Project_Germination
Idris Yusuf
2024-03-16
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Importing the dataset
library(readr)
Germination <- read_csv("C:/GGTUAN/DREAMS/Yankee/TSU/MSc_TSU/Spring_2024/CS-583 Data Minning/Project_Data/Germination.csv")## Rows: 21 Columns: 4
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## chr (1): seed
## dbl (3): rownames, n, y
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.View(Germination)
germ_df <- GerminationChecking out values of the datasets - Structures , NA etc
germ_df## # A tibble: 21 × 4
## rownames seed n y
## <dbl> <chr> <dbl> <dbl>
## 1 1 O75 39 10
## 2 2 O75 62 23
## 3 3 O75 81 23
## 4 4 O75 51 26
## 5 5 O75 39 17
## 6 6 O75 6 5
## 7 7 O75 74 53
## 8 8 O75 72 55
## 9 9 O75 51 32
## 10 10 O75 79 46
## # ℹ 11 more rowsView(germ_df)
str(germ_df)## spc_tbl_ [21 × 4] (S3: spec_tbl_df/tbl_df/tbl/data.frame)
## $ rownames: num [1:21] 1 2 3 4 5 6 7 8 9 10 ...
## $ seed : chr [1:21] "O75" "O75" "O75" "O75" ...
## $ n : num [1:21] 39 62 81 51 39 6 74 72 51 79 ...
## $ y : num [1:21] 10 23 23 26 17 5 53 55 32 46 ...
## - attr(*, "spec")=
## .. cols(
## .. rownames = col_double(),
## .. seed = col_character(),
## .. n = col_double(),
## .. y = col_double()
## .. )
## - attr(*, "problems")=<externalptr>dim(germ_df)## [1] 21 4germ_df$seed## [1] "O75" "O75" "O75" "O75" "O75" "O75" "O75" "O75" "O75" "O75" "O75" "O73"
## [13] "O73" "O73" "O73" "O73" "O73" "O73" "O73" "O73" "O73"#Change to factor to be sure it is just 2 as stated in the data notes
germ_df3 <- germ_df
germ_df3$seed <- factor(germ_df3$seed)
str(germ_df3$seed)## Factor w/ 2 levels "O73","O75": 2 2 2 2 2 2 2 2 2 2 ...is.na(germ_df)## rownames seed n y
## [1,] FALSE FALSE FALSE FALSE
## [2,] FALSE FALSE FALSE FALSE
## [3,] FALSE FALSE FALSE FALSE
## [4,] FALSE FALSE FALSE FALSE
## [5,] FALSE FALSE FALSE FALSE
## [6,] FALSE FALSE FALSE FALSE
## [7,] FALSE FALSE FALSE FALSE
## [8,] FALSE FALSE FALSE FALSE
## [9,] FALSE FALSE FALSE FALSE
## [10,] FALSE FALSE FALSE FALSE
## [11,] FALSE FALSE FALSE FALSE
## [12,] FALSE FALSE FALSE FALSE
## [13,] FALSE FALSE FALSE FALSE
## [14,] FALSE FALSE FALSE FALSE
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## [17,] FALSE FALSE FALSE FALSE
## [18,] FALSE FALSE FALSE FALSE
## [19,] FALSE FALSE FALSE FALSE
## [20,] FALSE FALSE FALSE FALSE
## [21,] FALSE FALSE FALSE FALSEsum(is.na(germ_df))## [1] 0table(germ_df$seed)##
## O73 O75
## 10 11prop.table(table(germ_df$seed))##
## O73 O75
## 0.4761905 0.5238095Scaling the dataset to suppress the impact skewness or high values on others
Here we use “euclidean” because it is best suited for our task which are in continous value.
