Bayes Factor comparison

Last updated: 2024-11-20

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Knit directory: survival-susie/

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Description:

Compare different types of Bayes factor under different levels of censoring, minor allele frequency, effect size and sample size.

censoring = 0.00, 0.20, 0.40, 0.60, 0.80, 0.99
MAF = 0.001, 0.01, 0.1
n = 5000, 50000, 500000
b1 = -1， -0.1, -0.01, 0.01, 0.1, 1
BFs:

Numerical BF using Gaussian quadrature, our gold standard. https://yunqiyang0215.github.io/survival-susie/gauss_quad.html
Laplace-MLE BF
Wakefeld BF
Wakefeld BF with zscore from coxph() corrected by saddle point approximation. https://yunqiyang0215.github.io/survival-susie/SPAcox.html

library(latex2exp)

res = readRDS("/project2/mstephens/yunqiyang/surv-susie/sim202406/bf_comparison0712/res_bf.rds")
dat_list = lapply(1:length(res$DSC), function(x) res$compute_bf.result[[x]])
dat = do.call(rbind, dat_list)
dat2 = cbind(res$simulate.censor_lvl, res$simulate.allele_freq, res$simulate.n, 
             res$simulate.b1, dat)
colnames(dat2)[1:4] = c("censor_lvl", "allele_freq", "n", "b1")
dat2 = as.data.frame(dat2)

censor_lvl = unique(res$simulate.censor_lvl)
af = unique(res$simulate.allele_freq)
print(censor_lvl)

[1] 0.00 0.20 0.40 0.60 0.80 0.99

print(af)

[1] 0.001 0.010 0.100 0.500

1.1 Plot BF difference

par(mfrow = c(6,4))
for (i in 1:length(censor_lvl)){
  for (j in 1:length(af)){
    indx = which(dat2$censor_lvl == censor_lvl[i] & dat2$allele_freq == af[j] & dat2$b1 == 1)
    diff1 = dat2$lbf.numerical - dat2$lbf.laplace
    diff2 = dat2$lbf.numerical - dat2$lbf.wakefeld
    ymin = min(diff1[indx], diff2[indx]) - 1
    ymax = max(diff1[indx], diff2[indx]) + 1
    plot(dat2$n[indx], diff1[indx], ylim = c(ymin, ymax),
         main = paste0("censor=",censor_lvl[i], ", MAF=", af[j]),
         xlab = "sample size", ylab = "difference in log(BF)", cex = 0.8, pch = 20)
    points(dat2$n[indx], diff2[indx], col = "#00CCFF", cex = 0.8, pch = 20)
    fit = lm(diff2[indx] ~ dat2$n[indx])
    abline(fit, col = "grey", lty = 2) 
    legend("bottomleft", legend = c("log(BF.lp)", 'log(BF.wf)'), col = c(1, "#00CCFF"), pch = 20, cex = 0.8)
  }
}

Version	Author	Date
c219afc	yunqi yang	2024-10-11

par(mfrow = c(6,4))
for (i in 1:length(censor_lvl)){
  for (j in 1:length(af)){
    indx = which(dat2$censor_lvl == censor_lvl[i] & dat2$allele_freq == af[j] & dat2$b1 == 0.01)
    diff1 = dat2$lbf.numerical - dat2$lbf.laplace
    diff2 = dat2$lbf.numerical - dat2$lbf.wakefeld
    ymin = min(diff1[indx], diff2[indx]) - 1
    ymax = max(diff1[indx], diff2[indx]) + 1
    plot(dat2$n[indx], diff1[indx], ylim = c(ymin, ymax),
         main = paste0("censor=",censor_lvl[i], ", MAF=", af[j]),
         xlab = "sample size", ylab = "difference in log(BF)", cex = 0.8, pch = 20)
    points(dat2$n[indx], diff2[indx], col = "#00CCFF", cex = 0.8, pch = 20)
    fit = lm(diff2[indx] ~ dat2$n[indx])
    abline(fit, col = "grey", lty = 2) 
    legend("bottomleft", legend = c("log(BF.lp)", 'log(BF.wf)'), col = c(1, "#00CCFF"), pch = 20, cex = 0.8)
  }
}

