nScree: NON GRAPHICAL CATTELL SCREE TEST

Description

The nScree function returns an analysis of the number of factors to retain in an exploratory principal components analysis. The function also return informations about the number of factors to retain with the Kaiser-Guttman rule and the parallel analysis.

Usage

nScree(eig, 
       aparallel = rep(1, length(eig))
       )

Arguments

eig

Eigenvalues to analyse

aparallel

Results of a parallel analysis (default eigenvalues fixed at 1.00 - the Kaiser-Guttman rule)

Value

ComponentsData frame for the number of components according to different rules
Components$nocNumber of factors to retain according to optimal coordinates
Components$nafNumber of factors to retain according to the acceleration factor
Components$npar.analysisNumber of factors to retain according to parallel analysis
Components$nkaiserNumber of factors to retain according to the Kaiser-Guttman rule
AnalysisData frame of vectors linked to the different rules
Analysis$EigenvaluesEigenvalues
Analysis$PropProportion of variance accounted by eigenvalues
Analysis$CumuCumulative proportion of variance accounted by eigenvalues
Analysis$Par.AnalysisCentiles of the random eigenvalues generated by the parallel analysis.
Analysis$Pred.eigPredicted eigenvalues by each optimal coordinate regression line
Analysis$OCCritical optimal coordinates
Analysis$Acc.factorAcceleration factor
Analysis$AFCritical acceleration factor
Otherwise, returns a summary of the analysis.

Details

The nScree function returns an analysis of the number of factors to retain in an exploratory principal components analysis. Different solutions are given. The classical ones are the Kaiser-Guttman 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 and the optimal coordinates index. 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-Guttman rule or a parallel analysis criterion (parallel) must also be simultaneously satisfied to retain the components. If $\lambda_i$ 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-Guttman rule is computed as: $$n_{Kaiser-Guttman} = \sum_{i} (\lambda_{i} \ge 1.00).$$ The parallel analysis is computed as: $$n_{parallel} = \sum_{i} (\lambda_{i} \ge LS_i).$$ The acceleration factor ($AF$) corresponds to a numeral 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 preceding eigenvalue by a regression line between the eignvalue coordinates and the last eigenvalue coordinate: $$n_{OC} = \sum_i \left[(\lambda_i \ge LS_i) \cap (\lambda_i \ge (\lambda_{i \ predicted}) \right].$$

References

Cattell, R. B. (1966). The scree test for the number of factors. Multivariate Behavioral Research, 1, 245-276. Guttman, L. (1954). Some necessary conditions for common factor analysis. Psychometrika, 19, 149-162. Horn, J. L. (965). 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/]

Examples

Run this code

## 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(eig       = eigenvalues,
                   aparallel = aparallel
                   )
                   
 results
 
## PLOT ACCORDING TO THE nScree CLASS 
 plotnScree(results)


## RESULTS OF THE nSree FUNCTION: AN EXAMPLE
## RESULTS CAN VARY ACCORDING TO RANDOM SIMULATION IN PARALLEL ANALYSIS

# $Components
#   noc naf nparallel nkeyser
# 1   2   1         2       5

# $Analysis
#    Eigenvalues        Prop      Cumu Par.Analysis  Pred.eig     OC Acc.factor     AF
# 1         5.54 0.369333333 0.3693333    2.8807740 2.6461538                NA (< AF)
# 2         2.46 0.164000000 0.5333333    2.3570618 1.9575000 (< OC)       2.43       
# 3         1.81 0.120666667 0.6540000    2.0132405 1.4690909              0.19       
# 4         1.35 0.090000000 0.7440000    1.7117006 1.0960000              0.11       
# 5         1.00 0.066666667 0.8106667    1.5115951 0.7400000              0.02       
# 6         0.67 0.044666667 0.8553333    1.3196573 0.6025000              0.20       
# 7         0.54 0.036000000 0.8913333    1.1188151 0.4857143              0.02       
# 8         0.43 0.028666667 0.9200000    0.9552613 0.3900000              0.02       
# 9         0.34 0.022666667 0.9426667    0.8249083 0.3640000              0.06       
# 10        0.31 0.020666667 0.9633333    0.6714989 0.2275000             -0.09       
# 11        0.19 0.012666667 0.9760000    0.5814073 0.1866667              0.08       
# 12        0.15 0.010000000 0.9860000    0.4473462 0.1450000              0.00       
# 13        0.11 0.007333333 0.9933333    0.3451530 0.0800000             -0.01       
# 14        0.06 0.004000000 0.9973333    0.2622424        NA              0.03       
# 15        0.04 0.002666667 1.0000000    0.1722911        NA                NA

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