Visual inspection

The above contingency table is not very large. Therefore, it’s easy to visually inspect and interpret row and column profiles:

• It’s evident that, the housetasks - Laundry, Main_Meal and Dinner - are more frequently done by the “Wife”.
  
• Repairs and driving are dominantly done by the husband
• Holidays are frequently associated with the column “jointly”

Visualize a contingency table using graphical matrix

It’s also possible to visualize a contingency table using the function balloonplot() [in gplots package]. This function draws a graphical matrix where each cell contains a dot whose size reflects the relative magnitude of the corresponding component.

To execute the R code below, you should install the package gplots: install.packages(“gplots”).

library("gplots")
# 1. convert the data as a table
dt <- as.table(as.matrix(housetasks))
# 2. Graph
balloonplot(t(dt), main ="housetasks", xlab ="", ylab="",
            label = FALSE, show.margins = FALSE)

Note that, row and column sums are printed by default in the bottom and right margins, respectively. These values can be hidden using the argument show.margins = FALSE.

Mosaic / association plots

The function mosaicplot() from the built-in R package garphics can be used also to visualize a contingency table.

library("graphics")
mosaicplot(dt, shade = TRUE, las=2,
           main = "housetasks")

• The argument shade is used to color the graph
• The argument las = 2 produces vertical labels

The surface of an element of the mosaic reflects the relative magnitude of its value.

• Blue color indicates that the observed value is higher than the expected value if the data were random
• Red color specifies that the observed value is lower than the expected value if the data were random

From this mosaic plot, it can be seen that the housetasks Laundry, Main_meal, Dinner and breakfeast (blue color) are mainly done by the wife in our example.

It’s also possible to use the package vcd to make a mosaic plot (function mosaic()) or an association plot (function assoc()).

# install.packages("vcd")
library("vcd")
# plot just a subset of the table
assoc(head(dt), shade = T, las=3)

Chi-square statistic

Another method to analyse a frequency table is to use the Chi-square test of independence. The Chi-square test evaluates whether there is a significant dependence between row and column categories.

Chi-square statistic can be easily computed using the function chisq.test() as follow:

chisq <- chisq.test(housetasks)
chisq

    Pearson's Chi-squared test

data:  housetasks
X-squared = 1944.456, df = 36, p-value < 2.2e-16

In our example, the row and the column variables are statistically significantly associated (p-value = 0).

Read more: correspondence analysis basics

Significance of the association between rows and columns

To interpret correspondence analysis, the first step is to evaluate whether there is a significant dependency between the rows and columns.

There are two methods to inspect the significance:

1. Using the trace
2. Using the Chi-square statistic

The trace is the the total inertia of the table (i.e, the sum of the eigenvalues). The square root of the trace is interpreted as the correlation coefficient between rows and columns.

The correlation coefficient is calculated as follow:

eig <- get_eigenvalue(res.ca)
trace <- sum(eig$eigenvalue) 
cor.coef <- sqrt(trace)
cor.coef

[1] 1.055907

Note that, as a rule of thumb 0.2 is the threshold above which the correlation can be considered as important (Bendixen 1995, 576; Healey 2013, 289-290).

In our example, the correlation coefficient is 1.0559074 indicating a strong association between row and column variables.

A more rigorous method is to use the chi-square statistic for examining the association. This appears at the top of the report generated by the function summary.CA(). A high chi-square statistic means strong link between row and column variables.

In our example, the association is highly significant (chi-square: 1944.456, p = 0).

# Chi-square statistics
chi2 <- trace*sum(as.matrix(housetasks))
chi2

[1] 1944.456

# Degree of freedom
df <- (nrow(housetasks) - 1) * (ncol(housetasks) - 1)
# P-value
pval <- pchisq(chi2, df = df, lower.tail = FALSE)
pval

[1] 0

Eigenvalues and scree plot

How many dimensions are sufficient for the data interpretation?

The number of dimensions to retain in the solution can be determined by examining the table of eigenvalues.

As mentioned above, trace is the total sum of eigenvalues. For a given axis, the ratio of the axis eigenvalue to the trace is called the percentage of variance (or total inertia or chi-square value) explained by that axis.

