princals: Categorical principal component analysis (PRINCALS).

In Gifi: Multivariate Analysis with Optimal Scaling

Categorical principal component analysis (PRINCALS).

Description

Fits a categorical PCA. The default is to take each input variable as ordinal but it works for mixed scale levels (incl. nominal) as well. Through a proper spline specification various continuous transformation functions can be specified: linear, polynomials, and (monotone) splines.

Usage

princals(data, ndim = 2, levels = "ordinal", ordinal, knots, ties = "s", 
degrees = 1, copies = 1, missing = "s", normobj.z = TRUE, active = TRUE,
itmax = 1000, eps = 1e-06, verbose = FALSE)

Arguments

data	Input data frame: n observations, m variables
ndim	Number of dimensions to be computed
levels	A vector of length m denoting basic scale levels ("nominal", "ordinal", "metric"; see details
ordinal	If knots are specified manually, a boolean vector of length m denotes which variables should be ordinally restricted or not (see details)
knots	Scale levels can be specified manually using splines (see knotsGifi). If knots is set, this overrides level (see details)
ties	How ties should be handled: primary ("p"), secondary ("s"), or tertiary ("t")
degrees	Spline degrees. If different degrees should be used across variables, a vector of length m can be specified.
copies	Number of copies for each variables (also as vector of length m)
missing	How missing values should be handled: multiple ("m"), single ("s"), or average ("a")
active	Which variables should be active or inactive (also as vector of length m)
normobj.z	If TRUE, object scores are z-scores, if FALSE, they are restriction to SS of 1.
itmax	Maximum number of iterations
eps	Convergence criterion
verbose	Iteration printout

Details

The measurement (or scale) levels of the variables are incorporated via spline transformations. If the user only needs simple scale levels like nominal, ordinal, and metric, a corresponding vector can be specified in the levels argument without setting knots and ordinal. The corresponding spline transformations (unrestricted, monotone, and linear) are then created internally. If all scale level transformations are the same, ordinal can be a single value. For more advanced transformations such as polynomial or more flexible splines, the knots and ordinal arguments need to be specified instead of levels.

Value

transform	Optimally transformed scores
rhat	Induced correlation matrix
evals	Eigenvalues of induced correlation matrix
objectscores	Object scores (rows)
scoremat	Optimally scaled data matrix (first dimension)
quantifications	Category quantifications
dmeasures	Discimination matrices
lambda	Average discrimination matrix
weights	Component weights
loadings	Component loadings
ntel	Number of iterations
f	Loss function value
data	Original data frame
datanum	Numerical data frame
ndim	Number of extracted dimensions
call	Function call

References

Gifi, A. (1990). Nonlinear Multivariate Analysis. New York: Wiley.

De Leeuw, J., Mair, P., Groenen, P. J. F. (2017). Multivariate Analysis with Optimal Scaling.

See Also

homals, plot.princals

Examples

ABC6 <- ABC[,6:11]

## ordinal PCA
fitord <- princals(ABC6)  ## ordinal PCA
fitord
summary(fitord)

plot(fitord, plot.type = "transplot")
plot(fitord, "loadplot", main = "Loadings Plot ABC Data")  ## aspect ratio = 1
plot(fitord, "biplot", main = "Biplot ABC Data")
plot(fitord, "screeplot")

## linear restrictions (mimics standard PCA)
abc_knots <- knotsGifi(ABC6, "E")     ## 0 interior knots
fitlin <- princals(ABC6, knots = abc_knots, ordinal = TRUE)  
fitlin 
fitlin$evals
plot(fitlin, plot.type = "transplot")

## compare with standard PCA
ABCnum <- makeNumeric(ABC6)
fitpca <- prcomp(ABCnum, scale = TRUE)
fitpca$sdev^2   

## more complicated specifications
## Not run: 
data(epi.bfi, package = "psychTools")
epi6 <- epi.bfi[,1:6]
fitepi1 <- princals(epi6, knots = knotsGifi(epi6, "Q"))    ## monotone splines (degree 2)
fitepi1
plot(fitepi1, "transplot")

## no interior knots vars 1 and 2; data knots vars 3 and 4; 5 
## interior percentile knots var 5; no interior knots var 6)
knotList <- c(knotsGifi(epi6[,1:2], "E"), 
              knotsGifi(epi6[,3:4], "D"), 
              knotsGifi(epi6[,5], "Q", n = 5),
              knotsGifi(epi6[,6], "E"))
knotList  
ordvec <- c(TRUE, FALSE, TRUE, FALSE, FALSE, TRUE) ## ordinal restrictions
degvec <- c(3, -1, 2, 2, 3, 1)                     ## spline degrees (second variable nominal)
fitepi2 <- princals(epi6, knots = knotList, ordinal = ordvec, degrees = degvec)
fitepi2
plot(fitepi2, "transplot")

## End(Not run)

Gifi documentation built on Aug. 27, 2025, 3:01 a.m.