faBounds {fungible}

Bounds on the Correlation Between an External Variable and a Common Factor

Description

This function computes the bounds on the correlation between an external variable and a common factor.

Usage

faBounds(Lambda, RX, rXY, alphaY = 1)

Arguments

Lambda

(matrix) A p x 1 matrix of factor loadings.

RX

(matrix) A p x p matrix of correlations for the factor indicators.

rXY

(vector) A p x 1 vector of correlations between the factor indicators (X) and the external variable (Y).

alphaY

(scalar) The reliability of Y. Default alphaY = 1.

Value

faBounds returns the following objects:

• Lambda (matrix) A p x 1 vector of factor loadings.

• RX (matrix) The indicator correlation matrix.

• rXY: (vector) The correlations between the factor indicators (X) and the external variable (Y).

• alphaY (integer) The reliability of the external variable.

• bounds (vector) A 2 x 1 vector that includes the lower and upper bounds for the correlation between an external variable and a common factor.

• rUiY (vector) Correlations between the unique factors and the external variable for the lower bound estimate.

• rUjY (vector) Correlations between the unique factors and the external variable for the upper bound estimate.

Author(s)

Niels G. Waller

References

Steiger, J. H. (1979). The relationship between external variables and common factors. Psychometrika, 44, 93-97.

Waller, N. G. (under review). New results on the relationship between an external variable and a common factor.

Examples

## Example 
## We wish to compute the bounds between the Speed factor from the 
## Holzinger (H) and Swineford data and a hypothetical external 
## variable, Y.

## RH = R matrix for *H*olzinger Swineford data
RH <- 
 matrix(c( 1.00,   0,    0,     0,     0,     0,
           .73, 1.00,    0,     0,     0,     0, 
           .70,  .72,  1.00,    0,     0,     0,
           .17,  .10,   .12,  1.00,    0,     0,
           .11,  .14,   .15,   .49,  1.00,    0,
           .21,  .23,   .21,   .34,   .45,  1.00), 6, 6)

RH <- RH + t(RH) - diag(6)
RX <- RH[4:6, 4:6]

## S-C = Straight-curved
 colnames(RX) <- rownames(RX) <-
        c("Addition", "Counting dots", "S-C capitals")
print( RX, digits = 2 ) 

## Extract 1 MLE factor  
fout <- faMain(R = RX, 
              numFactors = 1, 
              facMethod = "faml", 
              rotate="none")

## Lambda = factor loadings matrix  
Lambda <- fout$loadings
print( Lambda, digits = 3 ) 

## rXY = correlations between the factor indicators (X) and
## the external variable (Y)

 rXY = c(.1, .2, .3)
 
 # Assume that the reliability of Y = .75
 
 faBounds(Lambda, RX, rXY, alphaY = .75)