nFactors: Number of factor or components to retain in a factor analysis

Description

A package for determining the number of factor or components to retain in a factor analysis. The methods are all based on eigenvalues.

Author(s)

Gilles Raiche
Centre sur les Applications des Modeles de Reponses aux Items (CAMRI)
Universite du Quebec a Montreal
raiche.gilles@uqam.ca

References

Raiche, G., Walls, T. A., Magis, D., Riopel, M. and Blais, J.-G. (2013). Non-graphical solutions for Cattell's scree test. Methodology, 9(1), 23-29.

Bentler and Yuan's Computation of the LRT Index and the Linear Trend Coefficients

Description

This function computes the Bentler and Yuan's (1996, 1998) LRT index for the linear trend in eigenvalues of a covariance matrix. The related \chi^2 and p-value are also computed. This function is generally called from the nBentler function. But it could be of use for graphing the linear trend function and to study it's behavior.

Usage

bentlerParameters(
  x,
  N,
  nFactors,
  log = TRUE,
  cor = TRUE,
  minPar = c(min(lambda) - abs(min(lambda)) + 0.001, 0.001),
  maxPar = c(max(lambda), stats::lm(lambda ~ I(length(lambda):1))$coef[2]),
  resParx = c(0.01, 2),
  resPary = c(0.01, 2),
  graphic = TRUE,
  resolution = 30,
  typePlot = "wireframe",
  ...
)

Arguments

Details

The implemented Bentler and Yuan's procedure must be used with care because the minimized function is not always stable. In many cases, constraints must applied to obtain a solution. The actual implementation did, but the user can modify these constraints.

The hypothesis tested (Bentler and Yuan, 1996, equation 10) is:

(1) \qquad \qquad H_k: \lambda_{k+i} = \alpha + \beta x_i, (i = 1, \ldots, q)

The solution of the following simultaneous equations is needed to find (\alpha, \beta) \in

(2) \qquad \qquad f(x) = \sum_{i=1}^q \frac{ [ \lambda_{k+j} - N \alpha + \beta x_j ] x_j}{(\alpha + \beta x_j)^2} = 0

and \qquad \qquad g(x) = \sum_{i=1}^q \frac{ \lambda_{k+j} - N \alpha + \beta x_j x_j}{(\alpha + \beta x_j)^2} = 0

The solution to this system of equations was implemented by minimizing the following equation:

(3) \qquad \qquad (\alpha, \beta) \in \inf{[h(x)]} = \inf{\log{[f(x)^2 + g(x)^2}}]

The likelihood ratio test LRT proposed by Bentler and Yuan (1996, equation 7) follows a \chi^2 probability distribution with q-2 degrees of freedom and is equal to:

(4) \qquad \qquad LRT = N(k - p)\left\{ {\ln \left( {{n \over N}} \right) + 1} \right\} - N\sum\limits_{j = k + 1}^p {\ln \left\{ {{{\lambda _j } \over {\alpha + \beta x_j }}} \right\}} + n\sum\limits_{j = k + 1}^p {\left\{ {{{\lambda _j } \over {\alpha + \beta x_j }}} \right\}}

With p beeing the number of eigenvalues, k the number of eigenvalues to test, q the p-k remaining eigenvalues, N the sample size, and n = N-1. Note that there is an error in the Bentler and Yuan equation, the variables N and n beeing inverted in the preceeding equation 4.

A better strategy proposed by Bentler an Yuan (1998) is to use a minimized \chi^2 solution. This strategy will be implemented in a future version of the nFactors package.

Value

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

References

Bentler, P. M. and Yuan, K.-H. (1996). Test of linear trend in eigenvalues of a covariance matrix with application to data analysis. British Journal of Mathematical and Statistical Psychology, 49, 299-312.

Bentler, P. M. and Yuan, K.-H. (1998). Test of linear trend in the smallest eigenvalues of the correlation matrix. Psychometrika, 63(2), 131-144.

See Also

nBartlett, nBentler

Examples

## Not run: 
if(interactive()){
## ................................................
## SIMPLE EXAMPLE OF THE BENTLER AND YUAN PROCEDURE
## #' @importFrom graphics abline

# Bentler (1996, p. 309) Table 2 - Example 2 .............
n=649
bentler2<-c(5.785, 3.088, 1.505, 0.582, 0.424, 0.386, 0.360, 0.337, 0.303,
            0.281, 0.246, 0.238, 0.200, 0.160, 0.130)

results  <- nBentler(x=bentler2, N=n,  details=TRUE)
results

# Two different figures to verify the convergence problem identified with
# the 2th component
bentlerParameters(x=bentler2, N=n, nFactors= 2, graphic=TRUE,
                  typePlot="contourplot",
                  resParx=c(0,9), resPary=c(0,9), cor=FALSE)

bentlerParameters(x=bentler2, N=n, nFactors= 4, graphic=TRUE, drape=TRUE,
                  resParx=c(0,9), resPary=c(0,9),
                  scales = list(arrows = FALSE) )

plotuScree(x=bentler2, model="components",
  main=paste(results$nFactors,
  " factors retained by the Bentler and Yuan's procedure (1996, p. 309)",
  sep=""))
# ........................................................

# Bentler (1998, p. 140) Table 3 - Example 1 .............
n        <- 145
example1 <- c(8.135, 2.096, 1.693, 1.502, 1.025, 0.943, 0.901, 0.816,
              0.790,0.707, 0.639, 0.543,0.533, 0.509, 0.478, 0.390,
              0.382, 0.340, 0.334, 0.316, 0.297,0.268, 0.190, 0.173)

results  <- nBentler(x=example1, N=n,  details=TRUE)
results

# Two different figures to verify the convergence problem identified with
# the 10th component
bentlerParameters(x=example1, N=n, nFactors= 10, graphic=TRUE,
                  typePlot="contourplot",
                  resParx=c(0,0.4), resPary=c(0,0.4))

bentlerParameters(x=example1, N=n, nFactors= 10, graphic=TRUE, drape=TRUE,
                  resParx=c(0,0.4), resPary=c(0,0.4),
                  scales = list(arrows = FALSE) )

plotuScree(x=example1, model="components",
   main=paste(results$nFactors,
   " factors retained by the Bentler and Yuan's procedure (1998, p. 140)",
   sep=""))
# ........................................................
 }

## End(Not run)

Principal Component Analysis With Only n First Components Retained

Description

The componentAxis function returns a principal component analysis with the first n components retained.

Usage

componentAxis(R, nFactors = 2)

Arguments

Value

Author(s)

Gilles Raiche
Centre sur les Applications des Modeles de Reponses aux Items (CAMRI)
Universite du Quebec a Montreal
raiche.gilles@uqam.ca

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

principalComponents, iterativePrincipalAxis, rRecovery

Examples

## Not run: 
if(interactive()){
# .......................................................
# Example from Kim and Mueller (1978, p. 10)
# Simulated sample: lower diagnonal
 R <- matrix(c( 1.000, 0.560, 0.480, 0.224, 0.192, 0.16,
                0.560, 1.000, 0.420, 0.196, 0.168, 0.14,
                0.480, 0.420, 1.000, 0.168, 0.144, 0.12,
                0.224, 0.196, 0.168, 1.000, 0.420, 0.35,
                0.192, 0.168, 0.144, 0.420, 1.000, 0.30,
                0.160, 0.140, 0.120, 0.350, 0.300, 1.00),
                nrow=6, byrow=TRUE)

# Factor analysis: Selected principal components - Kim and Mueller
# (1978, p. 20)
 componentAxis(R, nFactors=2)

# .......................................................
 }

## End(Not run)

Insert Communalities in the Diagonal of a Correlation or a Covariance Matrix

Description

This function inserts communalities in the diagonal of a correlation/covariance matrix.

Usage

corFA(R, method = "ginv")

Arguments

Value

A correlation matrix with coerced variables with communalities in the diagonal.

