| Type: | Package |
| Title: | Three-Way Correspondence Analysis Variants |
| Version: | 3.3.1 |
| Date: | 2022-10-10 |
| Author: | Rosaria Lombardo, Eric J Beh and Michel van de Velden |
| Maintainer: | Rosaria Lombardo <rosaria.lombardo@unicampania.it> |
| Description: | Provides four variants of three-way correspondence analysis (ca): three-way symmetrical ca, three-way non-symmetrical ca, three-way ordered symmetrical ca and three-way ordered non-symmetrical ca. |
| Depends: | R (> 3.0.1) |
| Imports: | ggplot2, plotly, tools, ggforce, gridExtra, ggrepel, multichull, utils |
| LazyData: | true |
| License: | GPL (> 2) |
| URL: | https://www.R-project.org |
| NeedsCompilation: | no |
| Packaged: | 2022-10-10 15:45:00 UTC; rosar |
| Repository: | CRAN |
| Date/Publication: | 2022-10-10 16:20:02 UTC |
Correspondence Analysis variants for three-way contingency tables
Description
This function performs four variants of three-way correspondence analysis (CA). It does the three-way symmetrical CA, when ca3type = "CA3", and three-way non-symmetrical
CA, when ca3type = "NSCA3", by using the Tucker3 decomposition.
It also performs ordered three-way symmetrical CA, when ca3type = "OCA3", and ordered
three-way non-symmetrical CA, when ca3type = "ONSCA3",
by using the Trivariate Moment Decomposition. The non-symmetrical variants consider the three
variables asymmetrically related, such that one of the variables is the response to be predicted
given the other two variables. It calculates the coordinates and inertia values of the chosen analyses.
Furthermore, it allows to look at the index (Pearson's chi-squared or Marcotorchino's tau) partition.
Usage
CA3variants(Xdata, dims = c(p, q, r), ca3type = "CA3", test = 10^-6,
resp = "row", norder = 3, sign = TRUE)
Arguments
Xdata |
The three-way data. It can be a |
dims |
The number of components for the first, second and third mode. By default, no |
ca3type |
The specification of the analysis to be performed.
If |
test |
Threshold used in the algorithm for stopping it after the convergence of the solutions. |
resp |
The input parameter for specifying in non-symmetrical three-way correspondence analysis variants ( |
norder |
The input parameter for specifying the number of ordered variable when |
sign |
The input parameter for changing the sign to the components according to the core sign. |
Details
This function recall internally many other functions, depending on the setting of the input parameters.
After performing three-way symmetric or non-symmetric correspondence analysis, it recall two functions for printing and plotting the results.
These two important functions are print.CA3variants and plot.CA3variants.
Value
The value of output returned depends on the kind of analysis performed.
For a detailed description of the output one can see:
the output value of ca3basic if the input parameter is ca3type="CA3";
the output value of nsca3basic if the input parameter is ca3type="NSCA3";
the output value of oca3basic if the input parameter is ca3type="OCA3"
the output value of onsca3basic if the input parameter is ca3type="ONSCA3"
Author(s)
Rosaria Lombardo, Eric J Beh and Michel van de Velden.
References
Beh EJ and Lombardo R (2014) Correspondence Analysis, Theory, Practice and New Strategies. John Wiley & Sons.
Kroonenberg PM (1994) The TUCKALS line: a suite of programs for three-way data analysis. Computational Statistics and Data Analysis, 18, 73–96.
Lombardo R, Beh EJ and Kroonenberg PM (2021) Symmetrical and Non-Symmetrical Variants of Three-Way Correspondence Analysis for Ordered Variables.
Statistical Science, 36 (4), 542-561.
Examples
data(ratrank)
CA3variants(Xdata = ratrank, dims = c(p=2,q=2,r=1), ca3type = "CA3")
data(happy)
CA3variants(Xdata = happy, dims = c(p=2,q=2,r=2), ca3type = "NSCA3")
CA3variants(Xdata = happy, dims = c(p=3,q=5,r=4), ca3type = "OCA3")
CA3variants(Xdata = happy, dims = c(p=3,q=5,r=4), ca3type = "ONSCA3")
Kronecker product
Description
The function performs the Kronecker product. Starting from two matrices of dimension
IxP and JxQ the resulting matrix will be of dimension IxJ,PxQ.
Usage
Kron(a, b)
Arguments
a |
The first matrix of dimension IxP involved in the kronecker product. |
b |
The second matrix of dimension JxQ involved in the kronecker product. |
Details
This function is utilised from several other functions like CA3variants,
newcomp3, step.g3 and reconst3.
Author(s)
Rosaria Lombardo, Eric J Beh, Ida Camminatiello.
Three-way Symmetrical Correspondence Analysis
Description
This function is used in the main function CA3variants when the input parameter is ca3type="CA3".
It performs the three-way symmetrical correspondence analysis by TUCKALS3 algorithm.
Usage
ca3basic(x, p, q, r, test = 10^-6, ctr = T, std = T, sign = TRUE)
Arguments
x |
The three-way contingency table. |
p |
The number of components of the first mode. |
q |
The number of components of the second mode. |
r |
The number of components of the third mode. |
test |
The treshold used in the algorithm TUCKALS3. |
ctr |
The flag parameter (T or F), if F the analysis is not centered. |
std |
The flag parameter (T or F) if F the analysis is not standardized. |
sign |
The input parameter for changing the sign to the components according to the core sign. |
Value
x |
The original three-way contingency table. |
xs |
The weighted three-way contingency table. |
xhat |
Three-way contingency table reconstructed after Tuckals3 by principal components and core array. |
nxhat2 |
The inertia of three-way symmetric correspondence analysis (Three-way Pearson ratio). |
prp |
The proportion of inertia reconstructed using the p, q, r principal components and the core array to the total inertia. To select the model dimensions (number of principal components), we examine the inertia explained by the p, q, r principal components with respect to the overall fit. |
a |
The row principal components. |
b |
The column principal coordinates. |
cc |
The tube principal coordinates. |
g |
The core array calculated by using the Tuckals3 algorithm and can be interpreted as generalised singular value table. They help to explain the strength of the association between the three principal components. |
iteration |
The number of iterations that are required for the TUCKALS3 algorithm to converge. |
Author(s)
Rosaria Lombardo, Eric J Beh.
Row isometric biplot or Column isometric biplot
Description
This function is used in the main plot function when the plot type parameter is
plottype = "biplot" and the variants of three-way CA are not ordered. It can produce a row or a column biplot.
Usage
ca3plot(frows, gcols, firstaxis, lastaxis, inertiapc, size1, size2, biptype, addlines)
Arguments
frows |
The row principal or standard coordinates. |
gcols |
The column principal or standard coordinates. |
firstaxis |
The first axis number. |
lastaxis |
The second axis number. |
inertiapc |
The percentage of the explained inertia by each dimension. |
size1 |
The size of the plotted symbol for categories in biplots. |
size2 |
The size of the plotted text for categories in biplots. |
biptype |
The input parameter for specifying what kind of biplots is requested. By default,
it is equal to |
addlines |
The input parameter for plotting lines in biplot. By default, it is equal to |
Note
This function depends on the R library plotly.
Author(s)
Rosaria Lombardo, Eric J. Beh and Michel van de Velden.
Three dimensional correspondence plot
Description
This function is used in the plot function plot.CAvariants when the logical parameter is
plot3d = TRUE.
It produces a 3-dimensional visualization of the association.
Usage
caplot3d(coordR, coordC, inertiaper, firstaxis = 1, lastaxis = 2, thirdaxis = 3)
Arguments
coordR |
The row principal or standard coordinates. |
coordC |
The column principal or standard coordinates. |
inertiaper |
The percentage of the total inertia explained inertia by each dimension. |
firstaxis |
The first axis number. By default, |
lastaxis |
The second axis number. By default, |
thirdaxis |
The third axis number. By default, |
Note
This function depends on the R library plotly.
Author(s)
Rosaria Lombardo and Eric J. Beh
The partition of the Pearson three-way index
Description
When three categorical variables are symmetrically related, we can analyse the strength of
the association using the three-way Pearson mean square contingency coefficient, named the chi-squared index.
The function chi3 partitions the Pearson phi-squared statistic when in CA3variants
we set the parameter ca3type = "CA3".
Usage
chi3(f3, digits = 3)
Arguments
f3 |
The three-way contingency array given as an input parameter in CA3variants. |
digits |
The number of decimal digits. By default digits=3. |
Value
The partition of the Pearson index into three two-way association terms and one three-way association term. It also shows the explained inertia, the degrees of freedom and p-value of each term of the partition.
Author(s)
Rosaria Lombardo, Eric J Beh, Ida Camminatiello.
References
Beh EJ and Lombardo R (2014) Correspondence Analysis, Theory, Practice and New Strategies. John Wiley & Sons.
Carlier A and Kroonenberg PM (1996) Decompositions and biplots in three-way correspondence analysis. Psychometrika, 61, 355-373.
Examples
data(happy)
chi3(f3=happy, digits=3)
The partition of the Pearson three-way index.
Description
When three categorical variables are symmetrically related, we can analyse the strength of
the symmetrical association using the three-way Pearson statistic.
The function chi3ordered partitions the Pearson phi-squared statistic using orthogonal polynomials
when, in CA3variants, we set the parameter ca3type = "OCA3".
Usage
chi3ordered(f3, digits = 3)
Arguments
f3 |
The three-way contingency array given as an input parameter in |
digits |
The number of decimal digits. By default digits=3. |
Value
The partition of the Pearson index into three two-way association terms and one three-way association term. It also shows the polynomial componets of inertia, the percentage of explained inertia, the degrees of freedom and p-value of each term of the partition.
Author(s)
Rosaria Lombardo, Eric J Beh, Ida Camminatiello.
References
Lombardo R, Beh EJ and Kroonenberg PM (2021) Symmetrical and Non-Symmetrical Variants of Three-Way Correspondence Analysis for Ordered Variables. Statistical Science, 36 (4), 542-561.
