nFactors (version 2.3.1)

principalAxis: Principal Axis Analysis

Description

The PrincipalAxis function return a principal axis analysis without iterated communalities estimates. Three different choices of communalities estimates are given: maximum corelation, multiple correlation or estimates based on the sum of the sqared principal component analysis loadings. Generally statistical packages initialize the the communalities at the multiple correlation value (usual inverse or generalized inverse). Unfortunately, this strategy cannot deal with singular correlation or covariance matrices. If a generalized inverse, the maximum correlation or the estimated communalities based on the sum of loading are used insted, then a solution can be computed.

Usage

principalAxis(R,
               nFactors=2,
               communalities="component")

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")

Value

• values
  numeric: variance of each component/factor
• varExplained
  numeric: variance explained by each component/factor
• varExplained
  numeric: cumulative variance explained by each component/factor
• loadings
  numeric: loadings of each variable on each component/factor

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, iterativePrincipalAxis, 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
# without iterated communalities -
# Kim and Mueller (1978, p. 21)
# Replace upper diagonal by lower diagonal
 RU <- diagReplace(R, upper=TRUE)
 principalAxis(RU, nFactors=2, communalities="component")
 principalAxis(RU, nFactors=2, communalities="maxr")
 principalAxis(RU, nFactors=2, communalities="multiple")
# Replace lower diagonal by upper diagonal
 RL <- diagReplace(R, upper=FALSE)
 principalAxis(RL, nFactors=2, communalities="component")
 principalAxis(RL, nFactors=2, communalities="maxr")
 principalAxis(RL, nFactors=2, communalities="multiple")
# .......................................................