Last updated: 2025-09-08

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Here we want to investigate why using out-of-sample LD matrix can cause many false discovery by looking that the result after every training iteration. Recall that SuSiE_rss work by iteratively fitting Single Effect Model to the residuals: \[\overline{r} = X^{\top} y - X^{\top} X \overline{b} = v - R \overline{b} \] where \(\overline{b} = \sum_{\ell=1}^{L} \overline{b}_{\ell}\) is the sum of the posterior mean of all effects.

We expect that after detecting all ``true’’ casual SNP, \(\overline{r}\) will be a vector of \(J\) equally small numbers so that the PIP of over-fitted \(\ell\) will be close to 0 (diffused). Hence the purity of over-fitted \(\ell\)-th CS is small and is not reported.

However, it can be seen that because of mis-specified \(R\), after controlling for all casual SNP, the residual \(\overline{r}\) of other SNPs can ``increase’’, leading to large PIP, thus creates false discovery.

We will also see that our method (projected \(R\)) will experience this bad behavior much less.

In-sample Covariance matrix

Out-sample Covariance matrix

## 2. Out-sample covariance matrix
V_xx = R_hat
sigma2 = 1
sigma02 = 1
L = 3
max_iter = 10

b_bar = matrix(0, nrow = L, ncol = J)
b_bar2 = matrix(0, nrow = L, ncol = J)
alphas = matrix(0, nrow = L, ncol = J)
mus = matrix(0, nrow = L, ncol = J)
sigmas = matrix(0, nrow = L, ncol = J)


for (iter in (1:max_iter)){
  par(mfrow = c(3, 3))
  V_xy_bar = V_xy - V_xx %*% colSums(b_bar)  ## residual signal
  
  for (ell in 1:L){
    V_xy_bar_ell = V_xy_bar + V_xx %*% b_bar[ell, ]  ## add back ell-th signal
    
    susie_plot(V_xy_bar_ell, y = "z", b=beta)
    
    ret = SER(V_xx, V_xy_bar_ell, n, sigma2, sigma02)
    alphas[ell, ] = ret$alpha
    mus[ell, ] = ret$mus
    sigmas[ell, ] = ret$sigma12
    b_bar[ell, ] = alphas[ell, ] * mus[ell, ]
    b_bar2[ell, ] = alphas[ell, ] * (mus[ell, ]^2 + sigmas[ell, ])
    V_xy_bar = V_xy_bar_ell - V_xx %*% b_bar[ell, ]
    
    susie_plot(ret$alpha, y = "PIP", b=beta)
    susie_plot(V_xy_bar, y = "z", b=beta)
    
  }
  mtext(paste0("Out-sample R, Iteration ", iter), outer = TRUE, cex = 1.5)
}

Projected Covariance matrix

## 3. Projected R
ret = proj_Dykstra(R=R_hat, v=V_xy)
V_xx = ret$R
sigma2 = 1
sigma02 = 1
L = 3
max_iter = 10

b_bar = matrix(0, nrow = L, ncol = J)
b_bar2 = matrix(0, nrow = L, ncol = J)
alphas = matrix(0, nrow = L, ncol = J)
mus = matrix(0, nrow = L, ncol = J)
sigmas = matrix(0, nrow = L, ncol = J)


for (iter in (1:max_iter)){
  par(mfrow = c(3, 3))
  V_xy_bar = V_xy - V_xx %*% colSums(b_bar)  ## residual signal
  
  for (ell in 1:L){
    V_xy_bar_ell = V_xy_bar + V_xx %*% b_bar[ell, ]  ## add back ell-th signal
    
    susie_plot(V_xy_bar_ell, y = "z", b=beta)
    
    ret = SER(V_xx, V_xy_bar_ell, n, sigma2, sigma02)
    alphas[ell, ] = ret$alpha
    mus[ell, ] = ret$mus
    sigmas[ell, ] = ret$sigma12
    b_bar[ell, ] = alphas[ell, ] * mus[ell, ]
    b_bar2[ell, ] = alphas[ell, ] * (mus[ell, ]^2 + sigmas[ell, ])
    V_xy_bar = V_xy_bar_ell - V_xx %*% b_bar[ell, ]
    
    susie_plot(ret$alpha, y = "PIP", b=beta)
    susie_plot(V_xy_bar, y = "z", b=beta)
    
  }
  mtext(paste0("Projected R, Iteration ", iter), outer = TRUE, cex = 1.5)
}

Let us look at some other seeds.

