diff --git a/R/colocboost_output.R b/R/colocboost_output.R
index 4267891..b7a0265 100644
--- a/R/colocboost_output.R
+++ b/R/colocboost_output.R
@@ -507,14 +507,19 @@ get_ucos_summary <- function(cb_output, outcome_names = NULL, region_name = NULL
 }
 
 #' Extract CoS at different coverages
-#'
-#' @description `get_cos` get the colocalization confidence sets (CoS) with different coverage. If Genotype...provided...check purity
-#'
+#' 
+#' @description `get_cos` extracts colocalization confidence sets (CoS) at different coverage levels 
+#' from ColocBoost results. When genotype data (X) or correlation matrix (Xcorr) is provided, it 
+#' can also calculate and filter CoS based on purity statistics, ensuring that variants within 
+#' each CoS are sufficiently correlated.
+#' 
 #' @param cb_output Output object from `colocboost` analysis
 #' @param coverage A number between 0 and 1 specifying the \dQuote{coverage} of the estimated colocalization confidence sets (CoS) (default is 0.95).
 #' @param X Genotype matrix of values of the p variables. Used to compute correlations if Xcorr is not provided.
 #' @param Xcorr Correlation matrix of correlations between variables. Alternative to X.
 #' @param n_purity The maximum number of CoS variables used in calculating the correlation (\dQuote{purity}) statistics. 
+#' @param min_abs_corr The minimum absolute correlation value of variants in a CoS to be considered pass (\dQuote{purity}) statistics.
+#' @param median_abs_corr The median absolute correlation value of variants in a CoS to be considered pass (\dQuote{purity}) statistics.
 #' When the number of variables included in the CoS is greater than this number, the CoS variables are randomly subsampled.
 #'
 #' @return A list of indices of variables in each CoS.
diff --git a/man/get_cos.Rd b/man/get_cos.Rd
index 3b277b6..9dbcee0 100644
--- a/man/get_cos.Rd
+++ b/man/get_cos.Rd
@@ -23,14 +23,21 @@ get_cos(
 
 \item{Xcorr}{Correlation matrix of correlations between variables. Alternative to X.}
 
-\item{n_purity}{The maximum number of CoS variables used in calculating the correlation (\dQuote{purity}) statistics.
+\item{n_purity}{The maximum number of CoS variables used in calculating the correlation (\dQuote{purity}) statistics.}
+
+\item{min_abs_corr}{The minimum absolute correlation value of variants in a CoS to be considered pass (\dQuote{purity}) statistics.}
+
+\item{median_abs_corr}{The median absolute correlation value of variants in a CoS to be considered pass (\dQuote{purity}) statistics.
 When the number of variables included in the CoS is greater than this number, the CoS variables are randomly subsampled.}
 }
 \value{
 A list of indices of variables in each CoS.
 }
 \description{
-\code{get_cos} get the colocalization confidence sets (CoS) with different coverage. If Genotype...provided...check purity
+\code{get_cos} extracts colocalization confidence sets (CoS) at different coverage levels
+from ColocBoost results. When genotype data (X) or correlation matrix (Xcorr) is provided, it
+can also calculate and filter CoS based on purity statistics, ensuring that variants within
+each CoS are sufficiently correlated.
 }
 \examples{
 # colocboost example
diff --git a/tests/testthat/test_colocboost.R b/tests/testthat/test_colocboost.R
index e7e7c25..7e2afab 100644
--- a/tests/testthat/test_colocboost.R
+++ b/tests/testthat/test_colocboost.R
@@ -49,11 +49,19 @@ test_that("colocboost runs with individual data", {
   X_list <- list(test_data$X, test_data$X)
   
   # Run colocboost with minimal parameters
-  result <- colocboost(
-    X = X_list, 
-    Y = Y_list,
-    M = 10,  # Small number of iterations for testing
-    output_level = 2  # More detailed output for testing
+  warnings <- capture_warnings({
+    result <- colocboost(
+      X = X_list, 
+      Y = Y_list,
+      M = 10,  # Small number of iterations for testing
+      output_level = 2  # More detailed output for testing
+    )
+  })
+
+  # Check if any of the expected warning patterns are present
+  expect_true(
+    any(grepl("smallest number of variables", warnings)) || 
+    any(grepl("did not coverage", warnings))
   )
   
