# Multiple Correspondence Analysis in R

### MCA

In this Rpub I'll show five different options of doing Multiple Correspondence Analysis in R (don't confuse it with Correspondence Analysis). Put in very simple terms, Multiple Correspondence Analysis (MCA) is to qualitative data, as Principal Component Analysis (PCA) is to quantitative data. Well, maybe I'm oversimplifying a little bit because MCA has some special features that make it mathematically different from PCA, but they both share a lot of things in common from a data analysis standpoint. As with PCA and Correspondence Analysis, MCA is just another tool in our kit of multivariate methods that allows us to analyze the systematic patterns of variations with categorical data. Keep in mind that MCA applies to tables in which the observations are described by a set of qualitative (i.e. categorical) variables. This means that in R you must have your table in the form of a data frame with factors (observations in the rows, qualitative variables in the columns).

### MCA in R

In R, there are several functions from different packages that allow us to apply Multiple Correspondence Analysis. In this post I'll show you 5 different ways to perform MCA using the following functions (with their corresponding packages in parentheses):

• MCA (FactoMineR)
• mca (MASS)
• mjca (ca)
• homals (homals)

No matter what function you decide to use for MCA, the typical results should consist of a set of eigenvalues, a table with the row coordinates, and a table with the column coordinates. Compared to the eigenvalues obtained from a PCA or a CA, the eigenvalues in a MCA can be much more smaller. This is important to know because if you just consider the eigenvalues, you might be tempted to conclude that MCA sucks. Which is absolutely false. Personally, I think that the real meat and potatoes of MCA relies in its dimension reduction properties that let us visualize our data, among other things. Besides the eigenvalues, the row coordinates provide information about the structure of the rows in the analyzed table. In turn, the column coordinates provide information about the structure of the analyzed variables and their corresponding categories.

### The Data

We'll use the dataset tea that comes in the R package FactoMineR. It's a data frame (of factors) containing the answers of a questionnaire on tea consumption for 300 individuals. Although the data contains 36 columns (i.e. variables), for demonstration purposes I will only consider the following columns:

1. What kind of tea do you drink (black, green, flavored)
2. How do you drink it (alone, w/milk, w/lemon, other)
3. What kind of presentation do you buy (tea bags, loose tea, both)
4. Do you add sugar (yes, no)
5. Where do you buy it (supermarket, shops, both)
6. Do you always drink tea (always, not always)
``````# load packages
require(FactoMineR)
require(ggplot2)
data(tea)
# select these columns
newtea = tea[, c("Tea", "How", "how", "sugar", "where", "always")]
# take a look
``````
``````##         Tea   How     how    sugar       where     always
## 1     black alone tea bag    sugar chain store Not.always
## 2     black  milk tea bag No.sugar chain store Not.always
## 3 Earl Grey alone tea bag No.sugar chain store Not.always
## 4 Earl Grey alone tea bag    sugar chain store Not.always
## 5 Earl Grey alone tea bag No.sugar chain store     always
## 6 Earl Grey alone tea bag No.sugar chain store Not.always
``````
``````# number of categories per variable
cats = apply(newtea, 2, function(x) nlevels(as.factor(x)))
cats
``````
``````##    Tea    How    how  sugar  where always
##      3      4      3      2      3      2
``````

