This vignette demonstrates diagnostic plot for consistency between summary statistics and reference LD matrix.

The `susie_rss`

assumes the LD matrix accurately estimate
the correlations among SNPs from the original GWAS genotype data.
Typically, the LD matrix comes from some public database of genotypes in
a suitable reference population. The inaccurate LD information leads to
unreliable fine-mapping result.

The diagnostic for consistency between summary statistics and refenrence LD matrix is based on the RSS model under the null with regularized LD matrix. \[ \hat{z} | R, \lambda \sim N(0, (1-\lambda)R + \lambda I), 0<\lambda<1 \] The parameter \(\lambda\) is estimated by maximum likelihood. A larger \(\lambda\) means a greater inconsistency between summary statistics and the LD matrix. The expected z score is computed for each SNP, \(E(\hat{z}_j | \hat{z}_{-j})\), and plotted against the observed z scores.

```
library(susieR)
library(curl)
```

We demonstrate the diagnostic plot in a simple case, the LD matrix is estimated from the original genotype data. We use the same simulated data as in fine mapping vignette.

```
data("N3finemapping")
= nrow(N3finemapping$X)
n = N3finemapping$true_coef[,1]
b <- univariate_regression(N3finemapping$X, N3finemapping$Y[,1])
sumstats <- sumstats$betahat / sumstats$sebetahat
z_scores = cor(N3finemapping$X)
Rin attr(Rin, "eigen") = eigen(Rin, symmetric = TRUE)
susie_plot(z_scores, y = "z", b=b)
```

The estimated \(\lambda\) is

```
= estimate_s_rss(z_scores, Rin, n=n)
lambda
lambda# [1] 6.474096e-09
```

The plot for the observed z scores vs the expected z scores is

```
= kriging_rss(z_scores, Rin, n=n)
condz_in $plot condz_in
```

Summary of SuSiE Credible Sets:

```
<- susie_rss(z_scores, Rin, n=n, estimate_residual_variance = TRUE)
fit # 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.
susie_plot(fit,y = "PIP", b=b)
```

We use another simulated data where the LD matrix is estimated from a reference panel. There is one signal in the simulated data (red point). There is one SNP with mismatched reference and alternative allele between summary statistics and the reference panel (yellow point).

```
<- tempfile(fileext = ".RData")
data_file <- paste0("https://raw.githubusercontent.com/stephenslab/susieR/",
data_url "master/inst/datafiles/SummaryConsistency1k.RData")
curl_download(data_url,data_file)
load(data_file)
= SummaryConsistency$z
zflip = SummaryConsistency$ldref
ld =10000
n= numeric(length(zflip))
b $signal_id] = zflip[SummaryConsistency$signal_id]
b[SummaryConsistencyplot(zflip, pch = 16, col = "#767676", main = "Marginal Associations",
xlab="SNP", ylab = "z scores")
points(SummaryConsistency$signal_id, zflip[SummaryConsistency$signal_id], col=2, pch=16)
points(SummaryConsistency$flip_id, zflip[SummaryConsistency$flip_id], col=7, pch=16)
```

Using the data with misaligned allele, SuSiE-RSS identifies a true positive CS containing the true effect SNP; and a false positive CS that incorrectly contains the mismatched SNP.

```
= susie_rss(zflip, ld, n=n)
fit susie_plot(fit, y='PIP', b=b)
points(SummaryConsistency$flip_id, fit$pip[SummaryConsistency$flip_id], col=7, pch=16)
```

The estimated \(\lambda\) is

```
= estimate_s_rss(zflip, ld, n=n)
lambda
lambda# [1] 0.005005011
```

In the diagnostic plot, the mismatched SNP shows the largest difference between observed and expected z-scores, and therefore appears furthest away from the diagonal.

```
= kriging_rss(zflip, ld, n=n)
condz $plot condz
```

After fixing the allele encoding, SuSiE-RSS identifies a single true positive CS containing the true-effect SNP, and the formerly mismatched SNP is (correctly) not included in a CS.

```
= zflip
z $flip_id] = -z[SummaryConsistency$flip_id]
z[SummaryConsistency= susie_rss(z, ld, n=n)
fit susie_plot(fit, y='PIP', b=b)
```

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] curl_4.3 Matrix_1.4-2 L0Learn_1.2.0 susieR_0.12.27
#
# loaded via a namespace (and not attached):
# [1] tidyselect_1.1.1 xfun_0.29 bslib_0.3.1 reshape2_1.4.3
# [5] purrr_0.3.4 lattice_0.20-38 colorspace_1.4-1 vctrs_0.3.8
# [9] generics_0.0.2 htmltools_0.5.2 yaml_2.2.0 utf8_1.1.4
# [13] rlang_0.4.11 mixsqp_0.3-46 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 evaluate_0.14 labeling_0.3 knitr_1.37
# [29] fastmap_1.1.0 irlba_2.3.3 parallel_3.6.2 fansi_0.4.0
# [33] Rfast_2.0.3 highr_0.8 Rcpp_1.0.8 scales_1.1.0
# [37] jsonlite_1.7.2 farver_2.0.1 ggplot2_3.3.6 digest_0.6.23
# [41] stringi_1.4.3 dplyr_1.0.7 grid_3.6.2 tools_3.6.2
# [45] magrittr_2.0.1 sass_0.4.0 tibble_3.1.3 crayon_1.4.1
# [49] pkgconfig_2.0.3 ellipsis_0.3.2 assertthat_0.2.1 rmarkdown_2.11
# [53] reshape_0.8.8 R6_2.4.1 compiler_3.6.2
```