Introduction

Here, we plot power curves under the assumptions of different sample sizes for an eQTL study. We also look at eQTL effect sizes from previously published response-eQTL studies to contextualize the power curves.

Power Curve derivations based on Abhishek Sarkar’s calculations: https://users.rcc.uchicago.edu/~aksarkar/hypoxia/reqtl.html

Load libraries

library(ggplot2)
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(tidyverse)

── Attaching packages ────────────────────────────────── tidyverse 1.3.0 ──

✓ tibble  3.0.4     ✓ purrr   0.3.4
✓ tidyr   1.1.2     ✓ stringr 1.4.0
✓ readr   1.3.1     ✓ forcats 0.4.0

── Conflicts ───────────────────────────────────── tidyverse_conflicts() ──
x dplyr::filter() masks stats::filter()
x dplyr::lag()    masks stats::lag()

library(ashr)

Plot a power curve

First, we need to specify the heritability for our model.

heritability <- 0.16 # median cis-heritability based on Gusev et al. 2016, Wheeler et al. 2016
lambda <- sqrt(heritability / (1-heritability))

Next, we make the plot for sample sizes ranging from 3 to 100. Here, vertical lines representing the average .99 quantile cdf eQTL effect sizes across conditions in each of the three external studies analyzed below are also plotted.

colors <- cbPalette <- c("#999999", "#E69F00", "#56B4E9", "#009E73", "#F0E442", "#0072B2", "#D55E00", "#CC79A7")
samp_size <- c(3, 10, 20, 40, 58, 100) # sample sizes for power curve
alpha <- 5e-6 #for FWER of 0.05 and a list of 10000 genes, we have an adjusted alpha (based on bonferroni correction) of 5e-6 

power_function <- function(x, n){ 
     pnorm(qnorm(alpha/2) + x * sqrt(n))
}

#The False positive rate for calling an eQTL discovered in one condition (out of two total condition) as a condition specific eQTL (a.k.a. a "response eQTL") is computed by this function.
FPR_function <- function(x, n){ 
     1 - pnorm(qnorm(alpha/2) + x * sqrt(n))
}

p <- ggplot(data.frame(x = c(0, 2)), aes(x = x)) +
     stat_function(fun = power_function, args = list(n = 3),
                      aes(colour = "3", linetype = "Power"), size = 1.5) +
     stat_function(fun = FPR_function, args = list(n = 3),
                      aes(colour = "3", linetype = "dynamic QTL FPR"), size = 1, alpha = 0.7) +
     stat_function(fun = power_function, args = list(n = 10),
                      aes(colour = "10", linetype = "Power"), size = 1.5) +
     stat_function(fun = FPR_function, args = list(n = 10),
                      aes(colour = "10", linetype = "dynamic QTL FPR"), size = 1, alpha = 0.7) +
     stat_function(fun = power_function, args = list(n = 30),
                      aes(colour = "30", linetype = "Power"), size = 1.5) +
     stat_function(fun = FPR_function, args = list(n = 30),
                      aes(colour = "30", linetype = "dynamic QTL FPR"), size = 1, alpha = 0.7) +
     stat_function(fun = power_function, args = list(n = 58),
                      aes(colour = "58", linetype = "Power"), size = 1.5) +
     stat_function(fun = FPR_function, args = list(n = 58),
                      aes(colour = "58", linetype = "dynamic QTL FPR"), size = 1, alpha = 0.7) +
     stat_function(fun = power_function, args = list(n = 100),
                      aes(colour = "100", linetype = "Power"), size = 1.5) +
     stat_function(fun = FPR_function, args = list(n = 100),
                      aes(colour = "100", linetype = "dynamic QTL FPR"), size = 1, alpha = 0.7) +
     