seed = 1
set.seed(seed)
print(paste("This is seed number", seed))
[1] "This is seed number 1"
# Remove SNPs with MAF < 0.01
maf = apply(gtex, 2, function(x) sum(x)/2/length(x))
X0 = gtex[, maf > 0.01]
dim(X0)
[1]  574 7154
X = na.omit(X0)
dim(X)
[1]  574 7154
snp_total = ncol(X0)
n = nrow(X0)
J = 200
# Start from a random point on the genome
indx_start = sample(1: (snp_total - J), 1)
X = X0[, indx_start:(indx_start + J -1)]
# View(cor(X)[1:10, 1:10])

## sub-sample into two
out_sample = sample(1:n, 100)
X_out = X[out_sample, ]
X_in = X[setdiff(1:n, out_sample), ]
sum(is.na(X_out))
[1] 0
rm_p = c(which(diag(cov(X_in))==0), which(diag(cov(X_out))==0))
length(rm_p)
[1] 0
indx_p = setdiff(1:J, rm_p)
X_in = X_in[, indx_p]
X_out = X_out[, indx_p]

## Standardize both sample matrices
X_in <- scale(X_in)
X_out <- scale(X_out)

## out-sample LD matrix
R_hat = cor(X_out)
R = cor(X_in)


## generate data from in-sample X matrix
J = ncol(X_in)
beta <- rep(0, J)
n = nrow(X_in)
num_casual_SNP = 1
gamma = sample(c(1:J), size = num_casual_SNP, replace = FALSE)
b = rnorm(num_casual_SNP) * 3
beta = rep(0, J)
beta[gamma] = b
y = X_in %*% beta + rnorm(n)
y = scale(y) 

V_xy = t(X_in) %*% y / (n-1)

## 1. In-sample covariance matrix
V_xx = R
sigma2 = 1
sigma02 = 1
L = 3
max_iter = 10

b_bar = matrix(0, nrow = L, ncol = J)
b_bar2 = matrix(0, nrow = L, ncol = J)
alphas = matrix(0, nrow = L, ncol = J)
mus = matrix(0, nrow = L, ncol = J)
sigmas = matrix(0, nrow = L, ncol = J)


for (iter in (1:max_iter)){
  par(mfrow = c(3, 3))
  V_xy_bar = V_xy - V_xx %*% colSums(b_bar)  ## residual signal
  
  for (ell in 1:L){
    V_xy_bar_ell = V_xy_bar + V_xx %*% b_bar[ell, ]  ## add back ell-th signal
    
    susie_plot(V_xy_bar_ell, y = "z", b=beta)
    
    ret = SER(V_xx, V_xy_bar_ell, n, sigma2, sigma02)
    alphas[ell, ] = ret$alpha
    mus[ell, ] = ret$mus
    sigmas[ell, ] = ret$sigma12
    b_bar[ell, ] = alphas[ell, ] * mus[ell, ]
    b_bar2[ell, ] = alphas[ell, ] * (mus[ell, ]^2 + sigmas[ell, ])
    V_xy_bar = V_xy_bar_ell - V_xx %*% b_bar[ell, ]
    
    susie_plot(ret$alpha, y = "PIP", b=beta)
    susie_plot(V_xy_bar, y = "z", b=beta)
    
  }
  mtext(paste0("In-sample R, Iteration ", iter), outer = TRUE, cex = 1.5)
}

## 2. Out-sample covariance matrix
V_xx = R_hat
sigma2 = 1
sigma02 = 1
L = 3
max_iter = 10

b_bar = matrix(0, nrow = L, ncol = J)
b_bar2 = matrix(0, nrow = L, ncol = J)
alphas = matrix(0, nrow = L, ncol = J)
mus = matrix(0, nrow = L, ncol = J)
sigmas = matrix(0, nrow = L, ncol = J)


for (iter in (1:max_iter)){
  par(mfrow = c(3, 3))
  V_xy_bar = V_xy - V_xx %*% colSums(b_bar)  ## residual signal
  
  for (ell in 1:L){
    V_xy_bar_ell = V_xy_bar + V_xx %*% b_bar[ell, ]  ## add back ell-th signal
    
    susie_plot(V_xy_bar_ell, y = "z", b=beta)
    
    ret = SER(V_xx, V_xy_bar_ell, n, sigma2, sigma02)
    alphas[ell, ] = ret$alpha
    mus[ell, ] = ret$mus
    sigmas[ell, ] = ret$sigma12
    b_bar[ell, ] = alphas[ell, ] * mus[ell, ]
    b_bar2[ell, ] = alphas[ell, ] * (mus[ell, ]^2 + sigmas[ell, ])
    V_xy_bar = V_xy_bar_ell - V_xx %*% b_bar[ell, ]
    
    susie_plot(ret$alpha, y = "PIP", b=beta)
    susie_plot(V_xy_bar, y = "z", b=beta)
    
