FCAk {PTAk}

Generalisation of Correspondence Analysis for k-way tables

Description

Performs a particular PTAk data as a ratio Observed/Expected under complete independence with metrics as margins of the multiple contingency table (in frequencies).

Usage

 FCAk(X,nbPT=3,nbPT2=1,minpct=0.01,
               smoothing=FALSE,smoo=rep(list(
                       function(u)ksmooth(1:length(u),u,kernel="normal",
                       bandwidth=3,x.points=(1:length(u)))$y),length(dim(X))),
                     verbose=getOption("verbose"),file=NULL,
                       modesnam=NULL,addedcomment="",chi2=TRUE,E=NULL, ...)

Arguments

X

a multiple contingency table (array) of order k

nbPT

a number or a vector of dimension (k-2)

nbPT2

if 0 no 2-modes solutions will be computed, 1 =all, >1 otherwise

minpct

numerical 0-100 to control of computation of future solutions at this level and below

smoothing

see SVDgen

smoo

see SVDgen

verbose

control printing

file

output printed at the prompt if NULL, or printed in the given ‘file’

modesnam

character vector of the names of the modes, if NULL "mo 1" ..."mo k"

addedcomment

character string printed if printt after the title of the analysis

chi2

print the chi2 information when computing margins in FCAmet

E

if not NULL is an array with the same dimensions as X

...

any other arguments passed to SVDGen or other functions

Details

Gives the SVD-kmodes decomposition of the 1+\chi^2/N of the multiple contingency table of full count N=\sum X_{ijk...}, i.e. complete independence + lack of independence (including marginal independences) as shown for example in Lancaster(1951)(see reference in Leibovici(2000)). Noting P=X/N, a PTAk of the (k+1)-uple is done, e.g. for a three way contingency table k=3 the 4-uple data and metrics is:

((D_I^{-1} \otimes D_J^{-1} \otimes D_K^{-1})P, \quad D_I, \quad D_J, \quad D_K)

where the metrics are diagonals of the corresponding margins. For full description of arguments see PTAk. If E is not NULL an FCAk-modes relatively to a model is done (see Escoufier(1985) and therin reference Escofier(1984) for a 2-way derivation), e.g. for a three way contingency table k=3 the 4-tuple data and metrics is:

((D_I^{-1} \otimes D_J^{-1} \otimes D_K^{-1})(P-E), \quad D_I, \quad D_J, \quad D_K)

If E was the complete independence (product of the margins) then this would give an AFCk but without looking at the marginal dependencies (i.e. for a three way table no two-ways lack of independence are looked for).

Value

a FCAk (inherits PTAk) object

Author(s)

Didier G. Leibovici

References

Escoufier Y (1985) L'Analyse des correspondances : ses propri<e9>t<e9>s et ses extensions. ISI 45th session Amsterdam.

Leibovici D(1993) Facteurs <e0> Mesures R<e9>p<e9>t<e9>es et Analyses Factorielles : applications <e0> un suivi <e9>pid<e9>miologique. Universit<e9> de Montpellier II. PhD Thesis in Math<e9>matiques et Applications (Biostatistiques).

Leibovici D (2000) Multiway Multidimensional Analysis for Pharmaco-EEG Studies.http://www.fmrib.ox.ac.uk/analysis/techrep/tr00dl2/tr00dl2.pdf

Leibovici DG (2010) Spatio-temporal Multiway Decomposition using Principal Tensor Analysis on k-modes:the R package PTAk. Journal of Statistical Software, 34(10), 1-34. doi:10.18637/jss.v034.i10

Leibovici DG and Birkin MH (2013) Simple, multiple and multiway correspondence analysis applied to spatial census-based population microsimulation studies using R. NCRM Working Paper. NCRM-n^o 07/13, Id-3178 https://eprints.ncrm.ac.uk/id/eprint/3178

See Also

PTAk, FCAmet, summary.FCAk

Examples

   # try the demo
   # demo.FCAk()