Extract the eigenvalues

Eigenvalues measure the amount of variation retained by a principal axis :

summary(res.ca)

Class: coa dudi
Call: dudi.coa(df = housetasks, scannf = FALSE, nf = 5)

Total inertia: 1.115

Eigenvalues:
    Ax1     Ax2     Ax3 
 0.5429  0.4450  0.1270 

Projected inertia (%):
    Ax1     Ax2     Ax3 
  48.69   39.91   11.40 

Cumulative projected inertia (%):
    Ax1   Ax1:2   Ax1:3 
  48.69   88.60  100.00 

You can also use the function get_eigenvalue() [in factoextra package] to extract the eigenvalues :

eig.val <- get_eigenvalue(res.ca)
head(eig.val)

      eigenvalue variance.percent cumulative.variance.percent
Dim.1  0.5428893         48.69222                    48.69222
Dim.2  0.4450028         39.91269                    88.60491
Dim.3  0.1270484         11.39509                   100.00000

Make a scree plot using ade4 base graphics

The function screeplot() can be used to draw the amount of inertia (variance) retained by the dimensions.

A simplified format is:

screeplot(x, ncps = length(x$eig), type = c("barplot", "lines"))

Example of usage :

screeplot(res.ca, main ="Screeplot - Eigenvalues")

~89% of the information contained in the data are retained by the first two dimensions.

Make the scree plot using factoextra

It’s also possible to use the function fviz_screeplot() [in factoextra] to make the scree plot. In the R code below, we’ll draw the percentage of variances retained by each component :

fviz_screeplot(res.ca, ncp=3)

Read more about eigenvalues and screeplot: Eigenvalues data visualization

Coordinates of rows

The coordinates of the rows on the factor map are :

head(res.ca$li)

               Axis1      Axis2       Axis3
Laundry    0.9918368 -0.4953220 -0.31672897
Main_meal  0.8755855 -0.4901092 -0.16406487
Dinner     0.6925740 -0.3081043 -0.20741377
Breakfeast 0.5086002 -0.4528038  0.22040453
Tidying    0.3938084  0.4343444 -0.09421375
Dishes     0.1889641  0.4419662  0.26694926

Use the function fviz_ca_row() [in factoextra package] to visualize only row points:

# Default plot
fviz_ca_row(res.ca)

s.label(res.ca$li, xax = 1, yax = 2)

Contribution of rows to the dimensions

The cos2 and the contributions of rows / columns are calculated using the function inertia.dudi() as follow :

inertia <- inertia.dudi(res.ca, row.inertia = TRUE,
                        col.inertia = TRUE)

Note that, the contributions and the cos2 are printed in 1/10 000. The sign is the sign of the coordinates.

The contributions can be printed in % as follow :

# absolute contribution of columns
contrib <- inertia$col.abs/100
head(contrib)

            Comp1 Comp2 Comp3
Wife        44.46 10.31 10.82
Alternating  0.10  2.78 82.55
Husband     54.23 17.79  6.13
Jointly      1.20 69.12  0.50

Recall that, as mentioned above, the simplest way is to use the function get_ca_row() [in factoextra package]. It provides a list of matrices containing all the results for the active rows(coordinates, squared cosine and contributions).

row <- get_ca_row(res.ca)
row

Correspondence Analysis - Results for rows
 ===================================================
  Name       Description                
1 "$coord"   "Coordinates for the rows" 
2 "$cos2"    "Cos2 for the rows"        
3 "$contrib" "contributions of the rows"
4 "$inertia" "Inertia of the rows"      

# Row contributions
row$contrib

           Dim.1 Dim.2 Dim.3
Laundry    18.29  5.56  7.97
Main_meal  12.39  4.74  1.86
Dinner      5.47  1.32  2.10
Breakfeast  3.82  3.70  3.07
Tidying     2.00  2.97  0.49
Dishes      0.43  2.84  3.63
Shopping    0.18  2.52  2.22
Official    0.52  0.80 36.94
Driving     8.08  7.65 18.60
Finances    0.88  5.56  0.06
Insurance   6.15  4.02  5.25
Repairs    40.73 15.88 16.60
Holidays    1.08 42.45  1.21

The row category with the largest value, contribute the most to the definition of the dimensions.

The function fviz_contrib()[in factoextra] can be used to visualize the most important row variables:

# Contributions of rows on Dim.1
fviz_contrib(res.ca, choice = "row", axes = 1)

The row items Repairs, Laundry, Main_meal and Driving contribute the most in the definition of the first axis.