  }
  mtext(paste0("Out-sample R, Iteration ", iter), outer = TRUE, cex = 1.5)
}

## 3. Projected R
ret = proj_Dykstra(R=R_hat, v=V_xy)
V_xx = ret$R
sigma2 = 1
sigma02 = 1
L = 3
max_iter = 10

b_bar = matrix(0, nrow = L, ncol = J)
b_bar2 = matrix(0, nrow = L, ncol = J)
alphas = matrix(0, nrow = L, ncol = J)
mus = matrix(0, nrow = L, ncol = J)
sigmas = matrix(0, nrow = L, ncol = J)


for (iter in (1:max_iter)){
  par(mfrow = c(3, 3))
  V_xy_bar = V_xy - V_xx %*% colSums(b_bar)  ## residual signal
  
  for (ell in 1:L){
    V_xy_bar_ell = V_xy_bar + V_xx %*% b_bar[ell, ]  ## add back ell-th signal
    
    susie_plot(V_xy_bar_ell, y = "z", b=beta)
    
    ret = SER(V_xx, V_xy_bar_ell, n, sigma2, sigma02)
    alphas[ell, ] = ret$alpha
    mus[ell, ] = ret$mus
    sigmas[ell, ] = ret$sigma12
    b_bar[ell, ] = alphas[ell, ] * mus[ell, ]
    b_bar2[ell, ] = alphas[ell, ] * (mus[ell, ]^2 + sigmas[ell, ])
    V_xy_bar = V_xy_bar_ell - V_xx %*% b_bar[ell, ]
    
    susie_plot(ret$alpha, y = "PIP", b=beta)
    susie_plot(V_xy_bar, y = "z", b=beta)
    
  }
  mtext(paste0("Projected R, Iteration ", iter), outer = TRUE, cex = 1.5)
}

seed = 2
set.seed(seed)
print(paste("This is seed number", seed))
[1] "This is seed number 2"
# Remove SNPs with MAF < 0.01
maf = apply(gtex, 2, function(x) sum(x)/2/length(x))
X0 = gtex[, maf > 0.01]
dim(X0)
[1]  574 7154
X = na.omit(X0)
dim(X)
[1]  574 7154
snp_total = ncol(X0)
n = nrow(X0)
J = 200
# Start from a random point on the genome
indx_start = sample(1: (snp_total - J), 1)
X = X0[, indx_start:(indx_start + J -1)]
# View(cor(X)[1:10, 1:10])

## sub-sample into two
out_sample = sample(1:n, 100)
X_out = X[out_sample, ]
X_in = X[setdiff(1:n, out_sample), ]
sum(is.na(X_out))
[1] 0
rm_p = c(which(diag(cov(X_in))==0), which(diag(cov(X_out))==0))
length(rm_p)
[1] 3
indx_p = setdiff(1:J, rm_p)
X_in = X_in[, indx_p]
X_out = X_out[, indx_p]

## Standardize both sample matrices
X_in <- scale(X_in)
X_out <- scale(X_out)

## out-sample LD matrix
R_hat = cor(X_out)
R = cor(X_in)


## generate data from in-sample X matrix
J = ncol(X_in)
beta <- rep(0, J)
n = nrow(X_in)
num_casual_SNP = 1
gamma = sample(c(1:J), size = num_casual_SNP, replace = FALSE)
b = rnorm(num_casual_SNP) * 3
beta = rep(0, J)
beta[gamma] = b
y = X_in %*% beta + rnorm(n)
y = scale(y) 

V_xy = t(X_in) %*% y / (n-1)

