---
title: "CA and MCA"
output:
rmarkdown::html_vignette:
toc: true
number_sections: true
bibliography: references.bib
vignette: >
%\VignetteIndexEntry{CA and MCA}
%\VignetteEncoding{UTF-8}
%\VignetteEngine{knitr::rmarkdown}
editor_options:
markdown:
wrap: 72
---
```{r, include = FALSE}
knitr::opts_chunk$set(
fig.height = 6, fig.width = 7,
collapse = TRUE,
comment = "#>"
)
```
```{r setup, include=FALSE}
library(biplotEZ)
```
# What is a Correspondence Analysis?
In its simplest form Correspondence Analysis (CA) aims to expose the
association between two categorical variables by utilising a two-way
frequency table. Numerous variants of CA are available for the
application to diverse problems, the interested reader is referred to:
@UB2011, @BehLombardo2014.
In this vignette, focus will be placed on three *EZ*-to-use versions based on the Pearson residuals (@UB2011 :300).
Now, the two-way frequency table is also referred to as the data matrix:
$\mathbf{X}:r\times c$. This data matrix is different from the
continuous case used for the `PCA()` and `CVA()` examples, as it
represents the cross-tabulations of two categorical variables (i.e.
factors), each with a finite number of levels (i.e response values). The
elements of the data matrix represent the frequency of the co-occurrence
of two particular levels of the two variables. Consider the
`HairEyeColor` data set in `R`, which summarises the hair and eye color
of male and female statistics students. For the purpose of this example
only the male students will be considered:
```{r}
X <- HairEyeColor[,,2]
X
```
The grand total of the table $N$ is obtained from the total of all
frequencies:
$$
\sum_{r=1}^{R}\sum_{c=1}^{C}x_{rc}=N
$$
```{r}
N <- sum(X)
N
```
It is common to work with the proportions rather than the frequencies
in terms of the correspondence matrix, $\mathbf{P}$:
$$
\mathbf{P}=\frac{\mathbf{X}}{N}
$$
```{r}
P <- X/N
P
```
Other useful summaries of $\mathbf{P}$ include the row and column masses (for arbitrary row and column $r$ and $c$, respectively), also expressed as diagonal matrices:
$$
\mathbf{r}_r = \sum_{c=1}^{C}p_{rc}; \hspace{0.5 cm}
\mathbf{c}_c = \sum_{r=1}^{R}p_{rc}\\
\mathbf{r}=\mathbf{P1}; \hspace{0.5 cm} \mathbf{c}=\mathbf{P}^\prime\mathbf{1}
$$
```{r}
rMass <- rowSums(P)
rMass
cMass <- colSums(P)
cMass
```
Diagonal matrices:
$$
\mathbf{D_r}=\text{diag}(\mathbf{r}); \hspace{0.5 cm} \mathbf{D_c}=\text{diag}(\mathbf{c})
$$
```{r}
Dr <- diag(apply(P, 1, sum))
Dr
Dc <- diag(apply(P, 2, sum))
Dc
```
In order to obtain the first form of the row and column coordinates, the singular value decomposition (SVD) of the matrix of standardised Pearson residuals ($\mathbf{S}$) is computed:
$$
\begin{aligned}
\text{SVD}(\mathbf{S}) &= \text{SVD}\left(\mathbf{D_r^{-\frac{1}{2}}}(\mathbf{P}-\mathbf{rc^\prime})\mathbf{D_c^{-\frac{1}{2}}}\right)\\&= \mathbf{U\Lambda V^\prime}
\end{aligned}
$$
The return value for the Standardised pearson residuals is `Smat` and the singular value decomposition, `SVD`.
```{r}
Smat <- sqrt(solve(Dr))%*%(P-(Dr %*%matrix(1, nrow = nrow(X),
ncol = ncol(X)) %*% Dc))%*%sqrt(solve(Dc))
svd.out <- svd(Smat)
svd.out
```
This is linked to the $\chi^2$-statistic to determine whether the two categorical variables (i.e. the rows and columns of the contingency table) are independent. The expected frequencies represented by the product of the row and column masses ($\mathbf{rc^\prime}$).
