nScree {nFactors} | R Documentation |
The nScree
function returns an analysis of the number of components or
factors to retain in an exploratory
principal components or factor analysis. The function also return
informations about the number of components/factors to retain with the Kaiser
rule and the parallel analysis.
nScree(eig=NULL, x=eig, aparallel=NULL, cor=TRUE, model="components", criteria=NULL, ...)
eig |
depreciated parameter (use x instead): eigenvalues to analyse |
x |
numeric: a vector of eigenvalues, a matrix of
correlations or of covariances or a data.frame of data |
aparallel |
numeric: results of a parallel analysis. Defaults eigenvalues fixed at λ >= bar{λ} (Kaiser and related rule) or λ >= 0 (CFA analysis) |
cor |
logical: if TRUE computes eigenvalues from a correlation
matrix, else from a covariance matrix |
model |
character: "components" or "factors" |
criteria |
numeric: by default fixed at hat{λ}. When the λs are computed prom a principal components analysis on a correlation matrix, it corresponds to the usual Kaiser λ >= 1 rule. On a covariance matrix or from a factor analysis, it is simply the mean. To apply the λ >= 0 sometimes used with factor analysis, fixed the criteria to 0. |
... |
variabe: additionnal parameters to give to the cor or
cov functions |
The nScree
function returns an analysis of the number of components/factors to retain in an exploratory
principal components 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 let the observed eigenvalue
be over 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. Would it be fot the
acceleration factor, or for the optimal coordinates, the Kaiser rule or a parallel analysis criterion (parallel
)
must also be simultaneously satisfied to retain the components/factors.
If λ_i is the i^{th} eigenvalue, and LS_i is a location statistics like the mean or a centile (generally the following: 1^{st}, 5^{th}, 95^{th}, or 99^{th}).
The Kaiser rule is computed as:
n_{Kaiser} = sum_{i} (λ_{i} >= bar{λ}).
Note that bar{λ} is equal to 1 when a correlation matrix is used.
The parallel analysis is computed as:
n_{parallel} = sum_{i} (λ_{i} >= LS_i).
The acceleration factor (AF) corresponds to a numeral solution to the elbow of the scree plot:
n_{AF} equiv If <=ft[ (λ_{i} >= LS_i) and max(AF_i) right].
The optimal coordinates (OC) corresponds to an extrapolation of the preceding eigenvalue by a regression line between the eignvalue coordinates and the last eigenvalue coordinate:
n_{OC} = sum_i <=ft[(λ_i >= LS_i) cap (λ_i >= (λ_{i predicted}) right].
Components |
Data frame for the number of components/factors according to different rules |
Components$noc |
Number of components/factors to retain according to optimal coordinates oc |
Components$naf |
Number of components/factors to retain according to the acceleration factor af |
Components$npar.analysis |
Number of components/factors to retain according to parallel analysis |
Components$nkaiser |
Number of components/factors to retain according to the Kaiser rule |
Analysis |
Data frame of vectors linked to the different rules |
Analysis$Eigenvalues |
Eigenvalues |
Analysis$Prop |
Proportion of variance accounted by eigenvalues |
Analysis$Cumu |
Cumulative proportion of variance accounted by eigenvalues |
Analysis$Par.Analysis |
Centiles of the random eigenvalues generated by the parallel analysis. |
Analysis$Pred.eig |
Predicted eigenvalues by each optimal coordinate regression line |
Analysis$OC |
Critical optimal coordinates oc |
Analysis$Acc.factor |
Acceleration factor af |
Analysis$AF |
Critical acceleration factor af |
Otherwise, returns a summary of the analysis.
Gilles Raiche
Centre sur les Applications des Modeles de Reponses aux Items (CAMRI)
Universite du Quebec a Montreal
raiche.gilles@uqam.ca, http://www.er.uqam.ca/nobel/r17165/
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
[http://doyenne.com/Software/files/PA_for_PCA_vs_FA.pdf]
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., Riopel, M. and Blais, J.-G. (2006). Non graphical solutions for the Cattell's scree test. Paper presented at the International Annual meeting of the Psychometric Society, Montreal. [http://www.er.uqam.ca/nobel/r17165/RECHERCHE/COMMUNICATIONS/]
plotuScree
,
plotnScree
,
parallel
,
plotParallel
,
## 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 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)