Correspondence analysis, also called reciprocal averaging, is a useful data science visualization technique for finding out and displaying the relationship between categories. It is a multivariate statistical tool that was first proposed in 1935 by Herman Otto Harley which uses a graph that plots data, visually showing the outcome of two or more data points.
It is conceptually similar to principle component analysis , but applies to categorical rather than continuous data. In a similar manner to principal component analysis, it provides a means of displaying or summarizing a set of data in two-dimensional graphical form.
A correspondence analysis uses a contingency table (a table of frequencies that shows how variables distribute categories). The data in the table undergoes a series of transformations in relation to the data around it to produce relational data. The resulting data is then graphed to show those relationships visually.
Description of the data
‘’smoke data set’’ which contains 5 rows (staff group) and 4 columns (smoking categories), giving the frequencies of smoking categories in each staff group in a fictional organization.
| none | light | medium | heavy | |
| SM | 4 | 2 | 3 | 2 |
| JM | 4 | 3 | 7 | 4 |
| SE | 25 | 10 | 12 | 4 |
| JE | 18 | 24 | 33 | 13 |
| SC | 10 | 6 | 7 | 2 |
SM: senior managers
JM: junior managers
SE: senior employees
JE: junior employees
SC: secretaries
library(ca)
library(ggrepel)Loading required package: ggplot2library(FactoMineR)
library(factoextra)Welcome! Want to learn more? See two factoextra-related books at https://goo.gl/ve3WBalibrary(ade4)
Attaching package: 'ade4'The following object is masked from 'package:FactoMineR':
reconstlibrary(prettyGraphs)
data(smoke)
data(smoke, package="ca")
ca_smoke<-ca(smoke)names(ca_smoke) [1] "sv" "nd" "rownames" "rowmass" "rowdist"
[6] "rowinertia" "rowcoord" "rowsup" "colnames" "colmass"
[11] "coldist" "colinertia" "colcoord" "colsup" "N"
[16] "call" summary(ca_smoke)
Principal inertias (eigenvalues):
dim value % cum% scree plot
1 0.074759 87.8 87.8 **********************
2 0.010017 11.8 99.5 ***
3 0.000414 0.5 100.0
-------- -----
Total: 0.085190 100.0
Rows:
name mass qlt inr k=1 cor ctr k=2 cor ctr
1 | SM | 57 893 31 | -66 92 3 | -194 800 214 |
2 | JM | 93 991 139 | 259 526 84 | -243 465 551 |
3 | SE | 264 1000 450 | -381 999 512 | -11 1 3 |
4 | JE | 456 1000 308 | 233 942 331 | 58 58 152 |
5 | SC | 130 999 71 | -201 865 70 | 79 133 81 |
Columns:
name mass qlt inr k=1 cor ctr k=2 cor ctr
1 | none | 316 1000 577 | -393 994 654 | -30 6 29 |
2 | lght | 233 984 83 | 99 327 31 | 141 657 463 |
3 | medm | 321 983 148 | 196 982 166 | 7 1 2 |
4 | hevy | 130 995 192 | 294 684 150 | -198 310 506 |plot(ca_smoke)
plot(ca_smoke, mass = TRUE, contrib = "absolute",map = "rowgreen", arrows = c(FALSE, TRUE))
According to the above plot,we can clearly see that most of the senior managers are non smokers and junior managers are heavy smokers while junior employees seems to be medium smokers.
library(rgl)
plot3d.ca(ca(smoke, nd=3))Multiple Correspondence Analysis (MCA) is an extension of simple CA of a single cross-tabulation to more than two categorical variables.
