Introduction

multistateQTL is a Bioconductor package for applying basic statistical tests (e.g., feature-wise FDR correction, calculating pairwise sharing), summarizing, and visualizing QTL summary statistics from multiple states (e.g., tissues, celltypes, environmental conditions). It works on the QTLExperiment (QTLE) object class, where rows represent features (e.g., genes, transcripts, genomic regions), columns represent states, and assays are the various summary statistics. It also provides wrapper implementations of a number of multi-test correction methods (e.g., mashr, meta-soft, etc), which result in a set of multi-test corrected summary statistics.

Installation

QTLExperiment and multistateQTL can be installed from GitHub:

if (!require("BiocManager", quietly=TRUE))
    install.packages("BiocManager")

BiocManager::install(c("QTLExperiment", "multistateQTL"), version="devel")

They are also available on GitHub:

devtools::install_git("https://github.com/dunstone-a/QTLExperiment", build_vignettes=TRUE)
devtools::install_git(https://github.com/dunstone-a/multistateQTL", build_vignettes=TRUE)

library(QTLExperiment)
library(multistateQTL)

Simulating data

Estimate parameters from GTEx

Provided with real QTL summary statistics as either a QTLE object or a named list with betas and error (and optionally pval or lfsr), key parameters are estimated that are used to simulate realistic multi-state QTL data. We demonstrate the parameter estimation function on publicly available summary statistics from GTEx (v8). Note that this data only contains tests that were called as significant by GTEx and vroom only loads the first chunk of results as it does not read .gz compressed objects well. This truncated dataset is used in the vignette for convenience, however, to estimate the default parameters in qtleParams() we downloaded QTL summary statistics for all associations tested for the 10 GTEx tissues with the largest sample sizes from Google Cloud). To speed up calculations we filtered this to include only associations on chromosome 1 and considered significant tests with pval < 0.05 and null tests with a pval > 0.1.

See QTLExperiment for more info on the sumstats2qtle function and for other approaches for reading in QTL summary statistics.

The parameters estimated here include:

Significant betas shape and rate: Define the gamma distribution used to sample mean effect sizes for each QTL that is significant in at least one state. These simulated mean effect sizes are used as the mean parameter in rnorm to sample an effect size for each QTL for each state. The variance parameter for rnorm is user defined (default = 0.1).
Significant coefficient of variation (cv) shape and rate: Define the gamma distribution used to sample the cv for each QTL in each state where that QTL is significant. The cv is multiplied by the simulated significant effect size for that test/state to get the simulated standard error values.
Null beta shape and rate: Define the gamma distribution used to sample the effect sizes for each QTL in each state where the effect is not significant.
Null beta cv shape and rate: Define the gamma distribution used to sample the cv for each QTL in each state where that QTL is not significant. This cv is then multiplied by the simulated null effect size for that test/state to get the simulated stand error values.

input_path <- system.file("extdata", package="multistateQTL")
state <- c("lung", "thyroid", "spleen", "blood")

input <- data.frame(
    state=state, 
    path=paste0(input_path, "/GTEx_tx_", state, ".tsv"))

gtex <- sumstats2qtle(
    input, 
    feature_id="molecular_trait_id",
    variant_id="rsid", 
    betas="beta", 
    errors="se",
    pvalues="pvalue", 
    verbose=TRUE)
gtex

## class: QTLExperiment 
## dim: 1163 4 
## metadata(0):
## assays(3): betas errors pvalues
## rownames(1163): ENST00000428771|rs554008981 ENST00000477976|rs554008981
##   ... ENST00000445118|rs368254419 ENST00000483767|rs368254419
## rowData names(2): variant_id feature_id
## colnames(4): lung thyroid spleen blood
## colData names(1): state_id

head(betas(gtex))

##                                   lung    thyroid   spleen     blood
## ENST00000428771|rs554008981 -0.1733690         NA 0.134913        NA
## ENST00000477976|rs554008981  0.1616170  0.3173110       NA        NA
## ENST00000483767|rs554008981 -0.4161480 -0.0483018       NA -0.204647
## ENST00000623070|rs554008981 -0.1137930         NA       NA        NA
## ENST00000669922|rs554008981 -0.1921680 -0.1067540 0.724622 -0.117424
## ENST00000428771|rs201055865 -0.0630909         NA       NA        NA