Examples
#data(happy)
chi3ordered(f3 = happy, digits = 3)
Check the sign of component values
Description
This function is called from signscore. It checks the negativity of the column of an array
AND the positivity of the columns of an array
If NegPtr = 1 then there is an entirely negative component
If PosPtr = 1 then there is an entirely positive component
If BigPtr = 1 then maximum neg. abs > max pos
Usage
chkneg(comp, nr, nc)
Arguments
comp |
One of three component matrix. |
nr |
The row number of the component matrix. |
nc |
The column number of the component matrix. |
Author(s)
Rosaria Lombardo and Pieter M Kroonenberg.
References
Kroonenberg PM (1994) The TUCKALS line: a suite of programs for three-way data analysis. Computational Statistics and Data Analysis, 18, 73–96.
The weighted components of the Tucker3 algorithm
Description
The function computes the weighted components from the Tucker3 algorithm (to take into account the different weight systems in row, column and tube spaces) for symmetrical three-way correspondence analysis.
Usage
coord(res, x)
Arguments
res |
The component matrices resulting from the Tucker3 algorithm. |
x |
The original three-way contingency table. |
Author(s)
Rosaria Lombardo, Eric J Beh.
The weighted components of the Tucker3 algorithm
Description
The function computes the weighted components from the Tucker3 algorithm (to take into account the different weight systems in row, column and tube spaces) for non-symmetrical three-way correspondence analysis.
Usage
coordnsc3(res, x)
Arguments
res |
The component matrices resulting from the Tucker3 algorithm. |
x |
The original three-way contingency table. |
Author(s)
Rosaria Lombardo, Eric J Beh.
The stopping criteria for the Tucker3 algorithm
Description
The function computes the stopping criteria needed for the Tucker3 algorithm.
Usage
criter(x, xhat)
Arguments
x |
The three-way contingency table |
xhat |
The reconstruction of the data array using the three component matrices and the core array. |
Value
The criterion used in Tuckals3 is the sum of squares of the differences between the weighted data array and the reconstructed data array.
Author(s)
Rosaria Lombardo, Eric J Beh.
References
Kroonenberg PM (1994) The TUCKALS line: a suite of programs for three-way data analysis. Computational Statistics and Data Analysis, 18, 73–96.
Criterion of the Tucker3 algorithm
Description
The function critera is used to define the criterion of the Tucker3 algorithm.
Usage
critera(aold, anew)
Arguments
aold |
The old component of the first mode. |
anew |
The new component of the first mode. |
Author(s)
Rosaria Lombardo, Eric J Beh
References
Kroonenberg PM (1994) The TUCKALS line: a suite of programs for three-way data analysis. Computational Statistics and Data Analysis, 18, 73–96.
Pointing to the columns of the component matrices.
Description
Given ICORE, i.e. the pointer to an element in CORE(p,q,r), this subroutine calculates the IA, IB and IC, pointing to the columns of the component matrices A, B and C that are responsible for the value in CORE(ICORE).
Usage
crptrs(icore, p, q, r)
Arguments
icore |
The pointer to the core elements whose sign should be reversed. |
p |
The dimension number of the first mode. |
q |
The dimension number of the second mode. |
r |
The dimension number of the third mode. |
Value
IA |
The pointer to the columns of the first component matrix, given the pointer to an element in core. |
IB |
The pointer to the columns of the second component matrix, given the pointer to an element in core. |
IC |
The pointer to the columns of the third component matrix, given the pointer to an element in core. |
Author(s)
Rosaria Lombardo and Pieter M Kroonenberg.
References
Kroonenberg PM (1994) The TUCKALS line: a suite of programs for three-way data analysis. Computational Statistics and Data Analysis, 18, 73–96.
Orthogonal polynomials
Description
This function is called from the function oca3basic when in CA3variants we set
ca3type = "OCA3". It allows the analyst to compute the orthogonal polynomials of
the ordered categorical variable. The number of the polynomials is equal to the variable category less one.
The function computes the polynomial transformation of the ordered categorical variable.
Usage
emerson.poly(mj, pj)
Arguments
mj |
The ordered scores of an ordered variable. By default |
pj |
The marginals, relative frequencies of the ordered variable. |
Value
Describe the value returned
B |
The matrix of the orthogonal polynomials without the trivial polynomial. |
Note
Note that the sum of the marginals of the ordered variables should be one.
Author(s)
Rosaria Lombardo and Eric J Beh.
References
Beh EJ and Lombardo R (2014) Correspondence Analysis: Theory, Practice and New Strategies. John Wiley & Sons.
Emerson PL (1968) Numerical construction of orthogonal polynomials from a general recurrence formula. Biometrics, 24 (3), 695-701.
Lombardo R Beh EJ (2016) Variants of Simple Correspondence Analysis. The R Journal, 8 (2), 167–184.
Flattened table
Description
The function flattens the three-way table into the concatenation of two-way matrices.
Usage
flatten(x)
Arguments
x |
The three-way contingency table. |
Details
It is utilised by a number of functions: CA3variants, reconst3, newcomp3 and
step.g3.
Value
x |
The flattened table of size I,JK where I, J and K are the number of the categories of rows, columns and tubes, respectively. |
Author(s)
Rosaria Lombardo, Eric J Beh.
Three-way contingency table
Description
This three-way contingency table was generated from the database of the European Social Survey 2016. The variables that we selected for our analysis are Education, Households and Happiness.
Usage
data(happy)Format
The format is: row names [1:4] "ED1", "ED2", "ED3", "ED45" col names [1:6] "HS1", "HS2", "HS3", "HS4", "HS5", "HS>5" tube names [1:4] "low", "middle", "high", "very-high"
References
Beh EJ and Lombardo R (2014) Correspondence Analysis: Theory, Practice and New Strategies. John Wiley & Sons.
Examples
happy <-
structure(c(325, 411, 793, 602, 239, 374, 827, 583, 63,
181, 361, 303, 42, 129, 229, 224, 16, 49, 89, 54,
11, 37, 31, 21, 357, 477, 1049, 929, 327, 610, 1447,
1446, 115, 303, 763, 832, 64, 250, 591, 638, 35, 105,
183, 185, 15, 56, 99, 71, 265, 327, 769, 928, 342,
565, 1461, 1808, 104, 314, 768, 1006, 69, 312, 729,
977, 21, 122, 215, 362, 14, 57, 126, 129, 214, 241,
554, 660, 419, 561, 1467, 1861, 130, 290, 786, 938,
89, 319, 741, 1022, 36, 121, 289, 408, 35, 87, 153,
171), .Dim = c(4, 6, 4), .Dimnames = list(c("ED1",
"ED2", "ED3", "ED45"), c("HS1", "HS2", "HS3", "HS4", "HS5", "HS>5"
), c("low", "middle", "high", "very-high")))
dim(happy)
Raw data: Three variables from a Dutch survey on happiness
Description
This raw data table represents a possible data set selected from a large survey on happiness. The rows are individuals. The first column concerns four level of happiness, the second column concerns the number of households in a family, and the third column their level of Education.
Usage
data(museum)Format
The format is: row names [1:4] "low", "middle", "high", "very-high" col names [1:5] "HS1", "HS2", "HS3", "HS4", ">HS5" tube names [1:4] "ED1", "ED2", "ED3", "ED45"
References
Lombardo R, van de Velden M and Beh E J (2022) Three-way Correspondence Analysis in R. (submitted)
Examples
happyNL<-structure(c(11L, 12L, 15L, 7L, 2L, 6L, 17L, 13L, 0L, 2L, 4L,
6L, 0L, 5L, 7L, 3L, 0L, 3L, 3L, 1L, 14L, 56L, 52L, 22L, 11L,
39L, 70L, 65L, 1L, 14L, 19L, 14L, 5L, 12L, 16L, 20L, 2L, 3L,
10L, 4L, 14L, 44L, 44L, 15L, 6L, 27L, 79L, 47L, 4L, 17L, 40L,
27L, 2L, 25L, 49L, 38L, 1L, 12L, 12L, 11L, 10L, 41L, 66L, 24L,
4L, 32L, 100L, 90L, 1L, 8L, 40L, 28L, 3L, 15L, 49L, 35L, 1L,
4L, 23L, 15L), .Dim = c(4L, 5L, 4L), .Dimnames = list(happy = c("low",
"middle", "high", "very-high"), hhmmb = c("HS1", "HS2", "HS3",
"HS4", ">HS5"), edulvla = c("ED1", "ED2", "ED3", "ED45")), class = "table")
dim(happyNL)
data(happyNL)
Initial components from the Tuckals3 algorithm
Description
The function is utilised from the function tucker to compute the
initial components for each of the three categorical variables.
Usage
init3(x, p, q, r)
Arguments
x |
The three-way contingency table. |
p |
The number of components of the first mode. |
q |
The number of components of the second mode. |
r |
The number of components of the third mode. |
Value
The initial components for each of the three categorical variables.
a |
The initial component derived from the Tucker3 decomposition for the first mode. |
b |
The initial component derived from the Tucker3 decomposition for the second mode. |
cc |
The initial component derived from the Tucker3 decomposition for the third mode. |
x |
The three-way contingency table |
Author(s)
Rosaria Lombardo, Eric J Beh, Ida Camminatiello.
Initial components from the Trivariate Moment Decomposition algorithm
Description
The function is utilised from the function tuckerordered to compute the
initial components for each of the three ordered categorical variables.
Usage
init3ordered(x, p, q, r, x0)
Arguments
x |
The three-way contingency table. |
p |
The number of components of the first mode. |
q |
The number of components of the second mode. |
r |
The number of components of the third mode. |
x0 |
The original three-way contingency table. |
Value
The initial components for each of the three categorical variables.
a |
The initial component derived from the Trivariate Moment Decomposition for the first mode. |
b |
The initial component derived from the Trivariate Moment Decomposition for the second mode. |
cc |
The initial component derived from the Trivariate Moment Decomposition for the third mode. |
x |
The three-way contingency table. |
Author(s)
Rosaria Lombardo, Eric J Beh, Ida Camminatiello.
Initial components from the Trivariate Moment Decomposition algorithm
Description
The function is utilised from the function tuckerORDERED to compute the
initial components for the first ordered categorical variables.
