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: Gower, Lubbe, and Roux (2011), Beh and Lombardo (2014).

In this vignette, focus will be placed on three *EZ*-to-use
versions based on the Pearson residuals (Gower,
Lubbe, and Roux (2011) :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:

```
X <- HairEyeColor[,,2]
X
#> Eye
#> Hair Brown Blue Hazel Green
#> Black 36 9 5 2
#> Brown 66 34 29 14
#> Red 16 7 7 7
#> Blond 4 64 5 8
```

The grand total of the table \(N\) is obtained from the total of all frequencies:

\[ \sum_{r=1}^{R}\sum_{c=1}^{C}x_{rc}=N \]

It is common to work with the proportions rather than the frequencies in terms of the correspondence matrix, \(\mathbf{P}\):

\[ \mathbf{P}=\frac{\mathbf{X}}{N} \]

```
P <- X/N
P
#> Eye
#> Hair Brown Blue Hazel Green
#> Black 0.115015974 0.028753994 0.015974441 0.006389776
#> Brown 0.210862620 0.108626198 0.092651757 0.044728435
#> Red 0.051118211 0.022364217 0.022364217 0.022364217
#> Blond 0.012779553 0.204472843 0.015974441 0.025559105
```

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:

\[ \mathbf{r}_r = \sum_{c=1}^{C}p_{rc}; \hspace{0.5 cm} \mathbf{c}_c = \sum_{r=1}^{R}p_{rc}\\ \mathbf{r}=\mathbf{P1}; \hspace{0.5 cm} \mathbf{c}=\mathbf{P}^\prime\mathbf{1} \]

Diagonal matrices:

\[ \mathbf{D_r}=\text{diag}(\mathbf{r}); \hspace{0.5 cm} \mathbf{D_c}=\text{diag}(\mathbf{c}) \]

```
Dr <- diag(apply(P, 1, sum))
Dr
#> [,1] [,2] [,3] [,4]
#> [1,] 0.1661342 0.000000 0.0000000 0.0000000
#> [2,] 0.0000000 0.456869 0.0000000 0.0000000
#> [3,] 0.0000000 0.000000 0.1182109 0.0000000
#> [4,] 0.0000000 0.000000 0.0000000 0.2587859
```

```
Dc <- diag(apply(P, 2, sum))
Dc
#> [,1] [,2] [,3] [,4]
#> [1,] 0.3897764 0.0000000 0.0000000 0.00000000
#> [2,] 0.0000000 0.3642173 0.0000000 0.00000000
#> [3,] 0.0000000 0.0000000 0.1469649 0.00000000
#> [4,] 0.0000000 0.0000000 0.0000000 0.09904153
```

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`

.

```
Smat <- sqrt(solve(Dr))%*%(P-(Dr %*%matrix(1, nrow = nrow(X),
ncol = ncol(X)) %*% Dc))%*%sqrt(solve(Dc))
svd.out <- svd(Smat)
svd.out
#> $d
#> [1] 5.499629e-01 1.806424e-01 7.541748e-02 7.966568e-17
#>
#> $u
#> [,1] [,2] [,3] [,4]
#> [1,] -0.3832205 0.7807477 -0.27828196 0.4075956
#> [2,] -0.3195599 -0.2982325 0.59335474 0.6759209
#> [3,] -0.1736837 -0.5332073 -0.75320188 0.3438181
#> [4,] 0.8490333 0.1310737 -0.05635808 0.5087101
#>
#> $v
#> [,1] [,2] [,3] [,4]
#> [1,] -0.61837991 0.4547976 -0.14487614 0.6243207
#> [2,] 0.75797511 0.2307003 0.08963172 0.6035041
#> [3,] -0.20612962 -0.5898626 0.68015284 0.3833600
#> [4,] 0.02430218 -0.6260980 -0.71300012 0.3147086
```

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. (Beh and Lombardo (2014), Greenacre (2017)).

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. (Gabriel (1971)) 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.

As presented in Gower, Lubbe, and Roux (2011) :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`

.

`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 (Gower, Lubbe, and Roux (2011) :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 (Gower, Lubbe, and Roux (2011) :299):

\[ diag(\mathbf{V}\mathbf{\Sigma}^2\mathbf{J}\mathbf{V}')[diag(\mathbf{V}\mathbf{\Sigma^2}\mathbf{V}')]^{-1} \]

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 is 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:

`Variant="Stand"`

To construct the CA biplot for row standard coordinates:

`Variant="Symmetric"`

To construct the symmetric CA map:

The `fit.mesaures()`

function should be utilised to obtain
the specific fit measures explained above.

```
ca.out <- biplot(HairEyeColor[,,2], center = FALSE) |>
CA(variant = "Symmetric") |> fit.measures()
print("Quality")
#> [1] "Quality"
```

```
ca.out$adequacy
#> Brown Blue Hazel Green
#> Dim 1 0.3824 0.5745 0.0425 0.0006
#> Dim 2 0.5892 0.6277 0.3904 0.3926
#> Dim 3 0.6102 0.6358 0.8530 0.9010
#> Dim 4 1.0000 1.0000 1.0000 1.0000
```

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 Greenacre (2017) 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.

The `sample()`

function should be utilised to specify the
colours, plotting characters and expansion of the samples.

Consider the South African Crime data set 2008, extracted from the South African police website (http://www.saps.gov.za/). Gower, Lubbe, and Roux (2011) :312.

```
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
#> Crimes
#> Provinces Arsn AGBH AtMr BNRs BRs CrJk CmAs CmRb DrgR InAs Mrd PubV
#> ECpe 1235 34479 2160 5946 29508 604 19875 7086 7929 764 3515 88
#> FrSt 432 16833 939 4418 15705 156 19885 4193 4525 427 879 59
#> Gaut 1824 46993 5257 15117 62703 7466 57153 22152 12348 1501 3674 174
#> KZN 1322 30606 4946 10258 37203 3889 29410 9264 24174 1197 4713 76
#> Limp 573 13670 722 5401 11857 203 11024 3760 3198 215 696 31
#> Mpml 588 16849 1271 4273 18855 664 12202 4752 1770 251 835 61
#> NWst 624 15861 881 4987 14722 291 10406 3863 7004 345 917 117
#> NCpe 169 9898 775 1956 4924 5 5431 1337 2201 213 422 32
#> WCpe 629 24915 1844 10639 42376 923 32663 8578 45985 1850 2836 257
#> Crimes
#> Provinces Rape RAC
#> ECpe 5499 8939
#> FrSt 2628 4501
#> Gaut 8073 50970
#> KZN 6502 24290
#> Limp 2816 2447
#> Mpml 2635 5907
#> NWst 3017 5528
#> NCpe 1020 1175
#> WCpe 4000 14555
```

Beh, E, and Rosaria Lombardo. 2014. “Correspondence
Analysis.” *Theory, Paractice and New Strategies*.

Gabriel, K. R. 1971. “The Biplot Graphic Display of Matrices with
Application to Principal Component Analysis.”
*Biometrika*, 453–67.

Gower, J. C., S. Lubbe, and N. J. le Roux. 2011. *Understanding
Biplots.* Wiley.

Greenacre, M. J. 2017. *Correspondence Analysis in Practice.* CRC
press.