Author(s)

Gilles Raiche, Universite du Quebec a Montreal (raiche.gilles@uqam.ca)

See Also

plotuScree, nScree, plotnScree, plotParallel

Examples

## Not run: 
if(interactive()){
## LOWER CORRELATION MATRIX WITH ZEROS ON UPPER PART
## From Gorsuch (table 1.3.1)
 gorsuch <- c(
 1,0,0,0,0,0,0,0,0,0,
 .6283, 1,0,0,0,0,0,0,0,0,
 .5631, .7353, 1,0,0,0,0,0,0,0,
 .8689, .7055, .8444, 1,0,0,0,0,0,0,
 .9030, .8626, .6890, .8874, 1,0,0,0,0,0,
 .6908, .9028, .9155, .8841, .8816, 1,0,0,0,0,
 .8633, .7495, .7378, .9164, .9109, .8572, 1,0,0,0,
 .7694, .7902, .7872, .8857, .8835, .8884, .7872, 1,0,0,
 .8945, .7929, .7656, .9494, .9546, .8942, .9434, .9000, 1,0,
 .5615, .6850, .8153, .7004, .6583, .7720, .6201, .6141, .6378, 1)

## UPPER CORRELATION MATRIX FILLED WITH UPPER CORRELATION MATRIX
 gorsuch <- makeCor(gorsuch)

## REPLACE DIAGONAL WITH COMMUNALITIES
 gorsuchCfa <- corFA(gorsuch)
 gorsuchCfa
 }

## End(Not run)

Eigenvalues from classical studies

Description

Classical examples of eigenvalues vectors used to study the number of factors to retain in the litterature. These examples generally give the number of subjects use to obtain these eigenvalues. The number of subjects is used with the parallel analysis.

Usage

dFactors

Format

A list of examples. For each example, a list is also used to give the eigenvalues vector and the number of subjects.

Bentler

$eigenvalues and $nsubjects

Buja

$eigenvalues and $nsubjects

Cliff1

$eigenvalues and $nsubjects

Cliff2

$eigenvalues and $nsubjects

Cliff3

$eigenvalues and $nsubjects

Hand

$eigenvalues and $nsubjects

Harman

$eigenvalues and $nsubjects

Lawley

$eigenvalues and $nsubjects

Raiche

$eigenvalues and $nsubjects

Tucker1

$eigenvalues and $nsubjects

Tucker2

$eigenvalues and $nsubjects

Details

Other datasets will be added in future versions of the package.

Source

Lawley and Hand dataset: Bartholomew et al. (2002, p. 123, 126)

Bentler dataset: Bentler and Yuan (1998, p. 139-140)

Buja datasets: Buja and Eyuboglu (1992, p. 516, 519) < Number of subjects not specified by Buja and Eyuboglu >

Cliff datasets: Cliff (1970, p. 165)

Raiche dataset: Raiche, Langevin, Riopel and Mauffette (2006)

Raiche dataset: Raiche, Riopel and Blais (2006, p. 9)

Tucker datasets: Tucker et al. (1969, p. 442)

References

Bartholomew, D. J., Steele, F., Moustaki, I. and Galbraith, J. I. (2002). The analysis and interpretation of multivariate data for social scientists. Boca Raton, FL: Chapman and Hall.

Bentler, P. M. and Yuan, K.-H. (1998). Tests for linear trend in the smallest eigenvalues of the correlation matrix. Psychometrika, 63(2), 131-144.

Buja, A. and Eyuboglu, N. (1992). Remarks on parallel analysis. Multivariate Behavioral Research, 27(4), 509-540.

Cliff, N. (1970). The relation between sample and population characteristic vectors. Psychometrika, 35(2), 163-178.

Hand, D. J., Daly, F., Lunn, A. D., McConway, K. J. and Ostrowski, E. (1994). A handbook of small data sets. Boca Raton, FL: Chapman and Hall.

Lawley, D. N. and Maxwell, A. E. (1971). Factor analysis as a statistical method (2nd edition). London: Butterworth.

Raiche, G., Langevin, L., Riopel, M. and Mauffette, Y. (2006). Etude exploratoire de la dimensionnalite et des facteurs expliques par une traduction francaise de l'Inventaire des approches d'enseignement de Trigwell et Prosser dans trois universite quebecoises. Mesure et Evaluation en Education, 29(2), 41-61.

Raiche, G., Walls, T. A., Magis, D., Riopel, M. and Blais, J.-G. (2013). Non-graphical solutions for Cattell's scree test. Methodology, 9(1), 23-29.

Tucker, L. D., Koopman, R. F. and Linn, R. L. (1969). Evaluation of factor analytic research procedures by mean of simulated correlation matrices. Psychometrika, 34(4), 421-459.

Zoski, K. and Jurs, S. (1993). Using multiple regression to determine the number of factors to retain in factor analysis. Multiple Linear Regression Viewpoint, 20(1), 5-9.

Examples

## Not run: 
if(interactive()){
# EXAMPLES FROM DATASET
 data(dFactors)

# COMMAND TO VISUALIZE THE CONTENT AND ATTRIBUTES OF THE DATASETS
 names(dFactors)
 attributes(dFactors)
 dFactors$Cliff1$eigenvalues
 dFactors$Cliff1$nsubjects

# SCREE PLOT OF THE Cliff1 DATASET
 plotuScree(dFactors$Cliff1$eigenvalues)
 }

## End(Not run)

Replacing Upper or Lower Diagonal of a Correlation or Covariance Matrix

Description

The diagReplace function returns a modified correlation or covariance matrix by replacing upper diagonal with lower diagonal, or lower diagonal with upper diagonal.

Usage

diagReplace(R, upper = TRUE)

Arguments

Value

Author(s)

Gilles Raiche
Centre sur les Applications des Modeles de Reponses aux Items (CAMRI)
Universite du Quebec a Montreal
raiche.gilles@uqam.ca

Examples

## Not run: 
if(interactive()){
# .......................................................
# 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)

# Replace upper diagonal with lower diagonal
 RU <- diagReplace(R, upper=TRUE)

# Replace lower diagonal with upper diagonal
 RL <- diagReplace(R, upper=FALSE)
# .......................................................
 }

## End(Not run)

Bootstrapping of the Eigenvalues From a Data Frame

Description

The eigenBootParallel function samples observations from a data.frame to produce correlation or covariance matrices from which eigenvalues are computed. The function returns statistics about these bootstrapped eigenvalues. Their means or their quantile could be used later to replace the eigenvalues inputted to a parallel analysis. The eigenBootParallel can also compute random eigenvalues from empirical data by column permutation (Buja and Eyuboglu, 1992).

Usage

eigenBootParallel(
  x,
  quantile = 0.95,
  nboot = 30,
  option = "permutation",
  cor = TRUE,
  model = "components",
  ...
)

Arguments

Value

Author(s)

Gilles Raiche
Centre sur les Applications des Modeles de Reponses aux Items (CAMRI)
Universite du Quebec a Montreal
raiche.gilles@uqam.ca

References

Buja, A. and Eyuboglu, N. (1992). Remarks on parallel analysis. Multivariate Behavioral Research, 27(4), 509-540.

Zwick, W. R. and Velicer, W. F. (1986). Comparison of five rules for determining the number of components to retain. Psychological bulletin, 99, 432-442.

See Also

principalComponents, iterativePrincipalAxis, rRecovery

Examples

## Not run: 
if(interactive()){
# .......................................................
# Example from the iris data
 eigenvalues <- eigenComputes(x=iris[,-5])

# Permutation parallel analysis distribution
 aparallel   <- eigenBootParallel(x=iris[,-5], quantile=0.95)$quantile

# Number of components to retain
 results     <- nScree(x = eigenvalues, aparallel = aparallel)
 results$Components
 plotnScree(results)
# ......................................................

# ......................................................
# Bootstrap distributions study of the eigenvalues from iris data
# with different correlation methods
 eigenBootParallel(x=iris[,-5],quantile=0.05,
                   option="bootstrap",method="pearson")
 eigenBootParallel(x=iris[,-5],quantile=0.05,
                   option="bootstrap",method="spearman")
 eigenBootParallel(x=iris[,-5],quantile=0.05,
                   option="bootstrap",method="kendall")
 }

## End(Not run)

Computes Eigenvalues According to the Data Type

Description

The eigenComputes function computes eigenvalues from the identified data type. It is used internally in many fonctions of the nFactors package in order to apply these to a vector of eigenvalues, a matrix of correlations or covariance or a data frame.

Usage

eigenComputes(x, cor = TRUE, model = "components", ...)