The proportion of variances retained by the different dimensions (axes) can be extracted using the function get_eigenvalue()[in factoextra] as follow :

eigenvalues <- get_eigenvalue(res.ca)
head(round(eigenvalues, 2))

      eigenvalue variance.percent cumulative.variance.percent
Dim.1       0.54            48.69                       48.69
Dim.2       0.45            39.91                       88.60
Dim.3       0.13            11.40                      100.00
Dim.4       0.00             0.00                      100.00

There is no “rule of thumb” to choose the number of dimension to keep for the data interpretation. It depends on the research question and the researcher’s need. For example, if you are satisfied with 80% of the total inertia explained then use the number of dimensions necessary to achieve that.

Another method is to visually inspect the scree plot in which dimensions are ordered decreasingly according the amount of explained inertia.

The function fviz_screeplot() [in factoextra package] can be used to draw the scree plot (the percentages of inertia explained by the CA dimensions):

fviz_screeplot(res.ca)

The point at which the scree plot shows a bend (so called “elbow”) can be considered as indicating an optimal dimensionality.

It’s also possible to calculate an average eigenvalue above which the axis should be kept in the solution.

Any axis with a contribution larger than the maximum of these two percentages should be considered as important and included in the solution for the interpretation of the data (see, Bendixen 1995, 577).

The R code below, draws the scree plot with a red dashed line specifying the average eigenvalue:

fviz_screeplot(res.ca) +
 geom_hline(yintercept=33.33, linetype=2, color="red")

According to the graph above, only dimensions 1 and 2 should be used in the solution. The dimension 3 explains only 11.4% of the total inertia which is below the average eigeinvalue (33.33%) and too little to be kept for further analysis.

Note that, you can use more than 2 dimensions. However, the supplementary dimensions are unlikely to contribute significantly to the interpretation of nature of the association between the rows and columns.

Dimensions 1 and 2 explain approximately 48.7% and 39.9% of the total inertia respectively. This corresponds to a cumulative total of 88.6% of total inertia retained by the 2 dimensions.

The higher the retention, the more subtlety in the original data is retained in the low-dimensional solution (Mike Bendixen, 2003).

Read more about eigenvalues and screeplot: Eigenvalues data visualization

CA scatter plot: Biplot of row and column variables

The function plot.CA()[in FactoMineR] can be used to plot the coordinates of rows and columns presented in the correspondence analysis output.

A simplified format is :

plot.CA(x, axes = c(1,2), col.row = "blue", col.col = "red")

FactoMineR base graph for CA:

plot(res.ca)

It’s also possible to use the function fviz_ca_biplot()[in factoextra package] to draw a nice looking plot:

fviz_ca_biplot(res.ca)

# Change the theme
fviz_ca_biplot(res.ca) +
  theme_minimal()

Read more about fviz_ca_biplot(): fviz_ca_biplot

The graph above is called symetric plot and shows a global pattern within the data. Rows are represented by blue points and columns by red triangles.

The distance between any row points or column points gives a measure of their similarity (or dissimilarity).

Row points with similar profile are closed on the factor map. The same holds true for column points.

The next step for the interpretation is to determine which row and column variables contribute the most in the definition of the different dimensions retained in the model.

Row variables

The function get_ca_row()[in factoextra] is used to extract the results for row variables. This function returns a list containing the coordinates, the cos2, the contribution and the inertia of row variables:

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"      

head(row$coord)

                Dim 1      Dim 2       Dim 3
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

# Default plot
fviz_ca_row(res.ca)

fviz_ca_row(res.ca, col.row="steelblue", shape.row = 15)

# Hide columns
plot(res.ca, invisible="col") 

head(row$contrib)

                Dim 1    Dim 2    Dim 3
Laundry    18.2867003 5.563891 7.968424
Main_meal  12.3888433 4.735523 1.858689
Dinner      5.4713982 1.321022 2.096926
Breakfeast  3.8249284 3.698613 3.069399
Tidying     1.9983518 2.965644 0.488734
Dishes      0.4261663 2.844117 3.634294

library("corrplot")
corrplot(row$contrib, is.corr=FALSE)