Usage
init3ordered1(x, p, q, r, x0)
Arguments
x |
The three-way contingency table. |
p |
The number of components of the first mode. |
q |
The number of components of the second mode. |
r |
The number of components of the third mode. |
x0 |
The original three-way contingency table. |
Value
The initial components for each of the three categorical variables.
a |
The initial component derived from the Trivariate Moment Decomposition for the first mode. |
b |
The initial component derived from the Trivariate Moment Decomposition for the second mode. |
cc |
The initial component derived from the Trivariate Moment Decomposition for the third mode. |
x |
The three-way contingency table. |
Author(s)
Rosaria Lombardo, Eric J Beh, Ida Camminatiello.
Initial components from the Trivriate Moment Decomposition algorithm
Description
The function is utilised from the function tuckerordered to compute the
initial components for each of the two ordered categorical variables.
Usage
init3ordered2(x, p, q, r, x0)
Arguments
x |
The three-way contingency table. |
p |
The number of components of the first mode. |
q |
The number of components of the second mode. |
r |
The number of components of the third mode. |
x0 |
The original three-way contingency table. |
Value
The initial components for each of the three categorical variables.
a |
The initial component derived from the Trivriate Moment Decomposition for the first mode. |
b |
The initial component derived from the Trivriate Moment Decomposition for the second mode. |
cc |
The initial component derived from the Trivriate Moment Decomposition for the third mode. |
x |
The three-way contingency table. |
Author(s)
Rosaria Lombardo, Eric J Beh, Ida Camminatiello.
Changing component sign
Description
This function changes the sign of the elements in column of the component matrix.
Usage
invcmp(comp, nr, nc, chgcomp)
Arguments
comp |
One of the three component matrices. |
nr |
The row number of the component matrix. |
nc |
The column number of the component matrix. |
chgcomp |
The pointers to the columns of the component matrix that are responsible for the value in the ordered core. |
Author(s)
Rosaria Lombardo and Pieter M Kroonenberg.
References
Kroonenberg PM (1994) The TUCKALS line: a suite of programs for three-way data analysis. Computational Statistics and Data Analysis, 18, 73–96.
Changing core sign
Description
This function is used from the function signscore. It changes the sign of the elements of core slice
Usage
invcor(core, p, q, r, chgmode, chgcomp)
Arguments
core |
The core array. |
p |
The dimension of the first mode. |
q |
The dimension of the second mode. |
r |
The dimension of the third mode. |
chgmode |
One of the three mode to change. |
chgcomp |
One of the three component to change. |
Author(s)
Rosaria Lombardo and Pieter M Kroonenberg.
References
Kroonenberg PM (1994) The TUCKALS line: a suite of programs for three-way data analysis. Computational Statistics and Data Analysis, 18, 73–96.
General loss criterion
Description
This function represents the general loss function on which is based Tuckals3 and calculates the difference between two arrays, x and xhat, where x is the three-way contingency table and xhat is the reconstruction of this table by means of components and core array.
Usage
loss1.3(param, comp.old)
Arguments
param |
The matrices of the row, column and tube components derived via the Tucker3 model. |
comp.old |
The matrices of the row, column and tube components derived in the foregoing iteration of the Tuckals3 algorithm. |
Value
The difference between three-way contingency table and its reconstruction from the Tucker3 model.
Author(s)
Rosaria Lombardo, Eric J Beh, Ida Camminatiello.
General loss criterion
Description
This function represents the general loss function on which is based the Trivariate Moment Decomposition and calculates the difference between two arrays, x and xhat, where x is the three-way contingency table and xhat is the reconstruction of this table by means of components and core array.
Usage
loss1.3ordered(param, comp.old)
Arguments
param |
The matrices of the row, column and tube components derived via the Trivariate Moment Decomposition model. |
comp.old |
The matrices of the row, column and tube components derived in the foregoing iteration of the Trivriate Moment Decomposition algorithm. |
Value
The difference between three-way contingency table and its reconstruction from the Trivariate Moment Decomposition model.
Author(s)
Rosaria Lombardo, Eric J Beh, Ida Camminatiello.
Difference between two successive components
Description
The function computes the difference between two successive components in the iteration of the Tuckals3 algorithm.
Usage
loss2(param, comp.old)
Arguments
param |
The matrices of the row, column and tube components derived via the Tucker3 model. |
comp.old |
The matrices of the row, column and tube components derived in the foregoing iteration of the Tuckals3 algorithm. |
Value
The difference between two successive components in the iteration of the Tuckals3 algorithm.
Author(s)
Rosaria Lombardo and Eric J Beh.
Make an Indicator matrix
Description
From a three-way contingency table (as can be used in CA3variants), it gives the N x total number of categories (rows+cols+tubs) indicator matrix
Usage
makeindicator(X)
Arguments
X |
The three-way data array. It must be an |
Value
Z |
Output: the N x total number of categories (rows+cols+tubs) indicator matrix |
Author(s)
Rosaria Lombardo, Eric J Beh.
Row marginals of a three-way contingency table
Description
This function computes the row marginals of the three-way contingency table specified by the input parameter.
Usage
margI(m)
Arguments
m |
The three-way contingency table. |
Value
The row marginals of the considered three-way contingency table.
Author(s)
Rosaria Lombardo, Eric J Beh, Ida Camminatiello.
Column marginals of a three-way contingency table
Description
The function computes the column marginals of the three-way contingency table specified by the input parameter.
Usage
margJ(m)
Arguments
m |
The three-way contingency table. |
Value
The column marginals of the considered three-way contingency table.
Author(s)
Rosaria Lombardo, Eric J Beh, Ida Camminatiello.
Tube marginals of a three-way contingency table
Description
The function computes the tube marginals of the three-way contingency table specified by the input parameter.
Usage
margK(m)
Arguments
m |
The three-way contingency table. |
Value
The tube marginals of the considered three-way contingency table.
Author(s)
Rosaria Lombardo, Eric J Beh, Ida Camminatiello.
Raw data: Three variables from a survey
Description
This raw data table represents a possible data set selected from a large survey on customer satisfacyion during museum visiting. The rows are individuals. The first column concerns the number of visits, the second column concerns if they like it, and the third column their satisfaction.
Usage
data(museum)Format
The format is:
num [1:223, 1:3] "often" "much" "excellent" ...
References
Beh EJ and Lombardo R (2014) Correspondence Analysis: Theory, Practice and New Strategies. John Wiley & Sons.
Examples
museum<-structure(list(nvis = structure(c(2L, 2L, 4L, 4L, 1L, 3L, 3L,
2L, 4L, 1L, 3L, 3L, 4L, 2L, 4L, 3L, 4L, 2L, 2L, 3L, 4L, 4L, 2L,
4L, 3L, 4L, 2L, 2L, 4L, 1L, 2L, 2L, 4L, 1L, 4L, 2L, 2L, 2L, 4L,
1L, 1L, 1L, 1L, 2L, 2L, 3L, 2L, 3L, 4L, 4L, 1L, 3L, 2L, 2L, 3L,
3L, 3L, 2L, 4L, 3L, 2L, 4L, 2L, 3L, 3L, 3L, 3L, 3L, 3L, 2L, 3L,
3L, 3L, 3L, 2L, 2L, 4L, 4L, 4L, 4L, 3L, 2L, 3L, 3L, 3L, 4L, 2L,
2L, 2L, 4L, 1L, 1L, 1L, 1L, 2L, 2L, 3L, 2L, 3L, 4L, 4L, 1L, 3L,
3L, 2L, 4L, 3L, 2L, 4L, 3L, 2L, 4L, 2L, 3L, 3L, 2L, 2L, 2L, 3L,
2L, 3L, 2L, 2L, 3L, 2L, 2L, 4L, 4L, 4L, 4L, 4L, 3L, 2L, 3L, 3L,
3L, 4L, 4L, 1L, 3L, 3L, 2L, 1L, 1L, 1L, 1L, 3L, 4L, 2L, 4L, 3L,
4L, 2L, 2L, 3L, 4L, 2L, 3L, 3L, 3L, 4L, 2L, 2L, 2L, 4L, 1L, 3L,
1L, 1L, 2L, 2L, 3L, 2L, 3L, 3L, 3L, 1L, 3L, 2L, 2L, 2L, 1L, 1L,
2L, 2L, 2L, 1L, 3L, 2L, 3L, 4L, 4L, 1L, 3L, 2L, 2L, 2L, 3L, 2L,
3L, 4L, 4L, 1L, 3L, 3L, 3L, 2L, 1L, 4L, 1L, 3L, 4L, 3L, 4L, 2L,
4L, 3L, 4L, 2L, 2L, 3L, 3L, 4L), .Label = c("no", "often", "some",
"voften"), class = "factor"), like = structure(c(2L, 2L, 2L,
2L, 2L, 3L, 3L, 2L, 2L, 2L, 3L, 3L, 2L, 3L, 1L, 3L, 2L, 3L, 3L,
1L, 3L, 2L, 3L, 2L, 3L, 2L, 2L, 3L, 2L, 3L, 3L, 3L, 3L, 3L, 2L,
2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 2L, 2L, 3L, 2L, 3L, 2L, 2L, 2L,
2L, 2L, 2L, 1L, 2L, 2L, 2L, 1L, 3L, 3L, 2L, 3L, 3L, 2L, 3L, 2L,
3L, 2L, 2L, 3L, 2L, 3L, 2L, 3L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 2L, 3L, 3L, 3L,
2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 2L, 3L, 3L, 2L, 3L, 3L, 2L,
3L, 2L, 3L, 3L, 2L, 3L, 2L, 3L, 2L, 2L, 2L, 2L, 3L, 3L, 2L, 2L,
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 2L, 1L, 2L, 2L, 3L, 3L,
2L, 3L, 1L, 2L, 2L, 3L, 3L, 1L, 3L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
2L, 2L, 3L, 2L, 2L, 3L, 2L, 2L, 3L, 2L, 3L, 2L, 2L, 2L, 2L, 2L,
2L, 2L, 2L, 2L, 2L, 3L, 2L, 2L, 3L, 2L, 3L, 2L, 2L, 2L, 2L, 2L,
2L, 2L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 2L, 1L, 2L, 2L, 3L,
3L, 3L, 2L, 3L, 1L, 3L, 2L, 3L, 3L, 1L, 3L, 3L), .Label = c("little",
"much", "some"), class = "factor"), satisfaction = structure(c(1L,