## 1. In-sample covariance matrix
V_xx = R
sigma2 = 1
sigma02 = 1
L = 3
max_iter = 10

b_bar = matrix(0, nrow = L, ncol = J)
b_bar2 = matrix(0, nrow = L, ncol = J)
alphas = matrix(0, nrow = L, ncol = J)
mus = matrix(0, nrow = L, ncol = J)
sigmas = matrix(0, nrow = L, ncol = J)


for (iter in (1:max_iter)){
  par(mfrow = c(3, 3))
  V_xy_bar = V_xy - V_xx %*% colSums(b_bar)  ## residual signal
  
  for (ell in 1:L){
    V_xy_bar_ell = V_xy_bar + V_xx %*% b_bar[ell, ]  ## add back ell-th signal
    
    susie_plot(V_xy_bar_ell, y = "z", b=beta)
    
    ret = SER(V_xx, V_xy_bar_ell, n, sigma2, sigma02)
    alphas[ell, ] = ret$alpha
    mus[ell, ] = ret$mus
    sigmas[ell, ] = ret$sigma12
    b_bar[ell, ] = alphas[ell, ] * mus[ell, ]
    b_bar2[ell, ] = alphas[ell, ] * (mus[ell, ]^2 + sigmas[ell, ])
    V_xy_bar = V_xy_bar_ell - V_xx %*% b_bar[ell, ]
    
    susie_plot(ret$alpha, y = "PIP", b=beta)
    susie_plot(V_xy_bar, y = "z", b=beta)
    
  }
  mtext(paste0("In-sample R, Iteration ", iter), outer = TRUE, cex = 1.5)
}

## 2. Out-sample covariance matrix
V_xx = R_hat
sigma2 = 1
sigma02 = 1
L = 3
max_iter = 10

b_bar = matrix(0, nrow = L, ncol = J)
b_bar2 = matrix(0, nrow = L, ncol = J)
alphas = matrix(0, nrow = L, ncol = J)
mus = matrix(0, nrow = L, ncol = J)
sigmas = matrix(0, nrow = L, ncol = J)


for (iter in (1:max_iter)){
  par(mfrow = c(3, 3))
  V_xy_bar = V_xy - V_xx %*% colSums(b_bar)  ## residual signal
  
  for (ell in 1:L){
    V_xy_bar_ell = V_xy_bar + V_xx %*% b_bar[ell, ]  ## add back ell-th signal
    
    susie_plot(V_xy_bar_ell, y = "z", b=beta)
    
    ret = SER(V_xx, V_xy_bar_ell, n, sigma2, sigma02)
    alphas[ell, ] = ret$alpha
    mus[ell, ] = ret$mus
    sigmas[ell, ] = ret$sigma12
    b_bar[ell, ] = alphas[ell, ] * mus[ell, ]
    b_bar2[ell, ] = alphas[ell, ] * (mus[ell, ]^2 + sigmas[ell, ])
    V_xy_bar = V_xy_bar_ell - V_xx %*% b_bar[ell, ]
    
    susie_plot(ret$alpha, y = "PIP", b=beta)
    susie_plot(V_xy_bar, y = "z", b=beta)
    
  }
  mtext(paste0("Out-sample R, Iteration ", iter), outer = TRUE, cex = 1.5)
}

## 3. Projected R
ret = proj_Dykstra(R=R_hat, v=V_xy)
V_xx = ret$R
sigma2 = 1
sigma02 = 1
L = 3
max_iter = 10

b_bar = matrix(0, nrow = L, ncol = J)
b_bar2 = matrix(0, nrow = L, ncol = J)
alphas = matrix(0, nrow = L, ncol = J)
mus = matrix(0, nrow = L, ncol = J)
sigmas = matrix(0, nrow = L, ncol = J)


for (iter in (1:max_iter)){
  par(mfrow = c(3, 3))
  V_xy_bar = V_xy - V_xx %*% colSums(b_bar)  ## residual signal
  
  for (ell in 1:L){
    V_xy_bar_ell = V_xy_bar + V_xx %*% b_bar[ell, ]  ## add back ell-th signal
    
    susie_plot(V_xy_bar_ell, y = "z", b=beta)
    
    ret = SER(V_xx, V_xy_bar_ell, n, sigma2, sigma02)
    alphas[ell, ] = ret$alpha
    mus[ell, ] = ret$mus
    sigmas[ell, ] = ret$sigma12
    b_bar[ell, ] = alphas[ell, ] * mus[ell, ]
    b_bar2[ell, ] = alphas[ell, ] * (mus[ell, ]^2 + sigmas[ell, ])
    V_xy_bar = V_xy_bar_ell - V_xx %*% b_bar[ell, ]
    
    susie_plot(ret$alpha, y = "PIP", b=beta)
    susie_plot(V_xy_bar, y = "z", b=beta)
    