Version	Author	Date
c219afc	yunqi yang	2024-10-11

# Sample size small 
par(mfrow = c(6,4))
for (i in 1:length(censor_lvl)){
  for (j in 1:length(af)){
    indx = which(dat2$censor_lvl == censor_lvl[i] & dat2$allele_freq == af[j] & dat2$n == 5e3)
    diff1 = dat2$lbf.numerical - dat2$lbf.laplace
    diff2 = dat2$lbf.numerical - dat2$lbf.wakefeld
    ymin = min(diff1[indx], diff2[indx]) - 0.5
    ymax = max(diff1[indx], diff2[indx]) + 0.5
    plot(dat2$bhat[indx], diff1[indx], ylim = c(ymin, ymax), xlab = "bhat", ylab = "difference in log(BF)",
         main = paste0("censor=",censor_lvl[i], ", MAF=", af[j]), cex = 0.8, pch = 20)
    points(dat2$bhat[indx], diff2[indx], col = "#00CCFF", cex = 0.8, pch = 20)
    legend("topleft", legend = c("log(BF.lp)", 'log(BF.wf)'), col = c(1, "#00CCFF"), pch = 20, cex = 0.8)
  }
}

Version	Author	Date
c219afc	yunqi yang	2024-10-11

# Large n
par(mfrow = c(6,4))
for (i in 1:length(censor_lvl)){
  for (j in 1:length(af)){
    indx = which(dat2$censor_lvl == censor_lvl[i] & dat2$allele_freq == af[j] & dat2$n == 5e5)
    diff1 = dat2$lbf.numerical - dat2$lbf.laplace
    diff2 = dat2$lbf.numerical - dat2$lbf.wakefeld
    ymin = min(diff1[indx], diff2[indx]) - 0.5
    ymax = max(diff1[indx], diff2[indx]) + 0.5
    plot(dat2$bhat[indx], diff1[indx], ylim = c(ymin, ymax), xlab = "bhat", ylab = "difference in log(BF)",
         main = paste0("censor=",censor_lvl[i], ", MAF=", af[j]), cex = 0.8, pch = 20)
    points(dat2$bhat[indx], diff2[indx], col = "#00CCFF", cex = 0.8, pch = 20)
    #legend("topright", legend = c("log(BF.lp)", 'log(BF.wf)'), col = c(1, "#00CCFF"), pch = 20, cex = 0.8)
  }
}

Version	Author	Date
c219afc	yunqi yang	2024-10-11
8a1ddda	yunqi yang	2024-10-07

1.2 Plot different BFs across N/bhat

# Setting: b1 = 1
par(mfrow = c(6,4))
for (i in 1:length(censor_lvl)){
  for (j in 1:length(af)){
    indx = which(dat2$censor_lvl == censor_lvl[i] & dat2$allele_freq == af[j] & dat2$b1 == 1)
    ymin = min(dat2$lbf.numerical[indx], dat2$lbf.laplace[indx], dat2$lbf.wakefeld[indx]) - 1
    ymax = max(dat2$lbf.numerical[indx], dat2$lbf.laplace[indx], dat2$lbf.wakefeld[indx]) + 1
    plot(log10(dat2$n[indx]), dat2$lbf.numerical[indx], ylim = c(ymin, ymax),
         main = paste0("censor=",censor_lvl[i], ", MAF=", af[j]),
         xlab = "log10(sample size)", ylab = "log_10(BF)s", cex = 0.8, pch = 20)
    points(log10(dat2$n[indx])-0.03, dat2$lbf.laplace[indx], col = "#00CCFF", cex = 0.8, pch = 20)
    points(log10(dat2$n[indx])+0.03, dat2$lbf.wakefeld[indx], col = 2, cex = 0.8, pch = 20)
    legend("topleft", legend = c("log(BF.numerical)", "log(BF.lp)", 'log(BF.wf)'), col = c(1, "#00CCFF", 2), pch = 20, cex = 0.8)
  }
}