Description of the data
This data frame(“wg93”) contains records of four questions on attitude towards science with responses on a five-point scale (1=agree strongly to 5=disagree strongly) and three demographic variables (sex, age and education).
data(wg93)
data(wg93, package="ca")
MCA_wg93<-MCA(wg93)
names(MCA_wg93)[1] "eig" "call" "ind" "var" "svd" summary(MCA_wg93)
Call:
MCA(X = wg93)
Eigenvalues
Dim.1 Dim.2 Dim.3 Dim.4 Dim.5 Dim.6 Dim.7
Variance 0.289 0.255 0.208 0.197 0.183 0.168 0.165
% of var. 7.483 6.604 5.400 5.117 4.732 4.350 4.281
Cumulative % of var. 7.483 14.087 19.488 24.605 29.337 33.687 37.968
Dim.8 Dim.9 Dim.10 Dim.11 Dim.12 Dim.13 Dim.14
Variance 0.160 0.157 0.151 0.147 0.144 0.139 0.133
% of var. 4.139 4.078 3.912 3.824 3.730 3.599 3.458
Cumulative % of var. 42.107 46.184 50.097 53.921 57.651 61.250 64.708
Dim.15 Dim.16 Dim.17 Dim.18 Dim.19 Dim.20 Dim.21
Variance 0.129 0.127 0.127 0.122 0.117 0.112 0.107
% of var. 3.357 3.297 3.280 3.163 3.023 2.893 2.764
Cumulative % of var. 68.065 71.362 74.643 77.806 80.829 83.722 86.486
Dim.22 Dim.23 Dim.24 Dim.25 Dim.26 Dim.27
Variance 0.098 0.094 0.092 0.085 0.082 0.069
% of var. 2.530 2.442 2.398 2.210 2.138 1.796
Cumulative % of var. 89.016 91.458 93.856 96.066 98.204 100.000
Individuals (the 10 first)
Dim.1 ctr cos2 Dim.2 ctr cos2 Dim.3 ctr cos2
1 | 0.115 0.005 0.005 | -0.278 0.035 0.028 | 0.377 0.078 0.051 |
2 | 0.410 0.067 0.049 | -0.499 0.112 0.072 | 0.363 0.073 0.038 |
3 | -0.409 0.067 0.072 | -0.298 0.040 0.038 | -0.112 0.007 0.005 |
4 | -0.138 0.008 0.008 | -0.318 0.045 0.042 | -0.233 0.030 0.022 |
5 | -0.059 0.001 0.001 | -0.945 0.403 0.289 | -0.243 0.033 0.019 |
6 | 0.364 0.053 0.043 | -0.093 0.004 0.003 | -0.032 0.001 0.000 |
7 | -0.133 0.007 0.007 | -0.501 0.113 0.096 | -0.039 0.001 0.001 |
8 | 0.578 0.133 0.112 | -0.428 0.082 0.061 | -0.105 0.006 0.004 |
9 | -0.358 0.051 0.030 | -0.158 0.011 0.006 | -0.537 0.159 0.069 |
10 | -0.117 0.005 0.005 | -0.630 0.179 0.155 | -0.572 0.181 0.128 |
Categories (the 10 first)
Dim.1 ctr cos2 v.test Dim.2 ctr cos2 v.test
A_1 | -0.967 6.325 0.148 -11.348 | 0.823 5.186 0.107 9.654 |
A_2 | -0.426 3.321 0.106 -9.624 | -0.071 0.105 0.003 -1.611 |
A_3 | 0.180 0.377 0.010 2.943 | -0.844 9.362 0.218 -13.772 |
A_4 | 0.769 5.976 0.152 11.491 | 0.228 0.594 0.013 3.403 |
A_5 | 1.639 7.324 0.157 11.672 | 1.183 4.323 0.082 8.424 |
B_1 | -1.466 8.671 0.191 -12.882 | 1.390 8.837 0.172 12.217 |
B_2 | -0.635 3.987 0.101 -9.358 | -0.335 1.256 0.028 -4.934 |
B_3 | -0.330 1.268 0.034 -5.399 | -0.515 3.505 0.082 -8.432 |
B_4 | 0.454 3.289 0.098 9.238 | -0.256 1.186 0.031 -5.211 |
B_5 | 1.105 9.712 0.234 14.262 | 0.979 8.645 0.184 12.641 |
Dim.3 ctr cos2 v.test
A_1 0.209 0.411 0.007 2.457 |
A_2 -0.082 0.170 0.004 -1.849 |
A_3 -0.069 0.076 0.001 -1.120 |
A_4 0.292 1.193 0.022 4.361 |
A_5 -0.760 2.183 0.034 -5.414 |
B_1 0.251 0.353 0.006 2.208 |
B_2 -0.067 0.062 0.001 -0.991 |
B_3 -0.064 0.067 0.001 -1.055 |
B_4 0.366 2.970 0.064 7.457 |
B_5 -0.685 5.169 0.090 -8.839 |
Categorical variables (eta2)