# Contributions of rows on Dim.2
fviz_contrib(res.ca, choice = "row", axes = 2)

Read more about fviz_contrib(): fviz_contrib

Using factoextra package, the color of rows can be automatically controlled by the value of their contributions

fviz_ca_row(res.ca, col.row="contrib")+
scale_color_gradient2(low="white", mid="blue", 
                      high="red", midpoint=10)+theme_minimal()

The graph above highlight the most important rows in the correspondence analysis solution.

Read more about fviz_ca_row(): fviz_ca_row

Cos2 : quality of representation of rows on the factor map

• A high cos2 indicates a good representation of the rows on the factor map.
  
• A low cos2 indicates that the variable is not perfectly represented by the principal dimensions.

The cos2 of the rows are (factoextra code) :

head(row$cos2)

            Dim.1  Dim.2  Dim.3
Laundry    0.7400 0.1846 0.0755
Main_meal  0.7416 0.2324 0.0260
Dinner     0.7766 0.1537 0.0697
Breakfeast 0.5049 0.4002 0.0948
Tidying    0.4398 0.5350 0.0252
Dishes     0.1181 0.6462 0.2357

Note that, the ade4 code is:

# relative contributions of rows
cos2 <- abs(inertia$row.rel/10000)
head(cos2)

The values of the cos2 are comprised between 0 and 1.

The function fviz_cos2()[in factoextra] can be used to draw a bar plot of rows cos2:

# Cos2 of rows on Dim.1 and Dim.2
fviz_cos2(res.ca, choice = "row", axes = 1:2)

Note that, all row points except Official are well represented by the first two dimensions. The position of the point corresponding the item Official on the scatter plot should be interpreted with some caution.

Using factoextra package, the color of rows can be automatically controlled by the value of their cos2.

fviz_ca_row(res.ca, col.row="cos2")+
scale_color_gradient2(low="white", mid="blue", 
      high="red", midpoint=0.5) + theme_minimal()

Read more about fviz_cos2(): fviz_cos2

Coordinates of columns

The coordinates of the columns on the factor maps can be extracted as follow :

# ade4 code
head(res.ca$co)

                  Comp1      Comp2       Comp3
Wife         0.83762154 -0.3652207 -0.19991139
Alternating  0.06218462 -0.2915938  0.84858939
Husband     -1.16091847 -0.6019199 -0.18885924
Jointly     -0.14942609  1.0265791 -0.04644302

Use the function fviz_ca_col() [in factoextra] to visualize only column points:

fviz_ca_col(res.ca)

s.label(res.ca$co, xax = 1, yax = 2)

Contribution of columns

The contributions can be printed in % as follow :

# absolute contributions of columns
# ade4 code
contrib <- inertia$col.abs/100
head(contrib)

            Comp1 Comp2 Comp3
Wife        44.46 10.31 10.82
Alternating  0.10  2.78 82.55
Husband     54.23 17.79  6.13
Jointly      1.20 69.12  0.50

It’s simple to use the function get_ca_col() [from factoextra package]. factoextra provides, a list of matrices containing all the results for the active columns (coordinates, squared cosine and contributions)./span>

columns <- get_ca_col(res.ca)
columns

Correspondence Analysis - Results for columns
 ===================================================
  Name       Description                   
1 "$coord"   "Coordinates for the columns" 
2 "$cos2"    "Cos2 for the columns"        
3 "$contrib" "contributions of the columns"
4 "$inertia" "Inertia of the columns"      

# Contributions of columns
head(columns$contrib)

            Dim.1 Dim.2 Dim.3
Wife        44.46 10.31 10.82
Alternating  0.10  2.78 82.55
Husband     54.23 17.79  6.13
Jointly      1.20 69.12  0.50

Use the function fviz_contrib()[factoextra package] to visualize the most contributing columns :

# Contributions of columns on Dim.1
fviz_contrib(res.ca, choice = "col", axes = 1)

# Contributions of columns on Dim.2
fviz_contrib(res.ca, choice = "col", axes = 2)

Read more about fviz_contrib(): fviz_contrib

Draw a scatter plot of column points and highlight columns according to the amount of their contributions. The function fviz_ca_col() [in factoextra] is used:

