Last updated: 2023-06-13

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###Load libraries
library(ggplot2)
library(cowplot)

repz <- 1:20
prefix <- "output/prediction_accuracy/ukb_caucasian_white_british_unrel_100000"
metric <- "r2"
traitz <- 1:5

Introduction

The goal of this analysis is to benchmark the newly developed mr.mash.rss (aka mr.mash with summary data) against already existing methods in the task of predicting phenotypes from genotypes using only summary data. To do so, we used real genotypes from the array data of the UK Biobank. We randomly sampled 105,000 nominally unrelated (\(r_A\) < 0.025 between any pair) individuals of European ancestry (i.e., Caucasian and white British fields). After retaining variants with minor allele frequency (MAF) > 0.01, minor allele count (MAC) > 5, genotype missing rate < 0.1 and Hardy-Weinberg Equilibrium (HWE) test p-value > \(1 *10^{-10}\), our data consisted of 595,071 genetic variants (i.e., our predictors). Missing genotypes were imputed with the mean genotype for the respective genetic variant.

The linkage disequilibrium (LD) matrices (i.e., the correlation matrices) were computed using 146,288 nominally unrelated (\(r_A\) < 0.025 between any pair) individuals of European ancestry (i.e., Caucasian and white British fields), that did not overlap with the 105,000 individuals used for the rest of the analyses.

For each replicate, we simulated 5 traits (i.e., our responses) by randomly sampling 5,000 variants (out of the total of 595,071) to be causal, with different effect sharing structures across traits (see below). The genetic effects explain 50% of the total per-trait variance – in genetics terminology this is called genomic heritability (\(h_g^2\)). The residuals are uncorrelated across traits.

We randomly sampled 5,000 (out of the 105,000) individuals to be the test set. The test set was only used to evaluate prediction accuracy. All the other steps were carried out on the training set of 100,000 individuals.

Summary statistics (i.e., effect size and its standard error) were obtained by univariate simple linear regression of each trait on each variant, one at a time. Traits and variants were not standardized.

Three different methods were fitted:

  • LDpred2 per-chromosome with the auto option, 500 iterations (after 500 burn-in iterations), \(h^2\) initialized as 0.5/22 and \(p\) initialized using the same grid as in the original paper.
  • LDpred2 genome-wide with the auto option, 500 iterations (after 500 burn-in iterations), \(h^2\) initialized as 0.5/22 (WRONG) and \(p\) initialized using the same grid as in the original paper.
  • mr.mash.rss per-chromosome, with only data-driven covariance matrices computed as described in the mr.mash paper, updating the (full rank) residual covariance and the mixture weights, without standardizing the variables. The residual covariance was initialized as in the mvSuSiE paper and the mixture weights were initialized as 90% of the weight on the null component and 10% of the weight split equally across the remaining components. The phenotypic covariance was computed
    as the sample covariance using the individual-level data.

Prediction accuracy was evaluated as the \(R^2\) of the regression of true phenotypes on the predicted phenotypes. This metric as the attractive property that its upper bound is \(h_g^2\).

20 replicates for each simulation scenario were run.

Equal effects scenario

In this scenario, the effects were drawn from a Multivariate Normal distribution with mean vector 0 and covariance matrix that achieves a per-trait variance of 1 and a correlation across traits of 1. This implies that the effects of the causal variants are equal across responses.

scenarioz <- "equal_effects_indep_resid"
methodz <- c("mr_mash_rss", "ldpred2_auto", "ldpred2_auto_gwide")

i <- 0

n_col <- 6
n_row <- length(repz) * length(scenarioz) * length(methodz) * length(traitz)
res <- as.data.frame(matrix(NA, ncol=n_col, nrow=n_row))
colnames(res) <- c("rep", "scenario", "method", "trait", "metric", "score")

for(sce in scenarioz){
  for(met in methodz){
    for(repp in repz){
      dat <- readRDS(paste0(prefix, "_", sce, "_", met, "_pred_acc_", repp, ".rds"))
      
      for(trait in traitz){
        i <- i + 1
        
        res[i, 1] <- repp
        res[i, 2] <- sce
        res[i, 3] <- met
        res[i, 4] <- trait
        res[i, 5] <- metric
        res[i, 6] <- dat$r2[trait]
      }
    }
  }
}

res <- transform(res, scenario=as.factor(scenario),
                      method=as.factor(method),
                      trait=as.factor(trait))

p_methods_shared <- ggplot(res, aes(x = trait, y = score, fill = method)) +
  geom_boxplot(color = "black", outlier.size = 1, width = 0.85) +
  stat_summary(fun=mean, geom="point", shape=23,
               position = position_dodge2(width = 0.87,   
                                          preserve = "single")) +
  ylim(0.3, 0.51) +
  scale_fill_manual(values = c("pink", "red", "green"))+ #, labels = c("g-lasso", "smt-lasso", "e-net")) +
  labs(x = "Trait", y = expression(italic(R)^2), fill="Method") +
  geom_hline(yintercept=0.5, linetype="dotted", linewidth=1, color = "black") +
  theme_cowplot(font_size = 18)

print(p_methods_shared)

Version Author Date
b4baad5 fmorgante 2023-06-13

In this scenario, there is a clear advantage to using multivariate methods. In fact, given that the effects are equal across traits and the residuals are uncorrelated, a multivariate analysis is roughly equivalent to having 5 times as many samples as in an univariate analysis. The results show that mr.mash.rss clearly does better than both flavors of LDpred2 auto. However, in this simulation there does not seem to be much of a difference between LDpred2 auto per-chromosome and genome-wide. Therefore, we will drop LDpred2 auto genome-wide from further analyses since it is more computationally intensive.


sessionInfo()
R version 4.1.2 (2021-11-01)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Rocky Linux 8.5 (Green Obsidian)

Matrix products: default
BLAS/LAPACK: /opt/ohpc/pub/libs/gnu9/openblas/0.3.7/lib/libopenblasp-r0.3.7.so

locale:
 [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
 [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8    
 [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
 [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
 [9] LC_ADDRESS=C               LC_TELEPHONE=C            
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] cowplot_1.1.1 ggplot2_3.4.1

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.10      highr_0.10       pillar_1.9.0     compiler_4.1.2  
 [5] bslib_0.4.2      later_1.3.0      jquerylib_0.1.4  git2r_0.31.0    
 [9] workflowr_1.7.0  tools_4.1.2      digest_0.6.31    gtable_0.3.3    
[13] jsonlite_1.8.4   evaluate_0.20    lifecycle_1.0.3  tibble_3.2.1    
[17] pkgconfig_2.0.3  rlang_1.1.0      cli_3.6.1        rstudioapi_0.14 
[21] yaml_2.3.7       xfun_0.37        fastmap_1.1.1    withr_2.5.0     
[25] dplyr_1.1.1      stringr_1.5.0    knitr_1.42       generics_0.1.3  
[29] fs_1.6.1         vctrs_0.6.1      sass_0.4.5       tidyselect_1.2.0
[33] rprojroot_2.0.3  grid_4.1.2       glue_1.6.2       R6_2.5.1        
[37] fansi_1.0.4      rmarkdown_2.20   farver_2.1.1     magrittr_2.0.3  
[41] whisker_0.4.1    scales_1.2.1     promises_1.2.0.1 htmltools_0.5.4 
[45] colorspace_2.1-0 httpuv_1.6.9     labeling_0.4.2   utf8_1.2.3      
[49] stringi_1.7.12   munsell_0.5.0    cachem_1.0.7