Dim.1 Dim.2 Dim.3
A | 0.471 0.349 0.059 |
B | 0.544 0.418 0.126 |
C | 0.519 0.486 0.203 |
D | 0.026 0.395 0.160 |
sex | 0.069 0.026 0.055 |
age | 0.158 0.030 0.391 |
edu | 0.233 0.080 0.464 |plot(MCA_wg93, cex = .7, col.var = "black", col.ind = "gray", invis = "ind")
library("factoextra")
print(MCA_wg93)**Results of the Multiple Correspondence Analysis (MCA)**
The analysis was performed on 871 individuals, described by 7 variables
*The results are available in the following objects:
name description
1 "$eig" "eigenvalues"
2 "$var" "results for the variables"
3 "$var$coord" "coord. of the categories"
4 "$var$cos2" "cos2 for the categories"
5 "$var$contrib" "contributions of the categories"
6 "$var$v.test" "v-test for the categories"
7 "$ind" "results for the individuals"
8 "$ind$coord" "coord. for the individuals"
9 "$ind$cos2" "cos2 for the individuals"
10 "$ind$contrib" "contributions of the individuals"
11 "$call" "intermediate results"
12 "$call$marge.col" "weights of columns"
13 "$call$marge.li" "weights of rows" The proportion of variances retained by the different dimensions (axes) can be extracted using the function get_eigenvalue()
eig.val <- get_eigenvalue(MCA_wg93)
eig.val eigenvalue variance.percent cumulative.variance.percent
Dim.1 0.28863097 7.483025 7.483025
Dim.2 0.25473881 6.604339 14.087365
Dim.3 0.20829251 5.400176 19.487541
Dim.4 0.19738826 5.117473 24.605014
Dim.5 0.18252871 4.732226 29.337240
Dim.6 0.16777120 4.349624 33.686864
Dim.7 0.16513859 4.281371 37.968235
Dim.8 0.15963130 4.138589 42.106824
Dim.9 0.15728148 4.077668 46.184492
Dim.10 0.15090458 3.912341 50.096833
Dim.11 0.14749027 3.823822 53.920654
Dim.12 0.14387158 3.730004 57.650658
Dim.13 0.13883734 3.599487 61.250145
Dim.14 0.13338181 3.458047 64.708192
Dim.15 0.12948212 3.356944 68.065135
Dim.16 0.12716579 3.296891 71.362026
Dim.17 0.12653319 3.280490 74.642516
Dim.18 0.12201043 3.163233 77.805750
Dim.19 0.11660840 3.023181 80.828931
Dim.20 0.11158366 2.892910 83.721840
Dim.21 0.10660185 2.763752 86.485592
Dim.22 0.09758714 2.530037 89.015629
Dim.23 0.09419675 2.442138 91.457767
Dim.24 0.09248798 2.397837 93.855604
Dim.25 0.08524996 2.210184 96.065788
Dim.26 0.08246419 2.137960 98.203748
Dim.27 0.06928400 1.796252 100.000000According to above result no dimension has explained significance amount of variance out of total. This can be clearly visualize by creating scree plot. To do that we can use the function “fviz_screeplot()”
fviz_screeplot(MCA_wg93, addlabels = TRUE, ylim = c(0, 45))
The function “fviz_mca_biplot()” is used to draw the biplot of individuals and variable categories
fviz_mca_biplot(MCA_wg93,
repel = TRUE, # Avoid text overlapping (slow if many point)
ggtheme = theme_minimal())Warning: ggrepel: 808 unlabeled data points (too many overlaps). Consider
increasing max.overlaps
The plot above shows a global pattern within the data. Rows (individuals) are represented by blue points and columns (variable categories) 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.
