FCAk: 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, ...)

Value

a FCAk (inherits PTAk) object

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

Author

Didier G. Leibovici

Details

Gives the SVD-kmodes decomposition of the $1 + χ^{2} / N$ of the multiple contingency table of full count $N = \sum X_{i j k . . .}$ , 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: $((D_{I}^{- 1} \otimes D_{J}^{- 1} \otimes D_{K}^{- 1}) P, D_{I}, D_{J}, D_{K})$ 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: $((D_{I}^{- 1} \otimes D_{J}^{- 1} \otimes D_{K}^{- 1}) (P - E), D_{I}, D_{J}, D_{K})$ 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).

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. tools:::Rd_expr_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

Examples

Run this code

   # try the demo
   # demo.FCAk()

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