Velicer's minimum average partial (MAP) test for determining the number of factors focuses on the common variance in a correlation matrix. It involves a complete principal components analysis followed by the examination of a series of matrices of partial correlations. Specifically, on the first step, the first principal component is partialled out of the correlations between the variables of interest, and the average squared coefficient in the off-diagonals of the resulting partial correlation matrix is computed. On the second step, the first two principal components are partialled out of the original correlation matrix and the average squared partial correlation is again computed. These computations are conducted for k (the number of variables) minus one steps. The average squared partial correlations from these steps are then lined up, and the number of components is determined by the step number in the analyses that resulted in the lowest average squared partial correlation. The average squared coefficient in the original correlation matrix is also computed, and if this coefficient happens to be lower than the lowest average squared partial correlation, then no components should be extracted from the correlation matrix. Statistically, components are retained as long as the variance in the correlation matrix represents systematic variance. Components are no longer retained when there is proportionately more unsystematic variance than systematic variance (see O'Connor, 2000, p. 397).
The MAP test is often more appropriate for factor analyses than it is for principal components analyses. In Velicer's words, "Component analysis has a variety of purposes. It can be used to find a parsimonious description of the total variance of the variables involved; in this case, the [MAP test] is not applicable. Principal component analysis is frequently used to express the variance shared among variables in a set; that is, it is used as kind of a factor analysis" (1976, p. 321). "... if component analysis is employed as an alternative to factor analysis or as a first-stage solution for factor analysis, the stopping rule proposed here would seem the most appropriate." (1976, p. 326)'
MAP(data, corkind, verbose)An all-numeric dataframe where the rows are cases & the columns are the variables,
or a correlation matrix with ones on the diagonal. The function internally determines
whether the data are a correlation matrix.
The kind of correlation matrix to be used if data is not a correlation matrix.
The options are 'pearson', 'kendall', 'spearman', and 'polychoric'. Required only if
the entered data is not a correlation matrix.
Should detailed results be displayed in console? The options are: TRUE (default) or FALSE.
A list with the following elements:
eigenvalues
Velicer's average squared correlations
number of factors according to the original (1976) MAP test
number of factors according to the revised (2000) MAP test
Velicer, W. F. (1976). Determining the number of components from the matrix of partial correlations. Psychometrika, 41, 321-327. Velicer, W. F., Eaton, C. A., and Fava, J. L. (2000). Construct explication through factor or component analysis: A review and evaluation of alternative procedures for determining the number of factors or components. In R. D. Goffin & E. Helmes, eds., Problems and solutions in human assessment (p.p. 41-71). Boston: Kluwer. O'Connor, B. P. (2000). SPSS and SAS programs for determining the number of components using parallel analysis and Velicer's MAP test. Behavior Research Methods, Instrumentation, and Computers, 32, 396-402.
# NOT RUN {
# MAP test on the Harman correlation matrix
MAP(data_Harman, corkind='pearson', verbose=TRUE)
# }
# NOT RUN {
# MAP test on the Rosenberg Self-Esteem Scale (RSE) using Pearson correlations
MAP(data_RSE, corkind='pearson', verbose=TRUE)
# MAP test on the Rosenberg Self-Esteem Scale (RSE) using polychoric correlations
MAP(data_RSE, corkind='polychoric', verbose=TRUE)
# MAP test on the NEO-PI-R data
MAP(data_NEOPIR, verbose=TRUE)
# }
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