Last updated: 2024-08-13

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###Load libraries
library(dplyr)

Attaching package: 'dplyr'
The following objects are masked from 'package:stats':

    filter, lag
The following objects are masked from 'package:base':

    intersect, setdiff, setequal, union
library(ggplot2)
library(cowplot)

repz <- c(1,2,3,4,5,6,7,9,10,11,12,13,14,15,16,17,18,19,20,22)
prefix <- "output/prediction_accuracy/ukb_caucasian_white_british_unrel_100000_external_LD"
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 503 individuals of European ancestry from the 1000G project, retaining variants with MAF > 0.05. A block diagonal strategy including denoising via SVD as suggested here was used.

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 (except for two scenario as explained below) – in genetics terminology this is called genomic heritability (\(h_g^2\)). The residuals are uncorrelated across traits. Each trait was quantile normalized before all the analyses were performed.

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. Variants were not standardized. Summary statistics were subjected to careful and stringent quality control following this tutorial.

A few different methods were fitted:

  • LDpred2 per-chromosome with the auto option, 1000 iterations (after 500 burn-in iterations), \(h^2\) initialized using an estimate from LD Score regression (LDSC), \(p\) initialized using the same grid as in the original paper, “shrink_corr=0.95”, and “allow_jump_sign=FALSE” to avoid convergence issues. NB this is a univariate method.
  • mr.mash.rss per-chromosome, with both canonical and data-driven covariance matrices computed as described in the mvSuSiE 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. NB this is a multivariate method.
  • BayesR per-chromosome, with 5000 iterations, 1000 burn-in iterations, thinning factor of 5, \(\pi\) (i.e., the proportion of causal variants) initialized as 0.0001, \(h^2\) initialized as 0.1, and other default parameters. NB this is a univariate method. We used the implementation in the qgg R package.
  • mvBayesC per-chromosome, with 5000 iterations, 1000 burn-in iterations, thinning factor of 5, \(\pi\) (i.e., the proportion of causal variants) initialized as 0.0001, \(h^2\) initialized as 0.1, and other default parameters. NB this is a multivariate method. We used the implementation in the qgg R package.
  • mvBayesCrest – a version of mvBayesC that only allows a variant to affect all or none of the traits – per-chromosome, with 5000 iterations, 1000 burn-in iterations, thinning factor of 5, \(\pi\) (i.e., the proportion of causal variants) initialized as 0.0001, \(h^2\) initialized as 0.1, and other default parameters. NB this is a multivariate method. We used the implementation in the qgg R package.
  • wMT-SBLUP per-chromosome, with \(M_{eff}=60000\) and SBLUP estimate obtained with a window size of 2 Mb (using the SumTool R package implementation). NB this is a multivariate method. We used the implementation in the qgg R package.
  • MTAG+LDpred2 with all variants. Because MTAG does not allow analyzing small indels and our data has a small numaber of those, we kept assigned the OLS estimates to those variants (instead of dropping them) before running LDpred2. LDpred2 was run with the same parameter as above. The effective sample size was estimated from the median \(\chi^2\) statistics of the OLS estimates and the MTAG estimates as done here. However, convergence issues still remained and some combinations of trait/replicate were dropped because of that.

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", "mvbayesC", "mvbayesC_rest", "wmt_sblup", "mtag_ldpred2_auto", "ldpred2_auto", "bayesR")

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 <- tryCatch(readRDS(paste0(prefix, "_", sce, "_", met, "_pred_acc_", repp, ".rds")), 
                     error = function(e) {
                       return(NULL)
                     },
                     warning = function(w) {
                       return(NULL)
                     }
      )
      
      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
        if(!is.null(dat)){
          res[i, 6] <- dat$r2[trait]
        } else {
          res[i, 6] <- NA
        }
        
      }
    }
  }
}

res1 <- transform(res, scenario=as.factor(scenario),
                      method=as.factor(method),
                      trait=as.factor(trait))

p_methods_shared <- ggplot(res1, 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, 0.51) +
  scale_fill_manual(values = c("pink", "red", "yellow", "orange", "green", "blue", "lightblue")) +
  labs(x = "Trait", y = expression(italic(R)^2), fill="Method", title="") +
  geom_hline(yintercept=0.5, linetype="dotted", linewidth=1, color = "black") +
  theme_cowplot(font_size = 18)

print(p_methods_shared)
Warning: Removed 48 rows containing non-finite outside the scale range
(`stat_boxplot()`).
Warning: Removed 48 rows containing non-finite outside the scale range
(`stat_summary()`).

Version Author Date
51935b3 fmorgante 2024-08-13 
res %>% group_by(method, trait) %>% summarise(n=sum(!is.na(score))) %>% filter(trait == 1) %>% select(method, n) %>% as.data.frame()
`summarise()` has grouped output by 'method'. You can override using the
`.groups` argument.
             method  n
1            bayesR 20
2      ldpred2_auto 20
3       mr_mash_rss 20
4 mtag_ldpred2_auto 11
5          mvbayesC 20
6     mvbayesC_rest 20
7         wmt_sblup 20

In this scenario, all the methods perform worse than with an LD matrix built using larger sample size. However, mr.mash.rss remains the best performing method, showing its robustness to the source of LD matrix. In any case, we do NOT recommend using LD matrices computed with such a small sample size with any method.


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_1.1.3 ggplot2_3.5.1 dplyr_1.1.4  

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.13       highr_0.11        pillar_1.9.0      compiler_4.1.2   
 [5] bslib_0.7.0       later_1.3.2       jquerylib_0.1.4   git2r_0.32.0     
 [9] workflowr_1.7.0   tools_4.1.2       digest_0.6.35     gtable_0.3.5     
[13] jsonlite_1.8.8    evaluate_0.23     lifecycle_1.0.4   tibble_3.2.1     
[17] pkgconfig_2.0.3   rlang_1.1.4       cli_3.6.2         rstudioapi_0.16.0
[21] yaml_2.3.8        xfun_0.44         fastmap_1.2.0     withr_3.0.0      
[25] stringr_1.5.1     knitr_1.47        generics_0.1.3    fs_1.6.4         
[29] vctrs_0.6.5       sass_0.4.9        grid_4.1.2        tidyselect_1.2.1 
[33] rprojroot_2.0.4   glue_1.7.0        R6_2.5.1          fansi_1.0.6      
[37] rmarkdown_2.27    farver_2.1.2      magrittr_2.0.3    whisker_0.4.1    
[41] scales_1.3.0      promises_1.3.0    htmltools_0.5.8.1 colorspace_2.1-0 
[45] httpuv_1.6.11     labeling_0.4.3    utf8_1.2.4        stringi_1.8.4    
[49] munsell_0.5.1     cachem_1.1.0