Version	Author	Date
c219afc	yunqi yang	2024-10-11
8a1ddda	yunqi yang	2024-10-07

# Setting: b1 = 0.1
par(mfrow = c(6,4))
for (i in 1:length(censor_lvl)){
  for (j in 1:length(af)){
    indx = which(dat2$censor_lvl == censor_lvl[i] & dat2$allele_freq == af[j] & dat2$b1 == 0.1)
    ymin = min(dat2$lbf.numerical[indx], dat2$lbf.laplace[indx], dat2$lbf.wakefeld[indx]) - 1
    ymax = max(dat2$lbf.numerical[indx], dat2$lbf.laplace[indx], dat2$lbf.wakefeld[indx]) + 1
    plot(log10(dat2$n[indx]), dat2$lbf.numerical[indx], ylim = c(ymin, ymax),
         main = paste0("censor=",censor_lvl[i], ", MAF=", af[j]),
         xlab = "log10(sample size)", ylab = "log_10(BF)s", cex = 0.8, pch = 20)
    points(log10(dat2$n[indx])-0.03, dat2$lbf.laplace[indx], col = "#00CCFF", cex = 0.8, pch = 20)
    points(log10(dat2$n[indx])+0.03, dat2$lbf.wakefeld[indx], col = 2, cex = 0.8, pch = 20)
    legend("topleft", legend = c("log(BF.numerical)", "log(BF.lp)", 'log(BF.wf)'), col = c(1, "#00CCFF", 2), pch = 20, cex = 0.8)
  }
}

Version	Author	Date
c219afc	yunqi yang	2024-10-11
8a1ddda	yunqi yang	2024-10-07

# Setting: b1 = 0.01
par(mfrow = c(6,4))
for (i in 1:length(censor_lvl)){
  for (j in 1:length(af)){
    indx = which(dat2$censor_lvl == censor_lvl[i] & dat2$allele_freq == af[j] & dat2$b1 == 0.01)
    ymin = min(dat2$lbf.numerical[indx], dat2$lbf.laplace[indx], dat2$lbf.wakefeld[indx]) - 1
    ymax = max(dat2$lbf.numerical[indx], dat2$lbf.laplace[indx], dat2$lbf.wakefeld[indx]) + 1
    plot(log10(dat2$n[indx]), dat2$lbf.numerical[indx], ylim = c(ymin, ymax),
         main = paste0("censor=",censor_lvl[i], ", MAF=", af[j]),
         xlab = "log10(sample size)", ylab = "log_10(BF)s", cex = 0.8, pch = 20)
    points(log10(dat2$n[indx])-0.03, dat2$lbf.laplace[indx], col = "#00CCFF", cex = 0.8, pch = 20)
    points(log10(dat2$n[indx])+0.03, dat2$lbf.wakefeld[indx], col = 2, cex = 0.8, pch = 20)
    legend("topleft", legend = c("log(BF.numerical)", "log(BF.lp)", 'log(BF.wf)'), col = c(1, "#00CCFF", 2), pch = 20, cex = 0.8)
  }
}