2L, 2L, 1L, 1L, 2L, 2L, 1L, 3L, 1L, 3L, 1L, 1L, 2L, 2L, 1L, 2L,
2L, 2L, 2L, 2L, 1L, 4L, 2L, 2L, 3L, 1L, 2L, 1L, 1L, 3L, 3L, 1L,
1L, 2L, 1L, 1L, 1L, 2L, 2L, 1L, 4L, 3L, 1L, 1L, 2L, 2L, 2L, 2L,
2L, 2L, 2L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 2L, 3L, 1L, 2L, 3L, 2L,
3L, 2L, 3L, 2L, 1L, 2L, 2L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 1L, 1L, 4L, 3L, 1L, 1L, 2L,
2L, 2L, 2L, 2L, 2L, 1L, 2L, 1L, 2L, 2L, 3L, 1L, 2L, 3L, 1L, 2L,
3L, 2L, 3L, 2L, 3L, 2L, 1L, 2L, 2L, 2L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 1L, 3L, 3L, 1L,
3L, 1L, 1L, 2L, 2L, 1L, 2L, 2L, 2L, 2L, 2L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 2L, 2L, 1L, 4L, 2L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 2L,
1L, 1L, 1L, 2L, 1L, 1L, 4L, 3L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 2L,
2L, 2L, 1L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 2L, 2L, 1L, 3L, 3L,
1L, 3L, 4L, 1L, 1L, 2L, 2L, 1L, 2L, 2L, 2L, 2L, 3L, 4L), .Label = c("excellent",
"good", "suff", "unsuff"), class = "factor")), class = "data.frame", row.names = c("1",
"2", "3", "5", "6", "8", "9", "10", "12", "13", "14", "16", "17",
"18", "19", "20", "21", "22", "23", "24", "25", "27", "30", "31",
"32", "33", "34", "35", "36", "37", "38", "39", "40", "41", "42",
"43", "44", "45", "46", "47", "48", "49", "50", "51", "52", "54",
"55", "56", "57", "58", "59", "60", "61", "64", "65", "66", "67",
"68", "69", "70", "71", "72", "73", "74", "75", "78", "80", "81",
"82", "84", "85", "86", "87", "88", "89", "90", "91", "92", "95",
"96", "97", "98", "99", "100", "101", "102", "104", "105", "106",
"107", "108", "109", "110", "111", "112", "113", "115", "116",
"117", "118", "119", "120", "121", "122", "123", "124", "125",
"126", "127", "128", "129", "130", "131", "132", "133", "136",
"138", "139", "140", "142", "143", "144", "145", "146", "147",
"148", "149", "150", "151", "153", "154", "155", "156", "157",
"158", "159", "160", "162", "163", "165", "166", "167", "168",
"169", "170", "171", "173", "174", "175", "176", "177", "178",
"179", "180", "181", "182", "183", "184", "185", "186", "187",
"189", "190", "191", "192", "193", "194", "195", "196", "197",
"198", "200", "201", "202", "203", "204", "205", "206", "207",
"208", "209", "210", "211", "212", "213", "214", "215", "217",
"218", "219", "220", "221", "222", "223", "224", "225", "227",
"228", "229", "230", "231", "232", "233", "234", "235", "236",
"237", "238", "239", "240", "241", "242", "243", "244", "245",
"246", "247", "248", "249", "250", "251", "252", "253"))
dim(museum)
data(museum)
Updated component matrices
Description
The function computes the updated component matrices of the Tucker3 decomposition.
Usage
newcomp3(param)
Arguments
param |
The initial matrices of the row, column and tube components derived via the |
Details
It is utilised from the function tucker.
Author(s)
Rosaria Lombardo, Eric J Beh, Ida Camminatiello.
Updated component matrices
Description
The function computes the updated component matrices of the Trivariate Moment Decomposition. It is supposed that the number of the ordered categorical variables is equal to 3.
Usage
newcomp3ordered(param)
Arguments
param |
The initial matrices of the row, column and tube components derived via the |
Details
It is utilised from the function tuckerORDERED.
Author(s)
Rosaria Lombardo, Eric J Beh, Ida Camminatiello.
Updated component matrices
Description
The function computes the updated component matrices of the Trivariate Moment Decomposition. It is supposed that the number of the ordered categorical variables is equal to 1.
Usage
newcomp3ordered1(param)
Arguments
param |
The initial matrices of the row, column and tube components derived via the |
Details
It is utilised from the function tuckerORDERED.
Author(s)
Rosaria Lombardo, Eric J Beh, Ida Camminatiello.
Updated component matrices
Description
The function computes the updated component matrices of the Trivariate Moment Decomposition. It is supposed that the number of the ordered categorical variables is equal to 2.
Usage
newcomp3ordered2(param)
Arguments
param |
The initial matrices of the row, column and tube components derived via the |
Details
It is utilised from the function tuckerORDERED.
Author(s)
Rosaria Lombardo, Eric J Beh, Ida Camminatiello.
Three-way Non-Symmetrical Correspondence Analysis
Description
This function is used in the main function CA3variants when the input parameter is
catype="NSCA3".
It decomposes the Marcotorchino index, computes principal axes, coordinates, weights of rows and columns,
total inertia (equal to the Marcotorchino index) and the rank of the matrix.
Usage
nsca3basic(x, p, q, r, test = 10^-6, ctr = T, std = T, sign = TRUE)
Arguments
x |
The three-way contingency table. |
p |
The number of components of the first mode. |
q |
The number of components of the second mode. |
r |
The number of components of the third mode. |
test |
The treshold used in the algorithm. |
ctr |
The flag parameter to center the data (T or F), if F the data are not centered. |
std |
The flag parameter to weight the data (T or F), if F the data are not weighted. |
sign |
The input parameter for changing the sign to the components according to the core sign. |
Value
x |
The original three-way contingency table. |
xs |
The weighted three-way contingency table. |
xhat |
The three-way contingency table reconstructed after Tuckals3 by means of the principal components and core array. |
nxhat2 |
The inertia of the three-way non-symmetrical correspondence analysis |
prp |
The proportion of inertia reconstructed using the principal components
and the core array to the total inertia. |
a |
The row principal components. |
b |
The column principal components. |
cc |
The tube principal components. |
g |
The core array (generalized singular values) calculated by using the Tuckals3 algorithm. |
iteration |
The number of iterations that are required for the TUCKALS3 algorithm to converge. |
Author(s)
Rosaria Lombardo, Eric J Beh.
References
Beh EJ and Lombardo R (2014) Correspondence Analysis, Theory, Practice and New Strategies. John Wiley & Sons.
Three-way Ordered Symmetrical Correspondence Analysis
Description
This function is used in the main function CA3variants when the input parameter is ca3type="OCA3".
It performs the three-way symmetric correspondence analysis by TUCKALS3.
Usage
oca3basic(x, p, q, r, test = 10^-6, ctr = T, std = T, norder = 3, sign = TRUE)
Arguments
x |
The three-way contingency table. |
p |
The number of components of the first mode. |
q |
The number of components of the second mode. |
r |
The number of components of the third mode. |
test |
The treshold used in the algorithm TUCKALS3. |
ctr |
The flag parameter (T or F), if F the analysis is not centered. |
std |
The flag parameter (T or F) if F the analysis is not standardized. |
norder |
The number of ordered variables considered. |
sign |
The input parameter for changing the sign to the components according to the core sign. |
Value
x |
The original three-way contingency table. |
xs |
The weighted three-way contingency table. |
xhat |
Three-way contingency table reconstructed after Tuckals3 by principal components and core array |
nxhat2 |
The inertia of three-way symmetric correspondence analysis (Three-way Pearson ratio). |
prp |
The proportion of inertia reconstructed using the p, q, r principal components and the core array to the total inertia. To select the model dimensions (number of principal components), we examine the inertia explained by the p, q, r principal components with respect to the overall fit. |
a |
The row principal components. |
b |
The column principal coordinates. |
cc |
The tube principal coordinates. |
g |
The core array calculated by using the Tuckals3 algorithm and can be interpreted as generalised singular value table. They help to explain the strength of the association between the three principal components. |
iteration |
The number of iterations that are required for the TUCKALS3 algorithm to converge. |
Author(s)
Rosaria Lombardo, Eric J Beh.
References
Lombardo R, Beh EJ and Kroonenberg PM (2021) Symmetrical and Non-Symmetrical Variants of Three-Way Correspondence Analysis for Ordered Variables. Statistical Science, 36 (4), 542-561.
Three-way contingency table
Description
This three-way contingency table represents an historical data set found in Agresti (1990).
Usage
data(olive)Format
The format is: row names [1:6] "A", "B", "C", "D", "E", "F" col names [1:3] "NW", "NE", "SW" tube names [1:2] "urban", "rural"
References
Beh EJ and Lombardo R (2014) Correspondence Analysis: Theory, Practice and New Strategies. John Wiley & Sons.
Examples
olive <-structure(c(20, 15, 12, 17, 16, 28, 18, 17, 18, 18,
6, 25, 12, 9, 23, 21, 19, 30, 30, 22, 21, 17, 8,
12, 23, 18, 20, 18, 10, 15, 11, 9, 26, 19, 17, 24
), .Dim = c(6L, 3L, 2L), .Dimnames = list(c("A", "B", "C", "D",
"E", "F"), c("NW", "NE", "SW"), c("urban", "rural")))
dim(olive)
data(olive)
Three-way Ordered Non-Symmetrical Correspondence Analysis
Description
This function is used in the main function CA3variants when the input parameter is ca3type="ONSCA3".
It performs the three-way symmetric correspondence analysis by TUCKALS3.
