Correspondence Analysis: Theory and Practice
This article presents the theory and the mathematical procedures behind correspondence Analysis. We write all the formula in a very simple format so that beginners can understand the methods.
Contents:
Related Book:
Practical Guide to Principal Component Methods in R
Required packages
FactoMineR
for computing CAfactoextra
for visualizing the results
Install:
install.packages("FactoMineR")
install.packages("factoextra")
Load:
library("FactoMineR")
library("factoextra")
Data format
 Demo data:
housetasks
[in factoextra package]  Data contents: housetasks repartition in the couple.
 rows are the different tasks
 values are the frequencies of the tasks done by the i) wife only, ii) alternatively, iii) husband only, iv) jointly.
 Data illustration:
Load the data:
library(factoextra)
data(housetasks)
# head(housetasks)
As the above contingency table is not very large, with a quick visual examination it can be seen that:
 The house tasks Laundry, Main_Meal and Dinner are dominant in the column Wife
 Repairs are dominant in the column Husband
 Holidays are dominant in the column Jointly
Visualize a contingency table
To easily interpret the contingency table, a graphical matrix can be drawn using the function balloonplot
() [gplots package]. In this graph, each cell contains a dot whose size reflects the relative magnitude of the value it contains.
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)
For a very large contingency table, the visual interpretation would be very hard. Other methods are required such as correspondence analysis.
Key terms
 Row margins: Row sums (
row.sum
)  Column margins: Column sums (
col.sum
)  grand total: Total sum of all values in the contingency table.
# Row margins
row.sum < apply(housetasks, 1, sum)
head(row.sum)
# Column margins
col.sum < apply(housetasks, 2, sum)
head(col.sum)
# grand total
n < sum(housetasks)
The contingency table with row and column margins are shown below:

Wife 
Alternating 
Husband 
Jointly 
TOTAL 
Laundry 
156 
14 
2 
4 
176 
Main_meal 
124 
20 
5 
4 
153 
Dinner 
77 
11 
7 
13 
108 
Breakfeast 
82 
36 
15 
7 
140 
Tidying 
53 
11 
1 
57 
122 
Dishes 
32 
24 
4 
53 
113 
Shopping 
33 
23 
9 
55 
120 
Official 
12 
46 
23 
15 
96 
Driving 
10 
51 
75 
3 
139 
Finances 
13 
13 
21 
66 
113 
Insurance 
8 
1 
53 
77 
139 
Repairs 
0 
3 
160 
2 
165 
Holidays 
0 
1 
6 
153 
160 
TOTAL 
600 
254 
381 
509 
1744 
 Row margins: light gray
 Column margins: light blue
 The grand total (the total of all values in the table): pink
Row variables
To compare rows, we can analyse their profiles in order to identify similar row variables.
Row profiles
The profile of a given row is calculated by taking each row point and dividing by its margin (i.e, the sum of all row points). The formula is:
\[ row.profile = \frac{row}{row.sum} \]
For example the profile of the row point Laundry/wife is P = 156/176 = 88.6%
.
The R code below can be used to compute row profiles:
row.profile < housetasks/row.sum
# head(row.profile)

Wife 
Alternating 
Husband 
Jointly 
TOTAL 
Laundry 
0.8864 
0.07955 
0.0114 
0.0227 
1 
Main_meal 
0.8105 
0.13072 
0.0327 
0.0261 
1 
Dinner 
0.7130 
0.10185 
0.0648 
0.1204 
1 
Breakfeast 
0.5857 
0.25714 
0.1071 
0.0500 
1 
Tidying 
0.4344 
0.09016 
0.0082 
0.4672 
1 
Dishes 
0.2832 
0.21239 
0.0354 
0.4690 
1 
Shopping 
0.2750 
0.19167 
0.0750 
0.4583 
1 
Official 
0.1250 
0.47917 
0.2396 
0.1562 
1 
Driving 
0.0719 
0.36691 
0.5396 
0.0216 
1 
Finances 
0.1150 
0.11504 
0.1858 
0.5841 
1 
Insurance 
0.0576 
0.00719 
0.3813 
0.5540 
1 
Repairs 
0.0000 
0.01818 
0.9697 
0.0121 
1 
Holidays 
0.0000 
0.00625 
0.0375 
0.9563 
1 
TOTAL 
0.3440 
0.14564 
0.2185 
0.2919 
1 
In the table above, the row TOTAL (in light blue) is called the average row profile (or marginal profile of columns or column margin)
The average row profile is computed as follow:
\[ average.rp = \frac{column.sum}{grand.total} \]
For example, the average row profile is : (600/1744, 254/1744, 381/1744, 509/1744). It can be computed in R as follow:
# Column sums
col.sum < apply(housetasks, 2, sum)
# average row profile = Column sums / grand total
average.rp < col.sum/n
average.rp
## Wife Alternating Husband Jointly
## 0.344 0.146 0.218 0.292
Distance (or similarity) between row profiles
If we want to compare 2 rows (row1 and row2), we need to compute the squared distance between their profiles as follow:
\[ d^2(row_1, row_2) = \sum{\frac{(row.profile_1  row.profile_2)^2}{average.profile}} \]
This distance is called Chisquare distance
For example the distance between the rows Laundry and Main_meal are:
\[ d^2(Laundry, Main\_meal) = \frac{(0.8860.810)^2}{0.344} + \frac{(0.07950.131)^2}{0.146} + ... = 0.036 \]
The distance between Laundry and Main_meal can be calculated as follow in R:
# Laundry and Main_meal profiles
laundry.p < row.profile["Laundry",]
main_meal.p < row.profile["Main_meal",]
# Distance between Laundry and Main_meal
d2 < sum(((laundry.p  main_meal.p)^2) / average.rp)
d2
## [1] 0.0368
The distance between Laundry and Driving is:
# Driving profile
driving.p < row.profile["Driving",]
# Distance between Laundry and Driving
d2 < sum(((laundry.p  driving.p)^2) / average.rp)
d2
## [1] 3.77
Note that, the rows Laundry and Main_meal are very close (d2 ~ 0.036, similar profiles) compared to the rows Laundry and Driving (d2 ~ 3.77)
You can also compute the squared distance between each row profile and the average row profile in order to view rows that are the most similar or different to the average row.