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

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

# Total contribution on Dim.1 and Dim.2
fviz_contrib(res.ca, choice = "row", axes = 1:2)

fviz_contrib(res.ca, choice = "row", axes = 1, top = 5)

# Control row point colors using their contribution
# Possible values for the argument col.row are :
  # "cos2", "contrib", "coord", "x", "y"
fviz_ca_row(res.ca, col.row = "contrib")

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

# Control the transparency of rows using their contribution
# Possible values for the argument alpha.var are :
  # "cos2", "contrib", "coord", "x", "y"
fviz_ca_row(res.ca, alpha.row="contrib")+
  theme_minimal()

# Select the top 5 contributing rows
fviz_ca_row(res.ca, alpha.row="contrib", select.row=list(contrib=5))

head(row$cos2)

               Dim 1     Dim 2      Dim 3
Laundry    0.7399874 0.1845521 0.07546047
Main_meal  0.7416028 0.2323593 0.02603787
Dinner     0.7766401 0.1537032 0.06965666
Breakfeast 0.5049433 0.4002300 0.09482670
Tidying    0.4398124 0.5350151 0.02517249
Dishes     0.1181178 0.6461525 0.23572969

library("corrplot")
corrplot(row$cos2, is.corr=FALSE)

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

Column varables

The function get_ca_col()[in factoextra] is used to extract the results for column variables. This function returns a list containing the coordinates, the cos2, the contribution and the inertia of columns variables:

col <- get_ca_col(res.ca)
col

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"      

The result for columns gives the same information as described for rows. For this reason, I’ll just displayed the result for columns in this section without commenting.

head(col$coord)

                  Dim 1      Dim 2       Dim 3
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

fviz_ca_col(res.ca)

# Hide rows
plot(res.ca, invisible="row") 

head(col$contrib)

                Dim 1     Dim 2      Dim 3
Wife        44.462018 10.312237 10.8220753
Alternating  0.103739  2.782794 82.5492464
Husband     54.233879 17.786612  6.1331792
Jointly      1.200364 69.118357  0.4954991

fviz_contrib(res.ca, choice = "col", axes = 1:2)

# 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()

# Control the transparency of rows using their contribution
# Possible values for the argument alpha.col are :
  # "cos2", "contrib", "coord", "x", "y"
fviz_ca_col(res.ca, alpha.col="contrib")

Cos2 : The quality of representation of columns

head(col$cos2)

                  Dim 1     Dim 2       Dim 3
Wife        0.801875947 0.1524482 0.045675847
Alternating 0.004779897 0.1051016 0.890118521
Husband     0.772026244 0.2075420 0.020431728
Jointly     0.020705858 0.9772939 0.002000236

Note that, the value of the cos2 is between 0 and 1. A cos2 closed to 1 corresponds to a column/row variables that are well represented on the factor map.

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.

Symmetric biplot

As mentioned above, the standard plot of correspondence analysis is a symmetric biplot in which both rows (blue points) and columns (red triangles) are represented in the same space using the principal coordinates. These coordinates represent the row and column profiles. In this case, only the distance between row points or the distance between column points can be really interpreted.

With symmetric plot, the inter-distance between rows and columns can’t be interpreted. Only a general statements can be made about the pattern.

fviz_ca_biplot(res.ca)+
  theme_minimal()

Remove the points from the graph, use texts only :

fviz_ca_biplot(res.ca, geom="text")

Note that, in order to interpret the distance between column points and row points, the simplest way is to make an asymmetric plot (Bendixen, 2003). This means that, the column profiles must be presented in row space or vice-versa.

Read more about fviz_ca_biplot(): fviz_ca_biplot

Asymmetric biplot for correspondence analysis

To make an asymetric plot, rows (or columns) points are plotted from the standard co-ordinates (S) and the profiles of the columns (or the rows) are plotted from the principale coordinates (P) (Bendixen 2003).

Depending on the situation, other types of display can be set using the argument map for the function fviz_ca_biplot()[in factoextra]. This is inspired from ca package (Michael Greenacre).

The allowed options for the argument map are:

1. “rowprincipal” or “colprincipal” - these are the so-called asymmetric biplots, with either rows in principal coordinates and columns in standard coordinates, or vice versa (also known as row-metric-preserving or column-metric-preserving respectively).
   