   # Test that we get a colocboost object
@@ -101,11 +109,19 @@ test_that("colocboost runs with summary statistics", {
   }
   
   # Run colocboost with summary statistics
-  result <- colocboost(
-    sumstat = sumstat_list,
-    LD = list(LD),
-    M = 10,  # Small number of iterations for testing
-    output_level = 2  # More detailed output for testing
+  warnings <- capture_warnings({
+    result <- colocboost(
+      sumstat = sumstat_list,
+      LD = list(LD),
+      M = 10,  # Small number of iterations for testing
+      output_level = 2  # More detailed output for testing
+    )
+  })
+
+  # Check if any of the expected warning patterns are present
+  expect_true(
+    any(grepl("smallest number of variables", warnings)) || 
+    any(grepl("did not coverage", warnings))
   )
   
   # Test that we get a colocboost object
@@ -124,12 +140,20 @@ test_that("colocboost handles focal outcome correctly", {
   X_list <- list(test_data$X, test_data$X)
   
   # Run colocboost with focal_outcome_idx = 1
-  result <- colocboost(
-    X = X_list, 
-    Y = Y_list,
-    focal_outcome_idx = 1,
-    M = 10,  # Small number of iterations for testing
-    output_level = 2  # More detailed output for testing
+  warnings <- capture_warnings({
+    result <- colocboost(
+      X = X_list, 
+      Y = Y_list,
+      focal_outcome_idx = 1,
+      M = 10,  # Small number of iterations for testing
+      output_level = 2  # More detailed output for testing
+    )
+  })
+
+  # Check if any of the expected warning patterns are present
+  expect_true(
+    any(grepl("smallest number of variables", warnings)) || 
+    any(grepl("did not coverage", warnings))
   )
   
   # Test that we get a colocboost object
@@ -148,11 +172,19 @@ test_that("get_cos_summary returns expected structure", {
   X_list <- list(test_data$X, test_data$X)
   
   # Run colocboost with minimal parameters
-  result <- colocboost(
-    X = X_list, 
-    Y = Y_list,
-    M = 10,  # Small number of iterations for testing
-    output_level = 2  # More detailed output for testing
+  warnings <- capture_warnings({
+    result <- colocboost(
+      X = X_list, 
+      Y = Y_list,
+      M = 10,  # Small number of iterations for testing
+      output_level = 2  # More detailed output for testing
+    )
+  })
+
+  # Check if any of the expected warning patterns are present
+  expect_true(
+    any(grepl("smallest number of variables", warnings)) || 
+    any(grepl("did not coverage", warnings))
   )
   
   # Get summary
@@ -182,11 +214,19 @@ test_that("colocboost_plot runs without error", {
   X_list <- list(test_data$X, test_data$X)
   
   # Run colocboost with minimal parameters
-  result <- colocboost(
-    X = X_list, 
-    Y = Y_list,
-    M = 10,  # Small number of iterations for testing
-    output_level = 2  # More detailed output for testing
+  warnings <- capture_warnings({
+    result <- colocboost(
+      X = X_list, 
+      Y = Y_list,
+      M = 10,  # Small number of iterations for testing
+      output_level = 2  # More detailed output for testing
+    )
+  })
+
+  # Check if any of the expected warning patterns are present
+  expect_true(
+    any(grepl("smallest number of variables", warnings)) || 
+    any(grepl("did not coverage", warnings))
   )
   
   # Basic plotting should not throw an error
@@ -202,11 +242,19 @@ test_that("get_robust_colocalization maintains colocboost structure", {
   X_list <- list(test_data$X, test_data$X)
   
   # Run colocboost with minimal parameters
-  result <- colocboost(
-    X = X_list, 
-    Y = Y_list,
-    M = 10,  # Small number of iterations for testing
-    output_level = 2  # More detailed output for testing
+  warnings <- capture_warnings({
+    result <- colocboost(
+      X = X_list, 
+      Y = Y_list,
+      M = 10,
+      output_level = 2
+    )
+  })
+
+  # Check if any of the expected warning patterns are present
+  expect_true(
+    any(grepl("smallest number of variables", warnings)) || 
+    any(grepl("did not coverage", warnings))
   )
   
   # Run get_strong_colocalization
@@ -234,5 +282,13 @@ test_that("colocboost handles missing/invalid inputs appropriately", {
   Y_list <- list(test_data$Y[,1], test_data$Y[,2])
   X_list <- list(X_bad, X_bad)
   