### MCA with FactoMineR

My preferred function to do multiple correspondence analysis is the MCA function that comes in the fabulous package FactoMineR Francois Husson, Julie Josse, Sebastien Le, and Jeremy Mazet. If you have seen my other posts you'll know that this is one of favorite packages and I strongly recommend other users to seriously take a look at it. It provides the most complete list of results with different calculations for interpretation and diagnosis.

``````# apply MCA
mca1 = MCA(newtea, graph = FALSE)

# table of eigenvalues
mca1\$eig
``````
``````##        eigenvalue percentage of variance cumulative percentage of variance
## dim 1     0.27976                 15.260                             15.26
## dim 2     0.25775                 14.059                             29.32
## dim 3     0.22014                 12.008                             41.33
## dim 4     0.18793                 10.251                             51.58
## dim 5     0.16876                  9.205                             60.78
## dim 6     0.16369                  8.928                             69.71
## dim 7     0.15289                  8.339                             78.05
## dim 8     0.13839                  7.548                             85.60
## dim 9     0.11569                  6.310                             91.91
## dim 10    0.08613                  4.698                             96.61
## dim 11    0.06221                  3.393                            100.00
``````
``````# column coordinates
``````
``````##              Dim 1   Dim 2    Dim 3    Dim 4    Dim 5
## black      0.44585  0.1434  1.12722  0.73788  0.17249
## Earl Grey -0.25031  0.1115 -0.48017 -0.06983 -0.07847
## green      0.46416 -0.9735  0.28058 -1.24626  0.07214
## alone     -0.02189 -0.2511  0.10326 -0.45737 -0.23317
## lemon      0.68232  0.4639 -1.26750  1.00191 -1.17060
## milk      -0.33099  0.2286  0.06944  0.64061  1.34609
``````
``````# row coordinates
``````
``````##     Dim 1   Dim 2    Dim 3    Dim 4   Dim 5
## 1 -0.3269 -0.2902  0.38114  0.40596 -0.2344
## 2 -0.2692 -0.1059  0.79969  0.55663  0.4269
## 3 -0.3911 -0.2739  0.24072 -0.17603 -0.3157
## 4 -0.5462 -0.3007 -0.18984  0.09543 -0.3362
## 5 -0.3387 -0.2150 -0.03242 -0.37509  0.2391
## 6 -0.3911 -0.2739  0.24072 -0.17603 -0.3157
``````
``````# data frames for ggplot
mca1_vars_df = data.frame(mca1\$var\$coord, Variable = rep(names(cats),
cats))
mca1_obs_df = data.frame(mca1\$ind\$coord)

# plot of variable categories
ggplot(data = mca1_vars_df, aes(x = Dim.1, y = Dim.2, label = rownames(mca1_vars_df))) +
geom_hline(yintercept = 0, colour = "gray70") + geom_vline(xintercept = 0,
colour = "gray70") + geom_text(aes(colour = Variable)) + ggtitle("MCA plot of variables using R package FactoMineR")
``````

In order to have a more interesting representation, we could superimpose a graphic display of both the observations and the categories. Moreover, since some individuals will be overlapped, we can use density curves to see those zones that are highly concentrated.

``````# MCA plot of observations and categories
ggplot(data = mca1_obs_df, aes(x = Dim.1, y = Dim.2)) + geom_hline(yintercept = 0,
colour = "gray70") + geom_vline(xintercept = 0, colour = "gray70") + geom_point(colour = "gray50",
alpha = 0.7) + geom_density2d(colour = "gray80") + geom_text(data = mca1_vars_df,
aes(x = Dim.1, y = Dim.2, label = rownames(mca1_vars_df), colour = Variable)) +
ggtitle("MCA plot of variables using R package FactoMineR") + scale_colour_discrete(name = "Variable")
``````

``````# default biplot in FactoMineR
plot(mca1)
``````