     # annotate("rect", xmin = 0.25340626, xmax = 0.2652244, ymin = -Inf, ymax = Inf, alpha = .2, fill = "blue") +
     annotate("segment", x = 0.2593153, xend = 0.2593153, y = -Inf, yend = Inf, alpha = 0.3, color = "blue", size = 1) + 
     # annotate("rect", xmin = 0.7232331, xmax = 0.8596644, ymin = -Inf, ymax = Inf, alpha = .2, fill = "purple") +
     annotate("segment", x = 0.793114125, xend = 0.793114125, y = -Inf, yend = Inf, alpha = 0.3, color = "purple", size = 1) + 
     #annotate("rect", xmin = 0.6035850, xmax = 0.6097121, ymin = -Inf, ymax = Inf, alpha = .2, fill = "brown") +
     annotate("segment", x = 0.605953025, xend = 0.605953025, y = -Inf, yend = Inf, alpha = 0.3, color = "brown", size = 1) + 
     # stat_function(fun = function(x) x^2/(1+x^2),
     #                  aes(colour = "heritability"), size = 1, alpha = 0.5) +
     scale_x_continuous(name = "Standardized Effect Size",
                              limits=c(0, 2)) +
     scale_y_continuous(name = "Power",
                           limits = c(0,1)) +
     ggtitle("eQTL Power Curves") +
     scale_colour_manual("Sample Size", breaks = c("3", "10", "30", "58", "100"), values = colors) +
     scale_linetype_manual("Curve Type", breaks = c("Power", "dynamic QTL FPR"), values = c("dynamic QTL FPR" = "dotted", "Power" = "solid")) +
     theme_bw() +
     geom_hline(yintercept = .8, linetype = "dashed", color = "red")
p

Past versions of curve-1.png

Import outside data on eQTLs to see distributions of standardized effect sizes

After plotting our power curve, we would like to get a sense of the range of standardized effect sizes in several previously published response-eQTL papers.

#This function creates an empirical cdf from a vector of effect sizes and prints out several quantiles of the ecdf
get_ecdf <- function(vector_of_SES){
     SES_ecdf <- ecdf(vector_of_SES)
     print(quantile(SES_ecdf, c(.75, .95, .99)))
     print(plot(SES_ecdf, ylim = c(-2,2)))
}

Ward et al hypoxia response eQTL study

#michelle hypoxia reQTL study
conditionA <- as_tibble(read.table('data/poweranalysis/ward_et_al_2020/A-adjust.txt.gz', fill = TRUE, stringsAsFactors = FALSE))
conditionB <- as_tibble(read.table('data/poweranalysis/ward_et_al_2020/B-adjust.txt.gz', fill = TRUE, stringsAsFactors = FALSE))
conditionC <- as_tibble(read.table('data/poweranalysis/ward_et_al_2020/C-adjust.txt.gz', fill = TRUE, stringsAsFactors = FALSE))
conditionD <- as_tibble(read.table('data/poweranalysis/ward_et_al_2020/D-adjust.txt.gz', fill = TRUE, stringsAsFactors = FALSE))

zz <- gzfile('data/poweranalysis/ward_et_al_2020/reqtls.txt.gz','rt')  
hypoxia <- as_tibble(read.table(zz, fill = TRUE, stringsAsFactors = FALSE))

names(hypoxia) <- hypoxia %>% 
     dplyr::slice(1) %>% 
     unlist()
hypoxia <- hypoxia %>% 
     dplyr::slice(-1)

## find SNP-gene pairs that were tested in all conditions and compute allelic effect sizes
common <- sort(Reduce(intersect, list(paste0(conditionA$TEST.SNP.CHROM, conditionA$TEST.SNP.POS, " ", conditionA$PHENO),
                           paste0(conditionB$TEST.SNP.CHROM, conditionB$TEST.SNP.POS, " ", conditionB$PHENO),
                           paste0(conditionC$TEST.SNP.CHROM, conditionC$TEST.SNP.POS, " ", conditionC$PHENO),
                           paste0(conditionD$TEST.SNP.CHROM, conditionD$TEST.SNP.POS, " ", conditionD$PHENO)
                           )))
length(common)

