diff --git a/R/checkSex.R b/R/checkSex.R
index 4d03468..e81c66b 100644
--- a/R/checkSex.R
+++ b/R/checkSex.R
@@ -13,13 +13,12 @@
 #' @return Depending on the value of `repair`, either a list containing validation results or a repaired dataframe is returned.
 #' @examples
 #' \dontrun{
-#'   ped <- data.frame(ID = c(1, 2, 3), sex = c("M", "F", "M"))
-#'   checkSex(ped, code_male = "M", verbose = TRUE, repair = FALSE, recode = TRUE)
+#' ped <- data.frame(ID = c(1, 2, 3), sex = c("M", "F", "M"))
+#' checkSex(ped, code_male = "M", verbose = TRUE, repair = FALSE, recode = TRUE)
 #' }
 #' @export
 #'
 checkSex <- function(ped, code_male = NULL, verbose = FALSE, repair = FALSE, recode = FALSE) {
-
   # Initialize a list to store validation results
   validation_results <- list()
 
@@ -71,8 +70,8 @@ checkSex <- function(ped, code_male = NULL, verbose = FALSE, repair = FALSE, rec
 #' @return A dataframe where the sex coding has been repaired.
 #' @examples
 #' \dontrun{
-#'   ped <- data.frame(ID = c(1, 2, 3), sex = c("M", "F", "M"))
-#'   repairSex(ped, code_male = "M", verbose = TRUE)
+#' ped <- data.frame(ID = c(1, 2, 3), sex = c("M", "F", "M"))
+#' repairSex(ped, code_male = "M", verbose = TRUE)
 #' }
 #' @export
 #'
@@ -90,6 +89,6 @@ repairSex <- function(ped, verbose = FALSE, code_male = NULL, recode = TRUE) {
 #' @return A modified version of the input data.frame \code{ped}, containing an additional or modified 'sex_recode' column where the 'sex' values are recoded according to \code{code_male}. NA values in the 'sex' column are preserved.
 #' @keywords internal
 #' @seealso \code{\link{plotPedigree}}
-recodeSex <- function(ped, verbose = FALSE,code_male = NULL) {
+recodeSex <- function(ped, verbose = FALSE, code_male = NULL) {
   checkSex(ped = ped, verbose = verbose, repair = FALSE, code_male = code_male, recode = TRUE)
 }
diff --git a/R/famSizeCal.R b/R/famSizeCal.R
index 1ef2b5a..6dde136 100644
--- a/R/famSizeCal.R
+++ b/R/famSizeCal.R
@@ -69,14 +69,14 @@ famSizeCal <- function(kpc, Ngen, marR) {
     )
     size <- sum(allGens)
   } else {
-    stop("You should never see this message. 
+    stop("You should never see this message.
     If you do, that means that famSizeCal is not working properly.")
   }
   return(size)
 }
 
 #' twinImpute
-#' A function to impute twins in the simulated pedigree \code{data.frame}. 
+#' A function to impute twins in the simulated pedigree \code{data.frame}.
 #' Twins can be imputed by specifying their IDs or by specifying the generation the twin should be imputed.
 #' This is a supplementary function for \code{simulatePedigree}.
 #' @param ped A \code{data.frame} in the same format as the output of \code{simulatePedigree}.
@@ -86,47 +86,47 @@ famSizeCal <- function(kpc, Ngen, marR) {
 #' @return Returns a \code{data.frame} with MZ twins infomration added as a new column.
 #' @export
 