Squared distance between each row profile and the average row profile
\[ d^2(row_i, average.profile) = \sum{\frac{(row.profile_i  average.profile)^2}{average.profile}} \]
The R code below computes the distance from the average profile for all the row variables:
d2.row < apply(row.profile, 1,
function(row.p, av.p){sum(((row.p  av.p)^2)/av.p)},
average.rp)
as.matrix(round(d2.row,3))
## [,1]
## Laundry 1.329
## Main_meal 1.034
## Dinner 0.618
## Breakfeast 0.512
## Tidying 0.353
## Dishes 0.302
## Shopping 0.218
## Official 0.968
## Driving 1.274
## Finances 0.456
## Insurance 0.727
## Repairs 3.307
## Holidays 2.140
The rows Repairs, Holidays, Laundry and Driving have the most different profiles from the average profile.
Distance matrix
In this section the squared distance is computed between each row profile and the other rows in the contingency table.
The result is a distance matrix (a kind of correlation or dissimilarity matrix).
The custom R function below is used to compute the distance matrix:
## data: a data frame or matrix;
## average.profile: average profile
dist.matrix < function(data, average.profile){
mat < as.matrix(t(data))
n < ncol(mat)
dist.mat< matrix(NA, n, n)
diag(dist.mat) < 0
for (i in 1:(n  1)) {
for (j in (i + 1):n) {
d2 < sum(((mat[, i]  mat[, j])^2) / average.profile)
dist.mat[i, j] < dist.mat[j, i] < d2
}
}
colnames(dist.mat) < rownames(dist.mat) < colnames(mat)
dist.mat
}
Compute and visualize the distance between row profiles. The package corrplot
is required for the visualization. It can be installed as follow: install.packages("corrplot")
.
# Distance matrix
dist.mat < dist.matrix(row.profile, average.rp)
dist.mat <round(dist.mat, 2)
# Visualize the matrix
library("corrplot")
corrplot(dist.mat, type="upper", is.corr = FALSE)
The size of the circle is proportional to the magnitude of the distance between row profiles.
When the data contains many categories, correspondence analysis is very useful to visualize the similarity between items.
Row mass and inertia
The Row mass (or row weight) is the total frequency of a given row. It’s calculated as follow:
\[ row.mass = \frac{row.sum}{grand.total} \]
row.sum < apply(housetasks, 1, sum)
grand.total < sum(housetasks)
row.mass < row.sum/grand.total
head(row.mass)
## Laundry Main_meal Dinner Breakfeast Tidying Dishes
## 0.1009 0.0877 0.0619 0.0803 0.0700 0.0648
The row inertia is calculated as the row mass multiplied by the squared distance between the row and the average row profile:
\[ row.inertia = row.mass * d^2(row) \]
 The inertia of a row (or a column) is the amount of information it contains.
 The total inertia is the total information contained in the data table. It’s computed as the sum of rows inertia (or equivalently, as the sum of columns inertia)
# Row inertia
row.inertia < row.mass * d2.row
head(row.inertia)
## Laundry Main_meal Dinner Breakfeast Tidying Dishes
## 0.1342 0.0907 0.0382 0.0411 0.0247 0.0196
# Total inertia
sum(row.inertia)
## [1] 1.11
The total inertia corresponds to the amount of the information the data contains.
Row summary
The result for rows can be summarized as follow:
row < cbind.data.frame(d2 = d2.row, mass = row.mass, inertia = row.inertia)
round(row,3)
## d2 mass inertia
## Laundry 1.329 0.101 0.134
## Main_meal 1.034 0.088 0.091
## Dinner 0.618 0.062 0.038
## Breakfeast 0.512 0.080 0.041
## Tidying 0.353 0.070 0.025
## Dishes 0.302 0.065 0.020
## Shopping 0.218 0.069 0.015
## Official 0.968 0.055 0.053
## Driving 1.274 0.080 0.102
## Finances 0.456 0.065 0.030
## Insurance 0.727 0.080 0.058
## Repairs 3.307 0.095 0.313
## Holidays 2.140 0.092 0.196
Column variables
Column profiles
These are calculated in the same way as the row profiles table.