• “rowprincipal”: columns are represented in row space
• “colprincipal”: rows are represented in column space

2. “symbiplot” - both rows and columns are scaled to have variances equal to the singular values (square roots of eigenvalues), which gives a symmetric biplot but does not preserve row or column metrics.

3. “rowgab” or “colgab”: Asymetric maps proposed by Gabriel & Odoroff (1990):

• “rowgab”: rows in principal coordinates and columns in standard coordinates multiplied by the mass.
• “colgab”: columns in principal coordinates and rows in standard coordinates multiplied by the mass.

4. “rowgreen” or “colgreen”: The so-called contribution biplots showing visually the most contributing points (Greenacre 2006b).

• “rowgreen”: rows in principal coordinates and columns in standard coordinates multiplied by square root of the mass.
• “colgreen”: columns in principal coordinates and rows in standard coordinates multiplied by the square root of the mass.

The R code below draw a standard asymetric biplot:

fviz_ca_biplot(res.ca, map ="rowprincipal", arrow = c(TRUE, TRUE))

The argument arrows is a vector of two logicals specifying if the plot should contain points (FALSE, default) or arrows (TRUE). First value sets the rows and the second value sets the columns.

Contribution biplot

In correspondence analysis, biplot is a graphical display of rows and columns in 2 or 3 dimensions.

In the standard symmetric biplot (mentioned in the previous sections), it’s difficult to know the most contributing points to the solution of the CA.

Michael Greenacre proposed a new scaling displayed (called contribution biplot) which incorporates the contribution of points. In this display, points that contribute very little to the solution, are close to the center of the biplot and are relatively unimportant to the interpretation.

A contribution biplot can be drawn using the argument map = “rowgreen” or map = “colgreen”.

Firstly, you have to decide whether to analyse the contributions of rows or columns to the definition of the axes.

In our example we’ll interpret the contribution of rows to the axes. The argument map =“colgreen” is used. In this case, remember that columns are in principal coordinates and rows in standard coordinates multiplied by the square root of the mass. For a given row, the square of the new coordinate on an axis i is exactly the contribution of this row to the inertia of the axis i.

fviz_ca_biplot(res.ca, map ="colgreen",
               arrow = c(TRUE, FALSE))

In the graph above, the position of the column profile points is unchanged relative to that in the conventional biplot. However, the distances of the row points from the plot origin are related to their contributions to the two-dimensional factor map.

The closer an arrow is (in terms of angular distance) to an axis the greater is the contribution of the row category on that axis relative to the other axis. If the arrow is halfway between the two, its row category contributes to the two axes to the same extent.

Plot rows or columns only

It’s also possible to draw the rows or columns only using the function fviz_ca_biplot() (instead of using fviz_ca_row() and fviz_ca_col)

Plot rows only by hiding the columns (invisible =“col”):

fviz_ca_biplot(res.ca, invisible = "col")+
  theme_minimal()

Plot columns only by hiding the rows (invisible =“row”):

fviz_ca_biplot(res.ca, invisible = "row")+
  theme_minimal()

Data

We’ll use the data set children [in FactoMineR package]. It contains 18 rows and 8 columns:

data(children)
# head(children)

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?

In CA terminology, our data contains :

CA with supplementary rows/columns

As mentioned above, supplementary rows and columns are not used for the definition of the principal dimensions. Their coordinates are predicted using only the informations provided by the performed CA on active rows/columns.

To specify supplementary rows/columns, the function CA()[in FactoMineR] can be used as follow :

CA(X,  ncp = 5, row.sup = NULL, col.sup = NULL,
   graph = TRUE)

Example of usage :

res.ca <- CA (children, row.sup = 15:18, col.sup = 6:8,
              graph = FALSE)

The summary of the CA is :

summary(res.ca, nb.dec = 2, ncp = 2)

Call:
rmarkdown::render("factominer-correspondance-analysis.Rmd", encoding = "UTF-8") 

The chi square of independence between the two variables is equal to 98.80159 (p-value =  9.748064e-05 ).