Arguments

Value

numeric: return a vector of eigenvalues

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

Examples

## Not run: 
if(interactive()){
# .......................................................
# Different data types
# Vector of eigenvalues
data(dFactors)
x1 <- dFactors$Cliff1$eigenvalues
eigenComputes(x1)

# Data from a data.frame
x2 <- data.frame(matrix(20*rnorm(100), ncol=5))
eigenComputes(x2, cor=TRUE,  use="everything")
eigenComputes(x2, cor=FALSE, use="everything")
eigenComputes(x2, cor=TRUE,  use="everything", method="spearman")
eigenComputes(x2, cor=TRUE,  use="everything", method="kendall")

x3 <- cov(x2)
eigenComputes(x3, cor=TRUE,  use="everything")
eigenComputes(x3, cor=FALSE, use="everything")

x4 <- cor(x2)
eigenComputes(x4, use="everything")
# .......................................................
 }

## End(Not run)

Identify the Data Type to Obtain the Eigenvalues

Description

The eigenFrom function identifies the data type from which to obtain the eigenvalues. The function is used internally in many functions of the nFactors package to be able to apply these to a vector of eigenvalues, a matrix of correlations or covariance or a data.frame.

Usage

eigenFrom(x)

Arguments

Value

character: return the data type to obtain the eigenvalues: "eigenvalues", "correlation" or "data"

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

Examples

## Not run: 
if(interactive()){
# .......................................................
# Different data types
# Examples of adequate data sources
# Vector of eigenvalues
data(dFactors)
x1 <- dFactors$Cliff1$eigenvalues
eigenFrom(x1)

# Data from a data.frame
x2 <- data.frame(matrix(20*rnorm(100), ncol=5))
eigenFrom(x2)

# From a covariance matrix
x3 <- cov(x2)
eigenFrom(x3)

# From a correlation matrix
x4 <- cor(x2)
eigenFrom(x4)

# Examples of inadequate data sources: not run because of errors generated
# x0 <- c(2,1)             # Error: not enough eigenvalues
# eigenFrom(x0)
# x2 <- matrix(x1, ncol=5) # Error: non a symetric covariance matrix
# eigenFrom(x2)
# eigenFrom(x3[,(1:2)])    # Error: not enough variables
# x6 <- table(x5)          # Error: not a valid data class
# eigenFrom(x6)
# .......................................................
 }

## End(Not run)

Generate a Factor Structure Matrix

Description

The generateStructure function returns a mjc factor structure matrix. The number of variables per major factor pmjc is equal for each factor. The argument pmjc must be divisible by nVar. The arguments are strongly inspired from Zick and Velicer (1986, p. 435-436) methodology.

Usage

generateStructure(var, mjc, pmjc, loadings, unique)

Arguments

Value

values numeric matrix: factor structure

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

References

Raiche, G., Walls, T. A., Magis, D., Riopel, M. and Blais, J.-G. (2013). Non-graphical solutions for Cattell's scree test. Methodology, 9(1), 23-29.

Zwick, W. R. and Velicer, W. F. (1986). Comparison of five rules for determining the number of components to retain. Psychological Bulletin, 99, 432-442.

See Also

principalComponents, iterativePrincipalAxis, rRecovery

Examples

## Not run: 
if(interactive()){
# .......................................................
# Example inspired from Zwick and Velicer (1986, table 2, p. 437)
## ...................................................................
unique=0.2; loadings=0.5
zwick1 <- generateStructure(var=36, mjc=6, pmjc= 6, loadings=loadings,
                           unique=unique)
zwick2 <- generateStructure(var=36, mjc=3, pmjc=12, loadings=loadings,
                           unique=unique)
zwick3 <- generateStructure(var=72, mjc=9, pmjc= 8, loadings=loadings,
                           unique=unique)
zwick4 <- generateStructure(var=72, mjc=6, pmjc=12, loadings=loadings,
                           unique=unique)
sat=0.8
## ...................................................................
zwick5 <- generateStructure(var=36, mjc=6, pmjc= 6, loadings=loadings,
                           unique=unique)
zwick6 <- generateStructure(var=36, mjc=3, pmjc=12, loadings=loadings,
                           unique=unique)
zwick7 <- generateStructure(var=72, mjc=9, pmjc= 8, loadings=loadings,
                           unique=unique)
zwick8 <- generateStructure(var=72, mjc=6, pmjc=12, loadings=loadings,
                          unique=unique)
## ...................................................................

# nsubjects <- c(72, 144, 180, 360)
# require(psych)
# Produce an usual correlation matrix from a congeneric model
nsubjects <- 72
mzwick5   <- psych::sim.structure(fx=as.matrix(zwick5), n=nsubjects)
mzwick5$r

# Factor analysis: recovery of the factor structure
iterativePrincipalAxis(mzwick5$model, nFactors=6,
                      communalities="ginv")$loadings
iterativePrincipalAxis(mzwick5$r    , nFactors=6,
                      communalities="ginv")$loadings
factanal(covmat=mzwick5$model,         factors=6)
factanal(covmat=mzwick5$r    ,         factors=6)

# Number of components to retain
eigenvalues  <- eigen(mzwick5$r)$values
aparallel    <- parallel(var      = length(eigenvalues),
                        subject  = nsubjects,
                        rep      = 30,
                        quantile = 0.95,
                        model="components")$eigen$qevpea
results <- nScree(x         = eigenvalues,
                 aparallel = aparallel)
results$Components
plotnScree(results)

# Number of factors to retain
eigenvalues.fa  <- eigen(corFA(mzwick5$r))$values
aparallel.fa    <- parallel(var      = length(eigenvalues.fa),
                           subject  = nsubjects,
                           rep      = 30,
                           quantile = 0.95,
                           model="factors")$eigen$qevpea
results.fa <- nScree(x      = eigenvalues.fa,
                    aparallel = aparallel.fa,
                    model     ="factors")
results.fa$Components
plotnScree(results.fa)
# ......................................................

 }

## End(Not run)

Utility Functions for nFactors Class Objects

Description

Utility functions for nFactors class objects.

Usage

is.nFactors(x)

## S3 method for class 'nFactors'
print(x, ...)

## S3 method for class 'nFactors'
summary(object, ...)

Arguments

Value

Generic functions for the nFactors class:

Author(s)

Gilles Raiche
Centre sur les Applications des Modeles de Reponses aux Items (CAMRI)
Universite du Quebec a Montreal
raiche.gilles@uqam.ca

References

Raiche, G., Walls, T. A., Magis, D., Riopel, M. and Blais, J.-G. (2013). Non-graphical solutions for Cattell's scree test. Methodology, 9(1), 23-29.

See Also

nBentler, nBartlett, nCng, nMreg, nSeScree

Examples

## Not run: 
if(interactive()){
## SIMPLE EXAMPLE
 data(dFactors)
 eig      <- dFactors$Raiche$eigenvalues
 N        <- dFactors$Raiche$nsubjects

 res <- nBartlett(eig,N); res; is.nFactors(res); summary(res, digits=2)
 res <- nBentler(eig,N);  res; is.nFactors(res); summary(res, digits=2)
 res <- nCng(eig);        res; is.nFactors(res); summary(res, digits=2)
 res <- nMreg(eig);       res; is.nFactors(res); summary(res, digits=2)
 res <- nSeScree(eig);    res; is.nFactors(res); summary(res, digits=2)

## SIMILAR RESULTS, BUT NOT A nFactors OBJECT
 res <- nScree(eig);      res; is.nFactors(res); summary(res, digits=2)

 }

## End(Not run)

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 instead, then a solution can be computed.

Usage

iterativePrincipalAxis(
  R,
  nFactors = 2,
  communalities = "component",
  iterations = 20,
  tolerance = 0.001
)

Arguments

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

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

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

Examples

## Not run: 
if(interactive()){
## ................................................
# 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)
# .......................................................
 }

## End(Not run)

Create a Full Correlation/Covariance Matrix from a Matrix With Lower Part Filled and Upper Part With Zeros

Description

This function creates a full correlation/covariance matrix from a matrix with lower part filled and upper part with zeros.