### mca with MASS

Another function for performing MCA is the mca function that comes in the MASS package by Brian Ripley et al.

``````require(MASS, quietly = TRUE)
``````
``````## Attaching package: 'MASS'
``````
``````## The following object(s) are masked _by_ '.GlobalEnv':
##
## cats
``````
``````# apply mca
mca2 = mca(newtea, nf = 5)

# eigenvalues
mca2\$d^2
``````
``````## [1] 0.2798 0.2577 0.2201 0.1879 0.1688
``````
``````# column coordinates
``````
``````##                        1         2         3        4         5
## Tea.black     -0.0081111  0.002719 -0.023118  0.01638  0.004040
## Tea.Earl Grey  0.0045538  0.002113  0.009848 -0.00155 -0.001838
## Tea.green     -0.0084442 -0.018452 -0.005754 -0.02766  0.001690
## How.alone      0.0003982 -0.004760 -0.002118 -0.01015 -0.005462
## How.lemon     -0.0124132  0.008793  0.025995  0.02224 -0.027419
## How.milk       0.0060215  0.004332 -0.001424  0.01422  0.031530
``````
``````# row coordiantes
``````
``````##          1         2         3          4         5
## 1 0.003145 -0.002793 -0.003668  0.0039064 -0.002256
## 2 0.002590 -0.001019 -0.007695  0.0053562  0.004107
## 3 0.003764 -0.002635 -0.002316 -0.0016939 -0.003038
## 4 0.005256 -0.002894  0.001827  0.0009183 -0.003235
## 5 0.003259 -0.002069  0.000312 -0.0036093  0.002301
## 6 0.003764 -0.002635 -0.002316 -0.0016939 -0.003038
``````
``````# data frame for ggplot
mca2_vars_df = data.frame(mca2\$cs, Variable = rep(names(cats), cats))

# MCA plot of variable categories
ggplot(data = mca2_vars_df, aes(x = X1, y = X2, label = rownames(mca2_vars_df))) +
geom_hline(yintercept = 0, colour = "gray70") + geom_vline(xintercept = 0,
colour = "gray70") + geom_text(aes(colour = Variable)) + ggtitle("MCA plot of variables using R package MASS")
``````

If you prefer not to use ggplot2, you can stay with the default plots (not for me)

``````# default biplot in MASS (kind of ugly)
plot(mca2)
``````

A third option to perform MCA is by using the function dudi.acm that comes with the package ade4 by Simon Penel et al (remember to install the package first).

``````# MCA with function dudi.acm
``````
``````## Attaching package: 'ade4'
``````
``````## The following object(s) are masked from 'package:FactoMineR':
##
## reconst
``````
``````## The following object(s) are masked from 'package:base':
##
## within
``````
``````# apply dudi.acm
mca3 = dudi.acm(newtea, scannf = FALSE, nf = 5)

# eigenvalues
mca3\$eig
``````
``````##  [1] 0.27976 0.25775 0.22014 0.18793 0.16876 0.16369 0.15289 0.13839
##  [9] 0.11569 0.08613 0.06221
``````
``````# column coordinates
``````
``````##                  Comp1   Comp2    Comp3    Comp4    Comp5
## Tea.black     -0.44585  0.1434  1.12722 -0.73788  0.17249
## Tea.Earl.Grey  0.25031  0.1115 -0.48017  0.06983 -0.07847
## Tea.green     -0.46416 -0.9735  0.28058  1.24626  0.07214
## How.alone      0.02189 -0.2511  0.10326  0.45737 -0.23317
## How.lemon     -0.68232  0.4639 -1.26750 -1.00191 -1.17060
## How.milk       0.33099  0.2286  0.06944 -0.64061  1.34609
``````
``````# row coordinates
``````
``````##    Axis1   Axis2    Axis3    Axis4   Axis5
## 1 0.3269 -0.2902  0.38114 -0.40596 -0.2344
## 2 0.2692 -0.1059  0.79969 -0.55663  0.4269
## 3 0.3911 -0.2739  0.24072  0.17603 -0.3157
## 4 0.5462 -0.3007 -0.18984 -0.09543 -0.3362
## 5 0.3387 -0.2150 -0.03242  0.37509  0.2391
## 6 0.3911 -0.2739  0.24072  0.17603 -0.3157
``````
``````# data frame for ggplot
mca3_vars_df = data.frame(mca3\$co, Variable = rep(names(cats), cats))

# MCA plot of variable categories
ggplot(data = mca3_vars_df, aes(x = Comp1, y = Comp2, label = rownames(mca3_vars_df))) +
geom_hline(yintercept = 0, colour = "gray70") + geom_vline(xintercept = 0,
colour = "gray70") + geom_text(aes(colour = Variable)) + ggtitle("MCA plot of variables using R package ade4")
``````