-# A function to impute twins in the simulated pedigree \code{data.frame}. 
+# A function to impute twins in the simulated pedigree \code{data.frame}.
 # Twins can be imputed by specifying their IDs or by specifying the generation the twin should be imputed.
-twinImpute <- function(ped, ID_twin1 = NA_integer_, ID_twin2 = NA_integer_, gen_twin = 2){
+twinImpute <- function(ped, ID_twin1 = NA_integer_, ID_twin2 = NA_integer_, gen_twin = 2) {
   # a support function
   resample <- function(x, ...) x[sample.int(length(x), ...)]
   # Check if the ped is the same format as the output of simulatePedigree
-  if(paste0(colnames(ped), collapse = "") != paste0(c("fam", "ID", "gen", "dadID", "momID", "spt", "sex"), collapse = "")){
+  if (paste0(colnames(ped), collapse = "") != paste0(c("fam", "ID", "gen", "dadID", "momID", "spt", "sex"), collapse = "")) {
     stop("The input pedigree is not in the same format as the output of simulatePedigree")
   }
   ped$MZtwin <- NA_integer_
   # Check if the two IDs are provided
-  if(is.na(ID_twin1) || is.na(ID_twin2)){
+  if (is.na(ID_twin1) || is.na(ID_twin2)) {
     # Check if the generation is provided
-    if(is.na(gen_twin)){
+    if (is.na(gen_twin)) {
       stop("You should provide either the IDs of the twins or the generation of the twins")
     } else {
       # Check if the generation is valid
-      if(gen_twin < 2 || gen_twin > max(ped$gen)){
+      if (gen_twin < 2 || gen_twin > max(ped$gen)) {
         stop("The generation of the twins should be an integer between 2 and the maximum generation in the pedigree")
       } else {
-        idx <- nrow(ped[ped$gen == gen_twin & !is.na(ped$dadID),]) 
-        usedID = c()
+        idx <- nrow(ped[ped$gen == gen_twin & !is.na(ped$dadID), ])
+        usedID <- c()
         # randomly loop through all the indivuduals in the generation until find an individual who is the same sex and shares the same dadID and momID with another individual
-        for (i in 1: idx) {
-          cat("loop",i)
+        for (i in 1:idx) {
+          cat("loop", i)
           # check if i is equal to the number of individuals in the generation
           usedID <- c(usedID, ID_twin1)
-          #print(usedID)
-          if(i < idx){
+          # print(usedID)
+          if (i < idx) {
             # randomly select one individual from the generation
             ID_twin1 <- resample(ped$ID[ped$gen == gen_twin & !(ped$ID %in% usedID) & !is.na(ped$dadID)], 1)
-            #cat("twin1", ID_twin1, "\n")
+            # cat("twin1", ID_twin1, "\n")
             # find one same sex sibling who has the same dadID and momID as the selected individual
-            twin2_Pool = ped$ID[ped$ID != ID_twin1 & ped$gen == gen_twin & ped$sex == ped$sex[ped$ID == ID_twin1] & ped$dadID == ped$dadID[ped$ID == ID_twin1] & ped$momID == ped$momID[ped$ID == ID_twin1]] 
+            twin2_Pool <- ped$ID[ped$ID != ID_twin1 & ped$gen == gen_twin & ped$sex == ped$sex[ped$ID == ID_twin1] & ped$dadID == ped$dadID[ped$ID == ID_twin1] & ped$momID == ped$momID[ped$ID == ID_twin1]]
             # if there is an non-NA value in the twin2_Pool, get rid of the NA value
             if (all(is.na(twin2_Pool))) {
               cat("twin2_Pool is all NA\n")
               next
             } else {
               twin2_Pool <- twin2_Pool[!is.na(twin2_Pool)]
-              ID_twin2 <- resample( twin2_Pool , 1)
+              ID_twin2 <- resample(twin2_Pool, 1)
               break
             }
             # test if the ID_twin2 is missing
@@ -136,10 +136,10 @@ twinImpute <- function(ped, ID_twin1 = NA_integer_, ID_twin2 = NA_integer_, gen_
           } else {
             # randomly select all males or females in the generation and put them in a vector
             selectGender <- ped$ID[ped$gen == gen_twin & ped$sex == resample(c("M", "F"), 1) & !is.na(ped$dadID) & !is.na(ped$momID)]
-            #print(selectGender)
+            # print(selectGender)
             # randomly select two individuals from the vector
             ID_DoubleTwin <- sample(selectGender, 2)
-            #print(ID_DoubleTwin)
+            # print(ID_DoubleTwin)
             # change the second person's dadID and momID to the first person's dadID and momID
             ped$dadID[ped$ID == ID_DoubleTwin[2]] <- ped$dadID[ped$ID == ID_DoubleTwin[1]]
             ped$momID[ped$ID == ID_DoubleTwin[2]] <- ped$momID[ped$ID == ID_DoubleTwin[1]]
@@ -147,16 +147,14 @@ twinImpute <- function(ped, ID_twin1 = NA_integer_, ID_twin2 = NA_integer_, gen_
             ID_twin1 <- ID_DoubleTwin[1]
             ID_twin2 <- ID_DoubleTwin[2]
             break
-
           }
-
+        }
+        # Impute the IDs of the twin in the MZtwin column
+        ped$MZtwin[ped$ID == ID_twin1] <- ID_twin2
+        ped$MZtwin[ped$ID == ID_twin2] <- ID_twin1
       }
-      # Impute the IDs of the twin in the MZtwin column
-      ped$MZtwin[ped$ID == ID_twin1] <- ID_twin2
-      ped$MZtwin[ped$ID == ID_twin2] <- ID_twin1
     }
-  }
-} else {
+  } else {
     # Impute the IDs of the twin in the MZtwin column
     ped$MZtwin[ped$ID == ID_twin1] <- ID_twin2
     ped$MZtwin[ped$ID == ID_twin2] <- ID_twin1
@@ -165,4 +163,3 @@ twinImpute <- function(ped, ID_twin1 = NA_integer_, ID_twin2 = NA_integer_, gen_
   cat("twin2", ID_twin2, "\n")
   return(ped)
 }
-
diff --git a/R/simulatePedigree.R b/R/simulatePedigree.R
index fc09f9f..ea83016 100644
--- a/R/simulatePedigree.R
+++ b/R/simulatePedigree.R
@@ -39,8 +39,11 @@ simulatePedigree <- function(kpc = 3,
   # Calculate the expected family size in each generations
   sizeGens <- allGens(kpc = kpc, Ngen = Ngen, marR = marR)
   famSizeIndex <- 1:sum(sizeGens)
-  if(verbose){print(
-    "Step 1: Let's build the connection within each generation first")}
+  if (verbose) {
+    print(
+      "Step 1: Let's build the connection within each generation first"
+    )
+  }
   for (i in 1:Ngen) {
     idGen <- as.numeric(paste(100, i, 1:sizeGens[i], sep = ""))
     # idGen <- ifelse(i==1,
@@ -169,8 +172,11 @@ simulatePedigree <- function(kpc = 3,
     }
   }
 