The profile of a given column is computed as follow:
\[ col.profile = \frac{col}{col.sum} \]
The R code below can be used to compute column profile:
col.profile < t(housetasks)/col.sum
col.profile < as.data.frame(t(col.profile))
# head(col.profile)

Wife 
Alternating 
Husband 
Jointly 
TOTAL 
Laundry 
0.2600 
0.05512 
0.00525 
0.00786 
0.1009 
Main_meal 
0.2067 
0.07874 
0.01312 
0.00786 
0.0877 
Dinner 
0.1283 
0.04331 
0.01837 
0.02554 
0.0619 
Breakfeast 
0.1367 
0.14173 
0.03937 
0.01375 
0.0803 
Tidying 
0.0883 
0.04331 
0.00262 
0.11198 
0.0700 
Dishes 
0.0533 
0.09449 
0.01050 
0.10413 
0.0648 
Shopping 
0.0550 
0.09055 
0.02362 
0.10806 
0.0688 
Official 
0.0200 
0.18110 
0.06037 
0.02947 
0.0550 
Driving 
0.0167 
0.20079 
0.19685 
0.00589 
0.0797 
Finances 
0.0217 
0.05118 
0.05512 
0.12967 
0.0648 
Insurance 
0.0133 
0.00394 
0.13911 
0.15128 
0.0797 
Repairs 
0.0000 
0.01181 
0.41995 
0.00393 
0.0946 
Holidays 
0.0000 
0.00394 
0.01575 
0.30059 
0.0917 
TOTAL 
1.0000 
1.00000 
1.00000 
1.00000 
1.0000 
In the table above, the column TOTAL is called the average column profile (or marginale profile of rows)
The average column profile is calculated as follow:
\[ average.cp = row.sum/grand.total \]
For example, the average column profile is : (176/1744, 153/1744, 108/1744, 140/1744, …). It can be computed in R as follow:
# Row sums
row.sum < apply(housetasks, 1, sum)
# average column profile= row sums/grand total
average.cp < row.sum/n
head(average.cp)
## Laundry Main_meal Dinner Breakfeast Tidying Dishes
## 0.1009 0.0877 0.0619 0.0803 0.0700 0.0648
Distance (similarity) between column profiles
If we want to compare columns, we need to compute the squared distance between their profiles as follow:
\[ d^2(col_1, col_2) = \sum{\frac{(col.profile_1  col.profile_2)^2}{average.profile}} \]
For example the distance between the columns Wife and Husband are:
\[ d^2(Wife, Husband) = \frac{(0.260.005)^2}{0.10} + \frac{(0.210.013)^2}{0.09} + ... + ... = 4.05 \]
The distance between Wife and Husband can be calculated as follow in R:
# Wife and Husband profiles
wife.p < col.profile[, "Wife"]
husband.p < col.profile[, "Husband"]
# Distance between Wife and Husband
d2 < sum(((wife.p  husband.p)^2) / average.cp)
d2
## [1] 4.05
You can also compute the squared distance between each column profile and the average column profile
Squared distance between each column profile and the average column profile
\[ d^2(col_i, average.profile) = \sum{\frac{(col.profile_i  average.profile)^2}{average.profile}} \]
The R code below computes the distance from the average profile for all the column variables
d2.col < apply(col.profile, 2,
function(col.p, av.p){sum(((col.p  av.p)^2)/av.p)},
average.cp)
round(d2.col,3)
## Wife Alternating Husband Jointly
## 0.875 0.809 1.746 1.078
Distance matrix
# Distance matrix
dist.mat < dist.matrix(t(col.profile), average.cp)
dist.mat <round(dist.mat, 2)
dist.mat
## Wife Alternating Husband Jointly
## Wife 0.00 1.71 4.05 2.93
## Alternating 1.71 0.00 2.67 2.58
## Husband 4.05 2.67 0.00 3.70
## Jointly 2.93 2.58 3.70 0.00
# Visualize the matrix
library("corrplot")
corrplot(dist.mat, type="upper", order="hclust", is.corr = FALSE)
column mass and inertia
The column mass(or column weight) is the total frequency of each column. It’s calculated as follow:
\[ col.mass = \frac{col.sum}{grand.total} \]
col.sum < apply(housetasks, 2, sum)
grand.total < sum(housetasks)
col.mass < col.sum/grand.total
head(col.mass)
## Wife Alternating Husband Jointly
## 0.344 0.146 0.218 0.292
The column inertia is calculated as the column mass multiplied by the squared distance between the column and the average column profile:
\[ col.inertia = col.mass * d^2(col) \]
col.inertia < col.mass * d2.col
head(col.inertia)
## Wife Alternating Husband Jointly
## 0.301 0.118 0.381 0.315
# total inertia
sum(col.inertia)
## [1] 1.11
Recall that the total inertia corresponds to the amount of the information the data contains. Note that, the total inertia obtained using column profile is the same as the one obtained when analyzing row profile. That’s normal, because we are analyzing the same data with just a different angle of view.
Column summary
The result for rows can be summarized as follow:
col < cbind.data.frame(d2 = d2.col, mass = col.mass,
inertia = col.inertia)
round(col,3)
## d2 mass inertia
## Wife 0.875 0.344 0.301
## Alternating 0.809 0.146 0.118
## Husband 1.746 0.218 0.381
## Jointly 1.078 0.292 0.315
Association between row and column variables
When the contingency table is not very large (as above), 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 more frequently taken jointly
Larger contingency table is complex to interpret visually and several methods are required to help to this process.
Another statistical method that can be applied to contingency table is the Chisquare test of independence.
Chisquare test
Chisquare test issued to examine whether rows and columns of a contingency table are statistically significantly associated.
 Null hypothesis (H0): the row and the column variables of the contingency table are independent.
 Alternative hypothesis (H1): row and column variables are dependent
For each cell of the table, we have to calculate the expected value under null hypothesis.
For a given cell, the expected value is calculated as follow:
\[ e = \frac{row.sum * col.sum}{grand.total} \]
The Chisquare statistic is calculated as follow:
\[ \chi^2 = \sum{\frac{(o  e)^2}{e}} \]
 o is the observed value
 e is the expected value
This calculated Chisquare statistic is compared to the critical value (obtained from statistical tables) with \(df = (r  1)(c  1)\) degrees of freedom and p = 0.05.
 r is the number of rows in the contingency table
 c is the number of column in the contingency table
If the calculated Chisquare statistic is greater than the critical value, then we must conclude that the row and the column variables are not independent of each other. This implies that they are significantly associated.
