var)
#   XtY = crossprod(X, Y)
#   lbf = numeric(J)
#   numerator = numeric(J)
#   for(j in 1:J){
#     numerator[j] = (crossprod(u, resid_inv %*% XtY[j,]))^2
#     lbf[j] = - 0.5 * log(1 + crossprod(u, resid_inv %*% u) * d[j]) + 0.5 * ((crossprod(u, resid_inv %*% XtY[j,]))^2) / (1 + crossprod(u, resid_inv %*% u) * d[j] )
#   }
#   return(list(lbf = lbf, numerator = numerator))
# }
# 
# compute_softmax = function(value) {
#     mvalue = max(value)
#     w = exp(value-mvalue)
#     return(as.vector(w / sum(w)))
# }
# 
# lbf_covy = compute_bf(dat$X, Y, u, dat$residual_var)
# lbf_diag = compute_bf(dat$X, Y, u, diag(diag(dat$residual_var)))
# 
# all.equal(compute_softmax(lbf_covy$lbf), m$alpha[1,])
# all.equal(compute_softmax(lbf_diag$lbf), m_diag$alpha[1,])

sessionInfo()
R version 3.6.3 (2020-02-29)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS Catalina 10.15.7

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.6/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] mmbr_0.0.1.0305 susieR_0.9.26   mashr_0.2.40    ashr_2.2-51    
[5] workflowr_1.6.2

loaded via a namespace (and not attached):
 [1] progress_1.2.2     tidyselect_1.1.0   xfun_0.19          purrr_0.3.4       
 [5] lattice_0.20-41    colorspace_1.4-1   vctrs_0.3.4        generics_0.1.0    
 [9] htmltools_0.5.0    yaml_2.2.1         rlang_0.4.8        mixsqp_0.3-46     
[13] later_1.1.0.1      pillar_1.4.6       glue_1.4.2         matrixStats_0.57.0
[17] lifecycle_0.2.0    plyr_1.8.6         stringr_1.4.0      munsell_0.5.0     
[21] gtable_0.3.0       mvtnorm_1.1-1      evaluate_0.14      knitr_1.30        
[25] httpuv_1.5.4       invgamma_1.1       irlba_2.3.3        Rcpp_1.0.5        
[29] promises_1.1.1     backports_1.2.0    scales_1.1.1       rmeta_3.0         
[33] truncnorm_1.0-8    abind_1.4-5        fs_1.5.0           hms_0.5.3         
[37] ggplot2_3.3.2      digest_0.6.27      stringi_1.5.3      dplyr_1.0.2       
[41] grid_3.6.3         rprojroot_1.3-2    tools_3.6.3        magrittr_1.5      
[45] tibble_3.0.4       crayon_1.3.4       whisker_0.4        pkgconfig_2.0.3   
[49] ellipsis_0.3.1     Matrix_1.2-18      prettyunits_1.1.1  SQUAREM_2020.5    
[53] reshape_0.8.8      assertthat_0.2.1   rmarkdown_2.5      rstudioapi_0.11   
[57] R6_2.5.0           git2r_0.27.1       compiler_3.6.3