PTAk (version 2.0.0)

FCA2: Correspondence Analysis for 2-way tables

Description

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

Usage

FCA2(X, nbdim =NULL, 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 = FALSE, E = NULL, ...)

Value

a FCA2 (inherits FCAk and PTAk) object

Arguments

X

a matrix table of positive values

nbdim

a number of dimension to retain, if NULL the default value of maximum possible number of dimensions is kept

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 a matrix with the same dimensions as X with the same margins

...

any other arguments passed to SVDGen or other functions

Author

Didier G. Leibovici

Details

Gives the SVD-2modes decomposition of the \(1+\chi^2/N\) of the contingency table of full count \(N=\sum X_{ij}\), i.e. complete independence + lack of independence (including marginal independences) as shown for example in Lancaster(1951)(see reference in Leibovici(1993 or 2000)). Noting \(P=X/N\), a SVD of the \((3)\)-uple is done, that is : $$ ((D_I^{-1} \otimes D_J^{-1})..P, \quad D_I, \quad D_J)$$ 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).

References

Escoufier Y (1985) L'Analyse des correspondances : ses propriétés et ses extensions. ISI 45th session Amsterdam.

Leibovici D(1993) Facteurs à Mesures Répétées et Analyses Factorielles : applications à un suivi Epidémiologique. Université de Montpellier II. PhD Thesis in Mathématiques 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

See Also

PTAk, FCAmet, summary.FCAk

Examples

 data(crimerate)
 cri.FCA2 <- FCA2(crimerate)
 summary(cri.FCA2)
  plot(cri.FCA2, mod = c(1,2), nb1 = 2, nb2 = 3) # unscaled
  plot(cri.FCA2, mod = c(1,2), nb1 = 2, nb2 = 3, coefi = 
  	list(c(0.130787,0.130787),c(0.104359,0.104359)) )# symmetric-map biplot
 CTR(cri.FCA2, mod = 1, solnbs = 2:4)
 CTR(cri.FCA2, mod = 2, solnbs = 2:4)
 COS2(cri.FCA2, mod = 2, solnbs = 2:4)