Version	Author	Date
c219afc	yunqi yang	2024-10-11
8a1ddda	yunqi yang	2024-10-07

# Sample size small 
par(mfrow = c(6,4))
for (i in 1:length(censor_lvl)){
  for (j in 1:length(af)){
    indx = which(dat2$censor_lvl == censor_lvl[i] & dat2$allele_freq == af[j] & dat2$n == 5e3)
    ymin = min(dat2$lbf.numerical[indx], dat2$lbf.laplace[indx], dat2$lbf.wakefeld[indx]) - 1
    ymax = max(dat2$lbf.numerical[indx], dat2$lbf.laplace[indx], dat2$lbf.wakefeld[indx]) + 1
    plot(dat2$bhat[indx], dat2$lbf.numerical[indx], ylim = c(ymin, ymax),
         main = paste0("censor=",censor_lvl[i], ", MAF=", af[j]),
         xlab = "bhat", ylab = "log_10(BF)s", cex = 0.8, pch = 20)
    points(dat2$bhat[indx], dat2$lbf.laplace[indx], col = "#00CCFF", cex = 0.8, pch = 20)
    points(dat2$bhat[indx], dat2$lbf.wakefeld[indx], col = 2, cex = 0.8, pch = 20)
    legend("bottomright", legend = c("log(BF.numerical)", "log(BF.lp)", 'log(BF.wf)'), col = c(1, "#00CCFF", 2), pch = 20, cex = 0.8)
  }
}

Version	Author	Date
410628d	yunqi yang	2024-10-11
c219afc	yunqi yang	2024-10-11

# Large n
par(mfrow = c(6,4))
for (i in 1:length(censor_lvl)){
  for (j in 1:length(af)){
    indx = which(dat2$censor_lvl == censor_lvl[i] & dat2$allele_freq == af[j] & dat2$n == 5e5)
    ymin = min(dat2$lbf.numerical[indx], dat2$lbf.laplace[indx], dat2$lbf.wakefeld[indx]) - 1
    ymax = max(dat2$lbf.numerical[indx], dat2$lbf.laplace[indx], dat2$lbf.wakefeld[indx]) + 1
    plot(dat2$bhat[indx], dat2$lbf.numerical[indx], ylim = c(ymin, ymax),
         main = paste0("censor=",censor_lvl[i], ", MAF=", af[j]),
         xlab = "bhat", ylab = "log_10(BF)s", cex = 0.8, pch = 20)
    points(dat2$bhat[indx], dat2$lbf.laplace[indx], col = "#00CCFF", cex = 0.8, pch = 20)
    points(dat2$bhat[indx], dat2$lbf.wakefeld[indx], col = 2, cex = 0.8, pch = 20)
    legend("bottomright", legend = c("log(BF.numerical)", "log(BF.lp)", 'log(BF.wf)'), col = c(1, "#00CCFF", 2), pch = 20, cex = 0.8)
  }
}

Version	Author	Date
410628d	yunqi yang	2024-10-11
c219afc	yunqi yang	2024-10-11

1.3 Plot numerical BF against wakefield/laplace BF

pdf("/project2/mstephens/yunqiyang/surv-susie/survival-susie/output/bf_comparison_b_1_n_500000.pdf", width = 8, height = 12)
par(mfrow = c(4,3))
censor_lvl <- c(0, 0.4, 0.8, 0.99)
af <- c(0.001, 0.010, 0.100)
for (i in 1:length(censor_lvl)){
  for (j in 1:length(af)){
    indx = which(res$simulate.censor_lvl == censor_lvl[i] & res$simulate.allele_freq == af[j]
                 & res$simulate.n == 5e5 & res$simulate.b1 == 1)
    dat = t(sapply(indx, function(x) res$compute_bf.result[[x]]))
    plot(x = dat[, 'lbf.numerical'], y = dat[, 'lbf.laplace'], pch = 20, main = paste0("censor=",censor_lvl[i], ", MAF=", af[j]),
         ylim = c(min(dat[,c(4:6)])-1, max(dat[, c(4:6)])+1), cex = 0.8, col = "blue",
         xlab = TeX("$log_{10}$(BF numerical)"), ylab = TeX("$log_{10}$($BF_{LM}$)"))
    abline(a = 0, b = 1, col = "grey")
    #legend("topleft", legend = c(TeX("$log_{10}$(BF.lp)"), TeX("$log_{10}$(BF.wf)")), col = c(1, "#26B300"), pch = 20, cex = 0.8)
  }
}