Usage

makeCor(x)

Arguments

Value

numeric: full correlation matrix

Author(s)

Gilles Raiche
Centre sur les Applications des Modeles de Reponses aux Items (CAMRI)
Universite du Quebec a Montreal
raiche.gilles@uqam.ca

See Also

plotuScree, nScree, plotnScree, plotParallel

Examples

## Not run: 
if(interactive()){
## ................................................
## LOWER CORRELATION MATRIX WITH ZEROS ON UPPER PART
## From Gorsuch (table 1.3.1)
gorsuch <- c(
 1,0,0,0,0,0,0,0,0,0,
 .6283, 1,0,0,0,0,0,0,0,0,
 .5631, .7353, 1,0,0,0,0,0,0,0,
 .8689, .7055, .8444, 1,0,0,0,0,0,0,
 .9030, .8626, .6890, .8874, 1,0,0,0,0,0,
 .6908, .9028, .9155, .8841, .8816, 1,0,0,0,0,
.8633, .7495, .7378, .9164, .9109, .8572, 1,0,0,0,
 .7694, .7902, .7872, .8857, .8835, .8884, .7872, 1,0,0,
 .8945, .7929, .7656, .9494, .9546, .8942, .9434, .9000, 1,0,
 .5615, .6850, .8153, .7004, .6583, .7720, .6201, .6141, .6378, 1)

## UPPER CORRELATION MATRIX FILLED WITH UPPER CORRELATION MATRIX
gorsuch <- makeCor(gorsuch)
gorsuch
 }

## End(Not run)

Statistical Summary of a Data Frame

Description

This function produces another summary of a data.frame. This function was proposed in order to apply some functions globally on a data.frame: quantile, median, min and max. The usual R version cannot do so.

Usage

moreStats(x, quantile = 0.95, show = FALSE)

Arguments

Value

numeric: data.frame of statistics: mean, median, quantile, standard deviation, minimum and maximum

Author(s)

Gilles Raiche
Centre sur les Applications des Modeles de Reponses aux Items (CAMRI)
Universite du Quebec a Montreal
raiche.gilles@uqam.ca

See Also

plotuScree, nScree, plotnScree, plotParallel

Examples

## Not run: 
if(interactive()){
## ................................................
## GENERATION OF A MATRIX OF 100 OBSERVATIONS AND 10 VARIABLES
x   <- matrix(rnorm(1000),ncol=10)

## STATISTICS
res <- moreStats(x, quantile=0.05, show=TRUE)
res
 }

## End(Not run)

Bartlett, Anderson and Lawley Procedures to Determine the Number of Components/Factors

Description

This function computes the Bartlett, Anderson and Lawley indices for determining the number of components/factors to retain.

Usage

nBartlett(
  x,
  N,
  alpha = 0.05,
  cor = TRUE,
  details = TRUE,
  correction = TRUE,
  ...
)

Arguments

Details

Note: the latex formulas are available only in the pdf version of this help file.

The hypothesis tested is:

(1) \qquad \qquad H_k: \lambda_{k+1} = \ldots = \lambda_p

This hypothesis is verified by the application of different version of a \chi^2 test with different values for the degrees of freedom. Each of these tests shares the computation of a V_k value:

(2) \qquad \qquad V_k = \prod\limits_{i = k + 1}^p \left[ {{\lambda _i} \over {1 \over q } \sum\limits_{i = k + 1}^p {\lambda _i }} \right]

p is the number of eigenvalues, k the number of eigenvalues to test, and q the p-k remaining eigenvalues. n is equal to the sample size minus 1 (n = N-1).

The Anderson statistic is distributed as a \chi^2 with (q + 2)(q - 1)/2 degrees of freedom and is equal to:

(3) \qquad \qquad - n\log (V_k ) \sim \chi _{(q + 2)(q - 1)/2}^2

An improvement of this statistic from Bartlett (Bentler, and Yuan, 1996, p. 300; Horn and Engstrom, 1979, equation 8) is distributed as a \chi^2 with (q)(q - 1)/2 degrees of freedom and is equal to:

(4) \qquad \qquad - \left[ {n - k - {{2q^2 q + 2} \over {6q}}} \right]\log (V_k ) \sim \chi _{(q + 2)(q - 1)/2}^2

Finally, Anderson (1956) and James (1969) proposed another statistic.

(5) \qquad \qquad - \left[ {n - k - {{2q^2 q + 2} \over {6q}} + \sum\limits_{i = 1}^k {{{\bar \lambda _q^2 } \over {\left( {\lambda _i - \bar \lambda _q } \right)^2 }}} } \right]\log (V_k ) \sim \chi _{(q + 2)(q - 1)/2}^2

Bartlett (1950, 1951) proposed a correction to the degrees of freedom of these \chi^2 after the first significant test: (q+2)(q - 1)/2.

Value

Author(s)

Gilles Raiche
Centre sur les Applications des Modeles de Reponses aux Items (CAMRI)
Universite du Quebec a Montreal
raiche.gilles@uqam.ca

References

Anderson, T. W. (1963). Asymptotic theory for principal component analysis. Annals of Mathematical Statistics, 34, 122-148.

Bartlett, M. S. (1950). Tests of significance in factor analysis. British Journal of Psychology, 3, 77-85.

Bartlett, M. S. (1951). A further note on tests of significance. British Journal of Psychology, 4, 1-2.

Bentler, P. M. and Yuan, K.-H. (1996). Test of linear trend in eigenvalues of a covariance matrix with application to data analysis. British Journal of Mathematical and Statistical Psychology, 49, 299-312.

Bentler, P. M. and Yuan, K.-H. (1998). Test of linear trend in the smallest eigenvalues of the correlation matrix. Psychometrika, 63(2), 131-144.

Horn, J. L. and Engstrom, R. (1979). Cattell's scree test in relation to Bartlett's chi-square test and other observations on the number of factors problem. Multivariate Behavioral Reasearch, 14(3), 283-300.

James, A. T. (1969). Test of equality of the latent roots of the covariance matrix. In P. K. Krishna (Eds): Multivariate analysis, volume 2.New-York, NJ: Academic Press.

Lawley, D. N. (1956). Tests of significance for the latent roots of covarianceand correlation matrix. Biometrika, 43(1/2), 128-136.

See Also

plotuScree, nScree, plotnScree, plotParallel

Examples

## Not run: 
if(interactive()){
## ................................................
## SIMPLE EXAMPLE OF A BARTLETT PROCEDURE

data(dFactors)
eig      <- dFactors$Raiche$eigenvalues

results  <- nBartlett(x=eig, N= 100, alpha=0.05, details=TRUE)
results

plotuScree(eig, main=paste(results$nFactors[1], ", ",
                           results$nFactors[2], " or ",
                           results$nFactors[3],
                           " factors retained by the LRT procedures",
                           sep=""))
 }

## End(Not run)

Bentler and Yuan's Procedure to Determine the Number of Components/Factors

Description

This function computes the Bentler and Yuan's indices for determining the number of components/factors to retain.

Usage

nBentler(
  x,
  N,
  log = TRUE,
  alpha = 0.05,
  cor = TRUE,
  details = TRUE,
  minPar = c(min(lambda) - abs(min(lambda)) + 0.001, 0.001),
  maxPar = c(max(lambda), stats::lm(lambda ~ I(length(lambda):1))$coef[2]),
  ...
)

Arguments

Details

The implemented Bentler and Yuan's procedure must be used with care because the minimized function is not always stable, as Bentler and Yan (1996, 1998) already noted. In many cases, constraints must applied to obtain a solution, as the actual implementation did, but the user can modify these constraints.

The hypothesis tested (Bentler and Yuan, 1996, equation 10) is:

(1) \qquad \qquad H_k: \lambda_{k+i} = \alpha + \beta x_i, (i = 1, \ldots, q)

The solution of the following simultaneous equations is needed to find (\alpha, \beta) \in

(2) \qquad \qquad f(x) = \sum_{i=1}^q \frac{ [ \lambda_{k+j} - N \alpha + \beta x_j ] x_j}{(\alpha + \beta x_j)^2} = 0

and \qquad \qquad g(x) = \sum_{i=1}^q \frac{ \lambda_{k+j} - N \alpha + \beta x_j x_j}{(\alpha + \beta x_j)^2} = 0

The solution to this system of equations was implemented by minimizing the following equation:

(3) \qquad \qquad (\alpha, \beta) \in \inf{[h(x)]} = \inf{\log{[f(x)^2 + g(x)^2}}]

The likelihood ratio test LRT proposed by Bentler and Yuan (1996, equation 7) follows a \chi^2 probability distribution with q-2 degrees of freedom and is equal to:

(4) \qquad \qquad LRT = N(k - p)\left\{ {\ln \left( {{n \over N}} \right) + 1} \right\} - N\sum\limits_{j = k + 1}^p {\ln \left\{ {{{\lambda _j } \over {\alpha + \beta x_j }}} \right\}} + n\sum\limits_{j = k + 1}^p {\left\{ {{{\lambda _j } \over {\alpha + \beta x_j }}} \right\}}

With p beeing the number of eigenvalues, k the number of eigenvalues to test, q the p-k remaining eigenvalues, N the sample size, and n = N-1. Note that there is an error in the Bentler and Yuan equation, the variables N and n beeing inverted in the preceeding equation 4.

