# Compare susie_rss variants

#### Peter Carbonetto

#### 2025-06-04

In this vignette, we briefly illustrate the different ways susie_rss can be called, and draw connections between running `susie_rss` on summary data, and running `susie` on individual-level data.

    library(susieR)

Simulate a data set with 200 samples and 1,000 variables, in which the only first 4 variables affect the outcome.

    set.seed(1)
    n <- 200
    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 = FALSE)
    y <- drop(X %*% beta + rnorm(n))

Compute summary statistics \(\hat{b}_j, \hat{s}_j\) and the correlation matrix, \({\bf R}\). These quantities will be provided as input to susie_rss.

    ss  <- univariate_regression(X,y)
    dat <- compute_suff_stat(X,y,standardize = FALSE)
    R   <- cov2cor(dat$XtX)

The susie and susie_rss analyses produce the exact same results when the summary statistics `bhat`, `shat`, `var_y` and `n` are provided to susie_rss (and when `R` is an “in sample” correlation estimate—that is, when it was computed from the same matrix `X` that was used to obtain the other statistics). If the covariate effects are removed from the genotypes in univariate regression, the in-sample LD matrix should compute from the genotype residuals where the covariate effects have been removed.

    res1 <- susie(X,y,L = 10)
    res2 <- susie_rss(bhat = ss$betahat,shat = ss$sebetahat,R = R,n = n,
                      var_y = var(y),L = 10,estimate_residual_variance = TRUE)
    # HINT: For estimate_residual_variance = TRUE, please check that R is the "in-sample" LD matrix; that is, the correlation matrix obtained using the exact same data matrix X that was used for the other summary statistics. Also note, when covariates are included in the univariate regressions that produced the summary statistics, also consider removing these effects from X before computing R.
    plot(coef(res1),coef(res2),pch = 20,xlab = "susie",ylab = "susie_rss")
    abline(a = 0,b = 1,col = "skyblue",lty = "dashed")

When some but not all of these statistics are provided, the results should be similar, but not exactly the same.

Next let’s compare the susie and susie_rss outputs when \({\bf X}, y\) are standardized before computing the summary statistics (by “standardize”, we mean that \(y\) and the columns of \({\bf X}\) are each divided by the sample standard deviation so that they each have the same standard deviation).

    ss  <- univariate_regression(scale(X),scale(y))
    dat <- compute_suff_stat(X,y,standardize = TRUE)
    R   <- cov2cor(dat$XtX)

Then we compute the z-scores:

    zhat <- ss$betahat/ss$sebetahat

When standardizing, providing susie_rss with summary data `z` (or `bhat`, `shat`), `R` and `n` is sufficient for susie_rss to recover the same results as susie:

    res1 <- susie(scale(X),scale(y),L = 10)
    res2 <- susie_rss(bhat = ss$betahat,shat = ss$sebetahat,R = R,n = n,
                      L = 10,estimate_residual_variance = TRUE)
    res3 <- susie_rss(zhat,R,n = n,L = 10,estimate_residual_variance = TRUE)
    layout(matrix(1:2,1,2))
    plot(coef(res1),coef(res2),pch = 20,xlab = "susie",
              ylab = "susie_rss(bhat,shat)")
    abline(a = 0,b = 1,col = "skyblue",lty = "dashed")
    plot(coef(res1),coef(res3),pch = 20,xlab = "susie",ylab = "susie_rss(z)")
    abline(a = 0,b = 1,col = "skyblue",lty = "dashed")

When the residual variance is not estimated in susie_rss, the susie_rss results may be close to susie, but may no longer be exactly the same:

    res4 <- susie_rss(zhat,R,n = n,L = 10)
    plot(coef(res1),coef(res4),pch = 20,xlab = "susie",ylab = "susie_rss")
    abline(a = 0,b = 1,col = "skyblue",lty = "dashed")

Whenever `R` is an “in sample” correlation matrix, we recommend estimating the residual variance.

Without providing the sample size, `n`, the coefficients are interpreted as the “noncentrality parameters” (NCPs), and are (roughly) related to the susie parameters by a factor of \(\sqrt{n}\):

    res5 <- susie_rss(zhat,R,L = 10)
    plot(coef(res1),coef(res5)/sqrt(n),pch = 20,xlab = "susie",
    ylab = "susie_rss/sqrt(n)")
    abline(a = 0,b = 1,col = "skyblue",lty = "dashed")

Whenever possible, the sample size, or a reasonable estimate of the sample size, should be provided.