### mjca with ca

Another interesting way for carrying out MCA is by using the function mjca from the package ca by Michael Greenacre and Oleg Nenadic.

``````# PCA with function mjca
require(ca, quietly = TRUE)
# apply mjca
mca4 = mjca(newtea, lambda = "indicator", nd = 5)

# eigenvalues
mca4\$sv^2
``````
``````##  [1] 0.27976 0.25775 0.22014 0.18793 0.16876 0.16369 0.15289 0.13839
##  [9] 0.11569 0.08613 0.06221
``````
``````# column coordinates
``````
``````##          [,1]    [,2]    [,3]    [,4]    [,5]    [,6]    [,7]    [,8]
## [1,] -0.84293  0.2825  2.4025 -1.7021  0.4199 -0.3094 -0.7239  0.6186
## [2,]  0.47324  0.2195 -1.0234  0.1611 -0.1910 -0.4697 -0.4730 -0.3718
## [3,] -0.87755 -1.9175  0.5980  2.8748  0.1756  3.4406  4.3895  0.7871
## [4,]  0.04138 -0.4947  0.2201  1.0550 -0.5676 -0.8043 -0.2693 -0.5657
## [5,] -1.29002  0.9138 -2.7015 -2.3112 -2.8495  1.2491  0.1538  4.7051
## [6,]  0.62577  0.4502  0.1480 -1.4777  3.2767  2.6284 -0.5791 -0.5741
##         [,9]    [,10]   [,11]
## [1,]  2.6024  0.17604  0.9394
## [2,] -1.0592 -0.38620 -0.3016
## [3,]  0.3589  1.86396 -0.3425
## [4,]  0.4204  0.23138  0.4805
## [5,]  0.4171  0.08123 -1.1769
## [6,] -0.9544 -0.48444 -0.5622
``````
``````# row coordinates
``````
``````##        [,1]    [,2]    [,3]    [,4]    [,5]    [,6]     [,7]    [,8]
## [1,] 0.6180 -0.5717  0.8123 -0.9365 -0.5706 -0.2387 -0.06383 -0.3996
## [2,] 0.5089 -0.2085  1.7044 -1.2840  1.0391  1.1337 -0.87521  0.5964
## [3,] 0.7395 -0.5394  0.5131  0.4061 -0.7684 -0.3465 -0.63620  0.1565
## [4,] 1.0327 -0.5923 -0.4046 -0.2201 -0.8185 -0.3048  0.04315 -0.8433
## [5,] 0.6403 -0.4235 -0.0691  0.8653  0.5820 -1.3092 -0.42242  1.0691
## [6,] 0.7395 -0.5394  0.5131  0.4061 -0.7684 -0.3465 -0.63620  0.1565
##          [,9]    [,10]     [,11]
## [1,]  1.87979  0.25331  1.153553
## [2,]  0.01721 -0.30176  0.086183
## [3,] -1.10331 -0.21454 -0.046316
## [4,]  0.08561 -0.06599  0.324294
## [5,] -0.65316  0.18058  0.006424
## [6,] -1.10331 -0.21454 -0.046316
``````
``````# data frame for ggplot
mca4_vars_df = data.frame(mca4\$colcoord, Variable = rep(names(cats),
cats))
rownames(mca4_vars_df) = mca4\$levelnames

# plot of variable categories
ggplot(data = mca4_vars_df, aes(x = X1, y = X2, label = rownames(mca4_vars_df))) +
geom_hline(yintercept = 0, colour = "gray70") + geom_vline(xintercept = 0,
colour = "gray70") + geom_text(aes(colour = Variable)) + ggtitle("MCA plot of variables using R package ca")
``````

``````# default plot in ca
plot(mca4)
``````