-  if(verbose){ print(
-    "Step 2: Let's try to build connection between each two generations")}
+  if (verbose) {
+    print(
+      "Step 2: Let's try to build connection between each two generations"
+    )
+  }
   df_Fam$ifparent <- FALSE
   df_Fam$ifson <- FALSE
   df_Fam$ifdau <- FALSE
@@ -280,8 +286,11 @@ simulatePedigree <- function(kpc = 3,
       df_Ngen <- df_Ngen[order(as.numeric(rownames(df_Ngen))), , drop = FALSE]
       df_Ngen <- df_Ngen[, -ncol(df_Ngen)]
       df_Fam[df_Fam$gen == i, ] <- df_Ngen
-      if(verbose){print(
-      "Step 2.2: mark a group of potential parents in the i-1 th generation")}
+      if (verbose) {
+        print(
+          "Step 2.2: mark a group of potential parents in the i-1 th generation"
+        )
+      }
       df_Ngen <- df_Fam[df_Fam$gen == i - 1, ]
       df_Ngen$ifparent <- FALSE
       df_Ngen$ifson <- FALSE
@@ -308,8 +317,11 @@ simulatePedigree <- function(kpc = 3,
 
       df_Ngen <- df_Ngen[order(as.numeric(rownames(df_Ngen))), , drop = FALSE]
       df_Fam[df_Fam$gen == i - 1, ] <- df_Ngen
-      if(verbose){print(
-      "Step 2.3: connect the i and i-1 th generation")}
+      if (verbose) {
+        print(
+          "Step 2.3: connect the i and i-1 th generation"
+        )
+      }
       if (i == 1) {
         next
       } else {
diff --git a/vignettes/articles/paper.pdf b/vignettes/articles/paper.pdf
index 0f52374..d8540c6 100644
Binary files a/vignettes/articles/paper.pdf and b/vignettes/articles/paper.pdf differ