A better strategy proposed by Bentler an Yuan (1998) is to used a minimized \chi^2 solution. This strategy will be implemented in a future version of the nFactors package.

Value

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

References

Bentler, P. M. and Yuan, K.-H. (1996). Test of linear trend in eigenvalues of a covariance matrix with application to data analysis. British Journal of Mathematical and Statistical Psychology, 49, 299-312.

Bentler, P. M. and Yuan, K.-H. (1998). Test of linear trend in the smallest eigenvalues of the correlation matrix. Psychometrika, 63(2), 131-144.

See Also

nBartlett, bentlerParameters

Examples

## Not run: 
if(interactive()){
## ................................................
## SIMPLE EXAMPLE OF THE BENTLER AND YUAN PROCEDURE

# Bentler (1996, p. 309) Table 2 - Example 2 .............
n=649
bentler2<-c(5.785, 3.088, 1.505, 0.582, 0.424, 0.386, 0.360, 0.337, 0.303,
            0.281, 0.246, 0.238, 0.200, 0.160, 0.130)

results  <- nBentler(x=bentler2, N=n)
results

plotuScree(x=bentler2, model="components",
    main=paste(results$nFactors,
    " factors retained by the Bentler and Yuan's procedure (1996, p. 309)",
    sep=""))
# ........................................................

# Bentler (1998, p. 140) Table 3 - Example 1 .............
n        <- 145
example1 <- c(8.135, 2.096, 1.693, 1.502, 1.025, 0.943, 0.901, 0.816, 0.790,
              0.707, 0.639, 0.543,
              0.533, 0.509, 0.478, 0.390, 0.382, 0.340, 0.334, 0.316, 0.297,
              0.268, 0.190, 0.173)

results  <- nBentler(x=example1, N=n)
results

plotuScree(x=example1, model="components",
   main=paste(results$nFactors,
   " factors retained by the Bentler and Yuan's procedure (1998, p. 140)",
   sep=""))
# ........................................................
 }

## End(Not run)

Cattell-Nelson-Gorsuch CNG Indices

Description

This function computes the CNG indices for the eigenvalues of a correlation/covariance matrix (Gorsuch and Nelson, 1981; Nasser, 2002, p. 400; Zoski and Jurs, 1993, p. 6).

Usage

nCng(x, cor = TRUE, model = "components", details = TRUE, ...)

Arguments

Details

Note that the nCng function is only valid when more than six eigenvalues are used and that these are obtained in the context of a principal component analysis. For a factor analysis, some eigenvalues could be negative and the function will stop and give an error message.

The slope of all possible sets of three adjacent eigenvalues are compared, so CNG indices can be applied only when more than six eigenvalues are used. The eigenvalue at which the greatest difference between two successive slopes occurs is the indicator of the number of components/factors to retain.

Value

Author(s)

Gilles Raiche
Centre sur les Applications des Modeles de Reponses aux Items (CAMRI)
Universite du Quebec a Montreal
raiche.gilles@uqam.ca

References

Gorsuch, R. L. and Nelson, J. (1981). CNG scree test: an objective procedure for determining the number of factors. Presented at the annual meeting of the Society for multivariate experimental psychology.

Nasser, F. (2002). The performance of regression-based variations of the visual scree for determining the number of common factors. Educational and Psychological Measurement, 62(3), 397-419.

Zoski, K. and Jurs, S. (1993). Using multiple regression to determine the number of factors to retain in factor analysis. Multiple Linear Regression Viewpoints, 20(1), 5-9.

See Also

plotuScree, nScree, plotnScree, plotParallel

Examples

## Not run: 
if(interactive()){
## SIMPLE EXAMPLE OF A CNG ANALYSIS

 data(dFactors)
 eig      <- dFactors$Raiche$eigenvalues

 results  <- nCng(eig, details=TRUE)
 results

 plotuScree(eig, main=paste(results$nFactors,
                            " factors retained by the CNG procedure",
                            sep=""))
 }

## End(Not run)

Multiple Regression Procedure to Determine the Number of Components/Factors

Description

This function computes the \beta indices, like their associated Student t and probability (Zoski and Jurs, 1993, 1996, p. 445). These three values can be used as three different indices for determining the number of components/factors to retain.

Usage

nMreg(x, cor = TRUE, model = "components", details = TRUE, ...)

Arguments

Details

When the associated Student t test is applied, the following hypothesis is considered:

(1) \qquad \qquad H_k: \beta (\lambda_1 \ldots \lambda_k) - \beta (\lambda_{k+1} \ldots \lambda_p), (k = 3, \ldots, p-3) = 0

Value

Author(s)

Gilles Raiche
Centre sur les Applications des Modeles de Reponses aux Items (CAMRI)
Universite du Quebec a Montreal
raiche.gilles@uqam.ca

References

Zoski, K. and Jurs, S. (1993). Using multiple regression to determine the number of factors to retain in factor analysis. Multiple Linear Regression Viewpoints, 20(1), 5-9.

Zoski, K. and Jurs, S. (1996). An objective counterpart to the visual scree test for factor analysis: the standard error scree test. Educational and Psychological Measurement, 56(3), 443-451.

See Also

plotuScree, nScree, plotnScree, plotParallel

Examples

## Not run: 
if(interactive()){
## SIMPLE EXAMPLE OF A MREG ANALYSIS

 data(dFactors)
 eig      <- dFactors$Raiche$eigenvalues

 results  <- nMreg(eig)
 results

 plotuScree(eig, main=paste(results$nFactors[1], ", ",
                            results$nFactors[2], " or ",
                            results$nFactors[3],
                            " factors retained by the MREG procedures",
                            sep=""))
 }

## End(Not run)

Non Graphical Cattel's Scree Test

Description

The nScree function returns an analysis of the number of component or factors to retain in an exploratory principal component or factor analysis. The function also returns information about the number of components/factors to retain with the Kaiser rule and the parallel analysis.

Usage

nScree(
  eig = NULL,
  x = eig,
  aparallel = NULL,
  cor = TRUE,
  model = "components",
  criteria = NULL,
  ...
)

Arguments

Details

The nScree function returns an analysis of the number of components/factors to retain in an exploratory principal component or factor analysis. Different solutions are given. The classical ones are the Kaiser rule, the parallel analysis, and the usual scree test (plotuScree). Non graphical solutions to the Cattell subjective scree test are also proposed: an acceleration factor (af) and the optimal coordinates index oc. The acceleration factor indicates where the elbow of the scree plot appears. It corresponds to the acceleration of the curve, i.e. the second derivative. The optimal coordinates are the extrapolated coordinates of the previous eigenvalue that allow the observed eigenvalue to go beyond this extrapolation. The extrapolation is made by a linear regression using the last eigenvalue coordinates and the k+1 eigenvalue coordinates. There are k-2 regression lines like this. The Kaiser rule or a parallel analysis criterion (parallel) must also be simultaneously satisfied to retain the components/factors, whether for the acceleration factor, or for the optimal coordinates.

If \lambda_i is the i^{th} eigenvalue, and LS_i is a location statistics like the mean or a centile (generally the followings: 1^{st}, \ 5^{th}, \ 95^{th}, \ or \ 99^{th}).

The Kaiser rule is computed as:

n_{Kaiser} = \sum_{i} (\lambda_{i} \ge \bar{\lambda}).

Note that \bar{\lambda} is equal to 1 when a correlation matrix is used.