[1] 1040874

conditionA <- conditionA %>% 
     dplyr::mutate(temp = paste0(TEST.SNP.CHROM, TEST.SNP.POS, " ", PHENO)) %>% 
     dplyr::filter(temp %in% common) %>% 
     dplyr::mutate(slope = log(ALPHA/BETA))%>% 
     dplyr::mutate(zscore = sign(ALPHA - BETA) * sqrt(qchisq(p = p_adjusted + 1e-30, df = 1))) %>% 
     dplyr::select(-temp)
conditionB <- conditionB %>% 
     dplyr::mutate(temp = paste0(TEST.SNP.CHROM, TEST.SNP.POS, " ", PHENO)) %>% 
     dplyr::filter(temp %in% common) %>% 
     dplyr::mutate(slope = log(ALPHA/BETA))%>% 
     dplyr::mutate(zscore = sign(ALPHA - BETA) * sqrt(qchisq(p = p_adjusted + 1e-30, df = 1))) %>% 
     dplyr::select(-temp)
conditionC <- conditionC %>% 
     dplyr::mutate(temp = paste0(TEST.SNP.CHROM, TEST.SNP.POS, " ", PHENO)) %>% 
     dplyr::filter(temp %in% common) %>% 
     dplyr::mutate(slope = log(ALPHA/BETA))%>% 
     dplyr::mutate(zscore = sign(ALPHA - BETA) * sqrt(qchisq(p = p_adjusted + 1e-30, df = 1))) %>% 
     dplyr::select(-temp)
conditionD <- conditionD %>% 
     dplyr::mutate(temp = paste0(TEST.SNP.CHROM, TEST.SNP.POS, " ", PHENO)) %>% 
     dplyr::filter(temp %in% common) %>% 
     dplyr::mutate(slope = log(ALPHA/BETA))%>% 
     dplyr::mutate(zscore = sign(ALPHA - BETA) * sqrt(qchisq(p = p_adjusted + 1e-30, df = 1))) %>% 
     dplyr::select(-temp)
dim(conditionA)

[1] 1040874      22

dim(conditionB)

[1] 1040874      22

dim(conditionC)

[1] 1040874      22

dim(conditionD)

[1] 1040874      22

# For each condiiton in the study, compute the empirical CDF and quantiles after obtaining the effect sizes by dividing the zscore by the sqrt of the number of individuals in the study (15).
#conditionA
get_ecdf(conditionA$zscore/sqrt(15))

      75%       95%       99% 
0.1552778 0.4150685 0.6053753 

Past versions of hypoxia-1.png

NULL

#conditionB
get_ecdf(conditionB$zscore/sqrt(15))

      75%       95%       99% 
0.1572876 0.4167222 0.6051397 

Past versions of hypoxia-2.png

NULL

#conditionC
get_ecdf(conditionC$zscore/sqrt(15))

      75%       95%       99% 
0.1564676 0.4154604 0.6035850 

Past versions of hypoxia-3.png

NULL

#conditionD
get_ecdf(conditionD$zscore/sqrt(15))

      75%       95%       99% 
0.1582968 0.4196211 0.6097121 

Past versions of hypoxia-4.png

NULL

cd data/poweranalysis/alasoo_et_al/

#load in data from eQTL analysis downloaded using wget from zenodo 
wget https://zenodo.org/record/1158560/files/RNA_FastQTL_SL1344_500kb_pvalues.sorted.txt.gz  
wget https://zenodo.org/record/1158560/files/RNA_FastQTL_naive_500kb_pvalues.sorted.txt.gz 
wget https://zenodo.org/record/1158560/files/RNA_FastQTL_IFNg_SL1344_500kb_pvalues.sorted.txt.gz 
wget https://zenodo.org/record/1158560/files/RNA_FastQTL_IFNg_500kb_pvalues.sorted.txt.gz 

# load data into R
n_sorted <- as_tibble(read.table("data/poweranalysis/alasoo_et_al/RNA_FastQTL_naive_500kb_pvalues.sorted.txt.gz"))
sal_sorted <- as_tibble(read.table("data/poweranalysis/alasoo_et_al/RNA_FastQTL_SL1344_500kb_pvalues.sorted.txt.gz"))
IFN_sorted <- as_tibble(read.table("data/poweranalysis/alasoo_et_al/RNA_FastQTL_IFNg_500kb_pvalues.sorted.txt.gz"))
salIFN_sorted <- as_tibble(read.table("data/poweranalysis/alasoo_et_al/RNA_FastQTL_IFNg_SL1344_500kb_pvalues.sorted.txt.gz"))

#for sorted data, 
#V1 = gene 
#V2 = chromosome
#V3 = location of SNP
#V4 = SNP
#V5 = distance to gene from SNP?
#V6 = p-value
#v7 = slope of regression

