FAsim: Simulate data according to the common factor analytic model
In FMradio: Factor Modeling for Radiomics Data

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

FAsim is a function that enables the simulation of data according to the common factor analytic model.

FAsim(p, m, n, simplestructure = TRUE, balanced = TRUE,
      loadingfix = TRUE, loadingnegative = TRUE,
      loadingvalue = .8, loadingvaluelow = .2, numloadings,
      loadinglowerH = .7, loadingupperH = .9, 
      loadinglowerL = .1, loadingupperL = .3)

`p`	A `numeric` integer or `integer` indicating the number of observed features.
`m`	A `numeric` integer or `integer` indicating the latent dimension of the factor solution (i.e., the number of factors).
`n`	A `numeric` integer or `integer` indicating the number of samples.
`simplestructure`	A `logical` indicating if the generating factor structure should be factorially pure.
`balanced`	A `logical` indicating if the high (i.e., qualitatively 'significant') loadings should be divided evenly over the respective factors.
`loadingfix`	A `logical` indicating if the loadings should have a fixed value.
`loadingnegative`	A `logical` indicating if, next to positive, also negative loadings should be present.
`loadingvalue`	A `numeric` indicating the value for high (i.e., qualitatively 'significant') loadings. Used when `loadingfix = TRUE`.
`loadingvaluelow`	A `numeric` indicating the value for low loadings. Used when `loadingfix = TRUE` & `simplestructure = FALSE`.
`numloadings`	A `numeric` vector with length equalling argument `m`, indicating the number of high (i.e., qualitatively 'significant') loadings per factor. Used when `balanced = FALSE`.
`loadinglowerH`	A `numeric` indicating the lower-bound of high (i.e., qualitatively 'significant') loadings. Used when `loadingfix = FALSE`.
`loadingupperH`	A `numeric` indicating the upper-bound of high (i.e., qualitatively 'significant') loadings. Used when `loadingfix = FALSE`.
`loadinglowerL`	A `numeric` indicating the lower-bound of low (i.e., qualitatively 'non-significant') loadings. Used when `loadingfix = FALSE` & `simplestructure = FALSE`.
`loadingupperL`	A `numeric` indicating the upper-bound of low (i.e., qualitatively 'non-significant') loadings. Used when `loadingfix = FALSE` & `simplestructure = FALSE`.

FAsim provides certain flexibility when generating data according to an orthogonal common factor-analytic model. It can produce data according to, for example, (i) factorially pure loadings structures, (ii) loadings-structures with only positive entries or both positive and negative loadings, (iii) loadings-structures with fixed values or varying values, (iv) balanced and unbalanced loadings-structures.

The function returns an object of class list:

`$data`	A standardized data `matrix` of size n \times p.
`$loadings`	Loadings `matrix` of size p \times m on which the data-generation was based.
`$Uniqueness`	A `numeric` vector of size p representing the uniquenesses on which the data-generation was based.
`$cormatrix`	A p \times p correlation `matrix` based on the generated data in slot `$data`.

A uniform distribution is assumed when generating draws between loadinglowerH and loadingupperH.
A uniform distribution is assumed when generating draws between loadinglowerL and loadingupperL.
The argument m cannot exceed the Ledermann-bound (Ledermann, 1937): \lfloor [2p + 1 - (8p + 1)^{1/2}]/2\rfloor, where p indicates the observed-feature dimension.

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

Ledermann, W. (1937). On the rank of the reduced correlational matrix in multiple factor analysis. Psychometrika, 2:85–93.

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

dimGB, mlFA, facScore

## Simulate some data according to a factor model with 3 latent factors
## Balanced and factorially pure loadings structure
simDAT <- FAsim(p = 24, m = 3, n = 40, loadingvalue = .9)
simDAT$loadings

## Simulate some data according to a factor model with 3 latent factors
## Unbalanced and factorially pure loadings structure
simDAT <- FAsim(p = 24, m = 3, n = 40, loadingvalue = .9,
                balanced = FALSE, numloadings = c(10,10,4))
simDAT$loadings

## Simulate some data according to a factor model with 3 latent factors
## Unbalanced and factorially non-pure loadings structure
simDAT <- FAsim(p = 24, m = 3, n = 40, loadingvalue = .9,
                balanced = FALSE, numloadings = c(10,10,4),
                simplestructure = FALSE)
simDAT$loadings

## Simulate some data according to a factor model with 3 latent factors
## Unbalanced and factorially non-pure loadings structure
## Non-fixed high and low loadings
simDAT <- FAsim(p = 24, m = 3, n = 40, loadingvalue = .9,
                balanced = FALSE, numloadings = c(10,10,4),
                simplestructure = FALSE, loadingfix = FALSE)
simDAT$loadings