Estimating parameters:

params <- qtleEstimate(gtex, threshSig=0.05, threshNull=0.5)
params

## $cv.sig.shape
## [1] 7.070567
## 
## $cv.sig.rate
## [1] 7.717952
## 
## $cv.null.shape
## [1] 0.007593863
## 
## $cv.null.rate
## [1] 1269.631
## 
## $betas.sig.shape
## [1] 3.470341
## 
## $betas.sig.rate
## [1] 16.27275
## 
## $betas.null.shape
## [1] 1.360216
## 
## $betas.null.rate
## [1] 11.03421

Looking at the distributions defined by these estimated parameters, the simulated effect sizes for significant QTL will tend to be larger, while the simulated coefficient of variation values will be smaller than for the non-significant QTL.

plotSimulationParams(params=params)

Gamma distributions defined by the parameters estimated by qtleEstimate.

The default parameters available through qtleParams() were estimated from the GTEx v8 tissue-level eQTL summary statistics from chromosome 1 using the 10 tissues with the largest sample sizes. From these data, significant QTL parameters were estimated from tests in the lowest p-value quantile, while null parameters were estimated from tests in the highest p-value quantile. Data for tests on chromosome 1 were included in all four tissues (n=32613).

Simulate multi-state QTL data

The simulation tool allows for the simulation of four types of associations: (1) Global, where the simulated effect size is approximately equal across all states; (2) Unique, where the association is only significant in one state; (3) Multi-state, where the association is significant in a subset of states (i.e., state-groups), and (4) Null, where the association has no significant effects in any state. First each test is randomly assigned as one of the above types according to the proportions specified by the user. For multi-state QTL, each state is assigned to a state-group, either randomly or according to user defined groups, then each multi-state QTL is assigned randomly to one of the state-groups. For unique QTL, the QTL is randomly assigned to a single state.

Simulated mean effect sizes for all non-null QTL are sampled from gamma(beta.sig.shape, beta.sig.rate) and are randomly assigned a positive or negative effect direction. Then for each state where that QTL is significant, an effect size is sampled from N(mean effect size, σ), where σ is user defined (default=0.1). Effect sizes for null QTL are sampled from gamma(beta.null.shape, beta.null.rate) and are randomly assigned a positive or negative effect direction. Standard errors for each QTL for each state are simulated by sampling from gamma(cv.sig.shape, cv.sig.rate) or gamma(cv.null.shape, cv.null.rate) for significant and null QTL, respectively, and multiplying the sampled cv by the absolute value of the simulated beta for that QTL in that state.

Here is an example of a simple simulation with half of the simulated QTL tests having globally significant effects. This example uses the default parameters.

sim <- qtleSimulate(nTests=1000, nStates=6, global=0.5)
sim

## class: QTLExperiment 
## dim: 1000 6 
## metadata(0):
## assays(3): betas errors lfsrs
## rownames(1000): F0502|v65513 F0829|v28175 ... F0647|v42219 F0431|v72934
## rowData names(11): feature_id variant_id ... S5 S6
## colnames(6): S1 S2 ... S5 S6
## colData names(1): state_id

head(rowData(sim))

## DataFrame with 6 rows and 11 columns
##               feature_id  variant_id         QTL           id mean_beta
##              <character> <character> <character>  <character> <numeric>
## F0502|v65513       F0502      v65513        null F0502|v65513  0.000000
## F0829|v28175       F0829      v28175      global F0829|v28175 -0.646669
## F0680|v76569       F0680      v76569      global F0680|v76569  0.314879
## F0007|v72522       F0007      v72522        null F0007|v72522  0.000000
## F0637|v6704        F0637       v6704      global  F0637|v6704  0.658535
## F0924|v60732       F0924      v60732      global F0924|v60732  0.660113
##                     S1        S2        S3        S4        S5        S6
##              <logical> <logical> <logical> <logical> <logical> <logical>
## F0502|v65513     FALSE     FALSE     FALSE     FALSE     FALSE     FALSE
## F0829|v28175      TRUE      TRUE      TRUE      TRUE      TRUE      TRUE
## F0680|v76569      TRUE      TRUE      TRUE      TRUE      TRUE      TRUE
## F0007|v72522     FALSE     FALSE     FALSE     FALSE     FALSE     FALSE
## F0637|v6704       TRUE      TRUE      TRUE      TRUE      TRUE      TRUE
## F0924|v60732      TRUE      TRUE      TRUE      TRUE      TRUE      TRUE