Usage
onsca3basic(x, p, q, r, test = 10^-6, ctr = T, std = T, norder = 3, sign = TRUE)
Arguments
x |
The three-way contingency table. |
p |
The number of components of the first mode. |
q |
The number of components of the second mode. |
r |
The number of components of the third mode. |
test |
The treshold used in the algorithm TUCKALS3. |
ctr |
The flag parameter (T or F), if F the analysis is not centered. |
std |
The flag parameter (T or F) if F the analysis is not standardized. |
norder |
The number of ordered variables considered. |
sign |
The input parameter for changing the sign to the components according to the core sign. |
Value
x |
The original three-way contingency table. |
xs |
The weighted three-way contingency table. |
xhat |
Three-way contingency table reconstructed after Tuckals3 by principal components and core array |
nxhat2 |
The inertia of three-way symmetric correspondence analysis (Three-way Pearson ratio). |
prp |
The proportion of inertia reconstructed using the p, q, r principal components and the core array to the total inertia. To select the model dimensions (number of principal components), we examine the inertia explained by the p, q, r principal components with respect to the overall fit. |
a |
The row principal components. |
b |
The column principal coordinates. |
cc |
The tube principal coordinates. |
g |
The core array calculated by using the Tuckals3 algorithm and can be interpreted as generalised singular value table. They help to explain the strength of the association between the three principal components. |
iteration |
The number of iterations that are required for the TUCKALS3 algorithm to converge. |
Author(s)
Rosaria Lombardo, Eric J Beh.
References
Lombardo R, Beh EJ and Kroonenberg PM (2021) Symmetrical and Non-Symmetrical Variants of Three-Way Correspondence Analysis for Ordered Variables. Statistical Science, 36 (4), 542-561.
The external product in Tuckals3.
Description
The computation of external product between the principal components.
Usage
p.ext(x,y)
Arguments
x |
x matrix IxS |
y |
y matrix JxS |
Value
resultant matrix (IxJ),S with elements xis per yis
Author(s)
Rosaria Lombardo, Eric J Beh, Ida Camminatiello.
Graphical display resulting from CA3variants
Description
The function plot.CA3variants allows the analyst to graphically display
six types of biplots for symmetrical 3-way variants and two types of biplots for non-symmetrical 3-way variants.
The six types of biplots for CA3 and OCA3 are the following.
When the input parameter is biptype = "column-tube" (or biptype = "col-tube"), the function displays the column-tube
interactive biplot, where the column and tube variables are coded
interactively and have principal coordinates and the row variable has standard coordinates.
When the input parameter is biptype = "row", the function displays the row biplot, where the rows have principal coordinates.
When the input parameter is biptype = "col", the function displays the column biplot, where the columns have principal coordinates.
When the input parameter is biptype = "row-tube", the function displays the row-tube biplot, where the row-tubes have principal coordinates.
When the input parameter is biptype = "tube", the function displays the tube biplot, where the tubes have principal coordinates
When the input parameter is biptype = "row-column" (or biptype = "row-col"), the function displays the row-column
interactive biplot, where the row-columns have principal coordinates.
The two types of biplots for NSCA3 and ONSCA3 are the following.
When the input parameter is biptype = "pred", the function displays the biplot where the predictors are coded
interactively and have principal coordinates and the response has standard coordinates.
When the input parameter is biptype = "resp", the function displays the biplot where the response variable has principal coordinates
and the predictors (interactively coded) have standard coordinates.
By default, biptype = "column-tube".
Usage
## S3 method for class 'CA3variants'
plot(x, firstaxis = 1, lastaxis = 2, thirdaxis = 3, cex = 0.8,
biptype="column-tube", scaleplot = NULL, plot3d = FALSE, pos = 1,
size1 = 1, size2 = 3, addlines = TRUE,...)
Arguments
x |
The output parameters of the main function |
firstaxis |
The dimension reflected along the horizontal axis. |
lastaxis |
The dimension reflected along the vertical axis. |
thirdaxis |
The dimension reflected along the third axis when |
cex |
The parameter that specifies the size of character labels of points in graphical displays. By default, it is equal to 0.8. |
biptype |
The input parameter for specifying what kind of biplot is requested. By default,
it is equal to |
scaleplot |
The scaling parameter for biplots to pull points away from the origin (see gamma biplot in Gower et al 2011). By default, it is equal to the overall average for the sum of squares of the two sets of coordinates (principal and standard ones), because of the average sum of squares for the two sets of points is the same (see Van de Velden et al 2017). |
plot3d |
The logical parameter specifies whether a 3D plot is to be included
in the output or not. By default, |
pos |
The input parameter for changing the label position. By default, it is equal to 1. |
size1 |
The input parameter for specifying the size of pointers. By default, it is equal to 1. |
size2 |
The input parameter for specifying the label size. By default, it is equal to 2. |
addlines |
The input parameter for plotting lines in biplots (the points in standard coordinates
are represented using lines). By default, it is equal to |
... |
Further arguments passed to or from other methods. |
Details
It is utilised by the main function CA3variants and uses the secondary graphical function graph2poly.
Value
Graphical displays of three-way correspondence analysis variants. Interactive plots or biplots are the graphical results of this function.
Author(s)
Rosaria Lombardo, Eric J Beh and Michel van de Velden.
References
Beh EJ and Lombardo R (2014) Correspondence Analysis, Theory, Practice and New Strategies. John Wiley & Sons.
Van de Velden M, Iodice D'Enza A, Palumbo F (2017) Cluster Correspondence Analysis. Psychometrika, 82, 158–185.
Gower JC, Lubbe SG, and Le Roux, NJ (2011) Understanding biplots. New York: Wiley.
Lombardo R, Beh EJ and Kroonenberg PM (2021) Symmetrical and Non-Symmetrical Variants of Three-Way Correspondence Analysis for Ordered Variables. Statistical Science, 36 (4), 542-561.
Examples
data(happy)
res.ca3<-CA3variants(happy, dims = c(p = 2, q = 2, r = 2), ca3type = "CA3")
plot(res.ca3)
res.nsca3<-CA3variants(happy, dims = c(p = 2, q = 2, r = 2), ca3type = "NSCA3")
plot(res.nsca3, biptype = "resp", plot3d = TRUE)
res.oca3<-CA3variants(happy, dims = c(p = 3, q = 5, r = 4), ca3type = "OCA3", norder = 3)
plot(res.oca3, biptype = "tube",firstaxis=4,lastaxis=7)
res.onsca3<-CA3variants(happy, dims = c(p = 3, q = 5, r = 4), ca3type = "ONSCA3", norder = 3)
plot(res.onsca3,biptype="resp",firstaxis=6,lastaxis=7)
Graphical display resulting from tunelocal
Description
The function plot.tunelocal allows the analyst to graphically display
the optimal model dimension using a convex hull.
Usage
## S3 method for class 'tunelocal'
plot(x,...)
Arguments
x |
The results of the function |
... |
Further arguments passed to or from other methods. |
Value
Graphical displays of a convex hull computed using the original data and the bootstrapped data when in tunelocal the input parameter boot=TRUE computed bootstrapped data too.
Author(s)
Rosaria Lombardo, Michel van de Velden and Eric J. Beh.
References
Beh EJ and Lombardo R (2014) Correspondence Analysis, Theory, Practice and New Strategies. John Wiley & Sons.
Wilderjans TF, Ceulemans E, and Meers K (2013) CHull: A generic convex hull based model
selection method. Behavior Research Methods, 45, 1-15.
Ceulemans E, and Kiers H A L (2006) Selecting among three-mode principal component models
of different types and complexities: A numerical convex hull based method. British Journal of
Mathematical & Statistical Psychology, 59, 133-150.
Examples
res.tunelocal<-tunelocal(happy, ca3type = "CA3",boots = FALSE,
nboots = 0)
plot(res.tunelocal)
Print of three-way correspondence analysis results
Description
This function prints the results of three-way symmetrical or non-symmetrical correspondence analysis.
If the input parameter, in CA3variants, is ca3type="CA3", the function prints the results
of three-way symmetrical correspondence analysis.
If the input parameter, in CA3variants, is ca3type="NSCA3",
the function prints the results of three-way non-symmetrical correspondence analysis.
If the input parameter, in CA3variants, is ca3type="OCA3",
the function prints the results of ordered three-way symmetrical correspondence analysis.
If the input parameter, in CA3variants, is ca3type="ONSCA3",
the function prints the results of ordered three-way non-symmetrical correspondence analysis.
When the input parameter, in print.CA3variants, is digits = 3,
the function prints all the results using three digital numbers.
Usage
## S3 method for class 'CA3variants'
print(x, printall= FALSE, digits = 3,...)
Arguments
x |
The name of the output of the main function |
printall |
The logical parameter that specifies if to print all the results or some of them. By default, |
digits |
The input parameter specifying the digital number. By default, |
... |
Further arguments passed to or from other methods. |
Value
The value of output returned depends on the kind of three-way correspondence analysis variant performed. It also gives the number of the iteration of the algorithm to reach the convergence of the solution. Depending on the variant of three-way correspondence analysis performed, it gives the related weighted contingency table, the reconstructed table by the components and core array, the explained inertia, the total inertia, the inertia in percentage, the proportion of explained inertia given the defined number of the components, the row standard and principal coordinates, the interactive column-tube standard and principal coordinates, the inner-product matrix of coordinates, the core array and index partitioning. In detail:
CA3variants |
The output of the kind of three-way correspondence analysis analysis considered. |
Data |
The original three-way contingency table. |
xs |
The centred and weighted three-way contingency table when the input parameters are |
xhat |
The three-way contingency table approximated (reconstructed) by the three component matrices (of dimension Ixp, Jxq, and Kxr) and the core array. |
nxhat2 |
The sum of squares of the approximated contingency table. |
prp |
The ratio between the inertia of the complete contingency table and the inertia of the approximated contingency table. |
fi |
The principal row coordinates. |
fiStandard |
The standard row coordinates. |
gjk |
The principal colum-tube coordinates. |
gjkStandard |
The standard colum-tube coordinates. |
fj |
The principal column coordinates. |
fjStandard |
The standard column coordinates. |
gik |
The principal row-tube coordinates. |
gikStandard |
The standard row-tube coordinates. |
fk |
The principal tube coordinates. |
fkStandard |
The standard tube coordinates. |
gij |
The principal row-colum coordinates. |
gijStandard |
The standard row-colum coordinates. |
rows |
The row marginals of the three-way data table. |
cols |
The column marginals of the three-way data table. |
tubes |
The tube marginals of the three-way data table. |
flabels |
The row category labels. |
glabels |
The column category labels. |
maxaxes |
The maximum dimension to consider. |
inertia |
The total inertia of a variant of three-way correspondence analysis. |
inertiaRSS |
The residual inertia of a variant of three-way correspondence analysis. |
inertiapc |
The percentage contribution of the three components to the total variation. |
inertiacoltub |
The vector of the percentage contributions of the interactively coded colum-tube components to the total inertia, useful for making interactively coded biplots. |
inertiarow |
The vector of the percentage contributions of the row components to the total inertia, useful for making response biplots. |
iproduct |
The inner product between the standard row coordinates (fi) and the column-tube principal coordinates (gjk). |
g |
The core array (i.e. the generalized singular values) calculated by using the Tuckals3 algorithm. |
index3 |
When |
ca3type |
The specification of the analysis to be performed.