-  expect_error(colocboost(X = X_list, Y = Y_list), "Please provide the sample index of X and Y, since they do not have the same samples!")
+  expect_error(
+    suppressWarnings(
+      colocboost(
+        X = X_list, 
+        Y = Y_list
+      )
+    ),
+    "Please provide the sample index of X and Y, since they do not have the same samples!"
+  )
 })
\ No newline at end of file
diff --git a/tests/testthat/test_corner_cases.R b/tests/testthat/test_corner_cases.R
index dcd9d68..78756cf 100644
--- a/tests/testthat/test_corner_cases.R
+++ b/tests/testthat/test_corner_cases.R
@@ -94,10 +94,12 @@ test_that("colocboost handles missing values in Y", {
   
   # Run colocboost - should handle NAs automatically
   expect_error(
-    result <- colocboost(
-      X = X_list, 
-      Y = Y_list,
-      M = 5  # Small number of iterations for testing
+    suppressWarnings(
+      result <- colocboost(
+        X = X_list, 
+        Y = Y_list,
+        M = 5  # Small number of iterations for testing
+      )
     ),
     NA
   )
@@ -114,9 +116,11 @@ test_that("colocboost handles different sample sizes", {
   
   # Run colocboost - should error without sample indices
   expect_error(
-    colocboost(
-      X = test_data$X, 
-      Y = test_data$Y
+    suppressWarnings(
+      colocboost(
+        X = test_data$X, 
+        Y = test_data$Y
+      )
     )
   )
   
@@ -144,13 +148,17 @@ test_that("colocboost handles different variant sets", {
   test_data <- generate_edge_case_data("different_variants")
   
   # Run colocboost
-  expect_warning(
+  warnings <- capture_warnings({
     result <- colocboost(
       X = test_data$X, 
       Y = test_data$Y,
       dict_YX = dict_YX,
       M = 5  # Small number of iterations for testing
     )
+  })
+  # Check if any of the expected warning patterns are present
+  expect_true(
+    any(grepl("did not coverage", warnings))
   )
   
   # Test that we get a colocboost object
diff --git a/tests/testthat/test_model.R b/tests/testthat/test_model.R
index 23041ed..8a8848f 100644
--- a/tests/testthat/test_model.R
+++ b/tests/testthat/test_model.R
@@ -116,12 +116,18 @@ test_that("colocboost_workhorse performs boosting iterations", {
   
   # If the workhorse function is exported
   if (exists("colocboost_workhorse")) {
-    expect_error({
+    warnings <- capture_warnings({
       result <- colocboost_workhorse(
         cb_obj$cb_data,
         M = 5  # Small number for testing
       )
-    }, NA)
+    })
+    # Check if any of the expected warning patterns are present
+    expect_true(
+      any(grepl("did not coverage", warnings))
+    )
+    # Then test the result
+    expect_s3_class(result, "colocboost")
   } else {
     skip("colocboost_workhorse not directly accessible")
   }
diff --git a/tests/testthat/test_sumstats.R b/tests/testthat/test_sumstats.R
index 30e13ad..deb3aaa 100644
--- a/tests/testthat/test_sumstats.R
+++ b/tests/testthat/test_sumstats.R
@@ -102,11 +102,18 @@ test_sumstat_data <- generate_sumstat_test_data()
 # Test 1: Basic summary statistics input
 test_that("colocboost runs with basic summary statistics format", {
   # Run colocboost with sumstat and single LD matrix
-  result <- colocboost(
-    sumstat = test_sumstat_data$sumstat,
-    LD = test_sumstat_data$LD,
-    M = 10,  # Small number of iterations for testing
-    output_level = 2  # More detailed output for testing
+  warnings <- capture_warnings({
+    result <- colocboost(
+        sumstat = test_sumstat_data$sumstat,
+        LD = test_sumstat_data$LD,
+        M = 10,  # Small number of iterations for testing
+        output_level = 2  # More detailed output for testing
+    )
+  })
+
+  # Check if any of the expected warning patterns are present
+  expect_true(
+    any(grepl("did not coverage", warnings))
   )
   
   # Test that we get a colocboost object
@@ -124,13 +131,20 @@ test_that("colocboost runs with basic summary statistics format", {
 # Test 2: HyPrColoc compatible format
 test_that("colocboost runs with HyPrColoc compatible format", {
   # Run colocboost with effect matrices
-  result <- colocboost(
-    effect_est = test_sumstat_data$effect_est,
-    effect_se = test_sumstat_data$effect_se,
-    effect_n = test_sumstat_data$effect_n,
-    LD = test_sumstat_data$LD,
-    M = 10,  # Small number of iterations for testing
-    output_level = 2  # More detailed output for testing
+  warnings <- capture_warnings({
+    result <- colocboost(
+        effect_est = test_sumstat_data$effect_est,
+        effect_se = test_sumstat_data$effect_se,
+        effect_n = test_sumstat_data$effect_n,
+        LD = test_sumstat_data$LD,
+        M = 10,  # Small number of iterations for testing
+        output_level = 2  # More detailed output for testing
+    )
+  })
+
+  # Check if any of the expected warning patterns are present
+  expect_true(
+    any(grepl("did not coverage", warnings))
   )
   