The parallel analysis is computed as:

n_{parallel} = \sum_{i} (\lambda_{i} \ge LS_i).

The acceleration factor (AF) corresponds to a numerical solution to the elbow of the scree plot:

n_{AF} \equiv \ If \ \left[ (\lambda_{i} \ge LS_i) \ and \ max(AF_i) \right].

The optimal coordinates (OC) corresponds to an extrapolation of the preceeding eigenvalue by a regression line between the eigenvalue coordinates and the last eigenvalue coordinates:

n_{OC} = \sum_i \left[(\lambda_i \ge LS_i) \cap (\lambda_i \ge (\lambda_{i \ predicted}) \right].

Value

Otherwise, returns a summary of the analysis.

Author(s)

Gilles Raiche
Centre sur les Applications des Modeles de Reponses aux Items (CAMRI)
Universite du Quebec a Montreal
raiche.gilles@uqam.ca

References

Cattell, R. B. (1966). The scree test for the number of factors. Multivariate Behavioral Research, 1, 245-276.

Dinno, A. (2009). Gently clarifying the application of Horn's parallel analysis to principal component analysis versus factor analysis. Portland, Oregon: Portland Sate University.

Guttman, L. (1954). Some necessary conditions for common factor analysis. Psychometrika, 19, 149-162.

Horn, J. L. (1965). A rationale for the number of factors in factor analysis. Psychometrika, 30, 179-185.

Kaiser, H. F. (1960). The application of electronic computer to factor analysis. Educational and Psychological Measurement, 20, 141-151.

Raiche, G., Walls, T. A., Magis, D., Riopel, M. and Blais, J.-G. (2013). Non-graphical solutions for Cattell's scree test. Methodology, 9(1), 23-29.

See Also

plotuScree, plotnScree, parallel, plotParallel,

Examples

## Not run: 
if(interactive()){
## INITIALISATION
 data(dFactors)                      # Load the nFactors dataset
 attach(dFactors)
 vect         <- Raiche              # Uses the example from Raiche
 eigenvalues  <- vect$eigenvalues    # Extracts the observed eigenvalues
 nsubjects    <- vect$nsubjects      # Extracts the number of subjects
 variables    <- length(eigenvalues) # Computes the number of variables
 rep          <- 100                 # Number of replications for PA analysis
 cent         <- 0.95                # Centile value of PA analysis

## PARALLEL ANALYSIS (qevpea for the centile criterion, mevpea for the
## mean criterion)
 aparallel    <- parallel(var     = variables,
                          subject = nsubjects,
                          rep     = rep,
                          cent    = cent
                          )$eigen$qevpea  # The 95 centile

## NUMBER OF FACTORS RETAINED ACCORDING TO DIFFERENT RULES
 results      <- nScree(x=eigenvalues, aparallel=aparallel)
 results
 summary(results)

## PLOT ACCORDING TO THE nScree CLASS
 plotnScree(results)
 }

## End(Not run)

Standard Error Scree and Coefficient of Determination Procedures to Determine the Number of Components/Factors

Description

This function computes the seScree (S_{Y \bullet X}) indices (Zoski and Jurs, 1996) and the coefficient of determination indices of Nelson (2005) R^2 for determining the number of components/factors to retain.

Usage

nSeScree(
  x,
  cor = TRUE,
  model = "components",
  details = TRUE,
  r2limen = 0.75,
  ...
)

Arguments

Details

The Zoski and Jurs S_{Y \bullet X} index is the standard error of the estimate (predicted) eigenvalues by the regression from the (k+1, \ldots, p) subsequent ranks of the eigenvalues. The standard error is computed as:

(1) \qquad \qquad S_{Y \bullet X} = \sqrt{ \frac{(\lambda_k - \hat{\lambda}_k)^2} {p-2} }

A value of 1/p is choosen as the criteria to determine the number of components or factors to retain, p corresponding to the number of variables.

The Nelson R^2 index is simply the multiple regresion coefficient of determination for the k+1, \ldots, p eigenvalues. Note that Nelson didn't give formal prescriptions for the criteria for this index. He only suggested that a value of 0.75 or more must be considered. More is to be done to explore adequate values.

Value

Author(s)

Gilles Raiche
Centre sur les Applications des Modeles de Reponses aux Items (CAMRI)
Universite du Quebec a Montreal
raiche.gilles@uqam.ca

References

Nasser, F. (2002). The performance of regression-based variations of the visual scree for determining the number of common factors. Educational and Psychological Measurement, 62(3), 397-419.

Nelson, L. R. (2005). Some observations on the scree test, and on coefficient alpha. Thai Journal of Educational Research and Measurement, 3(1), 1-17.

Raiche, G., Walls, T. A., Magis, D., Riopel, M. and Blais, J.-G. (2013). Non-graphical solutions for Cattell's scree test. Methodology, 9(1), 23-29.

Zoski, K. and Jurs, S. (1993). Using multiple regression to determine the number of factors to retain in factor analysis. Multiple Linear Regression Viewpoints, 20(1), 5-9.

Zoski, K. and Jurs, S. (1996). An objective counterpart to the visuel scree test for factor analysis: the standard error scree. Educational and Psychological Measurement, 56(3), 443-451.

See Also

plotuScree, nScree, plotnScree, plotParallel

Examples

## Not run: 
if(interactive()){
## SIMPLE EXAMPLE OF SESCREE AND R2 ANALYSIS

 data(dFactors)
 eig      <- dFactors$Raiche$eigenvalues

 results  <- nSeScree(eig)
 results

 plotuScree(eig, main=paste(results$nFactors[1], " or ", results$nFactors[2],
                            " factors retained by the sescree and R2 procedures",
                            sep=""))
 }

## End(Not run)

Parallel Analysis of a Correlation or Covariance Matrix

Description

This function gives the distribution of the eigenvalues of correlation or a covariance matrices of random uncorrelated standardized normal variables. The mean and a selected quantile of this distribution are returned.

Usage

parallel(
  subject = 100,
  var = 10,
  rep = 100,
  cent = 0.05,
  quantile = cent,
  model = "components",
  sd = diag(1, var),
  ...
)

Arguments

Details

Note that if the decision is based on a quantile value rather than on the mean, care must be taken with the number of replications (rep). In fact, the smaller the quantile (cent), the bigger the number of necessary replications.

Value

Otherwise, returns a summary of the parallel analysis.

Author(s)

Gilles Raiche
Centre sur les Applications des Modeles de Reponses aux Items (CAMRI)
Universite du Quebec a Montreal
raiche.gilles@uqam.ca

References

Drasgow, F. and Lissak, R. (1983) Modified parallel analysis: a procedure for examining the latent dimensionality of dichotomously scored item responses. Journal of Applied Psychology, 68(3), 363-373.

Hoyle, R. H. and Duvall, J. L. (2004). Determining the number of factors in exploratory and confirmatory factor analysis. In D. Kaplan (Ed.): The Sage handbook of quantitative methodology for the social sciences. Thousand Oaks, CA: Sage.

Horn, J. L. (1965). A rationale and test of the number of factors in factor analysis. Psychometrika, 30, 179-185.

See Also

plotuScree, nScree, plotnScree, plotParallel

Examples

## Not run: 
if(interactive()){
## SIMPLE EXAMPLE OF A PARALLEL ANALYSIS
## OF A CORRELATION MATRIX WITH ITS PLOT
 data(dFactors)
 eig      <- dFactors$Raiche$eigenvalues
 subject  <- dFactors$Raiche$nsubjects
 var      <- length(eig)
 rep      <- 100
 quantile <- 0.95
 results  <- parallel(subject, var, rep, quantile)

 results

## IF THE DECISION IS BASED ON THE CENTILE USE qevpea INSTEAD
## OF mevpea ON THE FIRST LINE OF THE FOLLOWING CALL
 plotuScree(x    = eig,
            main = "Parallel Analysis"
            )

 lines(1:var,
       results$eigen$qevpea,
       type="b",
       col="green"
       )


## ANOTHER SOLUTION IS SIMPLY TO
 plotParallel(results)
 }

## End(Not run)

Plot a Parallel Analysis Class Object

Description

Plot a scree plot adding information about a parallel analysis.