When |
Author(s)
Rosaria Lombardo, Eric J Beh and Michel van de Velden.
References
Beh EJ and Lombardo R (2014) Correspondence Analysis, Theory, Practice and New Strategies. John Wiley & Sons.
Examples
data(happy)
ris.ca3<-CA3variants(happy, dims= c(p=2,q=2,r=2), ca3type = "CA3")
print(ris.ca3)
ris.nsca3<-CA3variants(happy, dims = c(p=2,q=2,r=2), ca3type = "NSCA3")
print(ris.nsca3)
ris.oca3<-CA3variants(happy, dims = c(p=3,q=5,r=4), ca3type = "OCA3",norder=3)
print(ris.oca3)
ris.onsca3<-CA3variants(happy, dims = c(p=3,q=5,r=4), ca3type = "ONSCA3",norder=3)
print(ris.onsca3)
Print of tunelocal function results
Description
This function prints the results of tunelocal for choosing the optimal model dimension of a variant of three-way correspondence analysis.
When boots = T the number of different models that is assessed is based on the size of the original data being analysed.
For example, for a 4 x 5 x 4, there are 80 different models that are assessed.
When boots = T, the number of different models that is assessed is based on the size of all models
obtained from the combination of dimensions of the bootstrapped data.
For example, for a 4 x 5 x 4 array, there are 800 different models that are assessed. By defaultnboots = 100,
you can change the parameter value in input of tunelocal function.
Usage
## S3 method for class 'tunelocal'
print(x, digits = 3,...)
Arguments
x |
The name of the output of the function |
digits |
The input parameter specifying the digital number. By default, |
... |
Further arguments passed to or from other methods. |
Value
The value of output returned depends on the kind of sampling chosen. The sampling for making the convex hull can be based on the original data or on the bootstrapped data samples. In detail:
XG |
The data samples used for assessing the optimal model dimension (original and/or bootstrapped). |
output1 |
The results of |
ca3type |
It gives information about the kind of variant of three-way CA considered. |
boots |
The flag parameter to perform the search of optimal dimensions using
bootstrap samples. By defaults, |
Author(s)
Rosaria Lombardo, Michel van de Velden and Eric J. Beh.
References
Wilderjans T F, Ceulemans E, and Meers K (2013) CHull: A generic convex hull based model
selection method. Behavior Research Methods, 45, 1-15.
Ceulemans E, and Kiers H A L (2006) Selecting among three-mode principal component models
of different types and complexities: A numerical convex hull based method. British Journal of
Mathematical & Statistical Psychology, 59, 133-150.
Examples
res.tunelocal<-tunelocal(happy, ca3type = "CA3")
print(res.tunelocal)
Products among arrays
Description
The function calculates the products among arrays.
Usage
prod3(m, a1, a2, a3)
Arguments
m |
The original three-way contingency table. |
a1 |
The weight matrix related to the rows of the table. |
a2 |
The weight matrix related to the columns of the table. |
a3 |
The weight matrix related to the tubes of the table. |
Details
It is utilised in standtab, rstand3 and rstand3delta in order to weight the
contingency table with respect to the three weigth matrices defined in the row, column and tube spaces
differently for the three variants of three-way correspondence analysis.
Value
The three-way contingency table weighted with respect the suitable weight matrices (depending on the analysis variants).
Author(s)
Rosaria Lombardo, Eric J Beh, Ida Camminatiello.
Rating-ranking data a three-way contingency table
Description
This three-way contingency table represents a known data set described in van Herk and van de Velden (2007). The three-way contingency table consists of nine rating values against nine ranking values given by the same participants across five European countries (France, Italy, Germany, UK and Spain).
Usage
data(ratrank)Format
The format is: row names [1:9] "1", "2", "3", "4", "5", "6","7","8","9" col names [1:9] "rank1", "rank2", "rank3", "rank4", "rank5", "rank6", "rank7", "rank8", "rank9" tube names [1:5] "F", "I","G","U","S"
References
van Herk H and van de Velden M (2007) Insight into the relative merits of rating and ranking
in a cross-national context using three-way correspondence analysis. Food Quality and Preference, 18, 1096–1105.
Examples
ratrank<-structure(c(766L, 128L, 38L, 10L, 12L, 3L, 2L, 5L, 9L, 619L,
234L, 67L, 16L, 15L, 5L, 2L, 8L, 7L, 512L, 277L, 109L, 22L, 23L,
5L, 11L, 7L, 7L, 385L, 291L, 152L, 64L, 41L, 9L, 12L, 7L, 12L,
297L, 251L, 192L, 82L, 96L, 17L, 12L, 6L, 20L, 187L, 203L, 259L,
105L, 119L, 44L, 19L, 8L, 29L, 143L, 144L, 209L, 140L, 170L,
54L, 51L, 22L, 40L, 77L, 100L, 152L, 148L, 215L, 73L, 62L, 56L,
90L, 47L, 45L, 84L, 119L, 200L, 82L, 98L, 67L, 231L, 859L, 101L,
53L, 18L, 18L, 9L, 7L, 2L, 16L, 733L, 205L, 53L, 23L, 21L, 13L,
11L, 6L, 18L, 622L, 224L, 124L, 41L, 27L, 8L, 12L, 6L, 19L, 547L,
248L, 102L, 78L, 45L, 19L, 11L, 11L, 22L, 466L, 243L, 139L, 76L,
76L, 25L, 21L, 9L, 28L, 357L, 239L, 168L, 105L, 95L, 61L, 20L,
14L, 24L, 293L, 192L, 165L, 128L, 133L, 42L, 58L, 28L, 44L, 215L,
162L, 161L, 127L, 148L, 60L, 65L, 54L, 91L, 140L, 121L, 142L,
128L, 157L, 69L, 76L, 75L, 175L, 1219L, 193L, 29L, 13L, 3L, 4L,
2L, 6L, 3L, 651L, 504L, 111L, 30L, 19L, 8L, 5L, 8L, 6L, 476L,
335L, 230L, 35L, 13L, 8L, 5L, 4L, 6L, 346L, 324L, 201L, 136L,
30L, 5L, 5L, 6L, 5L, 239L, 299L, 234L, 101L, 170L, 22L, 14L,
4L, 6L, 166L, 246L, 265L, 116L, 96L, 71L, 27L, 11L, 16L, 124L,
179L, 215L, 163L, 139L, 52L, 80L, 20L, 32L, 80L, 114L, 172L,
148L, 168L, 80L, 84L, 96L, 70L, 63L, 48L, 101L, 115L, 183L, 92L,
123L, 131L, 292L, 916L, 99L, 40L, 12L, 7L, 3L, 3L, 13L, 42L,
578L, 224L, 65L, 11L, 15L, 6L, 3L, 18L, 30L, 486L, 207L, 140L,
34L, 14L, 10L, 7L, 16L, 29L, 405L, 207L, 149L, 64L, 30L, 7L,
14L, 21L, 19L, 304L, 256L, 157L, 60L, 83L, 9L, 17L, 20L, 21L,
239L, 222L, 195L, 95L, 55L, 34L, 20L, 18L, 18L, 204L, 169L, 213L,
113L, 89L, 23L, 45L, 16L, 15L, 165L, 148L, 184L, 128L, 121L,
46L, 38L, 51L, 23L, 89L, 94L, 147L, 141L, 181L, 70L, 57L, 32L,
82L, 1086L, 89L, 37L, 10L, 12L, 6L, 9L, 6L, 24L, 501L, 251L,
55L, 11L, 14L, 7L, 7L, 7L, 11L, 415L, 139L, 188L, 22L, 14L, 8L,
4L, 4L, 12L, 359L, 148L, 111L, 101L, 21L, 7L, 7L, 3L, 15L, 278L,
158L, 128L, 49L, 127L, 9L, 12L, 6L, 13L, 240L, 162L, 130L, 48L,
58L, 49L, 11L, 5L, 12L, 185L, 113L, 148L, 78L, 84L, 26L, 52L,
7L, 16L, 128L, 91L, 119L, 110L, 118L, 37L, 38L, 35L, 28L, 83L,
50L, 67L, 89L, 165L, 47L, 66L, 46L, 120L), .Dim = c(9L, 9L, 5L
), .Dimnames = list(c("1", "2", "3", "4", "5","6", "7",
"8", "9"), c("rank1", "rank2", "rank3",
"rank4", "rank5", "rank6",
"rank7", "rank8", "rank9"),
c("F", "I", "G", "U", "S")))
dim(ratrank)
data(ratrank)
Reconstruction of the three-way centred profile table
Description
The function reconstructs the three-way centred profile table using the component matrices from the Tucker3 decomposition and the core array.
Usage
reconst3(param)
Arguments
param |
The matrices of the row, column and tube components derived via the Tucker3 model. |
Value
The three-way reconstructed table of centred profiles.
Author(s)
Rosaria Lombardo, Eric J Beh, Ida Camminatiello.
Weighted centred three-way table for three-way non-symmetric correspondence analysis
Description
The function computes the three-way weighted centred contingency table to perform three-way non-symmetric correspondence analysis with one response and two predictors.