### homals with homals

A fifth possibility is the homals function from the package homals by Jan de Leeuw and Patrick Mair.

``````# CA with function corresp
require(homals, quietly = TRUE)
# apply homals
mca5 = homals(newtea, ndim = 5, level = "nominal")
``````
``````## Warning: NAs introduced by coercion
``````
``````## Warning: NAs introduced by coercion
``````
``````## Warning: NAs introduced by coercion
``````
``````## Warning: NAs introduced by coercion
``````
``````## Warning: NAs introduced by coercion
``````
``````## Warning: NAs introduced by coercion
``````
``````
# eigenvalues
mca5\$eigenvalues
``````
``````## [1] 0.02331 0.02148 0.01834 0.01566 0.01405
``````
``````# column coordinates
``````
``````## \$Tea
##                  D1        D2        D3        D4        D5
## Earl Grey -0.005902  0.002623  0.011318  0.001646  0.002682
## black      0.010507  0.003388 -0.026569 -0.017393 -0.003410
## green      0.010955 -0.022939 -0.006613  0.029374 -0.008038
##
## \$How
##               D1        D2        D3       D4        D5
## alone -0.0005122 -0.005920 -0.002434  0.01078  0.006872
## lemon  0.0160756  0.010945  0.029875 -0.02361  0.024779
## milk  -0.0078048  0.005382 -0.001637 -0.01511 -0.035961
## other  0.0067866  0.050458 -0.045354 -0.04129  0.011981
##
## \$how
##                           D1        D2        D3        D4         D5
## tea bag            -0.014523 -0.007774 -0.001527 -0.005628  0.0005400
## tea bag+unpackaged  0.008722  0.023588  0.001330  0.015138 -0.0002901
## unpackaged          0.045806 -0.024877  0.003735 -0.012950 -0.0017924
##
## \$sugar
##                 D1         D2       D3        D4         D5
## No.sugar  0.005606  0.0009363 -0.01381  0.008044 -0.0004736
## sugar    -0.005993 -0.0010009  0.01476 -0.008599  0.0005063
##
## \$where
##                            D1        D2        D3         D4         D5
## chain store          -0.01257 -0.008094 -0.003299  0.0001402  0.0005466
## chain store+tea shop  0.01131  0.031415  0.003057  0.0024093  0.0009563
## tea shop              0.05103 -0.029878  0.013167 -0.0071612 -0.0059849
##
## \$always
##                   D1        D2        D3        D4        D5
## Not.always -0.001346 -0.001452 -0.006222 -0.004187  0.009383
## always      0.002574  0.002777  0.011901  0.008008 -0.017946
##
``````
``````# row coordinates
``````
``````##         D1        D2        D3        D4        D5
## 1 -0.01456 -0.013483 -0.019147 -0.022069  0.014290
## 2 -0.01199 -0.004923 -0.040173 -0.030272 -0.029165
## 3 -0.01742 -0.012725 -0.012093  0.009576  0.019355
## 4 -0.02433 -0.013977  0.009537 -0.005183  0.020316
## 5 -0.01509 -0.009991  0.001629  0.020390 -0.007502
## 6 -0.01742 -0.012725 -0.012093  0.009576  0.019355
``````
``````# data frame for ggplot
D1 = unlist(lapply(mca5\$catscores, function(x) x[, 1]))
D2 = unlist(lapply(mca5\$catscores, function(x) x[, 2]))
mca5_vars_df = data.frame(D1 = D1, D2 = D2, Variable = rep(names(cats),
cats))
rownames(mca5_vars_df) = unlist(sapply(mca5\$catscores, function(x) rownames(x)))

# MCA plot of variable categories
ggplot(data = mca5_vars_df, aes(x = D1, y = D2, label = rownames(mca5_vars_df))) +
geom_hline(yintercept = 0, colour = "gray70") + geom_vline(xintercept = 0,
colour = "gray70") + geom_text(aes(colour = Variable)) + ggtitle("MCA plot of variables using R package homals")
``````

``````# default plot in homals
plot(mca5)
``````