facScore: Compute factor scores
In FMradio: Factor Modeling for Radiomics Data

Description Usage Arguments Details Value Note Author(s) References See Also Examples

facScore is a function that computes factor scores, the score each person/object attains on each latent factor.

1	facScore(X, LM, UM, type = "thomson")

`X`	A (scaled and possibly subsetted) data `matrix`.
`LM`	A (rotated) loadings `matrix`. Usually the `$Loadings`-slot object from the `mlFA` function output.
`UM`	A diagonal uniquenesses `matrix`. Usually the `$Uniqueness`-slot object from the `mlFA` function output.
`type`	A `character` indicating the type of factor score to calculate. Must be one of: "thomson", "bartlett", "anderson".

Once a factor model is fitted one may desire an estimate of the score each object/individual would obtain on each of the latent factors. Such scores are referred to as factor scores. The facScore function provides several types of factor score estimates. The default are Thomson-type scores (Thomson, 1939). These may be viewed as (empirical) Bayesian-type scores. Bartlett-type scores (Bartlett, 1937) are unbiased but less efficient in terms of mean-squared error. Under the orthogonal model the latent factors are orthogonal in the population and, hence, the Thomson and Bartlett-type factor scores will be near orthogonal in the sample. Anderson and Rubin (1956) constructed an alternative estimator for the factor scores that enforces their orthogonality in the sample.

The function returns an object of class data.frame. Observations are represented in the rows. Each column represent a latent factor.

The input data (argument X) are assumed to be scaled (or at least centered). The UM matrix is assumed to be positive definite. The LM matrix is assumed to be of full column rank.

Carel F.W. Peeters <cf.peeters@vumc.nl>

Anderson, T.W., & Rubin, H. (1956). Statistical inference in factor analysis. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, volume 5: Contributions to Econometrics, Industrial Research, and Psychometry, pages 111–150. Berkeley, CA: University of California Press.

Bartlett, M.S. (1937). The statistical conception of mental factors. British Journal of Psychology, 28:97–104.

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

Thomson, G. (1939). The Factorial Analysis of Human Ability. London: University of Londen Press.

dimGB, mlFA, facSMC

## Simulate some data according to a factor model with 5 latent factors
## Simulate high-dimensional situation in the sense that p > n
## $cormatrix gives the correlation matrix on the generated data
simDAT <- FAsim(p = 50, m = 5, n = 40, loadingvalue = .9)
simDAT$cormatrix

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

## Evaluate the Guttman bounds
## First Guttman bound indicates to retain 5 latent factors
GB <- dimGB(RegR$optCor)
print(GB)

## Produce ML factor solution under 5 factors
## Print loadings structure of this solution
fit <- mlFA(RegR$optCor, 5)
print(fit$Loadings, digits = 2, cutoff = .3, sort = TRUE)

## Obtain factor-scores
scores <- facScore(scale(simDAT$data), fit$Loadings, fit$Uniqueness)
print(scores)