pdf("/project2/mstephens/yunqiyang/surv-susie/survival-susie/output/bf_comparison_b_0.1_n_500000.pdf", width = 8, height = 12)
par(mfrow = c(4,3))
censor_lvl <- c(0, 0.4, 0.8, 0.99)
af <- c(0.001, 0.010, 0.100)
for (i in 1:length(censor_lvl)){
  for (j in 1:length(af)){
    indx = which(res$simulate.censor_lvl == censor_lvl[i] & res$simulate.allele_freq == af[j]
                 & res$simulate.n == 5e5 & res$simulate.b1 == 0.1)
    dat = t(sapply(indx, function(x) res$compute_bf.result[[x]]))
    plot(x = dat[, 'lbf.numerical'], y = dat[, 'lbf.laplace'], pch = 20, main = paste0("censor=",censor_lvl[i], ", MAF=", af[j]),
         ylim = c(min(dat[,c(4:6)])-1, max(dat[, c(4:6)])+1), cex = 0.8,
         xlab = TeX("$log_{10}(BF^{GH})$"), ylab = TeX("Other $log_{10}$(BFs)"))
    points(dat[, 'lbf.numerical'], dat[,'lbf.wakefeld'], col = "#26B300", pch = 20, cex = 0.8)
    abline(a = 0, b = 1, col = "grey")
    legend("topleft", legend = c(TeX("$log_{10}(\\hat{BF})$"), TeX("$log_{10}(ABF)$")), col = c(1, "#26B300"), pch = 20, cex = 0.8)
  }
}

pdf("/project2/mstephens/yunqiyang/surv-susie/survival-susie/output/bf_comparison_b_0.01_n_500000.pdf", width = 8, height = 18)
par(mfrow = c(6,3))

for (i in 1:length(censor_lvl)){
  for (j in 1:length(af)){
    indx = which(res$simulate.censor_lvl == censor_lvl[i] & res$simulate.allele_freq == af[j]
                 & res$simulate.n == 5e5 & res$simulate.b1 == 0.01)
    dat = t(sapply(indx, function(x) res$compute_bf.result[[x]]))
    plot(x = dat[, 'lbf.numerical'], y = dat[, 'lbf.laplace'], pch = 20, main = paste0("censor=",censor_lvl[i], ", MAF=", af[j]),
         ylim = c(min(dat[,c(4:6)])-1, max(dat[,c(4:6)])+1), cex = 0.8,
         xlab = TeX("$log_{10}(BF^{GH})$"), ylab = TeX("Other $log_{10}$(BFs)"))
    points(dat[, 'lbf.numerical'], dat[,'lbf.wakefeld'], col = "#26B300", pch = 20, cex = 0.8)
    abline(a = 0, b = 1, col = "grey")
    legend("topleft", legend = c(TeX("$log_{10}(BF^L)$"), TeX("$log_{10}(BF^W)$")), col = c(1, "#26B300"), pch = 20, cex = 0.8)
  }
}

1. Check BF

dat.sub = dat2[dat2$censor_lvl == 0 & dat2$n == 5e5 & dat2$allele_freq == 0.1, ]

par(mfrow = c(2,2))
hist(dat.sub$zscore[dat.sub$b1 == -1], xlab = "zscore", main = "true b = -1; MAF = 0.1; n = 5e5; no censor")
hist(dat.sub$zscore[dat.sub$b1 == 1],  xlab = "zscore", main = "true b = 1; MAF = 0.1; n = 5e5; no censor")
hist(dat.sub$bhat[dat.sub$b1 == -1], xlab = "bhat", main = "true b = -1; MAF = 0.1; n = 5e5; no censor")
hist(dat.sub$bhat[dat.sub$b1 == 1], xlab = "bhat", main = "true b = 1; MAF = 0.1; n = 5e5; no censor")