Note that, Chisquare test should only be applied when the expected frequency of any cell is at least 5.
Chisquare statistic can be easily computed using the function chisq.test()
as follow:
chisq < chisq.test(housetasks)
chisq
##
## Pearson's Chisquared test
##
## data: housetasks
## Xsquared = 2000, df = 40, pvalue <2e16
In our example, the row and the column variables are statistically significantly associated (pvalue = 0)
Note that, while Chisquare test can help to establish dependence between rows and the columns, the nature of the dependency is unknown.
The observed and the expected counts can be extracted from the result of the test as follow:
# Observed counts
chisq$observed
## Wife Alternating Husband Jointly
## Laundry 156 14 2 4
## Main_meal 124 20 5 4
## Dinner 77 11 7 13
## Breakfeast 82 36 15 7
## Tidying 53 11 1 57
## Dishes 32 24 4 53
## Shopping 33 23 9 55
## Official 12 46 23 15
## Driving 10 51 75 3
## Finances 13 13 21 66
## Insurance 8 1 53 77
## Repairs 0 3 160 2
## Holidays 0 1 6 153
# Expected counts
round(chisq$expected,2)
## Wife Alternating Husband Jointly
## Laundry 60.5 25.6 38.5 51.4
## Main_meal 52.6 22.3 33.4 44.6
## Dinner 37.2 15.7 23.6 31.5
## Breakfeast 48.2 20.4 30.6 40.9
## Tidying 42.0 17.8 26.6 35.6
## Dishes 38.9 16.5 24.7 33.0
## Shopping 41.3 17.5 26.2 35.0
## Official 33.0 14.0 21.0 28.0
## Driving 47.8 20.2 30.4 40.6
## Finances 38.9 16.5 24.7 33.0
## Insurance 47.8 20.2 30.4 40.6
## Repairs 56.8 24.0 36.0 48.2
## Holidays 55.0 23.3 35.0 46.7
As mentioned above the Chisquare statistic is 1944.456.
Which are the most contributing cells to the definition of the total Chisquare statistic?
If you want to know the most contributing cells to the total Chisquare score, you just have to calculate the Chisquare statistic for each cell:
\[ r = \frac{o  e}{\sqrt{e}} \]
The above formula returns the socalled Pearson residuals (r) for each cell (or standardized residuals). Cells with the highest absolute standardized residuals contribute the most to the total Chisquare score.
Pearson residuals can be easily extracted from the output of the function chisq.test()
:
round(chisq$residuals, 3)
## Wife Alternating Husband Jointly
## Laundry 12.27 2.298 5.878 6.61
## Main_meal 9.84 0.484 4.917 6.08
## Dinner 6.54 1.192 3.416 3.30
## Breakfeast 4.88 3.457 2.818 5.30
## Tidying 1.70 1.606 4.969 3.58
## Dishes 1.10 1.859 4.163 3.49
## Shopping 1.29 1.321 3.362 3.38
## Official 3.66 8.563 0.443 2.46
## Driving 5.47 6.836 8.100 5.90
## Finances 4.15 0.852 0.742 5.75
## Insurance 5.76 4.277 4.107 5.72
## Repairs 7.53 4.290 20.646 6.65
## Holidays 7.42 4.620 4.897 15.56
Let’s visualize Pearson residuals using the package corrplot
:
library(corrplot)
corrplot(chisq$residuals, is.cor = FALSE)
For a given cell, the size of the circle is proportional to the amount of the cell contribution.
The sign of the standardized residuals is also very important to interpret the association between rows and columns as explained in the block below:
 Positive residuals are in blue. Positive values in cells specify an attraction (positive association) between the corresponding row and column variables.
 In the image above, it’s evident that there are an association between the column Wife and, the rows Laundry and Main_meal.
 There is a strong positive association between the column Husband and the row Repair
 Negative residuals are in red. This implies a repulsion (negative association) between the corresponding row and column variables. For example the column Wife are negatively associated (~ “not associated”) with the row Repairs. There is a repulsion between the column Husband and, the rows Laundry and Main_meal
Note that, correspondence analysis is just the singular value decomposition of the standardized residuals. This will be explained in the next section.
The contribution (in %) of a given cell to the total Chisquare score is calculated as follow:
\[ contrib = \frac{r^2}{\chi^2} \]
r
is the residual of the cell
# Contibution in percentage (%)
contrib < 100*chisq$residuals^2/chisq$statistic
round(contrib, 3)
## Wife Alternating Husband Jointly
## Laundry 7.738 0.272 1.777 2.246
## Main_meal 4.976 0.012 1.243 1.903
## Dinner 2.197 0.073 0.600 0.560
## Breakfeast 1.222 0.615 0.408 1.443
## Tidying 0.149 0.133 1.270 0.661
## Dishes 0.063 0.178 0.891 0.625
## Shopping 0.085 0.090 0.581 0.586
## Official 0.688 3.771 0.010 0.311
## Driving 1.538 2.403 3.374 1.789
## Finances 0.886 0.037 0.028 1.700
## Insurance 1.705 0.941 0.868 1.683
## Repairs 2.919 0.947 21.921 2.275
## Holidays 2.831 1.098 1.233 12.445
# Visualize the contribution
corrplot(contrib, is.cor = FALSE)
The relative contribution of each cell to the total Chisquare score give some indication of the nature of the dependency between rows and columns of the contingency table.
