R/susie_rss.R
susie_rss.Rdsusie_rss performs variable selection under a
sparse Bayesian multiple linear regression of \(Y\) on \(X\)
using the z-scores from standard univariate regression
of \(Y\) on each column of \(X\), an estimate, \(R\), of
the correlation matrix for the columns of \(X\), and optionally,
but strongly recommended, the sample size n. See
“Details” for other ways to call susie_rss
susie_rss(
z,
R,
n,
bhat,
shat,
var_y,
z_ld_weight = 0,
estimate_residual_variance = FALSE,
prior_variance = 50,
check_prior = TRUE,
...
)p-vector of z-scores.
p x p correlation matrix.
The sample size.
Alternative summary data giving the estimated effects
(a vector of length p). This, together with shat, may be
provided instead of z.
Alternative summary data giving the standard errors of
the estimated effects (a vector of length p). This, together with
bhat, may be provided instead of z.
The sample variance of y, defined as \(y'y/(n-1)\).
When the sample variance is not provided, the coefficients
(returned from coef) are computed on the
“standardized” X, y scale.
This parameter is included for backwards
compatibility with previous versions of the function, but it is no
longer recommended to set this to a non-zero value. When
z_ld_weight > 0, the matrix R is adjusted to be
cov2cor((1-w)*R + w*tcrossprod(z)), where w =
z_ld_weight.
The default is FALSE, the
residual variance is fixed to 1 or variance of y. If the in-sample
LD matrix is provided, we recommend setting
estimate_residual_variance = TRUE.
The prior variance(s) for the non-zero
noncentrality parameterss \(\tilde{b}_l\). It is either a scalar,
or a vector of length L. When the susie_suff_stat option
estimate_prior_variance is set to TRUE (which is
highly recommended) this simply provides an initial value for the
prior variance. The default value of 50 is simply intended to be a
large initial value. Note this setting is only relevant when
n is unknown. If n is known, the relevant option is
scaled_prior_variance in susie_suff_stat.
When check_prior = TRUE, it checks if the
estimated prior variance becomes unreasonably large (comparing with
100 * max(abs(z))^2).
Other parameters to be passed to
susie_suff_stat.
A "susie" object with the following
elements:
An L by p matrix of posterior inclusion probabilites.
An L by p matrix of posterior means, conditional on inclusion.
An L by p matrix of posterior second moments, conditional on inclusion.
log-Bayes Factor for each single effect.
log-Bayes Factor for each variable and single effect.
Prior variance of the non-zero elements of b.
The value of the variational lower bound, or “ELBO” (objective function to be maximized), achieved at each iteration of the IBSS fitting procedure.
Credible sets estimated from model fit; see
susie_get_cs for details.
A vector of length p giving the (marginal) posterior inclusion probabilities for all p covariates.
Number of IBSS iterations that were performed.
TRUE or FALSE indicating whether
the IBSS converged to a solution within the chosen tolerance
level.
In some applications, particularly genetic applications,
it is desired to fit a regression model (\(Y = Xb + E\) say,
which we refer to as "the original regression model" or ORM)
without access to the actual values of \(Y\) and \(X\), but
given only some summary statistics. susie_rss assumes
availability of z-scores from standard univariate regression of
\(Y\) on each column of \(X\), and an estimate, \(R\), of the
correlation matrix for the columns of \(X\) (in genetic
applications \(R\) is sometimes called the “LD matrix”).
With the inputs z, R and sample size n,
susie_rss computes PVE-adjusted z-scores z_tilde, and
calls susie_suff_stat with XtX = (n-1)R, Xty =
\(\sqrt{n-1} z_tilde\), yty = n-1, n = n. The
output effect estimates are on the scale of \(b\) in the ORM with
standardized \(X\) and \(y\). When the LD matrix
R and the z-scores z are computed using the same
matrix \(X\), the results from susie_rss are same as, or
very similar to, susie with standardized \(X\) and
\(y\).
Alternatively, if the user provides n, bhat (the
univariate OLS estimates from regressing \(y\) on each column of
\(X\)), shat (the standard errors from these OLS
regressions), the in-sample correlation matrix \(R =
cov2cor(crossprod(X))\), and the variance of \(y\), the results
from susie_rss are same as susie with \(X\) and
\(y\). The effect estimates are on the same scale as the
coefficients \(b\) in the ORM with \(X\) and \(y\).
In rare cases in which the sample size, \(n\), is unknown,
susie_rss calls susie_suff_stat with XtX = R
and Xty = z, and with residual_variance = 1. The
underlying assumption of performing the analysis in this way is
that the sample size is large (i.e., infinity), and/or the
effects are small. More formally, this combines the log-likelihood
for the noncentrality parameters, \(\tilde{b} = \sqrt{n} b\),
$$L(\tilde{b}; z, R) = -(\tilde{b}'R\tilde{b} -
2z'\tilde{b})/2,$$ with the “susie prior” on
\(\tilde{b}\); see susie and Wang et al
(2020) for details. In this case, the effect estimates returned by
susie_rss are on the noncentrality parameter scale.
The estimate_residual_variance setting is FALSE by
default, which is recommended when the LD matrix is estimated from
a reference panel. When the LD matrix R and the summary
statistics z (or bhat, shat) are computed
using the same matrix \(X\), we recommend setting
estimate_residual_variance = TRUE.
G. Wang, A. Sarkar, P. Carbonetto and M. Stephens (2020). A simple new approach to variable selection in regression, with application to genetic fine-mapping. Journal of the Royal Statistical Society, Series B 82, 1273-1300 https://doi.org/10.1101/501114.
Y. Zou, P. Carbonetto, G. Wang and M. Stephens (2021). Fine-mapping from summary data with the “Sum of Single Effects” model. bioRxiv https://doi.org/10.1101/2021.11.03.467167.
set.seed(1)
n = 1000
p = 1000
beta = rep(0,p)
beta[1:4] = 1
X = matrix(rnorm(n*p),nrow = n,ncol = p)
X = scale(X,center = TRUE,scale = TRUE)
y = drop(X %*% beta + rnorm(n))
input_ss = compute_suff_stat(X,y,standardize = TRUE)
ss = univariate_regression(X,y)
R = with(input_ss,cov2cor(XtX))
zhat = with(ss,betahat/sebetahat)
res = susie_rss(zhat,R, n=n)
#> WARNING: XtX is not symmetric; forcing XtX to be symmetric by replacing XtX with (XtX + t(XtX))/2
# Toy example illustrating behaviour susie_rss when the z-scores
# are mostly consistent with a non-invertible correlation matrix.
# Here the CS should contain both variables, and two PIPs should
# be nearly the same.
z = c(6,6.01)
R = matrix(1,2,2)
fit = susie_rss(z,R)
#> WARNING: Providing the sample size (n), or even a rough estimate of n, is highly recommended. Without n, the implicit assumption is n is large (Inf) and the effect sizes are small (close to zero).
print(fit$sets$cs)
#> $L1
#> [1] 1 2
#>
print(fit$pip)
#> [1] 0.4854079 0.5145921
# In this second toy example, the only difference is that one
# z-score is much larger than the other. Here we expect that the
# second PIP will be much larger than the first.
z = c(6,7)
R = matrix(1,2,2)
fit = susie_rss(z,R)
#> WARNING: Providing the sample size (n), or even a rough estimate of n, is highly recommended. Without n, the implicit assumption is n is large (Inf) and the effect sizes are small (close to zero).
print(fit$sets$cs)
#> $L1
#> [1] 2
#>
print(fit$pip)
#> [1] 0.001713869 0.998286131