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The idea here is to look at multiple regression with missing data.
Youxin Zou suggested we take a pseudo-data approach to fitting multivariate multiple regression (MMR). This approach unifies the full-data and summary data approaches, and could also deal with missing data. So I wanted to start by checking it out.
The basic MMR model can be written \[Y \sim MN(Xb, I, V)\] where \(MN\) denotes the matrix normal distribution.
It follows that: \[X'Y \sim MN(X'Xb, X'X, V)\] Now assume \(X\) has SVD \(X=UDV'\), so \(R=X'X\) has eigendecomposition \(R=UD^2U'\). Define transform \(T = D^{-1} U'\). Then
\[TX'Y = D^{-1}U'UDV'Y = V'Y\] \[TX'Xb = D^{-1}U'UD^2U'b = DU'b\] \[TX'XT' = D^{-1}U'UD^2U'UD^{-1} = I\]
So \[V'Y \sim MN(DU'b,I,V)\]
So we can fit the MMR model by using a regression model to outcome \[\tilde{Y} := V'Y\] and covariate \[\tilde{X} := DU'\]
With no missing data \(X'Y\) is sufficient for \(b\),so we should get exactly the same results (even from a Bayesian approach, because the likelihood for \(b\) is the same in each case).
Let’s try this out via simulation in the univariate case (no missingness).
library(susieR)
n= 100
p = 100
X = matrix(rnorm(n*p),nrow=n,ncol=p)
b = rep(0,p)
b[1] = 2
b[3] = 2
Y = X %*% b + rnorm(n)
Y.s = susieR::susie(X,Y,10,standardize = FALSE,intercept=FALSE)
## pseduo data
X.svd = svd(t(X))
U = X.svd$u
D = X.svd$d
V = X.svd$v
Ytilde = t(V) %*% Y
Xtilde = diag(D) %*% t(U)
Y.stilde = susieR::susie(Xtilde,Ytilde,10,standardize = FALSE, intercept =FALSE)
all.equal(susie_get_pip(Y.s),susie_get_pip(Y.stilde))[1] TRUETry the singular case:
set.seed(1)
n= 20
p = 100
X = matrix(rnorm(n*p),nrow=n,ncol=p)
b = rep(0,p)
b[1] = 2
b[3] = 2
Y = X %*% b + rnorm(n)
Y.s = susieR::susie(X,Y,10,standardize = FALSE,intercept=FALSE)
# Y.s = susieR::susie(X,Y,10,standardize = FALSE,intercept=FALSE,estimate_prior_variance=FALSE, scaled_prior_variance = 1/sd(Y)^2)
## pseduo data
X.svd = svd(t(X))
U = X.svd$u
D = X.svd$d
V = X.svd$v
Ytilde = t(V) %*% Y
Xtilde = diag(D) %*% t(U)
Y.stilde = susieR::susie(Xtilde,Ytilde,10,standardize = FALSE, intercept =FALSE)
all.equal(susie_get_pip(Y.s),susie_get_pip(Y.stilde))[1] TRUE
sessionInfo()R version 3.6.0 (2019-04-26)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS Mojave 10.14.6
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] susieR_0.9.0.0581
loaded via a namespace (and not attached):
[1] workflowr_1.6.0.9000 Rcpp_1.0.3 lattice_0.20-38
[4] rprojroot_1.3-2 digest_0.6.23 later_1.0.0
[7] grid_3.6.0 R6_2.4.1 backports_1.1.5
[10] git2r_0.26.1 magrittr_1.5 evaluate_0.14
[13] stringi_1.4.5 rlang_0.4.2 fs_1.3.1
[16] promises_1.1.0 whisker_0.4 Matrix_1.2-18
[19] rmarkdown_2.0 tools_3.6.0 stringr_1.4.0
[22] glue_1.3.1 httpuv_1.5.2 xfun_0.12
[25] yaml_2.2.0 compiler_3.6.0 htmltools_0.4.0
[28] knitr_1.26