Manhattan is good for binary
germ_df1 <- germ_df[,-2]
germ_df1## # A tibble: 21 × 3
## rownames n y
## <dbl> <dbl> <dbl>
## 1 1 39 10
## 2 2 62 23
## 3 3 81 23
## 4 4 51 26
## 5 5 39 17
## 6 6 6 5
## 7 7 74 53
## 8 8 72 55
## 9 9 51 32
## 10 10 79 46
## # ℹ 11 more rowsdim(germ_df1)## [1] 21 3View(germ_df1)
germ_df1_scale <- as.data.frame(scale(germ_df1))
str(germ_df1_scale)## 'data.frame': 21 obs. of 3 variables:
## $ rownames: num -1.612 -1.45 -1.289 -1.128 -0.967 ...
## $ n : num -0.0231 0.9071 1.6755 0.4622 -0.0231 ...
## $ y : num -0.634 0.175 0.175 0.361 -0.198 ...View(germ_df1_scale)
dist_mat <- dist(germ_df1_scale, method="euclidean")
View(dist_mat)Comparison of cluster methods and Silhouttes coefficients
library(fpc)## Warning: package 'fpc' was built under R version 4.3.3library (cluster)
library (vegan)## Warning: package 'vegan' was built under R version 4.3.3## Loading required package: permute## Warning: package 'permute' was built under R version 4.3.3## Loading required package: lattice## This is vegan 2.6-4h_germ_df1<- hclust(dist_mat, method='average')
h_germ_df1_single<- hclust(dist_mat, method='single')
h_germ_df1_complete<- hclust(dist_mat, method='complete')
#show summary of cluster for method='average'
h_germ_df1##
## Call:
## hclust(d = dist_mat, method = "average")
##
## Cluster method : average
## Distance : euclidean
## Number of objects: 21##fixing margin too large error
#par(mar)
par(mar = c(1, 1, 1, 1))
plot(h_germ_df1)
#show summary of cluster for method='single'
h_germ_df1_single##
## Call:
## hclust(d = dist_mat, method = "single")
##
## Cluster method : single
## Distance : euclidean
## Number of objects: 21par(mar = c(1, 1, 1, 1))
plot(h_germ_df1_single)
#show summary of cluster for method='complete'
h_germ_df1_complete##
## Call:
## hclust(d = dist_mat, method = "complete")
##
## Cluster method : complete
## Distance : euclidean
## Number of objects: 21par(mar = c(1, 1, 1, 1))
plot(h_germ_df1_complete)
### Comparison of silhouette coefficients
################
set.seed(1234)
#d <- dist(scale(iris[ , -5]))
d <- dist(germ_df1_scale, method="euclidean")
methds <- c('complete', 'single', 'average')
avgS <- matrix(NA, ncol=3, nrow=5,
dimnames=list(2:6, methds))
for(k in 2:6)
for(m in seq_along(methds)) {
h <- hclust(d, meth=methds[m])
c <- cutree(h, k)
s <- silhouette(c, d)
avgS[k-1, m] <- mean(s[ , 3])
}
avgS## complete single average
## 2 0.4465135 0.3844378 0.4465135
## 3 0.3734599 0.4561129 0.4561129
## 4 0.4560353 0.4560353 0.4560353
## 5 0.4597658 0.3847092 0.3847092
## 6 0.4340711 0.2971553 0.3738207####################3h_germ_df1<- hclust(dist_mat, method='average')
#show summary of cluster
h_germ_df1##
## Call:
## hclust(d = dist_mat, method = "average")
##
## Cluster method : average
## Distance : euclidean
## Number of objects: 21##fixing margin too large error
#par(mar)
par(mar = c(1, 1, 1, 1))
plot(h_germ_df1)
##Checking out cluster … In this case, it is just 2
hca <- h_germ_df1
plot(hca)
rect.hclust(hca, k = 2, border = "red")
x <- rect.hclust(hca, h = 2, which = c(1,3), border = 2:5)
x## [[1]]