#We can use the p-value to obtain the zscore (pvalue -> Zscore)
#then use slopes and zscores to obtain SE (slope/zscore = SE)
n_sorted <- n_sorted %>% 
     dplyr::filter(V6 != 1) %>% 
     dplyr::mutate(zscore = if_else(V7 < 0, true = qnorm(V6/2), false = qnorm(1-V6/2))) %>% 
     dplyr::mutate(SE = V7/zscore)
sal_sorted <- sal_sorted %>% 
     dplyr::filter(V6 != 1) %>% 
     dplyr::mutate(zscore = if_else(V7 < 0, true = qnorm(V6/2), false = qnorm(1-V6/2))) %>% 
     dplyr::mutate(SE = V7/zscore)
IFN_sorted <- IFN_sorted %>% 
     dplyr::filter(V6 != 1) %>% 
     dplyr::mutate(zscore = if_else(V7 < 0, true = qnorm(V6/2), false = qnorm(1-V6/2))) %>% 
     dplyr::mutate(SE = V7/zscore)
salIFN_sorted <- salIFN_sorted %>% 
     dplyr::filter(V6 != 1) %>% 
     dplyr::mutate(zscore = if_else(V7 < 0, true = qnorm(V6/2), false = qnorm(1-V6/2))) %>% 
     dplyr::mutate(SE = V7/zscore)


# use ashr to estimate true effect sizes from the data in order to compute SES
n_sorted_ash_out <- ash(betahat = n_sorted$V7, sebetahat = n_sorted$SE, df = 84)
sal_sorted_ash_out <- ash(betahat = sal_sorted$V7, sebetahat = sal_sorted$SE, df = 84)
IFN_sorted_ash_out <- ash(betahat = IFN_sorted$V7, sebetahat = IFN_sorted$SE, df = 84)
salIFN_sorted_ash_out <- ash(betahat = salIFN_sorted$V7, sebetahat = salIFN_sorted$SE, df = 84)

#add columns to datasets corresponding to the SES computed from ash betas and sds
n_sorted$SES <- n_sorted_ash_out$result$PosteriorMean/n_sorted_ash_out$result$PosteriorSD
sal_sorted$SES <- sal_sorted_ash_out$result$PosteriorMean/sal_sorted_ash_out$result$PosteriorSD
IFN_sorted$SES <- IFN_sorted_ash_out$result$PosteriorMean/IFN_sorted_ash_out$result$PosteriorSD
salIFN_sorted$SES <- salIFN_sorted_ash_out$result$PosteriorMean/salIFN_sorted_ash_out$result$PosteriorSD

#Run the ecdf function separately on each of the different conditions
#naive
get_ecdf(n_sorted$SES)

      75%       95%       99% 
0.1606588 0.4507322 0.8475118 

NULL

get_ecdf(n_sorted_ash_out$result$PosteriorMean)

        75%         95%         99% 
0.002599666 0.010322366 0.042398086 

NULL

#salmonella condition
get_ecdf(sal_sorted$SES)

      75%       95%       99% 
0.1341273 0.3958948 0.7232331 

NULL

#IFN treated condition
get_ecdf(IFN_sorted$SES)

      75%       95%       99% 
0.1537379 0.4549502 0.8596644 

NULL

#salmonella and ifn treated condition
get_ecdf(salIFN_sorted$SES)

      75%       95%       99% 
0.1370046 0.4083029 0.7420472 

NULL

rhinovirus_data <- as.tibble(read.table("data/poweranalysis/caliskan_et_al/eQTL_results_cis_response.txt.gz", 
                              sep = "\t",
                              header = TRUE))

Warning: `as.tibble()` is deprecated as of tibble 2.0.0.
Please use `as_tibble()` instead.
The signature and semantics have changed, see `?as_tibble`.
This warning is displayed once every 8 hours.
Call `lifecycle::last_warnings()` to see where this warning was generated.

#obtain standard errors by using beta and t.stat
rhinovirus_data <- rhinovirus_data %>% 
     dplyr::mutate(se = beta/t.stat)

#add column to denote which condition the association comes from; separate into two datasets
rhinovirus_data <- rhinovirus_data %>% 
     mutate(condition = ifelse(grepl("_f", SNP), "f", "r"))

rhinovirus_data_f <- rhinovirus_data %>% 
     dplyr::filter(condition == "f")

rhinovirus_data_r <- rhinovirus_data %>% 
     dplyr::filter(condition == "r")

# use ashr to estimate true effect sizes from the data in order to compute SES (separately for each half of the data)
rhinovirus_f_ashr_out <- ash(betahat = rhinovirus_data_f$beta, sebetahat = rhinovirus_data_f$se, df = 96)
rhinovirus_r_ashr_out <- ash(betahat = rhinovirus_data_r$beta, sebetahat = rhinovirus_data_r$se, df = 96)