  }
  mtext(paste0("Projected R, Iteration ", iter), outer = TRUE, cex = 1.5)
}

seed = 3
set.seed(seed)
print(paste("This is seed number", seed))
[1] "This is seed number 3"
# Remove SNPs with MAF < 0.01
maf = apply(gtex, 2, function(x) sum(x)/2/length(x))
X0 = gtex[, maf > 0.01]
dim(X0)
[1]  574 7154
X = na.omit(X0)
dim(X)
[1]  574 7154
snp_total = ncol(X0)
n = nrow(X0)
J = 200
# Start from a random point on the genome
indx_start = sample(1: (snp_total - J), 1)
X = X0[, indx_start:(indx_start + J -1)]
# View(cor(X)[1:10, 1:10])

## sub-sample into two
out_sample = sample(1:n, 100)
X_out = X[out_sample, ]
X_in = X[setdiff(1:n, out_sample), ]
sum(is.na(X_out))
[1] 0
rm_p = c(which(diag(cov(X_in))==0), which(diag(cov(X_out))==0))
length(rm_p)
[1] 1
indx_p = setdiff(1:J, rm_p)
X_in = X_in[, indx_p]
X_out = X_out[, indx_p]

## Standardize both sample matrices
X_in <- scale(X_in)
X_out <- scale(X_out)

## out-sample LD matrix
R_hat = cor(X_out)
R = cor(X_in)


## generate data from in-sample X matrix
J = ncol(X_in)
beta <- rep(0, J)
n = nrow(X_in)
num_casual_SNP = 1
gamma = sample(c(1:J), size = num_casual_SNP, replace = FALSE)
b = rnorm(num_casual_SNP) * 3
beta = rep(0, J)
beta[gamma] = b
y = X_in %*% beta + rnorm(n)
y = scale(y) 

V_xy = t(X_in) %*% y / (n-1)

## 1. In-sample covariance matrix
V_xx = R
sigma2 = 1
sigma02 = 1
L = 3
max_iter = 10

b_bar = matrix(0, nrow = L, ncol = J)
b_bar2 = matrix(0, nrow = L, ncol = J)
alphas = matrix(0, nrow = L, ncol = J)
mus = matrix(0, nrow = L, ncol = J)
sigmas = matrix(0, nrow = L, ncol = J)


for (iter in (1:max_iter)){
  par(mfrow = c(3, 3))
  V_xy_bar = V_xy - V_xx %*% colSums(b_bar)  ## residual signal
  
  for (ell in 1:L){
    V_xy_bar_ell = V_xy_bar + V_xx %*% b_bar[ell, ]  ## add back ell-th signal
    
    susie_plot(V_xy_bar_ell, y = "z", b=beta)
    
    ret = SER(V_xx, V_xy_bar_ell, n, sigma2, sigma02)
    alphas[ell, ] = ret$alpha
    mus[ell, ] = ret$mus
    sigmas[ell, ] = ret$sigma12
    b_bar[ell, ] = alphas[ell, ] * mus[ell, ]
    b_bar2[ell, ] = alphas[ell, ] * (mus[ell, ]^2 + sigmas[ell, ])
    V_xy_bar = V_xy_bar_ell - V_xx %*% b_bar[ell, ]
    
    susie_plot(ret$alpha, y = "PIP", b=beta)
    susie_plot(V_xy_bar, y = "z", b=beta)
    
  }
  mtext(paste0("In-sample R, Iteration ", iter), outer = TRUE, cex = 1.5)
}

## 2. Out-sample covariance matrix
V_xx = R_hat
sigma2 = 1
sigma02 = 1
L = 3
max_iter = 10

b_bar = matrix(0, nrow = L, ncol = J)
b_bar2 = matrix(0, nrow = L, ncol = J)
alphas = matrix(0, nrow = L, ncol = J)
mus = matrix(0, nrow = L, ncol = J)
sigmas = matrix(0, nrow = L, ncol = J)


for (iter in (1:max_iter)){
  par(mfrow = c(3, 3))
  V_xy_bar = V_xy - V_xx %*% colSums(b_bar)  ## residual signal
  
  for (ell in 1:L){
    V_xy_bar_ell = V_xy_bar + V_xx %*% b_bar[ell, ]  ## add back ell-th signal
    
    susie_plot(V_xy_bar_ell, y = "z", b=beta)
    
    ret = SER(V_xx, V_xy_bar_ell, n, sigma2, sigma02)
    alphas[ell, ] = ret$alpha
    mus[ell, ] = ret$mus
    sigmas[ell, ] = ret$sigma12
    b_bar[ell, ] = alphas[ell, ] * mus[ell, ]
    b_bar2[ell, ] = alphas[ell, ] * (mus[ell, ]^2 + sigmas[ell, ])
    V_xy_bar = V_xy_bar_ell - V_xx %*% b_bar[ell, ]
    
    susie_plot(ret$alpha, y = "PIP", b=beta)
    susie_plot(V_xy_bar, y = "z", b=beta)
    