Eigenvalues
                      Dim.1  Dim.2  Dim.3  Dim.4  Dim.5
Variance               0.04   0.01   0.01   0.01   0.00
% of var.             57.04  21.13  11.76  10.06   0.00
Cumulative % of var.  57.04  78.17  89.94 100.00 100.00

Rows (the 10 first)
                      Dim.1   ctr  cos2   Dim.2   ctr  cos2  
money               | -0.12  4.55  0.43 |  0.02  0.37  0.01 |
future              |  0.18 17.57  0.72 | -0.10 14.59  0.22 |
unemployment        | -0.21 22.62  0.87 | -0.07  6.78  0.10 |
circumstances       |  0.40  6.27  0.58 |  0.33 11.54  0.40 |
hard                | -0.25  2.99  0.88 |  0.07  0.59  0.06 |
economic            |  0.35 12.00  0.48 |  0.32 26.60  0.40 |
egoism              |  0.06  0.68  0.07 | -0.03  0.34  0.01 |
employment          | -0.14  2.62  0.16 |  0.22 17.55  0.41 |
finances            | -0.24  2.79  0.28 | -0.21  5.69  0.21 |
war                 |  0.22  2.17  0.75 | -0.07  0.69  0.09 |

Columns
                      Dim.1   ctr  cos2   Dim.2   ctr  cos2  
unqualified         | -0.21 25.11  0.68 | -0.08 10.08  0.10 |
cep                 | -0.14 18.30  0.64 |  0.06  8.08  0.11 |
bepc                |  0.11  6.76  0.31 | -0.03  1.25  0.02 |
high_school_diploma |  0.27 37.98  0.76 | -0.12 20.10  0.15 |
university          |  0.23 11.86  0.31 |  0.32 60.49  0.59 |

Supplementary rows
                      Dim.1 cos2   Dim.2 cos2  
comfort             |  0.21 0.07 |  0.70 0.78 |
disagreement        |  0.15 0.13 |  0.12 0.09 |
world               |  0.52 0.88 |  0.14 0.07 |
to_live             |  0.31 0.14 |  0.50 0.37 |

Supplementary columns
                      Dim.1  cos2   Dim.2  cos2  
thirty              |  0.11  0.14 | -0.06  0.04 |
fifty               | -0.02  0.01 |  0.05  0.09 |
more_fifty          | -0.18  0.29 | -0.05  0.02 |

For the supplementary rows/columns, the coordinates and the quality of representation (cos2) on the factor maps are displayed. They don’t contribute to the dimensions.

Make a biplot of rows and columns

FactomineR base graph:

plot(res.ca)

Use factoextra:

fviz_ca_biplot(res.ca) +
  theme_minimal()

It’s also possible to hide supplementary rows and columns using the argument invisible:

fviz_ca_biplot(res.ca, invisible = c("row.sup", "col.sup") ) +
  theme_minimal()

The argument invisible is also available in FactoMineR base graph.

Visualize supplementary rows

All the results (coordinates and cos2) for the supplementary rows can be extracted as follow :

res.ca$row.sup

$coord
                 Dim 1     Dim 2      Dim 3      Dim 4
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

$cos2
                  Dim 1      Dim 2       Dim 3      Dim 4
comfort      0.06892759 0.77524032 0.007928672 0.14790342
disagreement 0.13132177 0.08692632 0.179649183 0.60210272
world        0.87587685 0.06537746 0.022564054 0.03618163
to_live      0.13899699 0.36853645 0.396830367 0.09563620

Factor map for rows :

fviz_ca_row(res.ca) +
  theme_minimal()

Supplementary rows are shown in darkblue color.

Visualize supplementary columns

Factor map for columns:

fviz_ca_col(res.ca) +
  theme_minimal()

Supplementary columns are shown in darkred.

The results for supplementary columns can be extracted as follow :

res.ca$col.sup

$coord
                 Dim 1       Dim 2       Dim 3       Dim 4
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

$cos2
               Dim 1      Dim 2       Dim 3       Dim 4
thirty     0.1375601 0.04411543 0.131910759 0.060278490
fifty      0.0108695 0.08990298 0.009188167 0.006367804
more_fifty 0.2860989 0.02114509 0.092666735 0.066200714