## [1] 6 11 12 13 14 15 16 17 18 19 20 21
##
## [[2]]
## [1] 1 2 3 4 5 9library(dendextend)## Registered S3 method overwritten by 'dendextend':
## method from
## rev.hclust vegan##
## ---------------------
## Welcome to dendextend version 1.17.1
## Type citation('dendextend') for how to cite the package.
##
## Type browseVignettes(package = 'dendextend') for the package vignette.
## The github page is: https://github.com/talgalili/dendextend/
##
## Suggestions and bug-reports can be submitted at: https://github.com/talgalili/dendextend/issues
## You may ask questions at stackoverflow, use the r and dendextend tags:
## https://stackoverflow.com/questions/tagged/dendextend
##
## To suppress this message use: suppressPackageStartupMessages(library(dendextend))
## ---------------------##
## Attaching package: 'dendextend'## The following object is masked from 'package:permute':
##
## shuffle## The following object is masked from 'package:stats':
##
## cutreeavg_dendogram_obj <- as.dendrogram(h_germ_df1)
avg_col_dendogram <- color_branches(avg_dendogram_obj, h=2)
plot(avg_col_dendogram)
hac_cut <- cutree(h_germ_df1, k=2)Scale and compare original data to clustering classfication
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, uniongerm_df_cl <- mutate(germ_df1_scale, cluster=hac_cut)
germ_df_cl## rownames n y cluster
## 1 -1.6116459 -0.02311077 -0.6338642 1
## 2 -1.4504813 0.90709781 0.1747569 1
## 3 -1.2893167 1.67553098 0.1747569 1
## 4 -1.1281521 0.46221544 0.3613618 1
## 5 -0.9669876 -0.02311077 -0.1984528 1
## 6 -0.8058230 -1.35775786 -0.9448723 2
## 7 -0.6446584 1.39242402 2.0408057 1
## 8 -0.4834938 1.31153632 2.1652089 1
## 9 -0.3223292 0.46221544 0.7345716 1
## 10 -0.1611646 1.59464328 1.6053943 1
## 11 0.0000000 -1.07465091 -0.6338642 2
## 12 0.1611646 -0.95331935 -0.7582674 2
## 13 0.3223292 -0.38710543 -0.6338642 2
## 14 0.4834938 -0.46799314 -0.7582674 2
## 15 0.6446584 0.21955234 0.1747569 2
## 16 0.8058230 -1.43864557 -1.2558804 2
## 17 0.9669876 -1.11509476 -1.0692756 2
## 18 1.1281521 0.05777693 0.1125553 2
## 19 1.2893167 -0.38710543 -0.3228561 2
## 20 1.4504813 0.46221544 0.7345716 2
## 21 1.6116459 -1.31731401 -1.0692756 2## Count Cluster classification
count(germ_df_cl, cluster)## cluster n
## 1 1 9
## 2 2 12## compare with Original Dataset classification
count(germ_df, seed)## # A tibble: 2 × 2
## seed n
## <chr> <int>
## 1 O73 10
## 2 O75 11Draw the plot
library(ggplot2)
##ggplot(seeds_df_cl, aes(x=seeds_df_cl$area, y=seeds_df_cl$perimeter , color=factor(cluster))) + geom_point()
ggplot(germ_df_cl, aes(x=n , y=y , color=factor(cluster))) + geom_point()
#Use of `seeds_df_cl$perimeter` is discouraged.ℹ Use `perimeter` instead.
##ggplot(seeds_df_cl, aes(x=area, y=perimeter , color=factor(cluster))) + geom_point()
ggplot(germ_df, aes(x=n, y=y , color=factor(germ_df$seed ))) + geom_point()
compute Dunn’s index
#Install clvalid
library(clValid)
dunn(dist_mat, hac_cut) ## [1] 0.3412918dist_mat[1:21]## [1] 1.2430326 1.9087032 1.2082146 0.7779251 1.5897671 3.1768951 3.2998213
## [8] 1.9417764 3.1201447 1.9243542 2.0058966 1.9679309 2.1454619 2.4090792
## [15] 2.8696339 2.8339659 2.8408059 2.9402045 3.3889204 3.5005936 0.7851519###KNN
set.seed(1234)
