Example Analysis with CAUSE: LDL -> CAD

Introduction

This document will walk through a real genome-sized example of how to use CAUSE. Some of the steps will take 5-10 minutes. The LD pruning steps will benefit from access to a cluster or multiple cores. For steps that require long computation we also provide output files that can be downloaded to make it easier to run through the example.

We will be analyzing GWAS data for LDL cholesterol and for coronary artery disease to test for a causal relationship of LDL on CAD. The analysis will have the following steps:

1. Format the data for use with CAUSE
2. Calculate nuisance parameters
3. LD pruning
4. Fit CAUSE
5. Look at results

Step 3 will require LD information estimated from the 1000 Genomes CEU population using LDshrink here. LD data are about 11 Gb. The GWAS data we will use are about 320 Gb. However, in this tutorial you will be able to skip the large data steps and simply download the results.

Step 0: Install CAUSE

In R

devtools::install_github("jean997/cause")

Please be sure you are using mixsqp-0-97 which is currently the version available in CRAN *not the latest version available on GitHub.

Step 1: Format Data for CAUSE

We will use read_tsv to read in summary statistics for a GWAS of LDL cholesterol and a GWAS of coronary artery disease from the internet. We will then combine these and format them for use with CAUSE. First read in the data. For LDL Cholesterol, we use summary statistics from Willer et al (2013) [PMID: 24097068]. For CAD we use summary statistics from van der Harst et al. (2017) [PMID: 29212778]

library(readr)
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(cause)

X1 <- read_tsv("http://csg.sph.umich.edu/abecasis/public/lipids2013/jointGwasMc_LDL.txt.gz")
X2 <- read_tsv("ftp://ftp.ebi.ac.uk/pub/databases/gwas/summary_statistics/vanderHarstP_29212778_GCST005194/CAD_META.gz")

CAUSE needs the following information from each data set: SNP or variant ID, effect size, and standard error, effect allele and other allele. For convenience, we provide a simple function that will merge data sets and produce a cause_data object that can be used with later functions. This step and the rest of the analysis are done in R.

The function gwas_format_cause will try to merge two data sets and and align effect sizes to correspond to the same allele. It will remove variants with ambiguous alleles (G/C or A/T) or with alleles that do not match between data sets (e.g A/G in one data set and A/C in the other). It will not remove variants that are simply strand flipped between the two data sets (e. g. A/C in one data set, T/G in the other).

LDL column headers:

• SNP: rsid
• Effect: beta
• Standard Error: se
• Effect Allele: A1
• Other Allele: A2

CAD column headers:

• SNP: oldID
• Effect: Effect
• Standard Error: StdErr
• Effect Allele: Allele1
• Other Allele: Allele2

X <- gwas_format_cause(X1, X2, snp_name_cols = c("rsid", "oldID"), 
                       beta_hat_cols = c("beta", "Effect"), 
                       se_cols = c("se", "StdErr"), 
                       A1_cols = c("A1", "Allele1"), 
                       A2_cols = c("A2", "Allele2"))

Alternatively, you can download already formatted data here and read it in using readRDS.

system("mkdir example_data/")
download.file("https://github.com/jean997/cause/blob/master/example_data/LDL_CAD_merged.RDS", destfile = "example_data/LDL_CAD_merged.RDS")

X <- readRDS("example_data/LDL_CAD_merged.RDS")

head(X)

        snp beta_hat_1   seb1 beta_hat_2   seb2 A1 A2
1 rs4747841     0.0037 0.0052     0.0106 0.0056  a  G
2 rs4749917    -0.0033 0.0052    -0.0108 0.0056  a  G
3  rs737656     0.0099 0.0054     0.0196 0.0058  a  G
4  rs737657     0.0084 0.0054     0.0195 0.0058  a  G
5 rs7086391    -0.0075 0.0067     0.0115 0.0072  a  G
6  rs878177    -0.0073 0.0055    -0.0225 0.0059  a  G

There are likely more efficient ways to do this merge. If you would like to process the data yourself, you can construct a cause_data object from a data frame using the constructor new_cause_data(X) where X is any data frame that includes the columns snp, beta_hat_1, seb1, beta_hat_2, and seb2.