It can be seen that:
 The column “Wife” is strongly associated with Laundry, Main_meal, Dinner
 The column “Husband” is strongly associated with the row Repairs
 The column jointly is frequently associated with the row Holidays
From the image above, it can be seen that the most contributing cells to the Chisquare are Wife/Laundry (7.74%), Wife/Main_meal (4.98%), Husband/Repairs (21.9%), Jointly/Holidays (12.44%).
These cells contribute about 47.06% to the total Chisquare score and thus account for most of the difference between expected and observed values.
This confirms the earlier visual interpretation of the data. As stated earlier, visual interpretation may be complex when the contingency table is very large. In this case, the contribution of one cell to the total Chisquare score becomes a useful way of establishing the nature of dependency.
Total inertia
As mentioned above, the total inertia is the amount of the information contained in the data table.
It’s called \(\phi^2\) (squared phi) and is calculated as follow:
\[ \phi^2 = \frac{\chi^2}{grand.total} \]
phi2 < as.numeric(chisq$statistic/sum(housetasks))
phi2
## [1] 1.11
The square root of \(\phi^2\) are called trace and may be interpreted as a correlation coefficient (Bendixen, 2003). Any value of the trace > 0.2 indicates a significant dependency between rows and columns (Bendixen M., 2003)
Mosaic plot
Mosaic plot is used to visualize a contingency table in order to examine the association between categorical variables.
The function mosaicplot()
[graphics package] can be used.
library("graphics")
# Mosaic plot of observed values
mosaicplot(housetasks, las=2, col="steelblue",
main = "housetasks  observed counts")
# Mosaic plot of expected values
mosaicplot(chisq$expected, las=2, col = "gray",
main = "housetasks  expected counts")
In these plots, column variables are firstly spitted (vertical split) and then row variables are spited(horizontal split). For each cell, the height of bars is proportional to the observed relative frequency it contains:
\[ \frac{cell.value}{column.sum} \]
The blue plot, is the mosaic plot of the observed values. The gray one is the mosaic plot of the expected values under null hypothesis.
If row and column variables were completely independent the mosaic bars for the observed values (blue graph) would be aligned as the mosaic bars for the expected values (gray graph).
It’s also possible to color the mosaic plot according to the value of the standardized residuals:
mosaicplot(housetasks, shade = TRUE, # Color the graph
las = 2, # produces vertical labels
main = "housetasks")
 This plot clearly show you that Laundry, Main_meal, Dinner and Breakfeast are more often done by the “Wife”.
 Repairs are done by the Husband
Gtest: Likelihood ratio test
The G–test
of independence is an alternative to the chisquare test of independence, and they will give approximately the same conclusion.
The test is based on the likelihood ratio defined as follow:
\[ ratio = \frac{o}{e} \]
 o is the observed value
 e is the expected value under null hypothesis
This likelihood ratio, or its logarithm, can be used to compute a pvalue
. When the logarithm of the likelihood ratio is used, the statistic is known as a loglikelihood ratio statistic
.
This test is called Gtest (or likelihood ratio test or maximum likelihood statistical significance test) and can be used in situations where Chisquare tests were previously recommended.
The Gtest is generally defined as follow:
\[ G = 2 * \sum{o * log(\frac{o}{e})} \]
o
is the observed frequency in a celle
is the expected frequency under the null hypothesislog
is the natural logarithm The
sum
is taken over all nonempty cells.
The distribution of G is approximately a chisquared distribution, with the same number of degrees of freedom as in the corresponding chisquared test:
\[df = (r  1)(c  1)\]
r
is the number of rows in the contingency tablec
is the number of column in the contingency table
The commonly used Pearson Chisquare test is, in fact, just an approximation of the loglikelihood ratio on which the Gtests are based.
Remember that, the Chisquare formula is:
\[ \chi^2 = \sum{\frac{(o  e)^2}{e}} \]
The functions likelihood.test()
[Deducer package] or G.test()
[RVAideMemoire] can be used to perform a Gtest on a contingency table.
We’ll use the package RVAideMemoire
which can be installed as follow : install.packages("RVAideMemoire")
.
The function G.test()
works as chisq.test()
:
library("RVAideMemoire")
gtest < G.test(as.matrix(housetasks))
gtest
##
## Gtest
##
## data: as.matrix(housetasks)
## G = 2000, df = 40, pvalue <2e16
Interpret the association between rows and columns
To interpret the association between the rows and the columns of the contingency table, the likelihood ratio can be used as an index (i):
\[ ratio = \frac{o}{e} \]
For a given cell,
 If ratio > 1, there is an “attraction” (association) between the corresponding column and row
 If ratio < 1, there is a “repulsion” between the corresponding column and row
The ratio can be calculated as follow:
ratio < chisq$observed/chisq$expected
round(ratio,3)
## Wife Alternating Husband Jointly
## Laundry 2.576 0.546 0.052 0.078
## Main_meal 2.356 0.898 0.150 0.090
## Dinner 2.072 0.699 0.297 0.412
## Breakfeast 1.702 1.766 0.490 0.171
## Tidying 1.263 0.619 0.038 1.601
## Dishes 0.823 1.458 0.162 1.607
## Shopping 0.799 1.316 0.343 1.570
## Official 0.363 3.290 1.097 0.535
## Driving 0.209 2.519 2.470 0.074
## Finances 0.334 0.790 0.851 2.001
## Insurance 0.167 0.049 1.745 1.898
## Repairs 0.000 0.125 4.439 0.042
## Holidays 0.000 0.043 0.172 3.276
Note that, you can also use the R code : gtest\$observed/gtest\$expected
The package corrplot
can be used to make a graph of the likelihood ratio:
corrplot(ratio, is.cor = FALSE)
The image above confirms our previous observations:
 The rows Laundry, Main_meal and Dinner are associated with the column Wife
 Repairs are done more often by the Husband
 Holidays are taken Jointly
Let’s take the log(ratio) to see the attraction and the repulsion in different colors:
 If ratio < 1 => log(ratio) < 0 (negative values) => red color
 If ratio > 1 = > log(ratio) > 0 (positive values) => blue color
We’ll also add a small value (0.5) to all cells to avoid log(0):
corrplot(log2(ratio + 0.5), is.cor = FALSE)
Correspondence analysis
Correspondence analysis (CA) is required for large contingency table.