germ_knn <- kmeans( germ_df[,-2], center=2, iter.max=1)
germ_knn## K-means clustering with 2 clusters of sizes 12, 9
##
## Cluster means:
## rownames n y
## 1 12.750000 22.08333 9.25000
## 2 8.666667 62.88889 34.77778
##
## Clustering vector:
## [1] 1 2 2 2 1 1 2 2 2 2 1 1 1 1 2 1 1 1 1 2 1
##
## Within cluster sum of squares by cluster:
## [1] 2979.417 3186.444
## (between_SS / total_SS = 66.1 %)
##
## Available components:
##
## [1] "cluster" "centers" "totss" "withinss" "tot.withinss"
## [6] "betweenss" "size" "iter" "ifault"germ_knn$betweenss## [1] 12000.52germ_knn$withinss## [1] 2979.417 3186.444germ_knn$tot.withinss## [1] 6165.861table(germ_knn$cluster, germ_df$seed)##
## O73 O75
## 1 8 4
## 2 2 7cm <- table(germ_knn$cluster, germ_df$seed)
1 - sum(diag(cm))/sum(cm)## [1] 0.2857143#install.packages("clusterCrit")
#library(clusterCrit)
#dunns_coefficient <- dunn(germ_df[,-2], germ_knn$cluster)
# Print the result
#print(paste("Dunn's Coefficient:", dunns_coefficient))
#Compare with this
#Install clvalid
library(clValid)
dunn(dist_mat, hac_cut) ## [1] 0.3412918dist_mat[1:21]## [1] 1.2430326 1.9087032 1.2082146 0.7779251 1.5897671 3.1768951 3.2998213
## [8] 1.9417764 3.1201447 1.9243542 2.0058966 1.9679309 2.1454619 2.4090792
## [15] 2.8696339 2.8339659 2.8408059 2.9402045 3.3889204 3.5005936 0.7851519Note here that germ_df_train_labels1 and germ_df_train_labels2 are built to have both factors in our Train and Test Dataset because, with one factors it will not give real spectrum of test
####PREDICTION
prop.table(table(germ_df$seed))##
## O73 O75
## 0.4761905 0.5238095#summary( wd1$radius_mean, wd1$area_mean, wd1$smoothness_mean)
summary(germ_df1[c("rownames", "n", "y")])## rownames n y
## Min. : 1 Min. : 4.00 Min. : 0.00
## 1st Qu.: 6 1st Qu.:16.00 1st Qu.: 8.00
## Median :11 Median :39.00 Median :17.00
## Mean :11 Mean :39.57 Mean :20.19
## 3rd Qu.:16 3rd Qu.:51.00 3rd Qu.:26.00
## Max. :21 Max. :81.00 Max. :55.00normalize <- function (x){
return ((x - min(x))/(max(x) - min(x)))
}
germ_df1## # A tibble: 21 × 3
## rownames n y
## <dbl> <dbl> <dbl>
## 1 1 39 10
## 2 2 62 23
## 3 3 81 23
## 4 4 51 26
## 5 5 39 17
## 6 6 6 5
## 7 7 74 53
## 8 8 72 55
## 9 9 51 32
## 10 10 79 46
## # ℹ 11 more rowsgerm_df1_n <- as.data.frame(lapply(germ_df1,normalize))
germ_df1_n## rownames n y
## 1 0.00 0.45454545 0.18181818
## 2 0.05 0.75324675 0.41818182
## 3 0.10 1.00000000 0.41818182
## 4 0.15 0.61038961 0.47272727
## 5 0.20 0.45454545 0.30909091
## 6 0.25 0.02597403 0.09090909
## 7 0.30 0.90909091 0.96363636
## 8 0.35 0.88311688 1.00000000
## 9 0.40 0.61038961 0.58181818
## 10 0.45 0.97402597 0.83636364
## 11 0.50 0.11688312 0.18181818
## 12 0.55 0.15584416 0.14545455
## 13 0.60 0.33766234 0.18181818
## 14 0.65 0.31168831 0.14545455
## 15 0.70 0.53246753 0.41818182
## 16 0.75 0.00000000 0.00000000
## 17 0.80 0.10389610 0.05454545
## 18 0.85 0.48051948 0.40000000
## 19 0.90 0.33766234 0.27272727
## 20 0.95 0.61038961 0.58181818
## 21 1.00 0.03896104 0.05454545summary(germ_df1_n[c("rownames", "n", "y")])## rownames n y
## Min. :0.00 Min. :0.0000 Min. :0.0000
## 1st Qu.:0.25 1st Qu.:0.1558 1st Qu.:0.1455
## Median :0.50 Median :0.4545 Median :0.3091
## Mean :0.50 Mean :0.4620 Mean :0.3671
## 3rd Qu.:0.75 3rd Qu.:0.6104 3rd Qu.:0.4727
## Max. :1.00 Max. :1.0000 Max. :1.0000germ_df$seed <- factor(germ_df$seed, levels=c(073, 075), labels=c("073", "075"))
str(germ_df$seed)## Factor w/ 2 levels "073","075": NA NA NA NA NA NA NA NA NA NA ...dim(germ_df$seed)## NULLset.seed(1234)
