Velicer's minimum average partial (MAP) criterion

Computes Velicer's Minimum Average Partial (MAP) criterion for determining the number of factors/components to retain. The function implements the original MAP criterion (Velicer, 1976), expressed via the $\mathrm{TR2}$ representation, and the revised $\mathrm{TR4}$ variant proposed by Velicer, Eaton, and Fava (2000).

Usage

efa_map(
  x,
  use = c("pairwise.complete.obs", "all.obs", "complete.obs", "everything",
    "na.or.complete"),
  cor_method = c("pearson", "spearman", "kendall", "poly", "tetra")
)

Source

Auerswald, M., & Moshagen, M. (2019). How to determine the number of factors to retain in exploratory factor analysis: A comparison of extraction methods under realistic conditions. Psychological Methods, 24(4), 468–491. https://doi.org/10.1037/met0000200

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., & 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 Goffin, R. D. & Helmes, E. (Eds.), Problems and Solutions in Human Assessment: Honoring Douglas N. Jackson at Seventy (pp. 41–71). Boston: Kluwer.

Zwick, W. R., & Velicer, W. F. (1986). Comparison of five rules for determining the number of components to retain. Psychological Bulletin, 99, 432–442. https://doi.org/10.1037/0033-2909.99.3.432

Arguments

x: A numeric matrix or data.frame. Can be either (a) a correlation matrix, or (b) raw data (rows = observations, columns = variables) from which correlations are computed.
use: Character string specifying the treatment of missing values when computing correlations. Passed to stats::cor(). Defaults to "pairwise.complete.obs".
cor_method: Character string specifying the correlation coefficient to be computed if raw data are supplied. One of "pearson", "spearman", or "kendall" (passed to stats::cor()), or "poly" / "tetra" for polychoric / tetrachoric correlations of ordinal / binary data (a two-step estimator). Defaults to "pearson".

Value

An object of class efa_retention (see print.efa_retention() for the print method). MAP has no plot; plot.efa_retention() returns NULL with a message for it. Its main elements are:

n_factors: A named numeric vector ("TR2", "TR4") with the index $m$ that minimizes the original (TR2) and revised (TR4) MAP criterion.
results: A list with one record per criterion, each holding the criterion values over $m$ and, in m_last, the largest $m$ at which the criterion could be evaluated (see details).
settings: A list containing use and cor_method.

Details

MAP partials successive principal components out of the correlation matrix and, after removing $m$ components, summarizes the off-diagonal partial correlations $r^*_{ij}$ that remain in the $m$-th partial correlation matrix $M$ (which has a unit diagonal); the suggested number of factors is the $m$ that minimizes the criterion. Two criteria are returned, each rescaling the trace of a matrix power of $M$ by the number of off-diagonal cells $p(p-1)$:

TR2 (original MAP; Velicer, 1976): the average squared off-diagonal partial correlation, $$\mathrm{TR2}_m = \frac{\mathrm{tr}(M^2) - p}{p(p-1)} = \frac{\sum_{i \neq j} (r^*_{ij})^2}{p(p-1)},$$ where subtracting $p$ removes the $p$ unit diagonal entries.
TR4 (revised MAP; Velicer, Eaton, & Fava, 2000): the analogous fourth-power summary, formed from the trace of the fourth matrix power, $$\mathrm{TR4}_m = \frac{\mathrm{tr}(M^4) - p}{p(p-1)}.$$ Moving from the squared to the fourth power downweights the small partial correlations relative to the large ones, which can sharpen the minimum. Unlike TR2, $\mathrm{tr}(M^4)$ is not the sum of the fourth powers of the individual partial correlations; the matrix power is intended and is what Velicer, Eaton, and Fava (2000) describe.

Both criteria are returned for every call and they can suggest different numbers of factors on the same correlation matrix. Both are in use in the literature and neither is treated as the default here, so be sure to state which of the two you report, as you would for any other analysis choice.

MAP is most dependable when the components are well determined, that is with many indicators per factor and substantial loadings. It has a well-documented tendency to under-extract, particularly with few indicators per factor or weak loadings (Zwick & Velicer, 1986; Auerswald & Moshagen, 2019), so it is best read as a lower bound and paired with a criterion that errs in the other direction, such as the Kaiser-Guttman criterion (efa_kgc()).

The criterion is evaluated over $m = 0, \ldots, p - 1$. Each step standardizes the partial covariance matrix by its residual standard deviations, which requires every residual variance to stay positive. Partialling out all but one component leaves a rank-one residual, so the final point $m = p - 1$ is undefined for most correlation matrices and is routinely returned as NA. A residual variance can also reach zero earlier, most often on a near-singular matrix; the search then stops there, the criterion values that could be computed are kept, the remaining values stay NA, and a warning (class efa_map_truncated) reports how far the grid was searched. In that case the suggested $m$ is the minimum over the evaluated range only, so it should be read together with the returned series.

A non-positive-definite input correlation matrix (e.g. from sampling error) is smoothed with psych::cor.smooth().

Examples

## Example with raw data
res <- efa_map(GRiPS_raw)
#> ℹ `x` is not a correlation matrix; computing correlations from the raw data.
res
#> ── Minimum average partial ─────────────────────────────────────────────────────
#> 
#> • Original implementation (TR2): 1
#> • Revised implementation (TR4): 1

## Example with a correlation matrix
res2 <- efa_map(test_models$baseline$cormat)
res2
#> ── Minimum average partial ─────────────────────────────────────────────────────
#> 
#> • Original implementation (TR2): 1
#> • Revised implementation (TR4): 3