We can also generate more complex simulations, for example this simulation has 20% global, 40% multi-state, 20% unique, and 20% null QTL effects, where multi-state effects are assigned to one of two state-groups.

sim <- qtleSimulate(
    nStates=10, nFeatures=100, nTests=1000,
    global=0.2, multi=0.4, unique=0.2, k=2)

Here is a snapshot of the simulation key for QTL simulated as unique to a single state:

head(rowData(subset(sim, QTL == "unique")))

## DataFrame with 6 rows and 16 columns
##              feature_id  variant_id         QTL          id mean_beta       S01
##             <character> <character> <character> <character> <numeric> <logical>
## F056|v68405        F056      v68405      unique F056|v68405  1.132321     FALSE
## F089|v18603        F089      v18603      unique F089|v18603  0.641141     FALSE
## F001|v67681        F001      v67681      unique F001|v67681 -0.337594     FALSE
## F067|v31085        F067      v31085      unique F067|v31085 -0.300979     FALSE
## F072|v52705        F072      v52705      unique F072|v52705 -0.309610     FALSE
## F061|v86447        F061      v86447      unique F061|v86447  0.441208     FALSE
##                   S02       S03       S04       S05       S06       S07
##             <logical> <logical> <logical> <logical> <logical> <logical>
## F056|v68405     FALSE     FALSE     FALSE     FALSE     FALSE     FALSE
## F089|v18603     FALSE     FALSE     FALSE     FALSE     FALSE      TRUE
## F001|v67681      TRUE     FALSE     FALSE     FALSE     FALSE     FALSE
## F067|v31085     FALSE     FALSE     FALSE     FALSE     FALSE     FALSE
## F072|v52705     FALSE     FALSE     FALSE     FALSE     FALSE     FALSE
## F061|v86447      TRUE     FALSE     FALSE     FALSE     FALSE     FALSE
##                   S08       S09       S10 multistateGroup
##             <logical> <logical> <logical>     <character>
## F056|v68405     FALSE      TRUE     FALSE              NA
## F089|v18603     FALSE     FALSE     FALSE              NA
## F001|v67681     FALSE     FALSE     FALSE              NA
## F067|v31085     FALSE      TRUE     FALSE              NA
## F072|v52705      TRUE     FALSE     FALSE              NA
## F061|v86447     FALSE     FALSE     FALSE              NA

Here is a snapshot of the simulation key for QTL simulated as multi-state:

head(rowData(subset(sim, QTL == "multistate")))

## DataFrame with 6 rows and 16 columns
##              feature_id  variant_id         QTL          id mean_beta       S01
##             <character> <character> <character> <character> <numeric> <logical>
## F031|v89404        F031      v89404  multistate F031|v89404  0.531274      TRUE
## F081|v39490        F081      v39490  multistate F081|v39490 -0.976719     FALSE
## F015|v91714        F015      v91714  multistate F015|v91714 -0.306235      TRUE
## F014|v61533        F014      v61533  multistate F014|v61533  0.456947     FALSE
## F022|v3677         F022       v3677  multistate  F022|v3677 -0.480573     FALSE
## F018|v16936        F018      v16936  multistate F018|v16936 -0.487362     FALSE
##                   S02       S03       S04       S05       S06       S07
##             <logical> <logical> <logical> <logical> <logical> <logical>
## F031|v89404     FALSE     FALSE      TRUE      TRUE      TRUE      TRUE
## F081|v39490      TRUE      TRUE     FALSE     FALSE     FALSE     FALSE
## F015|v91714     FALSE     FALSE      TRUE      TRUE      TRUE      TRUE
## F014|v61533      TRUE      TRUE     FALSE     FALSE     FALSE     FALSE
## F022|v3677       TRUE      TRUE     FALSE     FALSE     FALSE     FALSE
## F018|v16936      TRUE      TRUE     FALSE     FALSE     FALSE     FALSE
##                   S08       S09       S10 multistateGroup
##             <logical> <logical> <logical>     <character>
## F031|v89404     FALSE     FALSE      TRUE          Group1
## F081|v39490      TRUE      TRUE     FALSE          Group2
## F015|v91714     FALSE     FALSE      TRUE          Group1
## F014|v61533      TRUE      TRUE     FALSE          Group2
## F022|v3677       TRUE      TRUE     FALSE          Group2
## F018|v16936      TRUE      TRUE     FALSE          Group2

message("Number of QTL specific to each state-group:")
table(rowData(subset(sim, QTL == "multistate"))$multistateGroup)

