Ledermann {IMIFA}

Ledermann Bound

Description

Returns the maximum number of latent factors in a factor analysis model for data of dimension P which actually achieves dimension reduction in terms of the number of covariance parameters. This Ledermann bound is given by the largest integer smaller than or equal to the solution k of (M - k)^2 \geq M + k.

Usage

Ledermann(P,
          isotropic = FALSE,
          int = TRUE)

Arguments

P

Integer number of variables in data set. This argument is vectorised.

isotropic

Logical indicating whether uniquenesses are constrained to be isotropic, in which case the bound is simply P-1. Defaults to FALSE.

int

Logical indicating if the result should be returned as an integer by applying the floor function. Defaults to TRUE.

Details

The Ledermann bound when istropic is FALSE is given by P + 0.5\left(1 - \sqrt{8P + 1}\right).

Value

The Ledermann bound, a non-negative integer obtained using floor, or a vector of length(P) such bounds.

Note

It has also been argued that the number of factors should not exceed floor((P - 1)/2), which is a necessarily stricter condition.

References

Anderson, T. W. and Rubin, H. (1956) Statistical inference in factor analysis. In Neyman, J. (Ed.), Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Volume 3.5: Contributions to Econometrics, Industrial Research, and Psychometry, University of California Press, Berkeley, CA, U.S.A., pp. 111-150.

Examples

Ledermann(c(25, 50, 100))
floor((c(25, 50, 100) - 1) / 2) # stricter bounds

data(olive)
P <- ncol(olive[,-(1:2)])
Ledermann(P)
Ledermann(P, int=FALSE)
floor((P - 1)/2) # stricter bound