   # Test that we get a colocboost object
@@ -147,11 +161,18 @@ test_that("colocboost runs with matched LD matrices", {
   LD_list <- list(test_sumstat_data$LD, test_sumstat_data$LD)
   
   # Run colocboost with summary statistics and multiple LD matrices
-  result <- colocboost(
-    sumstat = test_sumstat_data$sumstat,
-    LD = LD_list,
-    M = 10,  # Small number of iterations for testing
-    output_level = 2  # More detailed output for testing
+  warnings <- capture_warnings({
+    result <- colocboost(
+        sumstat = test_sumstat_data$sumstat,
+        LD = LD_list,
+        M = 10,  # Small number of iterations for testing
+        output_level = 2  # More detailed output for testing
+    )
+  })
+
+  # Check if any of the expected warning patterns are present
+  expect_true(
+    any(grepl("did not coverage", warnings))
   )
   
   # Test that we get a colocboost object
@@ -172,12 +193,19 @@ test_that("colocboost runs with dictionary-mapped LD matrices", {
   dict_sumstatLD <- matrix(c(1:2, 1:2), ncol = 2)
   
   # Run colocboost with dictionary mapping
-  result <- colocboost(
-    sumstat = test_sumstat_data$sumstat,
-    LD = LD_list,
-    dict_sumstatLD = dict_sumstatLD,
-    M = 10,  # Small number of iterations for testing
-    output_level = 2  # More detailed output for testing
+  warnings <- capture_warnings({
+    result <- colocboost(
+        sumstat = test_sumstat_data$sumstat,
+        LD = LD_list,
+        dict_sumstatLD = dict_sumstatLD,
+        M = 10,  # Small number of iterations for testing
+        output_level = 2  # More detailed output for testing
+    )
+  })
+
+  # Check if any of the expected warning patterns are present
+  expect_true(
+    any(grepl("did not coverage", warnings))
   )
   
   # Test that we get a colocboost object
@@ -191,10 +219,17 @@ test_that("colocboost runs with dictionary-mapped LD matrices", {
 # Test 5: Missing LD matrix (LD-free approach)
 test_that("colocboost runs with missing LD matrix", {
   # Run colocboost without LD matrix (should default to diagonal)
-  result <- colocboost(
-    sumstat = test_sumstat_data$sumstat,
-    M = 1,  # Only one iteration expected in LD-free case
-    output_level = 2  # More detailed output for testing
+  warnings <- capture_warnings({
+    result <- colocboost(
+        sumstat = test_sumstat_data$sumstat,
+        M = 1,  # Only one iteration expected in LD-free case
+        output_level = 2  # More detailed output for testing
+    )
+  })
+
+  # Check if any of the expected warning patterns are present
+  expect_true(
+    any(grepl("The smallest number of variables across outcomes is 20 < 100", warnings)) 
   )
   
   # Test that we get a colocboost object
@@ -229,14 +264,19 @@ test_that("colocboost handles missing sample size correctly", {
   })
   
   # Should run but give a warning about missing sample size
-  expect_warning(
+  warnings <- capture_warnings({
     result <- colocboost(
       sumstat = sumstat_no_n,
       LD = test_sumstat_data$LD,
       M = 10,
       output_level = 2
-    ),
-    "Providing the sample size"
+    )
+  })
+
+  # Check if any of the expected warning patterns are present
+  expect_true(
+    any(grepl("Providing the sample size", warnings)) || 
+    any(grepl("did not coverage", warnings))
   )
   
   # Still should get a valid result
diff --git a/tests/testthat/test_utils.R b/tests/testthat/test_utils.R
index 2a75356..e3bd354 100644
--- a/tests/testthat/test_utils.R
+++ b/tests/testthat/test_utils.R
@@ -175,14 +175,6 @@ test_that("get_cormat calculates correlation matrix correctly", {
   expect_equal(result, expected, tolerance = 1e-6)
 })
 
