Ledermann function

Ledermann Bound

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 $(M - k)^2 >= M + k$.

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.

Returns

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

Details

The Ledermann bound when istropic is FALSE is given by $P + 0.5 * (1 - sqrt(8 * P + 1))$.

Note

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

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

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.