fitted <- susie(X, Y[,1],
L = 10,
verbose = TRUE)
# [1] "objective:-1380.57545244487"
# [1] "objective:-1377.4866091747"
# [1] "objective:-1375.85777210115"
# [1] "objective:-1375.80892303931"
# [1] "objective:-1370.33949333171"
# [1] "objective:-1370.19677276994"
# [1] "objective:-1370.10919739202"
# [1] "objective:-1370.10918017469"
# [1] "objective:-1370.10901872278"
Credible sets
By default, we output 95% credible set:
print(fitted$sets)
# $cs
# $cs$L2
# [1] 653
#
# $cs$L1
# [1] 773 777
#
# $cs$L3
# [1] 362 365 372 373 374 379 381 383 384 386 387 388 389 391 392 396 397 398 399
# [20] 400 401 403 404 405 407 408 415
#
#
# $purity
# min.abs.corr mean.abs.corr median.abs.corr
# L2 1.0000000 1.0000000 1.0000000
# L1 0.9815726 0.9815726 0.9815726
# L3 0.8686309 0.9640176 0.9720711
#
# $cs_index
# [1] 2 1 3
#
# $coverage
# [1] 0.9998236 0.9988858 0.9539811
#
# $requested_coverage
# [1] 0.95
The 3 causal signals have been captured by the 3 CS reported here.
The 3rd CS contains many variables, including the true causal variable
403. The minimum absolute correlation is 0.86.
If we use the default 90% coverage for credible sets, we still
capture the 3 signals, but “purity” of the 3rd CS is now 0.91 and size
of the CS is also a bit smaller.
sets <- susie_get_cs(fitted,
X = X,
coverage = 0.9,
min_abs_corr = 0.1)
print(sets)
# $cs
# $cs$L2
# [1] 653
#
# $cs$L1
# [1] 773 777
#
# $cs$L3
# [1] 373 374 379 381 383 384 386 387 388 389 391 392 396 398 399 400 401 403 404
# [20] 405 407 408
#
#
# $purity
# min.abs.corr mean.abs.corr median.abs.corr
# L2 1.0000000 1.0000000 1.0000000
# L1 0.9815726 0.9815726 0.9815726
# L3 0.9119572 0.9726283 0.9765888
#
# $cs_index
# [1] 2 1 3
#
# $coverage
# [1] 0.9998236 0.9988858 0.9119917
#
# $requested_coverage
# [1] 0.9
Posterior inclusion probabilities
Previously we’ve determined that summing over 3 single effect
regression models is approperate for our application. Here we summarize
the variable selection results by posterior inclusion probability
(PIP):
susie_plot(fitted, y="PIP", b=b)
The true causal variables are colored red. The 95% CS identified are
circled in different colors. Of interest is the cluster around position
400. The true signal is 403 but apparently it does not have the highest
PIP. To compare ranking of PIP and original z-score in that CS:
i <- fitted$sets$cs[[3]]
z3 <- cbind(i,z_scores[i],fitted$pip[i])
colnames(z3) <- c('position', 'z-score', 'PIP')
z3[order(z3[,2], decreasing = TRUE),]
# position z-score PIP
# [1,] 396 5.189811 0.056704331
# [2,] 381 5.164794 0.100360243
# [3,] 386 5.164794 0.100360243
# [4,] 379 5.077563 0.054179507
# [5,] 391 5.068388 0.055952118
# [6,] 383 5.057053 0.052896918
# [7,] 384 5.057053 0.052896918
# [8,] 389 5.052519 0.042161265
# [9,] 405 5.039617 0.045761975
# [10,] 403 5.035949 0.031992848
# [11,] 387 5.013526 0.041041505
# [12,] 388 4.997955 0.039650079
# [13,] 408 4.994865 0.041551961
# [14,] 404 4.954407 0.032013339
# [15,] 374 4.948060 0.030571484
# [16,] 373 4.934410 0.023577221
# [17,] 362 4.894243 0.012145481
# [18,] 399 4.860780 0.026454056
# [19,] 392 4.856384 0.019741011
# [20,] 407 4.849285 0.014699313
# [21,] 400 4.827361 0.021659443
# [22,] 365 4.782770 0.006263425
# [23,] 398 4.751205 0.012907848
# [24,] 401 4.723184 0.014858460
# [25,] 397 4.716886 0.008690915
# [26,] 415 4.663208 0.009003129
# [27,] 372 4.581560 0.005886458
Choice of priors
Notice that by default SuSiE estimates prior effect size from data.
For fine-mapping applications, however, we sometimes have knowledge of
SuSiE prior effect size since it is parameterized as percentage of
variance explained (PVE) by a non-zero effect, which, in the context of
fine-mapping, is related to per-SNP heritability. It is possible to use
scaled_prior_variance to specify this PVE and explicitly
set estimate_prior_variance=FALSE to fix the prior effect
to given value.
In this data-set, SuSiE is robust to choice of priors. Here we set
PVE to 0.2, and compare with previous results:
fitted = susie(X, Y[,1],
L = 10,
estimate_residual_variance = TRUE,
estimate_prior_variance = FALSE,
scaled_prior_variance = 0.2)
susie_plot(fitted, y='PIP', b=b)
which largely remains unchanged.
A note on covariate adjustment
To include covariate Z in SuSiE, one approach is to
regress it out from both y and X, and then run
SuSiE on the residuals. The code below illustrates the procedure:
remove.covariate.effects <- function (X, Z, y) {
# include the intercept term
if (any(Z[,1]!=1)) Z = cbind(1, Z)
A <- forceSymmetric(crossprod(Z))
SZy <- as.vector(solve(A,c(y %*% Z)))
SZX <- as.matrix(solve(A,t(Z) %*% X))
y <- y - c(Z %*% SZy)
X <- X - Z %*% SZX
return(list(X = X,y = y,SZy = SZy,SZX = SZX))
}
out = remove.covariate.effects(X, Z, Y[,1])
fitted_adjusted = susie(out$X, out$y, L = 10)
Note that the covariates Z should have a column of ones
as the first column. If not, the above function
remove.covariate.effects will add such a column to
Z before regressing it out. Data will be centered as a
result. Also the scale of data is changed after regressing out
Z. This introduces some subtleties in terms of interpreting
the results. For this reason, we provide covariate adjustment procedure
as a tip in the documentation and not part of
susieR::susie() function. Cautions should be taken when
applying this procedure and interpreting the result from it.