Furthermore, since the weights of certain objects might be substantially different from others which could result in a distorted approximation in lower dimension, the $\chi^2$-distance, also referred to as the weighted Euclidean distance, is rather used to measure distances in CA.
This is an intuitive decision as it follows from the $\chi^2$-statistic to test the independence between two categorical variables, in this case the independence between the rows and columns of the contingency table. (@BehLombardo2014, @Greenacre2017).
# CA biplot
## Coordinates
In order to construct a biplot in which the distances between the row and column coordinates are meaningful an asymmetric display should be constructed. This means that the contribution of the singular values should be different for the row and column coordinates. (@Gabriel1971)
The standard coordinates are expressed by:
$$
\begin{aligned}
\text{Row standard coordinates:} \hspace{0.5 cm}&\mathbf{U}\\
\text{Column standard coordinates:} \hspace{0.5 cm}&\mathbf{V}
\end{aligned}
$$
The principal coordinates are expressed by:
$$
\begin{aligned}
\text{Row principal coordinates:} \hspace{0.5 cm}&\mathbf{U\Lambda}\\
\text{Column principal coordinates:} \hspace{0.5 cm}&\mathbf{V\Lambda}
\end{aligned}
$$
By including the singular values the magnitude of the association between the variables are incorporated in the scaling of the coordinates.
In the `ca()` function the argument `variant` allows the user to choose between three types of CA biplots: `Princ`, `Stand` and `Symmetric`.
$$
\begin{aligned}
\text{Row coordinates:} \hspace{0.5 cm}&\mathbf{U\Lambda^\gamma}\\
\text{Column coordinates:} \hspace{0.5 cm}&\mathbf{V\Lambda^{1-\gamma}}
\end{aligned}
$$
The row standard (i.e. column principal) coordinate biplot: `Stand`, results from $\gamma=0$.
The row principal (i.e. column standard) coordinate biplot: `Princ`, results from $\gamma=1$.
The symmetric plot in which row and column coordinates are scaled equally: `Symmetric`, results from $\gamma=0.5$.
The return value is `rowcoor` and `colcoor`, respectively.
```{r}
gamma <- 0
rowcoor <- svd.out[[2]]%*%(diag(svd.out[[1]])^gamma)
colcoor <- svd.out[[3]]%*%(diag(svd.out[[1]])^(1-gamma))
```
## Lambda scaling
As presented in @UB2011 :24, when constructing a biplot representing the rows of a coordinate matrix $\mathbf{A}$ and $\mathbf{B}^\prime$. Take note that the inner product is invariant when $\mathbf{A}$ and $\mathbf{B}$ are scaled inversely by $\lambda$.
$$
\mathbf{AB} = (\lambda\mathbf{A})(\mathbf{B}/\lambda)
$$
An arbitrary value of $\lambda$ can be selected or an *optimal* value could be to ensure that the average squared distance of the points in $\lambda\mathbf{A}$ and $\mathbf{B}/\lambda$ is equal.
$$
\lambda^4 =\frac{r}{c}\frac{||\mathbf{B}||^2}{||\mathbf{A}||^2}
$$
The default setting is to not apply lambda-scaling (i.e. $\lambda=1$).
The return value is `lambda.val`.
## Measures of fit
`quality`, `adequacy`, `row.predictivities` and `column.predictivities` are available for CA biplots.