germ_df_train1 <- germ_df1_n[1:9,]
germ_df_train2 <- germ_df1_n[14:21,]
germ_df_train <- rbind(germ_df_train1, germ_df_train2)
germ_df_test <- germ_df1_n[10:13,]
#table1 <- data.frame(ID = 1:3, Name = c("John", "Alice", "Bob"))
#table2 <- data.frame(ID = 4:6, Name = c("Charlie", "David", "Emily"))
#combined_table <- rbind(table1, table2)
#print(combined_table)
germ_df <- Germination
germ_df_train_labels1 <- germ_df[1:9,2]
germ_df_train_labels2 <- germ_df[14:21,2]
germ_df_train_labels <- rbind(germ_df_train_labels1, germ_df_train_labels2)
germ_df_test_labels <- germ_df[10:13,2]
germ_df_test_labels## # A tibble: 4 × 1
## seed
## <chr>
## 1 O75
## 2 O75
## 3 O73
## 4 O73unique(germ_df_train_labels)## # A tibble: 2 × 1
## seed
## <chr>
## 1 O75
## 2 O73germ_df_train_labels$seed## [1] "O75" "O75" "O75" "O75" "O75" "O75" "O75" "O75" "O75" "O73" "O73" "O73"
## [13] "O73" "O73" "O73" "O73" "O73"dim(germ_df_train_labels)## [1] 17 1head(germ_df_train_labels)## # A tibble: 6 × 1
## seed
## <chr>
## 1 O75
## 2 O75
## 3 O75
## 4 O75
## 5 O75
## 6 O75head(germ_df_test_labels)## # A tibble: 4 × 1
## seed
## <chr>
## 1 O75
## 2 O75
## 3 O73
## 4 O73germ_df_test_labels## # A tibble: 4 × 1
## seed
## <chr>
## 1 O75
## 2 O75
## 3 O73
## 4 O73germ_df[,2]## # A tibble: 21 × 1
## seed
## <chr>
## 1 O75
## 2 O75
## 3 O75
## 4 O75
## 5 O75
## 6 O75
## 7 O75
## 8 O75
## 9 O75
## 10 O75
## # ℹ 11 more rowsprint(germ_df$seed[1:21])## [1] "O75" "O75" "O75" "O75" "O75" "O75" "O75" "O75" "O75" "O75" "O75" "O73"
## [13] "O73" "O73" "O73" "O73" "O73" "O73" "O73" "O73" "O73"Comparing K values
### Explaination
### The team used 4 test and 3 passed while 1 prediction failed. the 2 "O73" were categorized as such but 1 "075" were categorized correctly as stated in the confussion matrix - CrossTable above.
### For the sake of the analysis we tried different values of K from 1, 2, 4 and got the same output at the CrossTable but when K was increased to 21 2 passed and 2 failed out our 4 tests
#K=21
library(class)
germ_df_test_pred <- knn(train=germ_df_train, test=germ_df_test,cl=germ_df_train_labels$seed, k=21)## Warning in knn(train = germ_df_train, test = germ_df_test, cl =
## germ_df_train_labels$seed, : k = 21 exceeds number 17 of patternsgerm_df_test_pred## [1] O75 O75 O75 O75
## Levels: O73 O75dim(germ_df_test_pred)## NULLgerm_df_train## rownames n y
## 1 0.00 0.45454545 0.18181818
## 2 0.05 0.75324675 0.41818182
## 3 0.10 1.00000000 0.41818182
## 4 0.15 0.61038961 0.47272727
## 5 0.20 0.45454545 0.30909091
## 6 0.25 0.02597403 0.09090909
## 7 0.30 0.90909091 0.96363636
## 8 0.35 0.88311688 1.00000000
## 9 0.40 0.61038961 0.58181818
## 14 0.65 0.31168831 0.14545455
## 15 0.70 0.53246753 0.41818182
## 16 0.75 0.00000000 0.00000000
## 17 0.80 0.10389610 0.05454545
## 18 0.85 0.48051948 0.40000000
## 19 0.90 0.33766234 0.27272727
## 20 0.95 0.61038961 0.58181818
## 21 1.00 0.03896104 0.05454545germ_df_test## rownames n y
## 10 0.45 0.9740260 0.8363636
## 11 0.50 0.1168831 0.1818182
## 12 0.55 0.1558442 0.1454545
## 13 0.60 0.3376623 0.1818182germ_df_train_labels$seed## [1] "O75" "O75" "O75" "O75" "O75" "O75" "O75" "O75" "O75" "O73" "O73" "O73"
## [13] "O73" "O73" "O73" "O73" "O73"#Instal package gmodels and import the library(gmodels)
germ_df_test_pred## [1] O75 O75 O75 O75