Version	Author	Date
c219afc	yunqi yang	2024-10-11
8a1ddda	yunqi yang	2024-10-07
6445624	yunqi yang	2024-09-18
5b1f3bf	yunqi yang	2024-09-15
546c2bf	yunqi yang	2024-08-26

par(mfrow = c(2,2))
hist(dat.sub$lbf.numerical[dat.sub$b1 == -1],  xlab = "log(BF.numerical)", main = "true b = -1; MAF = 0.1; n = 5e5; no censor")
hist(dat.sub$lbf.numerical[dat.sub$b1 == 1],  xlab = "log(BF.numerical)", main = "true b = 1; MAF = 0.1; n = 5e5; no censor")
hist(dat.sub$lbf.wakefeld[dat.sub$b1 == -1],  xlab = "log(BF.wakefield)", main = "true b = -1; MAF = 0.1; n = 5e5; no censor")
hist(dat.sub$lbf.wakefeld[dat.sub$b1 == 1],  xlab = "log(BF.wakefield)", main = "true b = 1; MAF = 0.1; n = 5e5; no censor")

Version	Author	Date
c219afc	yunqi yang	2024-10-11
8a1ddda	yunqi yang	2024-10-07
6445624	yunqi yang	2024-09-18
5b1f3bf	yunqi yang	2024-09-15
546c2bf	yunqi yang	2024-08-26

2. Check BF when flip binary x.

I experimented with two different sample sizes, n = 5e3 and 5e5.

res = readRDS("/project2/mstephens/yunqiyang/surv-susie/sim202406/bf_comparison0825/res_bf.rds")

dat_list = lapply(1:length(res$DSC), function(x) res$compute_bf.result[[x]])
dat = do.call(rbind, dat_list)
dat2 = cbind(res$simulate.allele_freq, res$simulate.n, res$simulate.b1, dat)
colnames(dat2)[1:3] = c("allele_freq", "n", "b1")
dat2 = as.data.frame(dat2)
dat2$flip = res$compute_bf.flip

1. n = 5e3

par(mfrow = c(2,2))
n = 5e3
plot(dat2$zscore[dat2$n == n & dat2$flip == "flip"], dat2$zscore[dat2$n == n & dat2$flip == "nonflip"],
     xlim = c(-23,-18), ylim = c(18, 23), main = "zscore", xlab = "x flipped", ylab = "x original")
abline(a = 0, b = -1, col = "red")

plot(dat2$bhat[dat2$n == n & dat2$flip == "flip"], dat2$bhat[dat2$n == n & dat2$flip == "nonflip"], main = "bhat",
     xlab = "x flipped", ylab = "x original")
abline(a = 0, b = -1, col = "red")

plot(dat2$lbf.wakefeld[dat2$n == n & dat2$flip == "flip"], dat2$lbf.wakefeld[dat2$n == n & dat2$flip == "nonflip"], main = "Wakefield log(BF)", xlab = "x flipped", ylab = "x original")
abline(a = 0, b = 1, col = "red")

plot(dat2$lbf.numerical[dat2$n == n & dat2$flip == "flip"], dat2$lbf.numerical[dat2$n == n & dat2$flip == "nonflip"], main = "Numerical log(BF)", xlab = "x flipped", ylab = "x original")
abline(a = 0, b = 1, col = "red")

Version	Author	Date
c219afc	yunqi yang	2024-10-11
8a1ddda	yunqi yang	2024-10-07
6445624	yunqi yang	2024-09-18
5b1f3bf	yunqi yang	2024-09-15

plot(dat2$bhat[dat2$n == n], dat2$lbf.wakefeld[dat2$n == n]-dat2$lbf.numerical[dat2$n == n], xlab = "bhat", ylab = "log(BF.wakefield) - log(BF.numerical)")