As explained in the biplotEZ vignette, the quality of the biplot is measured by the ratio of the variance explained (sum of the squared singular values of the utilised ($M$) components) and the total variance (sum of all squared singular values ($p$)).
$$
\frac{\sum_{m=1}^{M}\lambda_m^2}{\sum_{m=1}^{p}\lambda_m^2}
$$
The `adequacy` refers to the representation of the variables. In `CA()` the factor variable represented in the columns is treated as the variables and is calculated as explained in the biplotEZ vignette:
$$
\frac{diag(\mathbf{V}_r\mathbf{V}_r')}{diag(\mathbf{VV}')}= diag(\mathbf{V}_r\mathbf{V}_r')
$$
The predictivities provide a measure of how well the original values are recovered from the biplot. An element that is well represented will have a predictivity close to one, indicating that the row or column variable values from prediction is close to the observed values. If an element is poorly represented, the predicted values will be very different from the original values and the predictivity value will be close to zero.
The `row.predictivities` are calculated as follows (@UB2011 :299):
$$
diag(\mathbf{U}\mathbf{\Sigma}\mathbf{J}\mathbf{\Sigma}\mathbf{U}')[diag(\mathbf{U}\mathbf{\Sigma}\mathbf{\Sigma}\mathbf{U}')]^{-1}\\
=diag(\mathbf{U}\mathbf{\Sigma}^2\mathbf{J}\mathbf{U}')[diag(\mathbf{U}\mathbf{\Sigma^2}\mathbf{U}')]^{-1}
$$
The `col.predictivities` are calculated as follows (@UB2011 :299):
$$
diag(\mathbf{V}\mathbf{\Sigma}^2\mathbf{J}\mathbf{V}')[diag(\mathbf{V}\mathbf{\Sigma^2}\mathbf{V}')]^{-1}
$$
# The function CA()
The function `CA()` requires a two-way contingency table as input and will return an object of class `CA` and `biplot`. As this is not a standard data matrix as for `PCA` and `CVA`, scaling and centering are not allowed on the two-way contingency table and a warning will be given if either `scale` or `center` is specified as `TRUE` in biplot()`.
## `Variant="Princ"`
The default CA biplot is a row principal coordinate biplot:
```{r}
biplot(HairEyeColor[,,2], center = FALSE) |> CA() |> plot()
```
## `Variant="Stand"`
To construct the CA biplot for row standard coordinates:
```{r}
ca.out <- biplot(HairEyeColor[,,2], center = FALSE) |> CA(variant = "Stand") |>
plot()
```
## `Variant="Symmetric"`
To construct the symmetric CA map:
```{r}
ca.out <- biplot(HairEyeColor[,,2], center = FALSE) |>
CA(variant = "Symmetric") |> plot()
ca.out$lambda.val
```
## Measures of fit
The `fit.mesaures()` function should be utilised to obtain the specific fit measures explained above.
```{r}
ca.out <- biplot(HairEyeColor[,,2], center = FALSE) |>
CA(variant = "Symmetric") |> fit.measures()
print("Quality")
ca.out$quality
print("Adequacy")
ca.out$adequacy
print("Row predictivities")
ca.out$row.predictivities
print("Column predictivities")
ca.out$col.predictivities
```
## Interpolating new samples
Adding cross-tabulations of the two categorical variables to the plot is facilitated by the function `interpolate()`. Note that the additional variables to be interpolated did not contribute to the construction of the biplot. This is the reason why @Greenacre2017 term these supplementary points.
The function `interpolate()` accepts a matrix or data frame containing the samples and variables to be interpolated. The argument `newdata` containing the samples to be interpolated needs to have a similar structure to the data set sent to `biplot()`. If `biplot()` received a data frame, `newdata` can be either another data frame or a matrix containing the subset of numerical variables.
The function `newsamples()` operates similar to `samples()` and enables aesthetic changes to the new samples.
```{r}
biplot(HairEyeColor[,,2], center = FALSE) |> CA(variant = "Symmetric") |>
samples(pch = c(0,2)) |> interpolate(newdata = HairEyeColor[,,1]) |>
newsamples(col = c("orange","purple"), pch = c(15,17), label = TRUE) |>
plot()
```
## Aesthetics
The `sample()` function should be utilised to specify the colours, plotting characters and expansion of the samples.