## Levels: O73 O75germ_df_test_labels$seed## [1] "O75" "O75" "O73" "O73"library(gmodels)## Warning: package 'gmodels' was built under R version 4.3.3CrossTable(x=germ_df_test_labels$seed , y=germ_df_test_pred , prop.chisq=FALSE)##
##
## Cell Contents
## |-------------------------|
## | N |
## | N / Table Total |
## |-------------------------|
##
##
## Total Observations in Table: 4
##
##
## | germ_df_test_pred
## germ_df_test_labels$seed | O75 | Row Total |
## -------------------------|-----------|-----------|
## O73 | 2 | 2 |
## | 0.500 | |
## -------------------------|-----------|-----------|
## O75 | 2 | 2 |
## | 0.500 | |
## -------------------------|-----------|-----------|
## Column Total | 4 | 4 |
## -------------------------|-----------|-----------|
##
## # K = 2
library(class)
germ_df_test_pred <- knn(train=germ_df_train, test=germ_df_test,cl=germ_df_train_labels$seed, k=2)
germ_df_test_pred## [1] O75 O73 O73 O73
## Levels: O73 O75dim(germ_df_test_pred)## NULLgerm_df_train## rownames n y
## 1 0.00 0.45454545 0.18181818
## 2 0.05 0.75324675 0.41818182
## 3 0.10 1.00000000 0.41818182
## 4 0.15 0.61038961 0.47272727
## 5 0.20 0.45454545 0.30909091
## 6 0.25 0.02597403 0.09090909
## 7 0.30 0.90909091 0.96363636
## 8 0.35 0.88311688 1.00000000
## 9 0.40 0.61038961 0.58181818
## 14 0.65 0.31168831 0.14545455
## 15 0.70 0.53246753 0.41818182
## 16 0.75 0.00000000 0.00000000
## 17 0.80 0.10389610 0.05454545
## 18 0.85 0.48051948 0.40000000
## 19 0.90 0.33766234 0.27272727
## 20 0.95 0.61038961 0.58181818
## 21 1.00 0.03896104 0.05454545germ_df_test## rownames n y
## 10 0.45 0.9740260 0.8363636
## 11 0.50 0.1168831 0.1818182
## 12 0.55 0.1558442 0.1454545
## 13 0.60 0.3376623 0.1818182germ_df_train_labels$seed## [1] "O75" "O75" "O75" "O75" "O75" "O75" "O75" "O75" "O75" "O73" "O73" "O73"
## [13] "O73" "O73" "O73" "O73" "O73"#Instal package gmodels and import the library(gmodels)
germ_df_test_pred## [1] O75 O73 O73 O73
## Levels: O73 O75germ_df_test_labels$seed## [1] "O75" "O75" "O73" "O73"library(gmodels)
CrossTable(x=germ_df_test_labels$seed , y=germ_df_test_pred , prop.chisq=FALSE)##
##
## Cell Contents
## |-------------------------|
## | N |
## | N / Row Total |
## | N / Col Total |
## | N / Table Total |
## |-------------------------|
##
##
## Total Observations in Table: 4
##
##
## | germ_df_test_pred
## germ_df_test_labels$seed | O73 | O75 | Row Total |
## -------------------------|-----------|-----------|-----------|
## O73 | 2 | 0 | 2 |
## | 1.000 | 0.000 | 0.500 |
## | 0.667 | 0.000 | |
## | 0.500 | 0.000 | |
## -------------------------|-----------|-----------|-----------|
## O75 | 1 | 1 | 2 |
## | 0.500 | 0.500 | 0.500 |
## | 0.333 | 1.000 | |
## | 0.250 | 0.250 | |
## -------------------------|-----------|-----------|-----------|
## Column Total | 3 | 1 | 4 |
## | 0.750 | 0.250 | |
## -------------------------|-----------|-----------|-----------|
##
##