nFactors (version 2.4.1.1)

nSeScree: 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, ...)

Value

nFactors

numeric: number of components/factors retained by the seScree procedure.

details

numeric: matrix of the details for each index.

Arguments

x

numeric: eigenvalues.

cor

logical: if TRUE computes eigenvalues from a correlation matrix, else from a covariance matrix

model

character: "components" or "factors"

details

logical: if TRUE also returns details about the computation for each eigenvalue.

r2limen

numeric: criterion value retained for the coefficient of determination indices.

...

variable: additionnal parameters to give to the eigenComputes and cor or cov functions

Author

Gilles Raiche
Centre sur les Applications des Modeles de Reponses aux Items (CAMRI)
Universite du Quebec a Montreal

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.

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

## 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=""))