Step 2: Calculate nuisance parameters

The next step is to estimate the parameters that define the prior distribution of \(\beta_{M}\) and \(\theta\) and to estimate \(\rho\), the correlation between summary statistics that is due to sample overlap or population structure. We will do this with a random subset of 1,000,000 variants since our data set is large. est_cause_params estimates the nuisance parameters by finding the maximum a posteriori estimate of \(\rho\) and the mixing parameters when \(\gamma = \eta = 0\). This step takes a several minutes.

set.seed(100)
varlist <- with(X, sample(snp, size=1000000, replace=FALSE))
params <- est_cause_params(X, varlist)

Estimating CAUSE parameters with  1000000  variants.
1 0.1255649 
2 0.000503845 
3 6.082345e-06 
4 7.406906e-08 

The object params is of class cause_params and contains information about the fit as well as the maximum a posteriori estimates of the mixing parameters (\(\pi\)) and \(\rho\). The object params$mix_grid is a data frame defining the distribution of summary statistics. The column S1 is the variance for trait 1 (\(M\)), the column S2 is the variance for trait 2 (\(Y\)) and the column pi is the mixture proportion assigned to that variance combination.

class(params)

[1] "cause_params"

names(params)

 [1] "rho"       "pi"        "mix_grid"  "loglik"    "PIS"      
 [6] "RHO"       "LLS"       "converged" "prior"     "var"      

params$rho

[1] 0.06515036

head(params$mix_grid)

           S1          S2         pi
1 0.000000000 0.000000000 0.31994147
2 0.000000000 0.003419647 0.20495624
3 0.000000000 0.004836112 0.17805780
4 0.003533615 0.004836112 0.16065491
5 0.004997287 0.009672224 0.11635100
6 0.009994574 0.009672224 0.00188234

So, for example, in this case, we have estimated that 32% of variants have trait 1 variance and trait 2 equal to 0 meaning that they have no association with either trait.

Tip: Do not try to estimate the nuisance parameters with substantially fewer than 100,000 variants. This can lead to poor estimates of the mixing parameters whih leads to bad model comparisons.

Step 3: LD Pruning

We estimate CAUSE posterior distributions using an LD pruned set of variants, prioritizing variants with low trait \(M\) (LDL) \(p\)-values. The CAUSE R package contains a function to help do this. The function ld_prune uses a greedy algorithm that selects the variant wtih the lowest LDL p-value and removes all variants in LD with the selected variant and then repeats until no variants are left. This step requires LD estimates. You can download estimates made in the 1000 Genomes CEU population here. We first demonstrate use of the function for one chromosome and then show an example of how to parallelize the analysis over all 22 autosomes.

download.file("https://zenodo.org/record/1464357/files/chr22_AF0.05_0.1.RDS?download=1", destfile = "example_data/chr22_AF0.05_0.1.RDS")
download.file("https://zenodo.org/record/1464357/files/chr22_AF0.05_snpdata.RDS?download=1", destfile="example_data/chr22_AF0.05_snpdata.RDS")

ld <- readRDS("example_data/chr22_AF0.05_0.1.RDS")
snp_info <- readRDS("example_data/chr22_AF0.05_snpdata.RDS")

head(ld)

      rowsnp      colsnp        r2
1 rs62224609 rs376238049 0.9012642
2 rs62224609  rs62224614 0.9907366
3 rs62224609   rs7286962 0.9907366
4 rs62224609  rs55926024 0.1103000
5 rs62224609 rs117246541 0.9012642
6 rs62224609  rs62224618 0.9907366

head(snp_info)