It used to graphically visualize row points and column points in a low dimensional space.
CA is a dimensional reduction method applied to a contingency table. The information retained by each dimension is called eigenvalue.
The total information (or inertia) contained in the data is called phi (\(\phi^2\)) and can be calculated as follow:
\[ \phi^2 = \frac{\chi^2}{grand.total} \]
For a given axis, the eigenvalue (\(\lambda\)) is computed as follow:
\[ \lambda_{axis} = \sum{\frac{row.sum}{grand.total} * row.coord^2} \]
Or equivalently
\[ \lambda_{axis} = \sum{\frac{col.sum}{grand.total} * col.coord^2} \]
 row.coord and col.coord are the coordinates of row and column variables on the axis.
The association index between a row and column for the principal axes can be computed as follow:
\[ i = 1 + \sum{\frac{row.coord * col.coord}{\sqrt{\lambda}}} \]
 \(\lambda\) is the eigenvalue of the axes
 The sum denotes the sum for all axis
If there is an attraction the corresponding row and column coordinates have the same sign on the axes. If there is a repulsion the corresponding row and column coordinates have different signs on the axes. A high value indicates a strong attraction or repulsion
CA  Singular value decomposition of the standardized residuals
Correspondence analysis (CA) is used to represent graphically the table of distances between row variables or between column variables.
CA approach includes the following steps:
STEP 1. Compute the standardized residuals
The standardized residuals (S) is:
\[ S = \frac{o  e}{\sqrt{e}} \]
In fact, S is just the square roots of the terms comprising \(\chi^2\) statistic.
STEP II. Compute the singular value decomposition (SVD) of the standardized residuals.
Let M be: \(M = \frac{1}{sqrt(grand.total)} \times S\)
SVD means that we want to find orthogonal matrices U and V, together with a diagonal matrix \(\Delta\), such that:
\[ M = U \Delta V^T \]
(Phillip M. Yelland, 2010)
 \(U\) is a matrix containing row eigenvectors
 \(\Delta\) is the diagonal matrix. The numbers on the diagonal of the matrix are called singular values (SV). The eigenvalues are the squared SV.
 \(V\) is a matrix containing column eigenvectors
The eigenvalue of a given axis is:
\[ \lambda = \delta^2 \]
 \(\delta\) is the singular value
The coordinates of row variables on a given axis are:
\[ row.coord = \frac{U * \delta }{\sqrt{row.mass}} \]
The coordinates of columns are:
\[ col.coord = \frac{V * \delta }{\sqrt{col.mass}} \]
Compute SVD in R:
# Grand total
n < sum(housetasks)
# Standardized residuals
residuals < chisq$residuals/sqrt(n)
# Number of dimensions
nb.axes < min(nrow(residuals)1, ncol(residuals)1)
# Singular value decomposition
res.svd < svd(residuals, nu = nb.axes, nv = nb.axes)
res.svd
## $d
## [1] 7.37e01 6.67e01 3.56e01 1.01e16
##
## $u
## [,1] [,2] [,3]
## [1,] 0.4276 0.2359 0.2823
## [2,] 0.3520 0.2176 0.1363
## [3,] 0.2339 0.1149 0.1448
## [4,] 0.1956 0.1923 0.1752
## [5,] 0.1414 0.1722 0.0699
## [6,] 0.0653 0.1686 0.1906
## [7,] 0.0419 0.1586 0.1491
## [8,] 0.0722 0.0892 0.6078
## [9,] 0.2842 0.2765 0.4312
## [10,] 0.0935 0.2358 0.0248
## [11,] 0.2479 0.2005 0.2292
## [12,] 0.6382 0.3985 0.4074
## [13,] 0.1038 0.6516 0.1101
##
## $v
## [,1] [,2] [,3]
## [1,] 0.6668 0.321 0.3290
## [2,] 0.0322 0.167 0.9086
## [3,] 0.7364 0.422 0.2477
## [4,] 0.1096 0.831 0.0704
sv < res.svd$d[1:nb.axes] # singular value
u <res.svd$u
v < res.svd$v
Eigenvalues and screeplot
# Eigenvalues
eig < sv^2
# Variances in percentage
variance < eig*100/sum(eig)
# Cumulative variances
cumvar < cumsum(variance)
eig< data.frame(eig = eig, variance = variance,
cumvariance = cumvar)
head(eig)
## eig variance cumvariance
## 1 0.543 48.7 48.7
## 2 0.445 39.9 88.6
## 3 0.127 11.4 100.0
barplot(eig[, 2], names.arg=1:nrow(eig),
main = "Variances",
xlab = "Dimensions",
ylab = "Percentage of variances",
col ="steelblue")
# Add connected line segments to the plot
lines(x = 1:nrow(eig), eig[, 2],
type="b", pch=19, col = "red")
How many dimensions to retain?:
 The maximum number of axes in the CA is :
\[ nb.axes = min( r1, c1) \]
r and c are respectively the number of rows and columns in the table.