Variables of Multiple Correspondence Analysis can be extracted by using below code.
var <- get_mca_var(MCA_wg93)
varMultiple Correspondence Analysis Results for variables
===================================================
Name Description
1 "$coord" "Coordinates for categories"
2 "$cos2" "Cos2 for categories"
3 "$contrib" "contributions of categories"head(var$coord) Dim 1 Dim 2 Dim 3 Dim 4 Dim 5
A_1 -0.9671597 0.82273703 0.20937924 -0.42460797 0.07117658
A_2 -0.4260310 -0.07132741 -0.08186277 0.15141118 -0.05311328
A_3 0.1803910 -0.84426258 -0.06866201 -0.43681468 0.30997012
A_4 0.7686636 0.22766291 0.29174984 0.48798870 -0.66303991
A_5 1.6385852 1.18265182 -0.76001539 0.08379486 1.32124301
B_1 -1.4660261 1.39035983 0.25123381 -0.73875510 -0.14521786head(var$coord) Dim 1 Dim 2 Dim 3 Dim 4 Dim 5
A_1 -0.9671597 0.82273703 0.20937924 -0.42460797 0.07117658
A_2 -0.4260310 -0.07132741 -0.08186277 0.15141118 -0.05311328
A_3 0.1803910 -0.84426258 -0.06866201 -0.43681468 0.30997012
A_4 0.7686636 0.22766291 0.29174984 0.48798870 -0.66303991
A_5 1.6385852 1.18265182 -0.76001539 0.08379486 1.32124301
B_1 -1.4660261 1.39035983 0.25123381 -0.73875510 -0.14521786head(var$coord) Dim 1 Dim 2 Dim 3 Dim 4 Dim 5
A_1 -0.9671597 0.82273703 0.20937924 -0.42460797 0.07117658
A_2 -0.4260310 -0.07132741 -0.08186277 0.15141118 -0.05311328
A_3 0.1803910 -0.84426258 -0.06866201 -0.43681468 0.30997012
A_4 0.7686636 0.22766291 0.29174984 0.48798870 -0.66303991
A_5 1.6385852 1.18265182 -0.76001539 0.08379486 1.32124301
B_1 -1.4660261 1.39035983 0.25123381 -0.73875510 -0.14521786Correlation between variables and dimensions.
fviz_mca_var(MCA_wg93, choice = "mca.cor",
repel = TRUE, # Avoid text overlapping (slow)
ggtheme = theme_minimal())
The plot above helps to identify variables that are the most correlated with each dimension. The squared correlations between variables and the dimensions are used as coordinates.
It can be seen that, the variables Sex, Age and education are the most correlated with dimension 1. Similarly, the variables “record of question D” is the most correlated with dimension 2.
head(round(var$coord, 2), 4) Dim 1 Dim 2 Dim 3 Dim 4 Dim 5
A_1 -0.97 0.82 0.21 -0.42 0.07
A_2 -0.43 -0.07 -0.08 0.15 -0.05
A_3 0.18 -0.84 -0.07 -0.44 0.31
A_4 0.77 0.23 0.29 0.49 -0.66fviz_mca_var(MCA_wg93,
repel = TRUE, # Avoid text overlapping (slow)
ggtheme = theme_minimal())
fviz_mca_var(MCA_wg93, col.var="blue", shape.var = 15,
repel = TRUE)
head(var$cos2, 4) Dim 1 Dim 2 Dim 3 Dim 4 Dim 5
A_1 0.148021742 0.107115226 0.006937394 0.02853024 0.0008016843
A_2 0.106454957 0.002983984 0.003930578 0.01344620 0.0016545879
A_3 0.009952545 0.218001464 0.001441907 0.05835778 0.0293862375
A_4 0.151760731 0.013312859 0.021862913 0.06116547 0.1129187613Contribution of variable categories to the dimensions
The variable categories with the larger value, contribute the most to the definition of the dimensions. It can be obtained by using below code.
head(round(var$contrib,2), 4) Dim 1 Dim 2 Dim 3 Dim 4 Dim 5
A_1 6.33 5.19 0.41 1.78 0.05
A_2 3.32 0.11 0.17 0.61 0.08
A_3 0.38 9.36 0.08 3.23 1.76
A_4 5.98 0.59 1.19 3.52 7.03fviz_contrib(MCA_wg93, choice = "var", axes = 1, top = 15)
fviz_contrib(MCA_wg93, choice = "var", axes = 2, top = 15)
Variable categories that contribute the most to Dim.1 and Dim.2 are the most important in explaining the variability in the data set.