Version	Author	Date
c219afc	yunqi yang	2024-10-11
8a1ddda	yunqi yang	2024-10-07
6445624	yunqi yang	2024-09-18

2. n = 5e5

n = 5e5
plot(dat2$zscore[dat2$n == n & dat2$flip == "flip"], dat2$zscore[dat2$n == n & dat2$flip == "nonflip"],
     main = "zscore", xlab = "x flipped", ylab = "x original")
abline(a = 0, b = -1, col = "red")

Version	Author	Date
c219afc	yunqi yang	2024-10-11
8a1ddda	yunqi yang	2024-10-07

plot(dat2$bhat[dat2$n == n & dat2$flip == "flip"], dat2$bhat[dat2$n == n & dat2$flip == "nonflip"], main = "bhat")
abline(a = 0, b = -1, col = "red", xlab = "x flipped", ylab = "x original")

Version	Author	Date
c219afc	yunqi yang	2024-10-11

plot(dat2$lbf.wakefeld[dat2$n == n & dat2$flip == "flip"], dat2$lbf.wakefeld[dat2$n == n & dat2$flip == "nonflip"], main = "Wakefield log(BF)", xlab = "x flipped", ylab = "x original")
abline(a = 0, b = 1, col = "red")

Version	Author	Date
c219afc	yunqi yang	2024-10-11

plot(dat2$lbf.numerical[dat2$n == n & dat2$flip == "flip"], dat2$lbf.numerical[dat2$n == n & dat2$flip == "nonflip"], main = "Numerical log(BF)", xlab = "x flipped", ylab = "x original")
abline(a = 0, b = 1, col = "red")

Version	Author	Date
c219afc	yunqi yang	2024-10-11

plot(dat2$bhat[dat2$n == n], dat2$lbf.wakefeld[dat2$n == n]-dat2$lbf.numerical[dat2$n == n], xlab = "bhat", ylab = "log(BF.wakefield) - log(BF.numerical)")

Version	Author	Date
c219afc	yunqi yang	2024-10-11

3. Computation time of Gauss-Hermite vs. laplace BF

par(mfrow = c(1,2))
hist(dat2$time.lp)
hist(dat2$time.gh)

Version	Author	Date
c219afc	yunqi yang	2024-10-11

mean(dat2$time.lp)

[1] 0.0111275

mean(dat2$time.gh)

[1] 35.52377

sessionInfo()

R version 4.2.0 (2022-04-22)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: CentOS Linux 7 (Core)

Matrix products: default
BLAS/LAPACK: /software/openblas-0.3.13-el7-x86_64/lib/libopenblas_haswellp-r0.3.13.so

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

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

other attached packages:
[1] latex2exp_0.9.4 workflowr_1.7.0

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.12      highr_0.9        compiler_4.2.0   pillar_1.9.0    
 [5] bslib_0.3.1      later_1.3.0      git2r_0.30.1     jquerylib_0.1.4 
 [9] tools_4.2.0      getPass_0.2-2    digest_0.6.29    jsonlite_1.8.0  
[13] evaluate_0.15    lifecycle_1.0.4  tibble_3.2.1     pkgconfig_2.0.3 
[17] rlang_1.1.3      cli_3.6.2        rstudioapi_0.13  yaml_2.3.5      
[21] xfun_0.30        fastmap_1.1.0    httr_1.4.3       stringr_1.5.1   
[25] knitr_1.39       fs_1.5.2         vctrs_0.6.5      sass_0.4.1      
[29] rprojroot_2.0.3  glue_1.6.2       R6_2.5.1         processx_3.8.0  
[33] fansi_1.0.3      rmarkdown_2.14   callr_3.7.3      magrittr_2.0.3  
[37] whisker_0.4      ps_1.7.0         promises_1.2.0.1 htmltools_0.5.2 
[41] httpuv_1.6.5     utf8_1.2.2       stringi_1.7.6