# Control column point colors using their contribution
# Possible values for the argument col.col are :
  # "cos2", "contrib", "coord", "x", "y"
fviz_ca_col(res.ca, col.col="contrib")+
scale_color_gradient2(low="white", mid="blue", 
                      high="red", midpoint=24.5)+theme_minimal()

Cos2 : The quality of representation of columns

# relative contributions of columns
cos2 <- abs(inertia$col.rel)/10000
head(cos2)

             Comp1  Comp2  Comp3 con.tra
Wife        0.8019 0.1524 0.0457  0.2700
Alternating 0.0048 0.1051 0.8901  0.1057
Husband     0.7720 0.2075 0.0204  0.3421
Jointly     0.0207 0.9773 0.0020  0.2823

The function fviz_cos2()[in factoextra] can be used to draw a bar plot of columns cos2:

# Cos2 of columns on Dim.1 and Dim.2
fviz_cos2(res.ca, choice = "col", axes = 1:2)

Note that, only the column item Alternating is not very well displayed on the first two dimensions. The position of this item must be interpreted with caution in the space formed by dimensions 1 and 2.

Read more about fviz_cos2(): fviz_cos2

Data

We’ll use the data set children available on STHDA website. It contains 18 rows and 8 columns:

ff <- "http://www.sthda.com/sthda/RDoc/data/ca-children.txt"
children <- read.table(file = ff, sep ="\t", 
                       header = TRUE, row.names = 1)

The data used here is a contingency table describing the answers given by different categories of people to the following question: What are the reasons that can make hesitate a woman or a couple to have children? (source of the data: FactoMineR package)

In CA terminology, our data contains :

R functions

The functions suprow() and supcol() [in ade4 package] are used to calculate the coordinates of supplementary rows and columns, respectively.

The simplified formats are :

# For supplementary rows
suprow(x, Xsup)

# For supplementary columns
supcol(x, Xsup)

Supplementary rows

# Data for the supplementary rows
row.sup <- children[15:18, 1:5, drop = FALSE]
head(row.sup)

             unqualified cep bepc high_school_diploma university
comfort                2   4    3                   1          4
disagreement           2   8    2                   5          2
world                  1   5    4                   6          3
to_live                3   3    1                   3          4

STEP 1/2 - CA using active rows/columns:

d.active <- children[1:14, 1:5]
res.ca <- dudi.coa(d.active, scannf = FALSE, nf =5)

STEP 2/2 - Predict the coordinates of the supplementary rows:

row.sup.ca <- suprow(res.ca, row.sup)
names(row.sup.ca)

[1] "tabsup" "lisup" 

# coordinates 
row.sup.coord <- row.sup.ca$lisup
head(row.sup.coord)

                 Axis1     Axis2      Axis3      Axis4
comfort      0.2096705 0.7031677 0.07111168  0.3071354
disagreement 0.1462777 0.1190106 0.17108916 -0.3132169
world        0.5233045 0.1429707 0.08399269 -0.1063597
to_live      0.3083067 0.5020193 0.52093397  0.2557357

How to visualize supplementary rows on the factor map?

The function fviz_add() is used :

# Plot of active rows
p <- fviz_ca_row(res.ca)
# Add supplementary rows
fviz_add(p, row.sup.coord, color ="darkgreen")

Supplementary columns

# Data for the supplementary quantitative variables
col.sup <- children[1:14, 6:8, drop = FALSE]
head(col.sup)

              thirty fifty more_fifty
money             59    66         70
future           115   117         86
unemployment      79    88        177
circumstances      9     8          5
hard               2    17         18
economic          18    19         17

Recall that, rows 15:18 are supplementary rows. We don’t want them in this current analysis. This is why, I extracted only rows 1:14.

Predict the coordinates of the supplementary columns :

col.sup.ca <- supcol(res.ca, col.sup)
names(col.sup.ca)

[1] "tabsup" "cosup" 

# coordinates 
col.sup.coord <- col.sup.ca$cosup
head(col.sup.coord)

                 Comp1       Comp2       Comp3       Comp4
thirty      0.10541339 -0.05969594 -0.10322613  0.06977996
fifty      -0.01706444  0.04907657 -0.01568923 -0.01306117
more_fifty -0.17706810 -0.04813788  0.10077299 -0.08517528

Visualize supplementary columns on the factor map using factoextra :

# Plot of active columns
p <- fviz_ca_col(res.ca)
# Add supplementary active variables
fviz_add(p, col.sup.coord , color ="darkgreen")