lessR (version 2.5)

corCFA: Confirmatory Factor Analysis of Multiple Indicator Measurement Models

Description

Abbreviation: cfa

A multiple indicator measurement model partitions a set of observed variables, such as items, into mutually exclusive groups with one common factor per group of items. From the input correlation matrix of the observed variables, this procedure uses iterated centroid estimation to estimate the coefficients of the model, the factor pattern and factor-factor correlations, as well as the correlations of each factor with each item. The analysis is provided by Fortran code, adapted from John Hunter's (1970) program PACKAGE.

Corresponding scale reliabilities are provided, as well as the residuals, the difference between the observed item correlations and those predicted by the model. To visualize the relationships, a heat map of the re-ordered correlation matrix is also provided, with item communalities in the diagonal. To understand the meaning of each factor, the corresponding item content is displayed for each factor if the items have been read as variable labels.

Usage

corCFA(x=mycor, iter=15, resid=TRUE, item.cor=TRUE, sort=TRUE,
         main=NULL, heat.map=TRUE, bottom=3, right=3, 
         colors=c("blue", "gray", "rose", "green", "gold", "red"),
         pdf.file=NULL, pdf.width=5, pdf.height=5,
         F1=NULL, F2=NULL, F3=NULL, F4=NULL, F5=NULL,
         F6=NULL, F7=NULL, F8=NULL, F9=NULL, F10=NULL,
         F11=NULL, F12=NULL)
cfa(...)

Arguments

x

Correlation matrix.

iter

Number of iterations for communality estimates.

resid

If TRUE, then calculate and print the residuals.

item.cor

If TRUE, display the item correlations.

sort

If TRUE, re-order the output correlation matrix so that items within each factor are sorted by their factor loadings on their own factor.

main

Graph title of heat map. Set to main="" to turn off.

heat.map

If TRUE, display a heat map of the item correlations with item communalities in the diagonal.

bottom

Number of lines of bottom margin of heat map.

right

Number of lines of right margin of heat map.

colors

Sets the color palette for the heat map.

pdf.file

Name of the pdf file to which graphics are redirected.

pdf.width

Width of the pdf file in inches.

pdf.height

Height of the pdf file in inches.

F1

Variables, listed by ordinal position, that define Factor 1.

F2

Variables, listed by ordinal position, that define Factor 2.

F3

Variables, listed by ordinal position, that define Factor 3.

F4

Variables, listed by ordinal position, that define Factor 4.

F5

Variables, listed by ordinal position, that define Factor 5.

F6

Variables, listed by ordinal position, that define Factor 6.

F7

Variables, listed by ordinal position, that define Factor 7.

F8

Variables, listed by ordinal position, that define Factor 8.

F9

Variables, listed by ordinal position, that define Factor 9.

F10

Variables, listed by ordinal position, that define Factor 10.

F11

Variables, listed by ordinal position, that define Factor 11.

F12

Variables, listed by ordinal position, that define Factor 12.

...

Parameter values.

Value

• The function returns the last correlation matrix specified. If the default resid=TRUE is active, then the residual matrix is returned. If code{resid=FALSE}, then the confirmatory factor analysis is returned. If the default item.cor=TRUE remains, the returned value includes all the coefficients of the correlation matrix with all observed variables (items) and latent variables (factors) and communalities in the diagonal. If item.cor=FALSE, then just the factor loadings, the correlations of items and factors, and the factor-factor correlations are returned.

Details

OVERVIEW A multiple indicator measurement model defines one or more latent variables, called factors, in terms of mutually exclusive sets of observed variables, such as items from a questionnaire or survey. That is, each factor is defined by a unique set or group of items, and each item only contributes to the definition of one factor. Two sets of parameters are estimated by the model, the factor pattern coefficients, the lambda's, and the factor-factor correlations, the phi's. Also estimated here are the correlations of each item with the other factors.

INPUT The analysis requires the correlation matrix of the items and the specification of the groups of items, each of which defines a factor in the multiple indicator measurement model. The default name for the item correlation matrix is mycor, which is also the default name of the matrix produced by the lessR function Correlation that computes the correlations from the data, as well as the name of the matrix read by the lessR function corRead that reads the already computed correlation matrix from an external file.