```{r}
biplot(HairEyeColor[,,2], center = FALSE) |> CA(variant = "Princ") |>
samples(col = c("cyan","purple"), pch = c(15,17),
label.side = c("bottom","top"), label.cex = 1) |> plot()
```
```{r}
biplot(HairEyeColor[,,2], center = FALSE) |> CA(variant = "Symmetric") |>
samples(col = c("forestgreen","magenta"), pch = c(12,17),
label.side = c("top","bottom")) |> plot()
```
## Additional example
Consider the South African Crime data set 2008, extracted from the South African police website (http://www.saps.gov.za/). @UB2011 :312.
```{r}
SACrime <- matrix(c(1235,432,1824,1322,573,588,624,169,629,34479,16833,46993,30606,13670,
16849,15861,9898,24915,2160,939,5257,4946,722,1271,881,775,1844,5946,
4418,15117,10258,5401,4273,4987,1956,10639,29508,15705,62703,37203,
11857,18855,14722,4924,42376,604,156,7466,3889,203,664,291,5,923,19875,
19885,57153,29410,11024,12202,10406,5431,32663,7086,4193,22152,9264,3760,
4752,3863,1337,8578,7929,4525,12348,24174,3198,1770,7004,2201,45985,764,
427,1501,1197,215,251,345,213,1850,3515,879,3674,4713,696,835,917,422,2836,
88,59,174,76,31,61,117,32,257,5499,2628,8073,6502,2816,2635,3017,1020,4000,
8939,4501,50970,24290,2447,5907,5528,1175,14555),nrow=9, ncol=14)
dimnames(SACrime) <- list(paste(c("ECpe", "FrSt", "Gaut", "KZN", "Limp", "Mpml", "NWst", "NCpe",
"WCpe")), paste(c("Arsn", "AGBH", "AtMr", "BNRs", "BRs", "CrJk",
"CmAs", "CmRb", "DrgR", "InAs", "Mrd", "PubV",
"Rape", "RAC" )))
names(dimnames(SACrime))[[1]] <- "Provinces"
names(dimnames(SACrime))[[2]] <- "Crimes"
SACrime
```
```{r}
biplot(SACrime, center = FALSE) |>
CA(variant = "Symmetric", lambda.scal = TRUE) |>
samples(col = c("cyan","purple"), pch = c(15,17),
label.side = c("bottom","top"))|>
plot()
```
# What is a Multiple Correspondence Analysis?
Now we shift our attention to data sets with more than two categorical variables. Consider a data matrix $\mathbf{X}^{*}:n \times p$ containing data on $n$ objects and $p$ **categorical** variables. Before approximating this data matrix to be represented in a 2D biplot, we recode the variables as dummy variables in the form of an indicator matrix, $\mathbf{G}$. Each variable $p$ consists of $k$ category levels, therefore the indicator matrix $\mathbf{G}$ has $n$ rows and $\sum_{j=1}^pk_j$ columns. This indicator matrix is treated as the data matrix in CA, which includes the steps of obtaining a correspondence matrix, row and column masses expressed as diagonal matrices before obtaining the SVD of the matrix of standardised Pearson residuals ($\mathbf{S}$). since it is weighted by the row and column masses before SVD (@UB2011; @Greenacre2017):
$$
\begin{aligned}
\text{SVD}(\mathbf{S}) &= \text{SVD}\left(\mathbf{D_r^{-\frac{1}{2}}}(\mathbf{G}-\mathbf{rc^\prime})\mathbf{D_c^{-\frac{1}{2}}}\right)\\&= \mathbf{U\Lambda V^\prime}
\end{aligned}
$$
There are other known approaches to MCA, for example, utilising a Burt matrix in the place of the indicator matrix. Also, considering different distance metrics, such as the extended mathcing coefficient (@UB2011). These are not covered in the current version of `biplotEZ`.
