iterativePrincipalAxis function

Iterative Principal Axis Analysis

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.

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

Returns

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

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 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)
# .......................................................

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

Author(s)

Gilles Raiche

Centre sur les Applications des Modeles de Reponses aux Items (CAMRI)

Universite du Quebec a Montreal

raiche.gilles@uqam.ca

David Magis

Departement de mathematiques

Universite de Liege

David.Magis@ulg.ac.be