Define the constituent variables, the items, of each factor with a listing of each variable by its name in the correlation matrix. Each of the up to 12 factors is named F1, F2, etc. If the specified variables of a factor are in consecutive order in the input correlation matrix, the list can be specified by listing the first variable, a colon, and then the last variable. To specify multiple variables, a single variable or a list, separate each by a comma, then invoke the R combine or c function, preceded by the factor's name and an equals sign. For example, if the first factor is defined by variables in the input correlation matrix from m02 through m05, and the variable Anxiety, then define the factor in the corCFA function call according to F1=c(m02:m05,Anxiety).

OUTPUT The result of the analysis is the correlation matrix of the observed variables and resulting factors, plus the reliability analysis of the observed total scores or scale that corresponds to each factor. Each scale is defined as an unweighted composite.

VARIABLE LABELS To display the item content, first read the items as variable labels with the lessR function Read. If this labels data frame exists, then the corresponding variable labels, that is, items, are listed by factor. For more information, see Read.

HEAT MAP To help visualize the overall pattering of the correlations, the corresponding heat map of the item correlation matrix with communalities is produced when heat.map=TRUE, the default. As is true of the output correlation matrix, the correlations illustrated in the heat map are also sorted by their ordering within each factor. The corresponding color scheme is dictated by the system setting, according to the lessR function set. The default color scheme is blue.

ESTIMATION PROCEDURE The estimation procedure is centroid factor analysis, which defines each factor, parallel to the definition of each scale score, as the unweighted composite of the corresponding items for that scale. The latent variables are obtained by replacing the 1's in the diagonal of the observed variable correlation matrix with communality estimates. These estimates are obtained by iterating the solution to the specified number of iterations according to iter, which defaults to 15.

A communality is the percentage of the item's correlation attributable to, in this situation of a multiple indicator measurement model, its one underlying factor. As such, the communality is comparable to the item correlations for items within the same factor, which are also due only to the influence of the one common, underlying factor. A value of 0 for iter implies that the 1's remain in the observed variable correlation matrix, which then means that there are no latent factors defined. Instead the resulting correlation matrix is of the observed scale scores and the component items.

References

Gerbing, D. W., & Hamilton, J. G. (1994). The surprising viability of a simple alternate estimation procedure for the construction of large-scale structural equation measurement models. Structural Equation Modeling: A Multidisciplinary Journal, 1, 103-115.

Hunter, J. E., Gerbing, D. W., & Boster, F. J. (1982). Machiavellian beliefs and personality: The construct invalidity of the Machiavellian dimension. Journal of Personality and Social Psychology, 43, 1293-1305.

Hunter, John. (1970). PACKAGE. Michigan State University.

See Also

cr.

Examples

# perfect input correlation matrix for two-factor model
# Population Factor Pattern of the 3 items for each respective
#   Factor: 0.8, 0.6, 0.4
# Population Factor-Factor correlation: 0.3
mycor <- matrix(nrow=6, ncol=6, byrow=TRUE,
c(1.000,0.480,0.320,0.192,0.144,0.096,
  0.480,1.000,0.240,0.144,0.108,0.072,
  0.320,0.240,1.000,0.096,0.072,0.048,
  0.192,0.144,0.096,1.000,0.480,0.320,
  0.144,0.108,0.072,0.480,1.000,0.240,
  0.096,0.072,0.048,0.320,0.240,1.000))
colnames(mycor) <- c("V1", "V2", "V3", "V4", "V5", "V6")
rownames(mycor) <- colnames(mycor)

# the confirmatory factor analysis
# first three variables with first factor, last three with second
# default correlation matrix is mycor
# two ways to specify the items, with a colon and with commas
corCFA(F1=V1:V3, F2=c(V6,V5,V4))

# abbreviated form
cfa(F1=V1:V3, F2=V4:V6)

# produce a gray scale heat map of the item correlations
#   with communalities in the diagonal
# all subsequent graphics are in gray scale until changed
set(colors="gray")
corCFA(F1=V1:V3, F2=V4:V6)

# access the lessR data set called datMach4
data(datMach4)
# select the subset of the data that just contains Mach items
mach <- subset(datMach4, select=c(m01:m20))
# calculate the correlations and store in mycor
cr(mach)

# confirmatory factor analysis of 4-factor solution of Mach IV scale
# Hunter, Gerbing and Boster, 1982
corCFA(F1=c(m06,m07,m09,m10), F2=c(m15,m02),
       F3=c(m04,m11,m16), F4=c(m01,m05,m12,m13))


# bad model to illustrate indicator diagnostics
mycor <- corReflect(vars=c(m20))
corCFA(F1=c(m06,m09,m19), F2=c(m07),
       F3=c(m04,m11,m16), F4=c(m20,m18,m12,m15))