FMradio (version 1.1.1)

SMC: Compare squared multiple correlations with model-based communalities

Description

SMC is a function that compares the best lower-bound estimates to the communalities with the model-based communalities implied by a factor solution of dimension \(m\).

Usage

SMC(R, LM)

Arguments

R

(Regularized) correlation matrix.

LM

(Rotated) factor loadings matrix.

Value

The function returns a matrix. The first column (labeled 'SMC') contains the lower-bound estimates to the communalities. The second column (labeled 'Communalities') contains the retrieved estimated communalities under the \(m\)-factor model.

Details

This function can be used to qualitatively assess the choice of dimensionality (as well as the fit) in the \(m\)-factor model. This is done using the concept of communalities. The communality refers to the amount of variance of feature \(j\) explained by the latent features. It is then of interest to compare lower-bound estimates of the (population) communalities to the extracted communalities under the \(m\)-factor model.

Guttman (1956) gave the best possible lower-bound estimates to the communalities, which can essentially be considered squared multiple correlations: the proportion of variance in feature \(j\) that is explained by the remaining \(p - 1\) features. To assess a factor model, these might be compared to the retrieved estimated communalities under the \(m\)-factor model. When the chosen latent dimensionality is sufficient then one would expect that, for almost all features, the retrieved communality approximately equals or exceeds its corresponding lower-bound estimate. If this is not the case then one might have extracted too few factors.

References

Guttman, L. (1956). Best possible systematic estimates of communalities. Psychometrika, 21:273--285.

Peeters, C.F.W. et al. (2019). Stable prediction with radiomics data. arXiv:1903.11696 [stat.ML].

See Also

dimGB, FAsim, mlFA, dimVAR

Examples

# NOT RUN {
## Simulate some high-dimensional data according to the factor model
simDAT <- FAsim(p = 50, m = 5, n = 40)

## Regularize the correlation matrix
RegR <- regcor(simDAT$data)

## Fit 5-factor model to the regularized correlation matrix
fit <- mlFA(RegR$optCor, m = 5)

## Compare lower-bound estimates to communalities with model-implied ones
C <- SMC(RegR$optCor, fit$Loadings)
print(C)
# }