 Use elbow method
Row coordinates
We can use the function apply
to perform arbitrary operations on the rows and columns of a matrix.
A simplified format is:
apply(X, MARGIN, FUN, ...)
x
: a matrixMARGIN
: allowed values can be 1 or 2. 1 specifies that we want to operate on the rows of the matrix. 2 specifies that we want to operate on the column.FUN
: the function to be applied...
: optional arguments to FUN
# row sum
row.sum < apply(housetasks, 1, sum)
# row mass
row.mass < row.sum/n
# row coord = sv * u /sqrt(row.mass)
cc < t(apply(u, 1, '*', sv)) # each row X sv
row.coord < apply(cc, 2, '/', sqrt(row.mass))
rownames(row.coord) < rownames(housetasks)
colnames(row.coord) < paste0("Dim.", 1:nb.axes)
round(row.coord,3)
## Dim.1 Dim.2 Dim.3
## Laundry 0.992 0.495 0.317
## Main_meal 0.876 0.490 0.164
## Dinner 0.693 0.308 0.207
## Breakfeast 0.509 0.453 0.220
## Tidying 0.394 0.434 0.094
## Dishes 0.189 0.442 0.267
## Shopping 0.118 0.403 0.203
## Official 0.227 0.254 0.923
## Driving 0.742 0.653 0.544
## Finances 0.271 0.618 0.035
## Insurance 0.647 0.474 0.289
## Repairs 1.529 0.864 0.472
## Holidays 0.252 1.435 0.130
# plot
plot(row.coord, pch=19, col = "blue")
text(row.coord, labels =rownames(row.coord), pos = 3, col ="blue")
abline(v=0, h=0, lty = 2)
Column coordinates
# Coordinates of columns
col.sum < apply(housetasks, 2, sum)
col.mass < col.sum/n
# coordinates sv * v /sqrt(col.mass)
cc < t(apply(v, 1, '*', sv))
col.coord < apply(cc, 2, '/', sqrt(col.mass))
rownames(col.coord) < colnames(housetasks)
colnames(col.coord) < paste0("Dim", 1:nb.axes)
head(col.coord)
## Dim1 Dim2 Dim3
## Wife 0.8376 0.365 0.1999
## Alternating 0.0622 0.292 0.8486
## Husband 1.1609 0.602 0.1889
## Jointly 0.1494 1.027 0.0464
# plot
plot(col.coord, pch=17, col = "red")
text(col.coord, labels =rownames(col.coord), pos = 3, col ="red")
abline(v=0, h=0, lty = 2)
Biplot of rows and columns to view the association
xlim < range(c(row.coord[,1], col.coord[,1]))*1.1
ylim < range(c(row.coord[,2], col.coord[,2]))*1.1
# Plot of rows
plot(row.coord, pch=19, col = "blue", xlim = xlim, ylim = ylim)
text(row.coord, labels =rownames(row.coord), pos = 3, col ="blue")
# plot off columns
points(col.coord, pch=17, col = "red")
text(col.coord, labels =rownames(col.coord), pos = 3, col ="red")
abline(v=0, h=0, lty = 2)
You can interpret the distance between rows points or between column points but the distance between column points and row points are not meaningful.
Diagnostic
Recall that, the total inertia contained in the data is:
\[ \phi^2 = \frac{\chi^2}{n} = 1.11494 \]
Our twodimensional plot captures about 88% of the total inertia of the table.
Contribution of rows and columns
The contributions of a rows/columns to the definition of a principal axis are :
\[ row.contrib = \frac{row.mass * row.coord^2}{eigenvalue} \]
\[ col.contrib = \frac{col.mass * col.coord^2}{eigenvalue} \]
Contribution of rows in %
# contrib < row.mass * row.coord^2/eigenvalue
cc < apply(row.coord^2, 2, "*", row.mass)
row.contrib < t(apply(cc, 1, "/", eig[1:nb.axes,1])) *100
round(row.contrib, 2)
## 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
corrplot(row.contrib, is.cor = FALSE)
Contribution of columns in %
# contrib < col.mass * col.coord^2/eigenvalue
cc < apply(col.coord^2, 2, "*", col.mass)
col.contrib < t(apply(cc, 1, "/", eig[1:nb.axes,1])) *100
round(col.contrib, 2)
## Dim1 Dim2 Dim3
## Wife 44.5 10.31 10.82
## Alternating 0.1 2.78 82.55
## Husband 54.2 17.79 6.13
## Jointly 1.2 69.12 0.50
corrplot(col.contrib, is.cor = FALSE)
Quality of the representation
The quality of the representation is called COS2.