#add column for SES (based on ash output)
rhinovirus_data_f$SES <- rhinovirus_f_ashr_out$result$PosteriorMean/rhinovirus_f_ashr_out$result$PosteriorSD
rhinovirus_data_r$SES <- rhinovirus_r_ashr_out$result$PosteriorMean/rhinovirus_r_ashr_out$result$PosteriorSD

#Run the ecdf function separately on each of the different conditions
#uninfected
get_ecdf(rhinovirus_data_f$SES)

      75%       95%       99% 
0.0486464 0.1414978 0.2271107 

NULL

get_ecdf(rhinovirus_f_ashr_out$result$PosteriorMean)

         75%          95%          99% 
0.0001144159 0.0003825204 0.0007927461 

NULL

#rhinovirus
get_ecdf(rhinovirus_data_r$SES)

      75%       95%       99% 
0.0540109 0.1631119 0.2652244 

NULL

get_ecdf(rhinovirus_r_ashr_out$result$PosteriorMean)

         75%          95%          99% 
0.0001393432 0.0004605828 0.0009016914 

NULL

Number of genes meeting the SES threshold

For the supplementary table, we are interested in the number of eQTLs that meet the SES threshold in each study for each sample size.

# Compute the SES thresholds for 80% power at each sample size
alpha <- 5e-6 #for FWER of 0.05 (assuming 10,000 genes)
#for Power = 0.8, what is the Standardized effect size?
#n = 100
n100_ses_threshold <- (qnorm(.8) - qnorm(alpha/2))/sqrt(100)
#n = 58
n58_ses_threshold <- (qnorm(.8) - qnorm(alpha/2))/sqrt(58)
#n = 30
n30_ses_threshold <- (qnorm(.8) - qnorm(alpha/2))/sqrt(30)
#n = 10
n10_ses_threshold <- (qnorm(.8) - qnorm(alpha/2))/sqrt(10)

#Ward et al study
# See how many genes have at least one test that meets each of the 3 SES thresholds determined above in each condition. 
n_egenes_ward <- matrix(NA, nrow = 4, ncol = 4)
colnames(n_egenes_ward) <- c("n=100", "n=58", "n=30", "n=10")
rownames(n_egenes_ward) <- c("conditionA", "conditionB", "conditionC", "conditionD")
i <- 1
for(condition in list(conditionA, conditionB, conditionC, conditionD)){
     j <- 1
     for(threshold in c(n100_ses_threshold, n58_ses_threshold, n30_ses_threshold, n10_ses_threshold)){
         n_egenes_ward[i,j] <- length(unique(condition %>% dplyr::filter(zscore/sqrt(15) > threshold | zscore/sqrt(15) < -threshold) %>% pull(PHENO)))
         j <- j + 1
     }
     i <- i + 1
}


#Alasoo et al study
# See how many genes have at least one test that meets each of the 3 SES thresholds determined above in each condition. 
alpha <- 0.05/15786 #for FWER of 0.05 (assuming 15786 genes)
#for Power = 0.8, what is the SES?
#n = 100
n100_ses_threshold <- (qnorm(.8) - qnorm(alpha/2))/sqrt(100)
#n = 58
n58_ses_threshold <- (qnorm(.8) - qnorm(alpha/2))/sqrt(58)
#n = 30
n30_ses_threshold <- (qnorm(.8) - qnorm(alpha/2))/sqrt(30)
#n = 10
n10_ses_threshold <- (qnorm(.8) - qnorm(alpha/2))/sqrt(10)

n_egenes_alasoo <- matrix(NA, nrow = 4, ncol = 4)
colnames(n_egenes_alasoo) <- c("n=100", "n=58", "n=30", "n=10")
rownames(n_egenes_alasoo) <- c("naive", "IFN", "salmonella", "IFN + salmonella")
i <- 1
for(condition in list(n_sorted, IFN_sorted, sal_sorted, salIFN_sorted)){
     j <- 1
     for(threshold in c(n100_ses_threshold, n58_ses_threshold, n30_ses_threshold, n10_ses_threshold)){
         n_egenes_alasoo[i,j] <- length(unique(condition %>% dplyr::filter(SES > threshold | SES < -threshold) %>% pull(V1)))
         j <- j + 1
     }
     i <- i + 1
}