Usage
rstand3(x, std = T, ctr = T)
Arguments
x |
The original three-way contingency table. |
std |
The flag parameter for weighting the original table. If |
ctr |
The flag parameter for centering the original table. If |
Value
xs |
The weighted array with respect to the three associated metrics. It is used
when |
Author(s)
Rosaria Lombardo, Eric J Beh, Ida Camminatiello.
Selecting the mode.
Description
Select the mode in which the column has to be sign reversed. Below is an heuristic algorithm but a fully rational choice is hard to come by. Maximal number of sign reversals = p+q+r-2, but this number can be much smaller. Sign reverse a component, determine which if any of p, q and r is available for reversal. If one of them is wholly positieve way never choose it, else if one is wholly negative choose that one from A, B, C respectively; else if there is a component with a largest absolute value which is negative choose that one, or the one from A,B,C in that order; else choose the column of the longest mode; end if FreeA, FreeB, FreeC = 0 component is not available; = 1 component is available.
Usage
selmod(aptr, bptr, cptr, posptrA, negptrA, bigptrA, posptrB, negptrB, bigptrB,
posptrC, negptrC, bigptrC, IA, IB, IC, I, J, K, p, q, r, longest)
Arguments
aptr |
The pointer to the first component matrix. |
bptr |
The pointer to the second component matrix. |
cptr |
The pointer to the third component matrix. |
posptrA |
The pointer to the positive component of the first component matrix. |
negptrA |
The pointer to the negative component of the first component matrix. |
bigptrA |
The pointer to the biggest component (a larger negative positive value than a positive one) of the first component matrix. |
posptrB |
The pointer to the positive component of the first component matrix. |
negptrB |
The pointer to the negative component of the second component matrix. |
bigptrB |
The pointer to the biggest component (a larger negative positive value than a positive one) of the second component matrix. |
posptrC |
The pointer to the positive component of the third component matrix. |
negptrC |
The pointer to the negative component of the third component matrix. |
bigptrC |
The pointer to the biggest component (a larger negative positive value than a positive one) of the third component matrix. |
IA |
The pointer to the columns of the first component matrix, given the pointer to an element in core. |
IB |
The pointer to the columns of the second component matrix, given the pointer to an element in core. |
IC |
The pointer to the columns of the third component matrix, given the pointer to an element in core. |
I |
The row number of the three-way contingency table. |
J |
The column number of the three-way contingency table. |
K |
The tube number of the three-way contingency table. |
p |
The dimension number of the first mode. |
q |
The dimension number of the second mode. |
r |
The dimension number of the third mode. |
longest |
The component matrix of the longest mode to change sign (when no special reason for selection could be found). |
Value
success |
The flag variable to indicate if one of the components has to be sign reversed. |
chgmode |
Select the mode (1, 2 or 3) in which the column has to be sign reversed. |
chgcomp |
The pointer to the columns of the component matrices (A, B or C) |
Author(s)
Rosaria Lombardo and Pieter M Kroonenberg.
References
Kroonenberg PM (1994) The TUCKALS line: a suite of programs for three-way data analysis. Computational Statistics and Data Analysis, 18, 73–96.
Changing the sign of negative core values.
Description
This function makes the signs of the largest core elements positive
to facilitate interpretation. The appropriate columns of the component matrices for inversion
are determined and are reversed accordingly.
Usage
signscore(a, b, cc, I, J, K, p, q, r, core, IFIXA, IFIXB, IFIXC)
Arguments
a |
The first component matrix A. |
b |
The second component matrix B. |
cc |
The third component matrix C. |
I |
The row number of the first mode of the three-way contingency table. |
J |
The column number of the second mode of the three-way contingency table. |
K |
The tube number of the third mode of the three-way contingency table. |
p |
The dimension number of the first mode. |
q |
The dimension number of the second mode. |
r |
The dimension number of the third mode. |
core |
The core array (generalized singular values). |
IFIXA |
The flag parameter to indicate whether the first component (A) belongs to a fixed mode. |
IFIXB |
The flag parameter to indicate whether the second component (B) belongs to a fixed mode. |
IFIXC |
The flag parameter to indicate whether the third component (C) belongs to a fixed mode. |
Value
g |
The core array. |
gord |
the core array ordered with respect to the largest values (descending order). |
a |
The first matrix of components. |
aord |
The ordered first matrix of the components. |
b |
The second matrix of the components. |
bord |
The ordered second matrix of the components. |
cc |
The third matrix of the components. |
ccord |
The ordered third matrix of the components. |
Author(s)
Rosaria Lombardo and Pieter M Kroonenberg.
References
Kroonenberg PM (1994) The TUCKALS line: a suite of programs for three-way data analysis. Computational Statistics and Data Analysis, 18, 73–96.
Generation of parametric bootstrap samples
Description
This function allows to generate parametric bootstrap samples in order to check the optimal dimension number of three-way correspondence analysis. The boostrap samples have the same marginal proportions and the total number of the original table. The adopted sampling scheme is simple.
Usage
simulabootsimple(Xtable,nboots=100,resamptype=1)Arguments
Xtable |
The three-way data array. It must be an |
nboots |
The number of bootstrap samples to generate when |
resamptype |
Set value of resamptype according to two methods: |
Author(s)
Michel van de Velden, Rosaria Lombardo and Eric J Beh.
References
Beh EJ and Lombardo R (2014) Correspondence Analysis, Theory, Practice and New Strategies. John Wiley & Sons.
Generation of parametric bootstrap samples
Description
This function allows to generate parametric bootstrap samples in order to check the optimal dimension number of three-way correspondence analysis. The boostrap samples have the same marginal proportions and total number of the original table. Te adopted sampling scheme is stratified.
Usage
simulabootstrat(Xtable,nboots=100,resamptype=1)Arguments
Xtable |
The three-way data array. It must be an |
nboots |
The number of bootstrap samples to generate when |
resamptype |
Set value of resamptype according to two methods: |
Author(s)
Rosaria Lombardo, Michel van de Velden, Eric J Beh.
References
Beh EJ and Lombardo R (2014) Correspondence Analysis, Theory, Practice and New Strategies. John Wiley & Sons.
Sort the core array
Description
This function sorts (a copy of) the core matrix and returns CORPTR, an p*q*r integer array holding the pointers to the greatest absolute values in CORE.
Usage
srtcor(core, p, q, r)
Arguments
core |
The core array (generalized singular values). |
p |
The dimension number of the first mode. |
q |
The dimension number of the second mode. |
r |
The dimension number of the third mode. |
Value
coreptr |
The pointer to the ordered largest value of the core array. |
Author(s)
Rosaria Lombardo and Pieter M Kroonenberg.
References
Kroonenberg PM (1994) The TUCKALS line: a suite of programs for three-way data analysis. Computational Statistics and Data Analysis, 18, 73–96.
Three-way centred column profile table for the three-way symmetric correspondence analysis
Description
The function computes the three-way centred column profile table to perform three-way symmetric correspondence analysis.
Usage
standtab(x, std = T, ctr = T)
Arguments
x |
The original three-way contingency table. |
std |
The flag parameter for weighting the original table. If F the original contingency table is not weighted. |
ctr |
The flag parameter for centering the original table. If F the original array is not centered. |
Author(s)
Rosaria Lombardo, Eric J Beh, Ida Camminatiello.
References
Beh EJ and Lombardo R (2014) Correspondence Analysis, Theory, Practice and New Strategies. John Wiley & Sons.
The core array derived via the Tucker3 model.
Description
The Tucker3 model involves the computation of principal components, which are derived for
each of the three categorical variables, and of the core array which is akin to the generalised
correlations between these components. The function step.g3 computes the core array.
Usage
step.g3(param)
Arguments
param |
The weighted three-way table and the matrices of the row, column and tube components derived via the Tuckals3 algorithm. |
Value
The core matrix whose the general element can be interpreted as a generalized singular value.
Author(s)
Rosaria Lombardo, Eric J Beh, Ida Camminatiello.
References
Beh EJ and Lombardo R (2014) Correspondence Analysis, Theory, Practice and New Strategies. John Wiley & Sons.
The core array derived via the Trivariate Moment Decomposition model.
Description
The Trivariate Moment Decomposition model involves the computation of
principal polynomial components, which are derived for each of the three categorical
variables, and of the polynomial core array which is akin to the generalised correlations
between these components. The function step.g3ordered computes the core array.
Usage
step.g3ordered(param)
Arguments
param |
The weighted three-way table and the matrices of the row, column and tube components derived via the Trivariate Moment Decomposition algorithm. |
Value
The core matrix whose the general element can be interpreted as a generalized singular value.
Author(s)
Rosaria Lombardo, Eric J Beh, Ida Camminatiello.
References
Beh EJ and Lombardo R (2014) Correspondence Analysis, Theory, Practice and New Strategies. John Wiley & Sons.
Component matrices from the Tucker3 decomposition
Description
The function computes the component matrices from the Tuckals3 algorithm.
Usage
stepi3(param)
Arguments
param |
The weighted contingency table and the matrices of the row, column and tube components derived via the Tucker3 model. |
Details
The functions newcomp3, stepi3, init3 and step.g3 compute the
component matrices and core array in the iterative steps of Tuckals3. They are all utilised from
the function tucker.
Value
Component matrices from the Tucker3 decomposition.
Author(s)
Rosaria Lombardo, Eric J Beh, Ida Camminatiello.
References
Beh EJ and Lombardo R (2014) Correspondence Analysis, Theory, Practice and New Strategies. John Wiley & Sons.
Component matrices from the Trivariate Moment Decomposition decomposition
Description
The function computes the polynomial component matrices from the Emerson's recurrence formula for the ordered categorical variables of the analysis.
Usage
stepi3ordered(param)
Arguments
param |
The weighted contingency table and the matrices of the row, column and tube components derived via the Trivariate Moment Decomposition model. |
Details
The functions newcomp3ordered, stepi3ordered, init3ordered and
step.g3ordered compute the polynomial component matrices and core array in the Trivariate Moment Decomposition.
They are all utilised from the function tuckerORDERED.
Value
Component matrices from the Trivariate Moment Decomposition decomposition.
