nFactors (version 2.3.1)

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 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

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