## 
## Group1 Group2 
##    192    208

Dealing with missing data

The multistateQTL toolkit provides two functions to help deal with missing data, getComplete and replaceNAs. The getComplete function is a smart subsetting function that remove QTL associations (rows) with more than an allowed amount of missing data. The replaceNAs function allows for NAs in each assay to be replaced with a constant or with the row mean or row median. For example, here is a snapshot of our simulated data from above with added NAs:

na_pattern <- sample(seq(1, ncol(sim)*nrow(sim)), 1000)

sim_na <- sim
assay(sim_na, "betas")[na_pattern] <- NA
assay(sim_na, "errors")[na_pattern] <- NA
assay(sim_na, "lfsrs")[na_pattern] <- NA

message("Number of simulated tests: ", nrow(sim_na))
head(betas(sim_na))

##                    S01        S02        S03        S04        S05        S06
## F056|v68405 -0.2069116 -0.1777617         NA         NA  0.1185643 -0.2044272
## F050|v48935  0.2019550 -0.3288880  0.1095000  0.6518384  0.4406165  0.4421207
## F031|v89404  0.4454272 -0.2475028 -0.1697007  0.5574706         NA  0.8846720
## F081|v39490  0.1993529 -0.9769058 -0.9641381  0.3075263 -0.3230863 -0.2608925
## F012|v22815 -0.6768760 -0.2111132 -0.1790861 -0.2369226  0.2726944 -0.1961095
## F056|v87470 -0.7515367 -0.9673488 -0.7851777 -0.8699182 -0.9189384 -0.8862677
##                    S07         S08        S09        S10
## F056|v68405  0.3645608  0.07241395  1.1229543  0.1795885
## F050|v48935  0.3286546  0.34967895 -0.2875481  0.1133443
## F031|v89404  0.4759706  0.28585838  0.2503787  0.3781954
## F081|v39490 -0.6649082 -0.94329947 -0.7270914         NA
## F012|v22815 -0.1972697  0.28779758  0.2598863         NA
## F056|v87470 -1.0427818 -0.80325057 -0.8584320 -0.8730868

First we can use getComplete to keep only the tests that have data for at least half of the states:

sim_na <- getComplete(sim_na, n=0.5, verbose=TRUE)

Then for the remaining QTL, we can fill in the missing values using the following scheme

sim_na <- replaceNAs(sim_na, verbose=TRUE)

head(betas(sim_na))

##                    S01        S02        S03        S04        S05        S06
## F056|v68405 -0.2069116 -0.1777617  0.0000000  0.0000000  0.1185643 -0.2044272
## F050|v48935  0.2019550 -0.3288880  0.1095000  0.6518384  0.4406165  0.4421207
## F031|v89404  0.4454272 -0.2475028 -0.1697007  0.5574706  0.0000000  0.8846720
## F081|v39490  0.1993529 -0.9769058 -0.9641381  0.3075263 -0.3230863 -0.2608925
## F012|v22815 -0.6768760 -0.2111132 -0.1790861 -0.2369226  0.2726944 -0.1961095
## F056|v87470 -0.7515367 -0.9673488 -0.7851777 -0.8699182 -0.9189384 -0.8862677
##                    S07         S08        S09        S10
## F056|v68405  0.3645608  0.07241395  1.1229543  0.1795885
## F050|v48935  0.3286546  0.34967895 -0.2875481  0.1133443
## F031|v89404  0.4759706  0.28585838  0.2503787  0.3781954
## F081|v39490 -0.6649082 -0.94329947 -0.7270914  0.0000000
## F012|v22815 -0.1972697  0.28779758  0.2598863  0.0000000
## F056|v87470 -1.0427818 -0.80325057 -0.8584320 -0.8730868