#Caliskan et al study
# See how many genes have at least one test that meets each of the 3 SES thresholds determined above in each condition. 
alpha <- 0.05/length(unique(rhinovirus_data_f$gene)) #for FWER of 0.05 (assuming 10893 genes)
#for Power = 0.8, what is the SES?
#n = 100
n100_ses_threshold <- (qnorm(.8) - qnorm(alpha/2))/sqrt(100)
#n = 58
n58_ses_threshold <- (qnorm(.8) - qnorm(alpha/2))/sqrt(58)
#n = 30
n30_ses_threshold <- (qnorm(.8) - qnorm(alpha/2))/sqrt(30)
#n = 10
n10_ses_threshold <- (qnorm(.8) - qnorm(alpha/2))/sqrt(10)

n_egenes_caliskan <- matrix(NA, nrow = 2, ncol = 4)
colnames(n_egenes_caliskan) <- c("n=100", "n=58", "n=30", "n=10")
rownames(n_egenes_caliskan) <- c("f", "r")
i <- 1
for(condition in list(rhinovirus_data_f, rhinovirus_data_r)){
     j <- 1
     for(threshold in c(n100_ses_threshold, n58_ses_threshold, n30_ses_threshold, n10_ses_threshold)){
         n_egenes_caliskan[i,j] <- length(unique(condition %>% dplyr::filter(SES > threshold | SES < -threshold) %>% pull(gene)))
         j <- j + 1
     }
     i <- i + 1
}

#These values populate the supplementary table
n_egenes_ward

           n=100 n=58 n=30 n=10
conditionA  7140 2854  151    0
conditionB  7171 2867  157    0
conditionC  7119 2846  168    1
conditionD  7179 2960  165    0

n_egenes_alasoo

                 n=100  n=58 n=30 n=10
naive            13291 11063 7621 3548
IFN              12704 10594 7379 3442
salmonella       11518  9128 5991 2752
IFN + salmonella 11518  9184 5931 2624

n_egenes_caliskan

  n=100 n=58 n=30 n=10
f   289   93   41   24
r   326  105   44   24

Session information

sessionInfo()

R version 3.6.1 (2019-07-05)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Scientific Linux 7.4 (Nitrogen)

Matrix products: default
BLAS/LAPACK: /software/openblas-0.2.19-el7-x86_64/lib/libopenblas_haswellp-r0.2.19.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] ashr_2.2-47     forcats_0.4.0   stringr_1.4.0   purrr_0.3.4    
 [5] readr_1.3.1     tidyr_1.1.2     tibble_3.0.4    tidyverse_1.3.0
 [9] dplyr_1.0.2     ggplot2_3.3.3  

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.5       lubridate_1.7.9  invgamma_1.1     lattice_0.20-41 
 [5] assertthat_0.2.1 rprojroot_2.0.2  digest_0.6.27    truncnorm_1.0-8 
 [9] R6_2.5.0         cellranger_1.1.0 backports_1.1.10 reprex_0.3.0    
[13] evaluate_0.14    httr_1.4.2       pillar_1.4.7     rlang_0.4.10    
[17] readxl_1.3.1     rstudioapi_0.13  irlba_2.3.3      whisker_0.3-2   
[21] Matrix_1.2-18    rmarkdown_1.13   labeling_0.4.2   munsell_0.5.0   
[25] mixsqp_0.3-43    broom_0.7.0      compiler_3.6.1   httpuv_1.5.1    
[29] modelr_0.1.8     xfun_0.8         pkgconfig_2.0.3  etrunct_0.1     
[33] SQUAREM_2020.4   htmltools_0.5.0  tidyselect_1.1.0 workflowr_1.6.2 
[37] fansi_0.4.1      crayon_1.3.4     dbplyr_1.4.2     withr_2.3.0     
[41] later_1.1.0.1    grid_3.6.1       jsonlite_1.7.2   gtable_0.3.0    
[45] lifecycle_0.2.0  DBI_1.1.0        git2r_0.26.1     magrittr_2.0.1  
[49] scales_1.1.1     cli_2.2.0        stringi_1.4.6    farver_2.0.3    
[53] fs_1.3.1         promises_1.1.1   xml2_1.3.2       ellipsis_0.3.1  
[57] generics_0.0.2   vctrs_0.3.6      tools_3.6.1      glue_1.4.2      
[61] hms_0.5.3        yaml_2.2.1       colorspace_2.0-0 rvest_0.3.6     
[65] knitr_1.23       haven_2.3.1