# MCA biplot
The MCA biplot consists of two types of coordinates and no axes. In total there are $n$ samples coordinates and $\sum_{j=1}^pk_j$ category level point (CLP) coordinates. Typically, these coordinates are illustrated by different plotting characters and colours to ease visual interpretation. The distances between the coordinates are meaningful and reveal association between objects, between responses (category levels), as well as inter-relationships between responses and objects. These associations are approximated in a two-dimensional display, as the quality of the display increases, the proximity of the coordinates reflect the true response profiles captured in the data matrix.
## Coordinates
Again, the coordinates representing the objects and variables should have different contribution of the singular values. Typically, the sample coordinates are presented in principal coordinates and the category level point (CLP) coordinates are presented in standard coordinates with some being located on the vertices of the plot (@Gabriel1971).
$$
\begin{aligned}
\text{Sample principal coordinates:} \hspace{0.5 cm}&\mathbf{D_r^{-\frac{1}{2}}}\mathbf{U}\\
\text{Category level standard coordinates:} \hspace{0.5 cm}&\mathbf{D_c^{-\frac{1}{2}}}\mathbf{V}
\end{aligned}
$$
The scaling of the coordinates by the row and column masses is a default approach adopted by @Greenacre2017. However, various valid alternative scaling options are available, but beyond the scope of `biplotEZ`.
## Measures of fit
`quality` is available for MCA biplots and is typically referred to as the total inertia in MCA literature. Again, it is measured by the ratio of the variance explained (sum of the squared singular values of the utilised ($M$) components) and the total variance (sum of all squared singular values ($p$)).
$$
\frac{\sum_{m=1}^{M}\lambda_m^2}{\sum_{m=1}^{p}\lambda_m^2}
$$
# The function MCA()
The function `MCA()` requires a multivariate categorical data matrix as input and will return an object of class `MCA` and `biplot`. If a matrix with both numeric and categorical variables are passed through the `biplot()` function, only the categorical variables will be considered for MCA. As the variables are categorical, scaling and centering are not suitable transformations and a warning will be given if either `scale` or `center` is specified as `TRUE` in biplot()`.
The full `HairEyeColor` is considered for the MCA examples.
```{r}
head(as.data.frame(HairEyeColor))
```
As the final column represents the frequency of the co-occurrences of the category elvels of the Hair, Eye and Sex variables, it is treated as numeric and will not be passed to the `MCA()` function.
```{r}
biplot(as.data.frame(HairEyeColor), center = FALSE) |> MCA()
```
The default arguments will result in the following biplot:
```{r}
biplot(as.data.frame(HairEyeColor), center = FALSE) |> MCA() |> plot()
```
Since, two types of coordinates are represented, changes to the sample aesthetics are crucial. The addition of plotting labels also enhance the interpretability of the plot.
```{r}
biplot(as.data.frame(HairEyeColor), center = FALSE) |>
MCA() |>
samples(col = c("forestgreen", "orange3"), pch = c(15, 17), label = TRUE,
label.col = "black", label.offset = 1) |>
plot()
```
Default arguments for `MCA` will suffice, the only argument that is worth noting controls the labels of the CLP coordinates in the display. In the first example `lvl.num = TRUE` was shown, which is the default value. Detail regarding the levels can also be displayed by changing this argument to `FALSE`:
```{r}
biplot(as.data.frame(HairEyeColor), center = FALSE) |>
MCA(lvl.num = FALSE) |>
samples(col = c("forestgreen", "orange3"), pch = c(15, 17), label = TRUE,
label.col = "black", label.offset = 1) |>
plot()
```
## Measure of fit
The `fit.mesaures()` function should be utilised to obtain the specific fit measures as is done in other biplot optpions available in `biplotEZ`:
```{r}
mca.out <- biplot(as.data.frame(HairEyeColor), center = FALSE) |>
MCA() |> fit.measures()
mca.out
```
# References