Calling significance

The multistateQTL toolkit also provides the callSignificance function, which calls QTL tests significant in each state using either a single or two-step threshold approach. For example, we can set a single lfsr threshold of 0.1 to call significance of our simulate QTL:

sim <- callSignificance(sim, assay="lfsrs", thresh=0.05)

message("Median number of significant tests per state: ", 
    median(colData(sim)$nSignificant))

Because we have the simulated ground-truth, we can compare these significance calls to what was simulated using the simPerformance function, which provides the following global (i.e. across all state) performance metrics:

sim <- callSignificance(sim, assay="lfsrs", thresh=0.001)
perf_metrics <- simPerformance(sim)
lapply(perf_metrics, FUN=function(x) {round(x, 2)})

## $accuracy
## [1] 0.42
## 
## $precision
## [1] 0.42
## 
## $recall
## [1] 1
## 
## $f1
## [1] 0.59
## 
## $cm
##          called
## simulated FALSE TRUE
##     FALSE    30 5786
##     TRUE     10 4174

As you can see the recall of TRUE significant QTL is quite low. However if we change our significance calling approach to be more flexible.

sim <- callSignificance(
    sim, mode="simple", assay="lfsrs",
    thresh=0.0001, secondThresh=0.0002)
simPerformance(sim)$recall

## [1] 0.9935468

Plotting global patterns of sharing

The multistateQTL package contains five functions for visualising multi-state eQTL data. These functions are based on the ggplot2 and ComplexHeatmap R packages.

plotPairwiseSharing: based on ComplexHeatmap::Heatmap
plotQTLClusters: based on ComplexHeatmap::Heatmap
plotUpSet: based on ComplexHeatmap::UpSet
plotCompareStates: based on ggplot2
plotSimulationParams: based on ggplot2

The functions built on ggplot2 are compatible with ggplot2 syntax such as the + operator.

Pairwise sharing

The function plotPairwiseSharing shows the degree of pairwise sharing of significant hits for each combination of two states. Column annotations can be added by specifying a valid column name from the colData of the object.

In the plot below, columns are ordered by the broad cell type (multistateGroup) of the states. We expect states belonging to the same multi-state group to be more related and have a greater degree of sharing of significant eQTLs. Column annotations are used to show the number of significant eQTLs for each state.

sim_sig <- getSignificant(sim)
sim_top <- getTopHits(sim_sig, assay="lfsrs", mode="state")
sim_top <- runPairwiseSharing(sim_top)
p1 <- plotPairwiseSharing(sim_top, annotate_cols=c("nSignificant", "multistateGroup"))

Upset plots

These plots show the set of tests that are significant, but not necessarily shared, by groups of states.

plotUpSet(sim_top, annotateColsBy=c("nSignificant", "multistateGroup"))

Characterizing multi-state QTL patterns

Categorizing multi-state QTL tests

Once multi-state test correction is performed, you will want to identify global, multi-state, and unique QTL.

Note that plotCompareStates returns a list with a ggplot2 object as the first element and a table as the second element. These can be accessed using the names “plot” or “counts”.

sim_top <- runTestMetrics(sim_top)

plotCompareStates(sim_top, x="S01", y="S02")

## $plot

## 
## $counts
## 
##            S01            S02 both_diverging    both_shared 
##              9              4            162            291

table(rowData(sim_top)$qtl_type)

## 
##     global_diverging        global_shared multistate_diverging 
##                  284                  116                   51 
##    multistate_shared 
##                   15

hist(rowData(sim_top)$nSignificant)

Visualizing multi-state QTL

The function plotQTLClusters can be used to produce a heatmap of the eQTL betas values for each state. Each row is a SNP-gene pair, and columns are states. Row and column annotations can be added by naming column names from the rowData or colData of the input QTLExperiment object.

sim_top_ms <- subset(sim_top, qtl_type_simple == "multistate")

plotQTLClusters(
    sim_top_ms, 
    annotateColsBy=c("multistateGroup"),
    annotateRowsBy=c("qtl_type", "mean_beta", "QTL"))

Session Info

sessionInfo()

