This vignette demonstrates `susieR`

in the context of
genetic fine-mapping. We use simulated data of expression level of a
gene (\(y\)) in \(N \approx 600\) individuals. We want to
identify with the genotype matrix \(X_{N\times
P}\) (\(P=1001\)) the genetic
variables that causes changes in expression level.

The simulated data set is simulated to have exactly 3 non-zero effects.

The loaded dataset contains regression data \(X\) and \(y\), along with some other relevant properties in the context of genetic studies. It also contains the “true” regression coefficent the data is simulated from.

Notice that we’ve simulated 2 sets of \(Y\) as 2 simulation replicates. Here we’ll focus on the first data-set.

```
dim(Y)
# [1] 574 2
```

Here are the 3 “true” signals in the first data-set:

```
b <- true_coef[,1]
plot(b, pch=16, ylab='effect size')
```

```
which(b != 0)
# [1] 403 653 773
```

So the underlying causal variables are 403, 653 and 773.

`univariate_regression`

function can be used to compute
summary statistics by fitting univariate simple regression variable by
variable. The results are \(\hat{\beta}\) and \(SE(\hat{\beta})\) from which z-scores can
be derived. Again we focus only on results from the first data-set:

```
sumstats <- univariate_regression(X, Y[,1])
z_scores <- sumstats$betahat / sumstats$sebetahat
susie_plot(z_scores, y = "z", b=b)
```

`susieR`

For starters, we assume there are at most 10 causal variables, i.e.,
set `L = 10`

, although SuSiE is robust to the choice of
`L`

.

The `susieR`

function call is:

```
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"
```

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
```

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
```

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.

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:

```
<- function (X, Z, y) {
remove.covariate.effects # include the intercept term
if (any(Z[,1]!=1)) Z = cbind(1, Z)
<- forceSymmetric(crossprod(Z))
A <- as.vector(solve(A,c(y %*% Z)))
SZy <- as.matrix(solve(A,t(Z) %*% X))
SZX <- y - c(Z %*% SZy)
y <- X - Z %*% SZX
X return(list(X = X,y = y,SZy = SZy,SZX = SZX))
}
= remove.covariate.effects(X, Z, Y[,1])
out = susie(out$X, out$y,
fitted_adjusted 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.

Here are some details about the computing environment, including the versions of R, and the R packages, used to generate these results.

```
sessionInfo()
# R version 3.6.2 (2019-12-12)
# Platform: x86_64-apple-darwin15.6.0 (64-bit)
# Running under: macOS Catalina 10.15.7
#
# 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_0.12.10
#
# loaded via a namespace (and not attached):
# [1] tidyselect_1.1.1 xfun_0.29 bslib_0.3.1 purrr_0.3.4
# [5] lattice_0.20-38 colorspace_1.4-1 vctrs_0.3.8 generics_0.0.2
# [9] htmltools_0.5.2 yaml_2.2.0 utf8_1.1.4 rlang_0.4.11
# [13] mixsqp_0.3-46 pkgdown_2.0.2 jquerylib_0.1.4 pillar_1.6.2
# [17] glue_1.4.2 DBI_1.1.0 RcppZiggurat_0.1.5 matrixStats_0.61.0
# [21] lifecycle_1.0.0 plyr_1.8.5 stringr_1.4.0 munsell_0.5.0
# [25] gtable_0.3.0 ragg_0.3.1 memoise_1.1.0 evaluate_0.14
# [29] knitr_1.37 fastmap_1.1.0 parallel_3.6.2 irlba_2.3.3
# [33] fansi_0.4.0 Rfast_2.0.3 highr_0.8 Rcpp_1.0.7
# [37] scales_1.1.0 backports_1.1.5 desc_1.2.0 jsonlite_1.7.2
# [41] systemfonts_1.0.2 fs_1.5.2 ggplot2_3.3.5 digest_0.6.23
# [45] stringi_1.4.3 dplyr_1.0.7 grid_3.6.2 rprojroot_1.3-2
# [49] tools_3.6.2 magrittr_2.0.1 sass_0.4.0 tibble_3.1.3
# [53] crayon_1.4.1 pkgconfig_2.0.3 ellipsis_0.3.2 Matrix_1.2-18
# [57] assertthat_0.2.1 rmarkdown_2.11 reshape_0.8.8 R6_2.4.1
# [61] compiler_3.6.2
```