# A tibble: 6 x 9
     AF SNP        allele    chr      pos  snp_id region_id   map ld_snp_id
  <dbl> <chr>      <chr>   <int>    <int>   <int>     <int> <dbl>     <int>
1 0.884 rs62224609 T,C        22 16051249  7.98e7        22     0  79758556
2 0.904 rs4965031  G,A        22 16052080  7.98e7        22     0  79758578
3 0.646 rs3756846… A,AAAAC    22 16052167  7.98e7        22     0  79758584
4 0.894 rs3762380… C,T        22 16052962  7.98e7        22     0  79758602
5 0.934 rs2007775… C,A        22 16052986  7.98e7        22     0  79758604
6 0.934 rs80167676 A,T        22 16053444  7.98e7        22     0  79758627

The ld data frame should contain the column names rowsnp, colsnp, and r2. The snp_info data frame contains information about all of the chromosome 22 variants with allele frequency greater than 0.05. The only piece of information we need from this data frame is the list of variants snp_info$SNP which provides the total list of variants used in the LD calculations.

variants <- X %>% mutate(pval1 = 2*pnorm(abs(beta_hat_1/seb1), lower.tail=FALSE))
pruned <- ld_prune(variants = variants, 
                            ld = ld, total_ld_variants = snp_info$SNP, 
                            pval_cols = c("pval1"), 
                            pval_thresh = c(1e-3))

You have suppplied information for  2023362  variants.
Of these,  22835  have LD information.

length(pruned)

[1] 15

library(parallel)
cores <- parallel::detectCores()-1
ld_files <- paste0("example_data/ld_data/chr", 1:22, "_AF0.05_0.1.RDS")
snp_info_files <- paste0("example_data/ld_data/chr", 1:22, "_AF0.05_snpdata.RDS")

cl <- makeCluster(cores, type="PSOCK")
clusterExport(cl, varlist=c("variants", "ld_files", "snp_info_files"))
clusterEvalQ(cl, library(cause))
system.time(
pruned <- parLapply(cl, seq_along(ld_files[20:22]), fun=function(i){
  ld <- readRDS(ld_files[i])
  snp_info <- readRDS(snp_info_files[i])
  ld_prune(variants = variants, 
          ld = ld, total_ld_variants = snp_info$SNP, 
          pval_cols = c("pval1"), 
          pval_thresh = c( 1e-3))
})
)
stopCluster(cl)

top_ldl_pruned_vars <- unlist(pruned)

download.file("https://github.com/jean997/cause/blob/master/example_data/top_ldl_pruned_vars.RDS", destfile = "example_data/top_ldl_pruned_vars.RDS")

Step 4: Fit CAUSE

Now that we have formatted data, an LD pruned set of variants, and nuisance parameters estimated, we can fit CAUSE! The function cause::cause will estimate posterior distributions under the confounding and causal models and calculate the elpd for both models as well as for the null model in which there is neither a causal or a confounding effect. This might take 5-10 minutes.

Note: To estimate the posterior distributions, we only need the variants that are most associated with the mediator. This is because other variants don’t add any information about the relationship between the traits. When we LD pruned, we used a \(p\)-value threshold of 0.001. The exact value of this threshold isn’t important as long as it is fairly lenient. Including additional variants may slow down computation but shouldn’t change the results

top_vars <- readRDS("example_data/top_ldl_pruned_vars.RDS")
res <- cause(X=X, variants = top_vars, param_ests = params)

Estimating CAUSE posteriors using  1215  variants.

Step 5: Look at Results

The resulting cause object contains an object for the partial sharing model fit (conf), and object for the causal model fit (full) and a table of ELPD results.

class(res)

[1] "cause"

names(res)

[1] "conf"    "full"    "elpd"    "loos"    "data"    "sigma_g" "qalpha" 
[8] "qbeta"  

res$elpd

  model1 model2 delta_elpd se_delta_elpd         z
1   null   conf -63.488710      7.978181 -7.957793
2   null   full -71.880159      9.175262 -7.834126
3   conf   full  -8.391449      1.285055 -6.530034

class(res$conf)

[1] "cause_post"

class(res$full)

[1] "cause_post"

The elpd table has the following columns:

• model1, model2: The models being compared
• delta_elpd: Estimated difference in elpd. If delta_elpd is negative, model 2 is a better fit
• se_delta_elpd: Estimated standard error of delta_elpd
• z: delta_elpd/se_delta_elpd. A z-score that can be compared to a normal distribution to test if the difference in model fit is significant.