Usage

plotParallel(
  parallel,
  eig = NA,
  x = eig,
  model = "components",
  legend = TRUE,
  ylab = "Eigenvalues",
  xlab = "Components",
  main = "Parallel Analysis",
  ...
)

Arguments

Details

If eig is FALSE the plot shows only the parallel analysis without eigenvalues.

Value

Nothing returned.

Author(s)

Gilles Raiche
Centre sur les Applications des Modeles de Reponses aux Items (CAMRI)
Universite du Quebec a Montreal
raiche.gilles@uqam.ca

References

Raiche, G., Walls, T. A., Magis, D., Riopel, M. and Blais, J.-G. (2013). Non-graphical solutions for Cattell's scree test. Methodology, 9(1), 23-29.

See Also

plotuScree, nScree, plotnScree, parallel

Examples

## Not run: 
if(interactive()){
## SIMPLE EXAMPLE OF A PARALLEL ANALYSIS
## OF A CORRELATION MATRIX WITH ITS PLOT
 data(dFactors)
 eig      <- dFactors$Raiche$eigenvalues
 subject  <- dFactors$Raiche$nsubjects
 var      <- length(eig)
 rep      <- 100
 cent     <- 0.95
 results  <- parallel(subject,var,rep,cent)

 results


## PARALLEL ANALYSIS SCREE PLOT
 plotParallel(results, x=eig)
 plotParallel(results)
 }

## End(Not run)

Scree Plot According to a nScree Object Class

Description

Plot a scree plot adding information about a non graphical nScree analysis.

Usage

plotnScree(
  nScree,
  legend = TRUE,
  ylab = "Eigenvalues",
  xlab = "Components",
  main = "Non Graphical Solutions to Scree Test"
)

Arguments

Value

Nothing returned.

Author(s)

Gilles Raiche
Centre sur les Applications des Modeles de Reponses aux Items (CAMRI)
Universite du Quebec a Montreal
raiche.gilles@uqam.ca

References

Raiche, G., Walls, T. A., Magis, D., Riopel, M. and Blais, J.-G. (2013). Non-graphical solutions for Cattell's scree test. Methodology, 9(1), 23-29.

See Also

plotuScree, nScree, plotParallel, parallel

Examples

## Not run: 
if(interactive()){
## INITIALISATION
 data(dFactors)                      # Load the nFactors dataset
 attach(dFactors)
 vect         <- Raiche              # Use the second example from Buja and Eyuboglu
                                     # (1992, p. 519, nsubjects not specified by them)
 eigenvalues  <- vect$eigenvalues    # Extract the observed eigenvalues
 nsubjects    <- vect$nsubjects      # Extract the number of subjects
 variables    <- length(eigenvalues) # Compute the number of variables
 rep          <- 100                 # Number of replications for the parallel analysis
 cent         <- 0.95                # Centile value of the parallel analysis

## PARALLEL ANALYSIS (qevpea for the centile criterion, mevpea for the mean criterion)
 aparallel    <- parallel(var     = variables,
                          subject = nsubjects,
                          rep     = rep,
                          cent    = cent)$eigen$qevpea  # The 95 centile

## NOMBER OF FACTORS RETAINED ACCORDING TO DIFFERENT RULES
 results <- nScree(eig       = eigenvalues,
                   aparallel = aparallel
                   )

 results

## PLOT ACCORDING TO THE nScree CLASS
 plotnScree(results)
 }

## End(Not run)

Plot of the Usual Cattell's Scree Test

Description

uScree plot a usual scree test of the eigenvalues of a correlation matrix.

Usage

plotuScree(
  Eigenvalue,
  x = Eigenvalue,
  model = "components",
  ylab = "Eigenvalues",
  xlab = "Components",
  main = "Scree Plot",
  ...
)

Arguments

Value

Nothing returned with this function.

Author(s)

Gilles Raiche
Centre sur les Applications des Modeles de Reponses aux Items (CAMRI)
Universite du Quebec a Montreal
raiche.gilles@uqam.ca

References

Cattell, R. B. (1966). The scree test for the number of factors. Multivariate Behavioral Research, 1, 245-276.

See Also

nScree, parallel

Examples

## Not run: 
if(interactive()){
## SCREE PLOT
 data(dFactors)
 attach(dFactors)
 eig = Cliff1$eigenvalues
 plotuScree(x=eig)
 }

## End(Not run)

Principal Axis Analysis

Description

The PrincipalAxis function returns 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 squared 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 instead, then a solution can be computed.

Usage

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

Arguments

Value

Author(s)

Gilles Raiche
Centre sur les Applications des Modeles de Reponses aux Items (CAMRI)
Universite du Quebec a Montreal
raiche.gilles@uqam.ca

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, iterativePrincipalAxis, rRecovery

Examples

## Not run: 
if(interactive()){
# .......................................................
# 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 with 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 with 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")
# .......................................................
 }

## End(Not run)

Principal Component Analysis

Description

The principalComponents function returns a principal component analysis. Other R functions give the same results, but principalComponents is customized mainly for the other factor analysis functions available in the nfactors package. In order to retain only a small number of components the componentAxis function has to be used.

Usage

principalComponents(R)

Arguments

Value

Author(s)

Gilles Raiche
Centre sur les Applications des Modeles de Reponses aux Items (CAMRI)
Universite du Quebec a Montreal
raiche.gilles@uqam.ca

References

Joliffe, I. T. (2002). Principal components analysis (2th Edition). New York, NJ: Springer-Verlag.

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, iterativePrincipalAxis, rRecovery

Examples

## Not run: 
if(interactive()){
# .......................................................
# 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 component -
# Kim et Mueller (1978, p. 21)
# Replace upper diagonal with lower diagonal
 RU <- diagReplace(R, upper=TRUE)
 principalComponents(RU)

# Replace lower diagonal with upper diagonal
 RL <- diagReplace(R, upper=FALSE)
 principalComponents(RL)
# .......................................................
 }

## End(Not run)

Test of Recovery of a Correlation or a Covariance matrix from a Factor Analysis Solution

Description

The rRecovery function returns a verification of the quality of the recovery of the initial correlation or covariance matrix by the factor solution.

Usage

rRecovery(R, loadings, diagCommunalities = FALSE)

Arguments

Value

Author(s)

Gilles Raiche
Centre sur les Applications des Modeles de Reponses aux Items (CAMRI)
Universite du Quebec a Montreal
raiche.gilles@uqam.ca

See Also

componentAxis, iterativePrincipalAxis, principalAxis

Examples

## Not run: 
if(interactive()){
# .......................................................
# 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)


# Replace upper diagonal with lower diagonal
 RU         <- diagReplace(R, upper=TRUE)
 nFactors   <- 2
 loadings   <- principalAxis(RU, nFactors=nFactors,
                             communalities="component")$loadings
 rComponent <- rRecovery(RU,loadings, diagCommunalities=FALSE)$cor

 loadings   <- principalAxis(RU, nFactors=nFactors,
                             communalities="maxr")$loadings
 rMaxr      <- rRecovery(RU,loadings, diagCommunalities=FALSE)$cor

 loadings   <- principalAxis(RU, nFactors=nFactors,
                             communalities="multiple")$loadings
 rMultiple  <- rRecovery(RU,loadings, diagCommunalities=FALSE)$cor

 round(c(rComponent = rComponent,
         rmaxr      = rMaxr,
         rMultiple  = rMultiple), 3)
# .......................................................

 }

## End(Not run)

Population or Simulated Sample Correlation Matrix from a Given Factor Structure Matrix

Description

The structureSim function returns a population and a sample correlation matrices from a predefined congeneric factor structure.

Usage

structureSim(
  fload,
  reppar = 30,
  repsim = 100,
  N,
  quantile = 0.95,
  model = "components",
  adequacy = FALSE,
  details = TRUE,
  r2limen = 0.75,
  all = FALSE
)

Arguments

Value

Author(s)

Gilles Raiche
Centre sur les Applications des Modeles de Reponses aux Items (CAMRI)
Universite du Quebec a Montreal
raiche.gilles@uqam.ca

References

Raiche, G., Walls, T. A., Magis, D., Riopel, M. and Blais, J.-G. (2013). Non-graphical solutions for Cattell's scree test. Methodology, 9(1), 23-29.

Zwick, W. R. and Velicer, W. F. (1986). Comparison of five rules for determining the number of components to retain. Psychological Bulletin, 99, 432-442.