  }
  mtext(paste0("Out-sample R, Iteration ", iter), outer = TRUE, cex = 1.5)
}

## 3. Projected R
ret = proj_Dykstra(R=R_hat, v=V_xy)
V_xx = ret$R
sigma2 = 1
sigma02 = 1
L = 3
max_iter = 10

b_bar = matrix(0, nrow = L, ncol = J)
b_bar2 = matrix(0, nrow = L, ncol = J)
alphas = matrix(0, nrow = L, ncol = J)
mus = matrix(0, nrow = L, ncol = J)
sigmas = matrix(0, nrow = L, ncol = J)


for (iter in (1:max_iter)){
  par(mfrow = c(3, 3))
  V_xy_bar = V_xy - V_xx %*% colSums(b_bar)  ## residual signal
  
  for (ell in 1:L){
    V_xy_bar_ell = V_xy_bar + V_xx %*% b_bar[ell, ]  ## add back ell-th signal
    
    susie_plot(V_xy_bar_ell, y = "z", b=beta)
    
    ret = SER(V_xx, V_xy_bar_ell, n, sigma2, sigma02)
    alphas[ell, ] = ret$alpha
    mus[ell, ] = ret$mus
    sigmas[ell, ] = ret$sigma12
    b_bar[ell, ] = alphas[ell, ] * mus[ell, ]
    b_bar2[ell, ] = alphas[ell, ] * (mus[ell, ]^2 + sigmas[ell, ])
    V_xy_bar = V_xy_bar_ell - V_xx %*% b_bar[ell, ]
    
    susie_plot(ret$alpha, y = "PIP", b=beta)
    susie_plot(V_xy_bar, y = "z", b=beta)
    
  }
  mtext(paste0("Projected R, Iteration ", iter), outer = TRUE, cex = 1.5)
}

It remains a question whether we can make it as good as the in-sample LD matrix…


sessionInfo()
R version 4.5.1 (2025-06-13)
Platform: aarch64-apple-darwin20
Running under: macOS Sequoia 15.6.1

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRblas.0.dylib 
LAPACK: /Library/Frameworks/R.framework/Versions/4.5-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.12.1

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

time zone: America/Chicago
tzcode source: internal

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

other attached packages:
[1] RSpectra_0.16-2 Matrix_1.7-3    susieR_0.14.2   workflowr_1.7.1

loaded via a namespace (and not attached):
 [1] sass_0.4.10        generics_0.1.4     stringi_1.8.7      lattice_0.22-7    
 [5] digest_0.6.37      magrittr_2.0.3     evaluate_1.0.4     grid_4.5.1        
 [9] RColorBrewer_1.1-3 fastmap_1.2.0      plyr_1.8.9         rprojroot_2.1.0   
[13] jsonlite_2.0.0     processx_3.8.6     whisker_0.4.1      reshape_0.8.10    
[17] ps_1.9.1           mixsqp_0.3-54      promises_1.3.3     httr_1.4.7        
[21] scales_1.4.0       jquerylib_0.1.4    cli_3.6.5          rlang_1.1.6       
[25] crayon_1.5.3       cachem_1.1.0       yaml_2.3.10        tools_4.5.1       
[29] dplyr_1.1.4        ggplot2_3.5.2      httpuv_1.6.16      vctrs_0.6.5       
[33] R6_2.6.1           matrixStats_1.5.0  lifecycle_1.0.4    git2r_0.36.2      
[37] stringr_1.5.1      fs_1.6.6           irlba_2.3.5.1      pkgconfig_2.0.3   
[41] callr_3.7.6        pillar_1.11.0      bslib_0.9.0        later_1.4.2       
[45] gtable_0.3.6       glue_1.8.0         Rcpp_1.1.0         xfun_0.52         
[49] tibble_3.3.0       tidyselect_1.2.1   rstudioapi_0.17.1  knitr_1.50        
[53] farver_2.1.2       htmltools_0.5.8.1  rmarkdown_2.29     compiler_4.5.1    
[57] getPass_0.2-4