In this case we see that the full (causal) model is significantly better than the confounding model from the thrid line of the table. The \(z\)-score is -6.53 corresponding to a p-value of 3.310^{-11}.

For each model (partial sharing and full) we can plot the posterior distributions of the parameters. Dotted lines show the prior distributions.

plot(res$conf)

Past versions of plot1-1.png

plot(res$full)

Past versions of plot1-2.png

The summary method will summarize the posterior medians and credible intervals.

summary(res, ci_size=0.95)

p-value testing that causal model is a better fit:  3.3e-11 
Posterior medians and  95 % credible intervals:
     model              gamma               eta                  
[1,] "Confounding Only" NA                  "0.38 (0.31, 0.45)"  
[2,] "Causal"           "0.34 (0.28, 0.39)" "-0.01 (-0.66, 0.54)"
     q                  
[1,] "0.78 (0.64, 0.88)"
[2,] "0.03 (0, 0.25)"   

The plot method applied to a cause object will arrange all of this information on one spread.

plot(res)

Past versions of plot2-1.png

The plot method can also produce scatter plots of the data showing for each model, the probability that each variant is acting through the confounder and the contribution of each variant to the ELPD test statistic.

plot(res, type="data")

Past versions of plot3-1.png

Session information

sessionInfo()

R version 3.6.1 (2019-07-05)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Ubuntu 18.04.2 LTS

Matrix products: default
BLAS:   /usr/lib/x86_64-linux-gnu/openblas/libblas.so.3
LAPACK: /usr/lib/x86_64-linux-gnu/libopenblasp-r0.2.20.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] cause_0.2.0.0126 dplyr_0.8.3      readr_1.3.1     

loaded via a namespace (and not attached):
 [1] tidyselect_0.2.5     xfun_0.8             ashr_2.2-32         
 [4] purrr_0.3.2          lattice_0.20-38      colorspace_1.4-1    
 [7] vctrs_0.2.0          htmltools_0.3.6      loo_2.1.0           
[10] yaml_2.2.0           utf8_1.1.4           rlang_0.4.0         
[13] mixsqp_0.1-97        pillar_1.4.2         glue_1.3.1          
[16] matrixStats_0.54.0   foreach_1.4.4        stringr_1.4.0       
[19] munsell_0.5.0        gtable_0.3.0         workflowr_1.4.0.9000
[22] codetools_0.2-16     evaluate_0.14        labeling_0.3        
[25] knitr_1.23           doParallel_1.0.14    pscl_1.5.2          
[28] parallel_3.6.1       fansi_0.4.0          highr_0.8           
[31] Rcpp_1.0.1           backports_1.1.4      scales_1.0.0        
[34] RcppParallel_4.4.3   truncnorm_1.0-8      fs_1.3.1            
[37] gridExtra_2.3        ggplot2_3.2.0        hms_0.5.0           
[40] digest_0.6.20        stringi_1.4.3        numDeriv_2016.8-1.1 
[43] grid_3.6.1           rprojroot_1.3-2      cli_1.1.0           
[46] tools_3.6.1          magrittr_1.5         lazyeval_0.2.2      
[49] tibble_2.1.3         crayon_1.3.4         whisker_0.3-2       
[52] tidyr_0.8.3          pkgconfig_2.0.2      zeallot_0.1.0       
[55] MASS_7.3-51.4        Matrix_1.2-17        SQUAREM_2017.10-1   
[58] assertthat_0.2.1     rmarkdown_1.13       iterators_1.0.10    
[61] R6_2.4.0             intervals_0.15.1     git2r_0.26.1        
[64] compiler_3.6.1