See Also

principalComponents, iterativePrincipalAxis, rRecovery

Examples

## Not run: 
if(interactive()){
# .......................................................
# Example inspired from Zwick and Velicer (1986, table 2, p. 437)
## ...................................................................
 nFactors  <- 3
 unique    <- 0.2
 loadings  <- 0.5
 nsubjects <- 180
 repsim    <- 30
 zwick     <- generateStructure(var=36, mjc=nFactors, pmjc=12,
                                loadings=loadings,
                                unique=unique)
## ...................................................................

# Produce statistics about a replication of a parallel analysis on
# 30 sampled correlation matrices

 mzwick.fa <-  structureSim(fload=as.matrix(zwick), reppar=30,
                            repsim=repsim, N=nsubjects, quantile=0.5,
                            model="factors")

 mzwick    <-  structureSim(fload=as.matrix(zwick), reppar=30,
                            repsim=repsim, N=nsubjects, quantile=0.5, all=TRUE)

# Very long execution time that could be used only with model="components"
# mzwick    <-  structureSim(fload=as.matrix(zwick), reppar=30,
#                            repsim=repsim, N=nsubjects, quantile=0.5, all=TRUE)

 par(mfrow=c(2,1))
 graphics::plot(x=mzwick,    nFactors=nFactors, index=c(1:14), cex.axis=0.7, col="red")
 graphics::plot(x=mzwick.fa, nFactors=nFactors, index=c(1:11), cex.axis=0.7, col="red")
 par(mfrow=c(1,1))

 par(mfrow=c(2,1))
 graphics::boxplot(x=mzwick,    nFactors=3, cex.axis=0.8, vLine="blue", col="red")
 graphics::boxplot(x=mzwick.fa, nFactors=3, cex.axis=0.8, vLine="blue", col="red",
         xlab="Components")
 par(mfrow=c(1,1))
# ......................................................
 }

## End(Not run)

Simulation Study from Given Factor Structure Matrices and Conditions

Description

The structureSim function returns statistical results from simulations from predefined congeneric factor structures. The main ideas come from the methodology applied by Zwick and Velicer (1986).

Usage

studySim(
  var,
  nFactors,
  pmjc,
  loadings,
  unique,
  N,
  repsim,
  reppar,
  stats = 1,
  quantile = 0.5,
  model = "components",
  r2limen = 0.75,
  all = FALSE,
  dir = NA,
  trace = TRUE
)

Arguments

Value

Author(s)

Gilles Raiche
Centre sur les Applications des Modeles de Reponses aux Items (CAMRI)
Universite du Quebec a Montreal
raiche.gilles@uqam.ca

References

Raiche, G., Walls, T. A., Magis, D., Riopel, M. and Blais, J.-G. (2013). Non-graphical solutions for Cattell's scree test. Methodology, 9(1), 23-29.

Zwick, W. R. and Velicer, W. F. (1986). Comparison of five rules for determining the number of components to retain. Psychological Bulletin, 99, 432-442.

See Also

generateStructure, structureSim

Examples

## Not run: 
# ....................................................................
# Example inspired from Zwick and Velicer (1986)
# Very long computimg time
# ...................................................................

# 1. Initialisation
# reppar    <- 30
# repsim    <- 5
# quantile  <- 0.50

# 2. Simulations
# X         <- studySim(var=36,nFactors=3, pmjc=c(6,12), loadings=c(0.5,0.8),
#                       unique=c(0,0.2), quantile=quantile,
#                       N=c(72,180), repsim=repsim, reppar=reppar,
#                       stats=c(1:6))

# 3. Results (first 10 results)
# print(X[1:10,1:14],2)
# names(X)

# 4. Study of the error done in the determination of the number
#    of components/factors. A positive value is associated to over
#    determination.
# results   <- X[X$stats=="mean",]
# residuals <- results[,c(11:25)] - X$nfactors
# BY        <- c("nsubjects","var","loadings")
# round(aggregate(residuals, by=results[BY], mean),0)
 
## End(Not run)

Utility Functions for nScree Class Objects

Description

Utility functions for nScree class objects. Some of these functions are already implemented in the nFactors package, but are easier to use with generic functions like these.

Usage

## S3 method for class 'nScree'
summary(object, ...)

## S3 method for class 'nScree'
print(x, ...)

## S3 method for class 'nScree'
plot(x, ...)

is.nScree(object)

Arguments

Value

Generic functions for the nScree class:

Author(s)

Gilles Raiche
Centre sur les Applications des Modeles de Reponses aux Items (CAMRI)
Universite du Quebec a Montreal
raiche.gilles@uqam.ca

References

Raiche, G., Walls, T. A., Magis, D., Riopel, M. and Blais, J.-G. (2013). Non-graphical solutions for Cattell's scree test. Methodology, 9(1), 23-29.

Examples

## Not run: 
if(interactive()){
## INITIALISATION
 data(dFactors)                      # Load the nFactors dataset
 attach(dFactors)
 vect         <- Raiche              # Use the example from Raiche
 eigenvalues  <- vect$eigenvalues    # Extract the observed eigenvalues
 nsubjects    <- vect$nsubjects      # Extract the number of subjects
 variables    <- length(eigenvalues) # Compute the number of variables
 rep          <- 100                 # Number of replications for the parallel analysis
 cent         <- 0.95                # Centile value of the parallel analysis

## PARALLEL ANALYSIS (qevpea for the centile criterion, mevpea for the mean criterion)
 aparallel    <- parallel(var     = variables,
                          subject = nsubjects,
                          rep     = rep,
                          cent    = cent
                          )$eigen$qevpea  # The 95 centile

## NOMBER OF FACTORS RETAINED ACCORDING TO DIFFERENT RULES
 results      <- nScree(x=eigenvalues, aparallel=aparallel)

 is.nScree(results)
 results
 summary(results)

## PLOT ACCORDING TO THE nScree CLASS
 plot(results)
 }

## End(Not run)

Utility Functions for nScree Class Objects

Description

Utility functions for structureSim class objects. Note that with the plot.structureSim a dotted black vertical line shows the median number of factors retained by all the different indices.

Usage

## S3 method for class 'structureSim'
summary(object, index = c(1:15), eigenSelect = NULL, ...)

## S3 method for class 'structureSim'
print(x, index = NULL, ...)

## S3 method for class 'structureSim'
boxplot(
  x,
  nFactors = NULL,
  eigenSelect = NULL,
  vLine = "green",
  xlab = "Factors",
  ylab = "Eigenvalues",
  main = "Eigen Box Plot",
  ...
)

## S3 method for class 'structureSim'
plot(x, nFactors = NULL, index = NULL, main = "Index Acuracy Plot", ...)

is.structureSim(object)

Arguments

Value

Generic functions for the structureSim class:

Author(s)

Gilles Raiche
Centre sur les Applications des Modeles de Reponses aux Items (CAMRI)
Universite du Quebec a Montreal
raiche.gilles@uqam.ca

References

Raiche, G., Walls, T. A., Magis, D., Riopel, M. and Blais, J.-G. (2013). Non-graphical solutions for Cattell's scree test. Methodology, 9(1), 23-29.

See Also

nFactors-package

Examples

## Not run: 
if(interactive()){
## INITIALISATION
 library(xtable)
 library(nFactors)
 nFactors  <- 3
 unique    <- 0.2
 loadings  <- 0.5
 nsubjects <- 180
 repsim    <- 10
 var       <- 36
 pmjc      <- 12
 reppar    <- 10
 zwick     <- generateStructure(var=var, mjc=nFactors, pmjc=pmjc,
                                loadings=loadings,
                                unique=unique)

## SIMULATIONS
mzwick    <-  structureSim(fload=as.matrix(zwick), reppar=reppar,
                           repsim=repsim, details=TRUE,
                           N=nsubjects, quantile=0.5)

## TEST OF structureSim METHODS
 is(mzwick)
 summary(mzwick, index=1:5, eigenSelect=1:10, digits=3)
 print(mzwick, index=1:10)
 plot(x=mzwick, index=c(1:10), cex.axis=0.7, col="red")
 graphics::boxplot(x=mzwick, nFactors=3, vLine="blue", col="red")
  }
 
## End(Not run)