The quality of the representation of a row on an axis is:
\[ row.cos2 = \frac{row.coord^2}{d^2} \]
 row.coord is the coordinate of the row on the axis
 \(d^2\) is the squared distance from the average profile
Recall that the distance between each row profile and the average row profile is:
\[ d^2(row_i, average.profile) = \sum{\frac{(row.profile_i  average.profile)^2}{average.profile}} \]
row.profile < housetasks/row.sum
head(round(row.profile, 3))
## Wife Alternating Husband Jointly
## Laundry 0.886 0.080 0.011 0.023
## Main_meal 0.810 0.131 0.033 0.026
## Dinner 0.713 0.102 0.065 0.120
## Breakfeast 0.586 0.257 0.107 0.050
## Tidying 0.434 0.090 0.008 0.467
## Dishes 0.283 0.212 0.035 0.469
average.profile < col.sum/n
head(round(average.profile, 3))
## Wife Alternating Husband Jointly
## 0.344 0.146 0.218 0.292
The R code below computes the distance from the average profile for all the row variables
d2.row < apply(row.profile, 1,
function(row.p, av.p){sum(((row.p  av.p)^2)/av.p)},
average.rp)
head(round(d2.row,3))
## Laundry Main_meal Dinner Breakfeast Tidying Dishes
## 1.329 1.034 0.618 0.512 0.353 0.302
The cos2 of rows on the factor map are:
row.cos2 < apply(row.coord^2, 2, "/", d2.row)
round(row.cos2, 3)
## Dim.1 Dim.2 Dim.3
## Laundry 0.740 0.185 0.075
## Main_meal 0.742 0.232 0.026
## Dinner 0.777 0.154 0.070
## Breakfeast 0.505 0.400 0.095
## Tidying 0.440 0.535 0.025
## Dishes 0.118 0.646 0.236
## Shopping 0.064 0.748 0.189
## Official 0.053 0.066 0.881
## Driving 0.432 0.335 0.233
## Finances 0.161 0.837 0.003
## Insurance 0.576 0.309 0.115
## Repairs 0.707 0.226 0.067
## Holidays 0.030 0.962 0.008
visualize the cos2:
corrplot(row.cos2, is.cor = FALSE)
Cos2 of columns
\[ col.cos2 = \frac{col.coord^2}{d^2} \]
col.profile < t(housetasks)/col.sum
col.profile < t(col.profile)
#head(round(col.profile, 3))
average.profile < row.sum/n
#head(round(average.profile, 3))
The R code below computes the distance from the average profile for all the column variables
d2.col < apply(col.profile, 2,
function(col.p, av.p){sum(((col.p  av.p)^2)/av.p)},
average.profile)
#round(d2.col,3)
The cos2 of columns on the factor map are:
col.cos2 < apply(col.coord^2, 2, "/", d2.col)
round(col.cos2, 3)
## Dim1 Dim2 Dim3
## Wife 0.802 0.152 0.046
## Alternating 0.005 0.105 0.890
## Husband 0.772 0.208 0.020
## Jointly 0.021 0.977 0.002
visualize the cos2:
corrplot(col.cos2, is.cor = FALSE)
Supplementary rows/columns
The supplementary row coordinates
\[ sup.row.coord = sup.row.profile * \frac{v}{\sqrt{col.mass}} \]
# Supplementary row
sup.row < as.data.frame(housetasks["Dishes",, drop = FALSE])
# Supplementary row profile
sup.row.sum < apply(sup.row, 1, sum)
sup.row.profile < sweep(sup.row, 1, sup.row.sum, "/")
# V/sqrt(col.mass)
vv < sweep(v, 1, sqrt(col.mass), FUN = "/")
# Supplementary row coord
sup.row.coord < as.matrix(sup.row.profile) %*% vv
sup.row.coord
## [,1] [,2] [,3]
## Dishes 0.189 0.442 0.267
## COS2 = coor^2/Distance from average profile
d2.row < apply(sup.row.profile, 1,
function(row.p, av.p){sum(((row.p  av.p)^2)/av.p)},
average.rp)
sup.row.cos2 < sweep(sup.row.coord^2, 1, d2.row, FUN = "/")
Practice in R
library(FactoMineR)
res.ca < CA(housetasks, graph = F)
# print
res.ca
## **Results of the Correspondence Analysis (CA)**
## The row variable has 13 categories; the column variable has 4 categories
## The chi square of independence between the two variables is equal to 1944 (pvalue = 0 ).
## *The results are available in the following objects:
##
## name description
## 1 "$eig" "eigenvalues"
## 2 "$col" "results for the columns"
## 3 "$col$coord" "coord. for the columns"
## 4 "$col$cos2" "cos2 for the columns"
## 5 "$col$contrib" "contributions of the columns"
## 6 "$row" "results for the rows"
## 7 "$row$coord" "coord. for the rows"
## 8 "$row$cos2" "cos2 for the rows"
## 9 "$row$contrib" "contributions of the rows"
## 10 "$call" "summary called parameters"
## 11 "$call$marge.col" "weights of the columns"
## 12 "$call$marge.row" "weights of the rows"
# eigenvalue
head(res.ca$eig)[, 1:2]
## eigenvalue percentage of variance
## dim 1 0.543 48.7
## dim 2 0.445 39.9
## dim 3 0.127 11.4
# barplot of percentage of variance
barplot(res.ca$eig[,2], names.arg = rownames(res.ca$eig))
# Plot row points
plot(res.ca, invisible ="col")
# Plot column points
plot(res.ca, invisible ="col")
# Biplot of rows and columns
plot(res.ca)
References
 Phillip M. Yelland. An introduction to correspondence analysis. Mathematica Journal. 2010. http://www.mathematicajournal.com/data/uploads/2010/09/Yelland.pdf
 Ricco RAKOTOMALALA (article in french). Analyse factorielle des correspondances. University Lyon 2. http://eric.univlyon2.fr/~ricco/cours/slides/AFC.pdf
 Bendixen M. 2003, A Practical Guide to the Use of Correspondence Analysis in Marketing Research, Marketing Bulletin, 2003, 14, Technical Note 2. http://marketingbulletin.massey.ac.nz/V14/MB_V14_T2_Bendixen.pdf