nFactors (version 2.4.1.2)

iterativePrincipalAxis: Iterative Principal Axis Analysis

Description

The iterativePrincipalAxis function returns a principal axis analysis with iterated communality estimates. Four different choices of initial communality estimates are given: maximum correlation, multiple correlation (usual and generalized inverse) or estimates based on the sum of the squared principal component analysis loadings. Generally, statistical packages initialize the communalities at the multiple correlation value. Unfortunately, this strategy cannot always deal with singular correlation or covariance matrices. If a generalized inverse, the maximum correlation or the estimated communalities based on the sum of loadings are used instead, then a solution can be computed.

Usage

iterativePrincipalAxis(
  R,
  nFactors = 2,
  communalities = "component",
  iterations = 20,
  tolerance = 0.001
)

Value

values numeric: variance of each component

varExplained numeric: variance explained by each component

varExplained numeric: cumulative variance explained by each component

loadings numeric: loadings of each variable on each component

iterations numeric: maximum number of iterations to obtain a solution

tolerance numeric: minimal difference in the estimated communalities after a given iteration

Arguments

R

numeric: correlation or covariance matrix

nFactors

numeric: number of factors to retain

communalities

character: initial values for communalities ("component", "maxr", "ginv" or "multiple")

iterations

numeric: maximum number of iterations to obtain a solution

tolerance

numeric: minimal difference in the estimated communalities after a given iteration

Author

Gilles Raiche
Centre sur les Applications des Modeles de Reponses aux Items (CAMRI)
Universite du Quebec a Montreal


David Magis
Departement de mathematiques
Universite de Liege

References

Kim, J.-O. and Mueller, C. W. (1978). Introduction to factor analysis. What it is and how to do it. Beverly Hills, CA: Sage.

Kim, J.-O. and Mueller, C. W. (1987). Factor analysis. Statistical methods and practical issues. Beverly Hills, CA: Sage.

See Also

componentAxis, principalAxis, rRecovery

Examples

if (FALSE) {
if(interactive()){
## ................................................
# Example from Kim and Mueller (1978, p. 10)
# Population: upper diagonal
# Simulated sample: lower diagnonal
R <- matrix(c( 1.000, .6008, .4984, .1920, .1959, .3466,
               .5600, 1.000, .4749, .2196, .1912, .2979,
               .4800, .4200, 1.000, .2079, .2010, .2445,
               .2240, .1960, .1680, 1.000, .4334, .3197,
               .1920, .1680, .1440, .4200, 1.000, .4207,
               .1600, .1400, .1200, .3500, .3000, 1.000),
            nrow=6, byrow=TRUE)

# Factor analysis: Principal axis factoring with iterated communalities
# Kim and Mueller (1978, p. 23)
# Replace upper diagonal with lower diagonal
RU         <- diagReplace(R, upper=TRUE)
nFactors   <- 2
fComponent <- iterativePrincipalAxis(RU, nFactors=nFactors,
                                     communalities="component")
fComponent
rRecovery(RU,fComponent$loadings, diagCommunalities=FALSE)

fMaxr      <- iterativePrincipalAxis(RU, nFactors=nFactors,
                                     communalities="maxr")
fMaxr
rRecovery(RU,fMaxr$loadings, diagCommunalities=FALSE)

fMultiple  <- iterativePrincipalAxis(RU, nFactors=nFactors,
                                     communalities="multiple")
fMultiple
rRecovery(RU,fMultiple$loadings, diagCommunalities=FALSE)
# .......................................................
 }
}