iterativePrincipalAxis {nFactors}R Documentation

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 insted, then a solution can be computed.

Usage

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

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

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

Author(s)

Gilles Raiche
Centre sur les Applications des Modeles de Reponses aux Items (CAMRI)
Universite du Quebec a Montreal
raiche.gilles@uqam.ca, http://www.er.uqam.ca/nobel/r17165/

References

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

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

See Also

componentAxis, principalAxis, rRecovery

Examples

# .......................................................
# 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 by 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)
# .......................................................
 

[Package nFactors version 2.3.1 Index]