## R version 4.4.2 (2024-10-31)
## Platform: x86_64-pc-linux-gnu
## Running under: Ubuntu 24.04.1 LTS
## 
## Matrix products: default
## BLAS:   /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3 
## LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.26.so;  LAPACK version 3.12.0
## 
## locale:
##  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
##  [3] LC_TIME=en_US.UTF-8        LC_COLLATE=C              
##  [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       
## 
## time zone: Etc/UTC
## tzcode source: system (glibc)
## 
## attached base packages:
## [1] grid      stats4    stats     graphics  grDevices utils     datasets 
## [8] methods   base     
## 
## other attached packages:
##  [1] multistateQTL_1.3.0         collapse_2.0.18            
##  [3] data.table_1.16.2           ComplexHeatmap_2.23.0      
##  [5] QTLExperiment_1.5.0         SummarizedExperiment_1.37.0
##  [7] Biobase_2.67.0              GenomicRanges_1.59.1       
##  [9] GenomeInfoDb_1.43.2         IRanges_2.41.1             
## [11] S4Vectors_0.45.2            BiocGenerics_0.53.3        
## [13] generics_0.1.3              MatrixGenerics_1.19.0      
## [15] matrixStats_1.4.1           BiocStyle_2.35.0           
## 
## loaded via a namespace (and not attached):
##  [1] gridExtra_2.3           rlang_1.1.4             magrittr_2.0.3         
##  [4] clue_0.3-66             GetoptLong_1.0.5        compiler_4.4.2         
##  [7] png_0.1-8               vctrs_0.6.5             pkgconfig_2.0.3        
## [10] shape_1.4.6.1           crayon_1.5.3            fastmap_1.2.0          
## [13] backports_1.5.0         XVector_0.47.0          labeling_0.4.3         
## [16] utf8_1.2.4              rmarkdown_2.29          tzdb_0.4.0             
## [19] UCSC.utils_1.3.0        purrr_1.0.2             bit_4.5.0              
## [22] xfun_0.49               zlibbioc_1.52.0         cachem_1.1.0           
## [25] jsonlite_1.8.9          DelayedArray_0.33.2     irlba_2.3.5.1          
## [28] parallel_4.4.2          cluster_2.1.6           R6_2.5.1               
## [31] bslib_0.8.0             RColorBrewer_1.1-3      SQUAREM_2021.1         
## [34] jquerylib_0.1.4         Rcpp_1.0.13-1           assertthat_0.2.1       
## [37] iterators_1.0.14        knitr_1.49              Matrix_1.7-1           
## [40] splines_4.4.2           tidyselect_1.2.1        viridis_0.6.5          
## [43] abind_1.4-8             yaml_2.3.10             doParallel_1.0.17      
## [46] codetools_0.2-20        lattice_0.22-6          tibble_3.2.1           
## [49] plyr_1.8.9              withr_3.0.2             evaluate_1.0.1         
## [52] survival_3.7-0          fitdistrplus_1.2-1      circlize_0.4.16        
## [55] pillar_1.9.0            BiocManager_1.30.25     checkmate_2.3.2        
## [58] foreach_1.5.2           vroom_1.6.5             invgamma_1.1           
## [61] truncnorm_1.0-9         ggplot2_3.5.1           munsell_0.5.1          
## [64] scales_1.3.0            ashr_2.2-63             glue_1.8.0             
## [67] maketools_1.3.1         tools_4.4.2             sys_3.4.3              
## [70] buildtools_1.0.0        mvtnorm_1.3-2           tidyr_1.3.1            
## [73] colorspace_2.1-1        mashr_0.2.79            GenomeInfoDbData_1.2.13
## [76] cli_3.6.3               fansi_1.0.6             viridisLite_0.4.2      
## [79] mixsqp_0.3-54           S4Arrays_1.7.1          dplyr_1.1.4            
## [82] gtable_0.3.6            sass_0.4.9              digest_0.6.37          
## [85] SparseArray_1.7.2       farver_2.1.2            rjson_0.2.23           
## [88] htmltools_0.5.8.1       lifecycle_1.0.4         rmeta_3.0              
## [91] httr_1.4.7              GlobalOptions_0.1.2     bit64_4.5.2            
## [94] MASS_7.3-61

multistateQTL: Orchestrating multi-state QTL analysis in R