-# Test for check_jk_jkeach function
-test_that("check_jk_jkeach identifies equivalent variants based on LD", {
-  skip("Internal function being tested through main function")
-  
-  # This function is harder to test directly as it relies on internal colocboost objects
-  # We'll test it indirectly through colocboost main function
-})
-
 # Test for get_between_purity function
 test_that("get_between_purity calculates correlation between sets", {
   # Create test data with known correlation structure
@@ -213,12 +205,6 @@ test_that("get_between_purity calculates correlation between sets", {
   expect_lt(result2[1], 0.5)
 })
 
-# Test for get_max_profile function
-test_that("get_max_profile updates check_null_max correctly", {
-  skip("Internal function being tested through main function")
-  
-  # This function modifies internal CB object - hard to test directly
-})
 
 # Test for get_merge_ordered_with_indices function
 test_that("get_merge_ordered_with_indices merges vectors", {
diff --git a/vignettes/ColocBoost_Wrapper_Pipeline.Rmd b/vignettes/ColocBoost_Wrapper_Pipeline.Rmd
index 4fb26e3..fdfa020 100644
--- a/vignettes/ColocBoost_Wrapper_Pipeline.Rmd
+++ b/vignettes/ColocBoost_Wrapper_Pipeline.Rmd
@@ -129,7 +129,7 @@ Example: for sQTL, `list(type_pattern = ".*clu_(\\d+_[+-?]).*", valid_pattern =
 - **`pip_cutoff_to_skip_sumstat`**: A vector of cutoff values for skipping analysis based on PIP values for each sumstat. Default is 0.
 - **`qc_method`**: Quality control method to use. Options are "rss_qc", "dentist", or "slalom" (default: "rss_qc").
 - **`impute`**: Logical; if TRUE, performs imputation for outliers identified in the analysis (default: TRUE).
-- **`impute_opts`**: A list of imputation options including rcond, R2_threshold, and minimum_ld (default: list(rcond = 0.01, R2_threshold = 0.6, minimum_ld = 5)).
+- **`impute_opts`**: A list of imputation options including rcond, R2_threshold, and minimum_ld (default: `list(rcond = 0.01, R2_threshold = 0.6, minimum_ld = 5)`).
 
 
 ```{r, colocboost-analysis}
diff --git a/vignettes/Interpret_ColocBoost_Output.Rmd b/vignettes/Interpret_ColocBoost_Output.Rmd
index 67aab7e..085cec7 100644
--- a/vignettes/Interpret_ColocBoost_Output.Rmd
+++ b/vignettes/Interpret_ColocBoost_Output.Rmd
@@ -145,6 +145,11 @@ res$data_info$outcome_info
 **`cos_details`** provides a detailed information for colocalization events identified by `colocboost`. 
 This section will provide a detailed explanation of the components in `cos_details`.
 
+```{r cos-details}
+names(res$cos_details)
+```
+
+
 #### 3.3.1. Colocalized variants for each CoS (**`cos`**)
 
 - **`cos`**: A list with a detailed information of colocalized variants for each CoS. 
@@ -201,7 +206,9 @@ res$cos_details$cos_top_variables
 ### 3.4. Model information (**`model_info`**)
 
 - **`model_coveraged`**: if the model is converged.
+- **`outcome_model_coveraged`**: if the trait-specific model is converged.
 - **`n_updates`**: number of boosting rounds
+- **`outcome_n_updates`**: number of boosting rounds for each trait.
 - **`jk_update`**: indices of the variants being updated in the model at each boosting round. 
 
 ```{r jk_update}
@@ -254,6 +261,7 @@ Y <- Ind_5traits$Y[1:3]
 X1 <- Heterogeneous_Effect$X[,1:3000]
 Y1 <- Heterogeneous_Effect$Y[,1,drop=F]
 res <- colocboost(X = c(X, list(X1)), Y = c(Y, list(Y1)), output_level = 2)
+names(res$ucos_details)
 ```
 
 #### 3.5.1. Trait-specific (uncolocalized) confidence sets (**`ucos`**)
diff --git a/vignettes/Summary_Statistics_Colocalization.Rmd b/vignettes/Summary_Statistics_Colocalization.Rmd
index 7824dd2..304b6bf 100644
--- a/vignettes/Summary_Statistics_Colocalization.Rmd
+++ b/vignettes/Summary_Statistics_Colocalization.Rmd
@@ -7,7 +7,7 @@ vignette: >
   %\VignetteEncoding{UTF-8}
 ---
 
-```{r, include = FALSE}s
+```{r, include = FALSE}
 knitr::opts_chunk$set(
   collapse = TRUE,
   comment = "#>"