Author(s)
Rosaria Lombardo, Eric J Beh, Ida Camminatiello.
References
Beh EJ and Lombardo R (2014) Correspondence Analysis, Theory, Practice and New Strategies. John Wiley & Sons.
Summary of three-way correspondence analysis results
Description
This function prints the summary of the results of three-way symmetrical or non-symmetrical correspondence analysis. In particular it gives information on core and squared core and on the explained inertia when reducing dimensions.
Usage
## S3 method for class 'CA3variants'
summary(object, digits=3, ...)
Arguments
object |
The name of the output of the main function |
digits |
The input parameter specifying the digital number. By default, |
... |
Further arguments passed to or from other methods. |
Value
The value of output returned in short depends on the kind of three-way correspondence analysis variant performed. It gives the core table, the squared core table, the explained inertia, the total inertia and its proportion.
Author(s)
Rosaria Lombardo, Eric J Beh and Michel van de Velden.
References
Beh EJ and Lombardo R (2014) Correspondence Analysis, Theory, Practice and New Strategies. John Wiley & Sons.
Examples
data(happy)
ris.ca3<-CA3variants(happy, dims= c(p=2,q=2,r=2), ca3type = "CA3")
summary(ris.ca3)
ris.nsca3<-CA3variants(happy, dims = c(p=2,q=2,r=2), ca3type = "NSCA3")
summary(ris.nsca3)
ris.oca3<-CA3variants(happy, dims = c(p=3,q=5,r=4), ca3type = "OCA3",norder=3)
summary(ris.oca3)
ris.onsca3<-CA3variants(happy, dims = c(p=3,q=5,r=4), ca3type = "ONSCA3",norder=3)
summary(ris.onsca3)
Partition of the Marcotorchino three-way index
Description
When the association among three categorical variables is asymmetric such that one variable is a
logical response variable to the other variables, we recommend calculating the non-symmetrical
three-way measure of predictability such as the Marcotorchino index (Marcotorchino, 1985).
The function tau3 partitions the Marcotorchino statistic when, in CA3variants,
we set the parameter ca3type = "NSCA3".
Usage
tau3(f3, digits = 3)
Arguments
f3 |
Three-way contingency array given as an input parameter in |
digits |
Number of decimal digits. By default digits=3. |
Value
z |
The partition of the Marcotorchino index into three two-way association terms and one three-way association term. It also shows the C statistic of each term, its degrees of freedom and p-value. |
CM |
the C statistic of the Marcotorchino index. |
devt |
The denominator of the Marcotorchino index. |
Author(s)
Rosaria Lombardo, Eric J Beh, Ida Camminatiello.
References
Beh EJ and Lombardo R (2014) Correspondence Analysis, Theory, Practice and New Strategies. John Wiley & Sons.
Examples
data(happy)
tau3(happy, digits = 3)
The partition of the Marcotorchino three-way index.
Description
When three categorical variables are symmetrically related, we can analyse the strength of
the association using the three-way Marcotorchino index.
The function chi3 partitions the Marcotorchino statistic using orthogonal polynomials
when, in CA3variants, we set the parameter ca3type = "ONSCA3".
Usage
tau3ordered(f3, digits = 3)
Arguments
f3 |
The three-way contingency array given as an input parameter in |
digits |
The number of decimal digits. By default digits=3. |
Value
The partition of the Marcotorchino index into three two-way non-symmetrical association terms and one three-way association term. It also shows the polynomial componets of inertia, the percentage of explained inertia, the degrees of freedom and p-value of each term of the partition.
Author(s)
Rosaria Lombardo, Eric J Beh.
References
Beh EJ and Lombardo R (2014) Correspondence Analysis, Theory, Practice and New Strategies. John Wiley & Sons.
Examples
#data(olive)
tau3ordered(f3 = olive, digits = 3)
Generation of non-parametric bootstrap samples
Description
This function allows to generate non-parametric bootstrap samples in order to check the
optimal dimension number of three-way correspondence analysis. The boostrap samples
have the same marginal proportions and the total number of the original table.
Do nboots bootstrap on the indicator matrix X (observations x (rows+cols+tubs) categories).
From a three-way contingency table, it makes the indicator using makeindicator.
The output is a list of three-way tables.
Usage
threewayboot(Xdata,nboots=100)Arguments
Xdata |
The three-way contingency array. It must be an |
nboots |
The number of bootstrap samples to generate when |
Author(s)
Rosaria Lombardo, Michel van de Velden, Eric J Beh.
References
Beh EJ and Lombardo R (2014) Correspondence Analysis, Theory, Practice and New Strategies. John Wiley & Sons.
Tucker3 decomposition of the three-way table.
Description
The Tucker3 model, originally proposed by psychologist Ledyard R. Tucker, involves the computation of principal components, which are derived for each of the three categorical variables, and of the core array which is akin to the generalised correlations between these components. The function represents the heart of the Tuckals3 algorithm to perform the Tucker3 decomposition of the three-way array x.
Usage
tucker(x, p, q, r, test = 10^-6)
Arguments
x |
The three-way contingency table. |
p |
The number of components of the first mode. |
q |
The number of components of the second mode. |
r |
The number of components of the third mode. |
test |
The treshold used in the algorithm. |
Details
The function tucker is utilised from the functions ca3basic, nsca3basic and
oca3basic.
Value
a |
The final component derived from the Tucker3 decomposition for the first mode. |
b |
The final component derived from the Tucker3 decomposition for the second mode. |
cc |
The final component derived from the Tucker3 decomposition for the third mode. |
g |
The core array. |
x |
The three-way contingency table. |
cont |
The number of iterations that are required for the Tucker3 algorithm to converge. |
Author(s)
Rosaria Lombardo, Eric J Beh.
References
Kroonenberg PM (1994) The TUCKALS line: a suite of programs for three-way data analysis. Computational Statistics and Data Analysis, 18, 73–96.
Trivariate moment decomposition of the three-way table.
Description
The Trivariate moment decomposition (TMD) represents the heart of a new algorithm to perform the decomposition of the three-way ordered contingency tables. It is based on the orthogonal polynomials (Emerson 1968) computed for each categorical ordered variable.
Usage
tuckerORDERED(x, p, q, r, test = 10^-6,xi, norder=3)
Arguments
x |
The three-way contingency table. |
p |
The number of components of the first mode. |
q |
The number of components of the second mode. |
r |
The number of components of the third mode. |
test |
The treshold used in the algorithm. |
xi |
The original three-way contingency table. |
norder |
The number of ordered variables. |
Details
The function tuckerORDERED is utilised from the function oca3basic.
Value
a |
The final component derived from the TMD decomposition for the first mode. |
b |
The final component derived from the TMD decomposition for the second mode. |
cc |
The final component derived from the TMD decomposition for the third mode. |
g |
The core array. |
x |
The three-way contingency table. |
cont |
The number of iterations that are required for the TMD algorithm to converge. If all variables are ordered, the convergence is reached in one step, differently if we have mixed variables. Indeed, the decmposition will become hybrid, a mix of TMD algorithm and Tuckals3 algorithm. |
Author(s)
Rosaria Lombardo, Eric J Beh.
References
Emerson PL (1968) Numerical construction of orthogonal polynomials from a general recurrence formula. Biometrics, 24 (3), 695-701.
Lombardo R Beh EJ (2016) Variants of Simple Correspondence Analysis. The R Journal, 8 (2), 167–184.
Lombardo R Beh EJ and Kroonenberg PM (2016) Modelling Trends in Ordered Correspondence Analysis Using Orthogonal Polynomials. Psychometrika, 81(2), 325–349.
Dimension selection for three-dimensional correspondence biplot using convex hull.
Description
This function allows to select the optimal dimension number
for correspondence biplot, given the set of possible dimension combination
of the original data. It determines the models that are located on the boundary of the convex hull
and selects an optimal model by means of the scree test values (st).
For exploring, it is also possible to check the optimal model dimension by using
boostrap samples which have the same marginal proportions and the total number
of the original table. When the input parameter boots = T, it does bootstrap sampling.
There are three kinds of possible bootstrap sampling.
When boottype = "bootnp" it performs a non parametric bootstrap sampling.
When boottype = "bootpsimple" it performs a parametric simple bootstrap sampling.
When boottype = "bootpstrat", it performs a parametric stratified bootstrap sampling.
In particular in case of parametric bootstrap types,
when resamptype=1 it considers a multinomial distribution, and when resamptype = 2
it considers a poisson distribution.
Usage
tunelocal(Xdata, ca3type = "CA3", resp = "row", norder = 3, digits = 3, boots = FALSE,
nboots = 0, boottype= "bootpsimple", resamptype = 1, PercentageFit = 0.01)
Arguments
Xdata |
The three-way data. It can be a |
ca3type |
The specification of the analysis to be performed.
If |
resp |
The input parameter for specifying in non-symmetrical three-way correspondence analysis variants ( |
norder |
The input parameter for specifying the number of ordered variable when |
digits |
The input parameter specifying the digital number. By default, |
boots |
The flag parameter to perform the search of optimal dimensions using
bootstrap samples. By defaults, |
nboots |
The number of bootstrap samples to generate when |
boottype |
The specification of the kind of bootstrap sampling to be performed.
If |
resamptype |
When the kind of bootstrap is parametric you can set the data distribution using
the input parameter |
PercentageFit |
Required proportion of increase in fit of a more complex model. By default,
|
Value
output1 |
Chi-square criterion and df of models on the convex hull.
It gives the criterion values of the models that are located on the boundary of the convex hull
and selects the optimal model by means of the scree test values (st).
When using |
Author(s)
Rosaria Lombardo, Michel van de Velden, Eric J Beh.
References
Wilderjans T F, Ceulemans E, and Meers K (2013) CHull: A generic convex hull based model
selection method. Behavior Research Methods, 45, 1-15.
Ceulemans E, and Kiers H A L (2006) Selecting among three-mode principal component models
of different types and complexities: A numerical convex hull based method. British Journal of
Mathematical & Statistical Psychology, 59, 133-150.
Examples
tunelocal(Xdata = happy, ca3type = "CA3")