Sum of Single Effects (SuSiE) Regression using Summary Statistics

susie_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

Usage

susie_rss(
  z,
  R,
  n,
  bhat,
  shat,
  var_y,
  z_ld_weight = 0,
  estimate_residual_variance = FALSE,
  prior_variance = 50,
  check_prior = TRUE,
  ...
)

Arguments

z

p-vector of z-scores.

R

p x p correlation matrix.

n

The sample size.

bhat

Alternative summary data giving the estimated effects (a vector of length p). This, together with shat, may be provided instead of z.

shat

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.

var_y

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.

z_ld_weight

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.

estimate_residual_variance

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.

prior_variance

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.

check_prior

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.

Value

A "susie" object with the following elements:

alpha

An L by p matrix of posterior inclusion probabilites.

mu

An L by p matrix of posterior means, conditional on inclusion.

mu2

An L by p matrix of posterior second moments, conditional on inclusion.

lbf

log-Bayes Factor for each single effect.

lbf_variable

log-Bayes Factor for each variable and single effect.

V

Prior variance of the non-zero elements of b.

elbo

The value of the variational lower bound, or “ELBO” (objective function to be maximized), achieved at each iteration of the IBSS fitting procedure.

sets

Credible sets estimated from model fit; see susie_get_cs for details.

pip

A vector of length p giving the (marginal) posterior inclusion probabilities for all p covariates.

niter

Number of IBSS iterations that were performed.

converged

TRUE or FALSE indicating whether the IBSS converged to a solution within the chosen tolerance level.

Details

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.

References

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 doi:10.1101/501114 .

Y. Zou, P. Carbonetto, G. Wang, G and M. Stephens (2022). Fine-mapping from summary data with the “Sum of Single Effects” model. PLoS Genetics 18, e1010299. doi:10.1371/journal.pgen.1010299 .

Examples

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