simul.nfac.prior {BayesFM}

Simulate prior distribution of number of latent factors

Description

This function produces a sample from the prior distribution of the number of latent factors. It depends on the prior parameters used for the distribution of the indicators, on the size of the model (number of manifest variables and maximum number of latent factors), and on the identification restriction (minimum number of manifest variables dedicated to each factor).

Usage

simul.nfac.prior(nvar, Kmax, Nid = 3, kappa = 1/Kmax, nrep = 10^6)

Arguments

nvar

Number of manifest variables.

Kmax

Maximum number of latent factors.

Nid

Minimum number of manifest variables dedicated to each latent factor for identification.

kappa

Concentration parameter of the Dirichlet prior distribution on the indicators.

nrep

Number of Monte Carlo replications.

Details

This function simulates the prior distribution of the number of latent factors for models that fulfill the identification restriction restriction that at least Nid manifest variables (or no variables) are loading on each latent factor. Several (scalar) parameters kappa can be passed to the function to simulate the prior for different prior parameter values and compare the results.

An accept/reject sampling scheme is used: a vector of probabilities is drawn from a Dirichlet distribution with concentration parameter kappa, and the nvar manifest variables are randomly allocated to the Kmax latent factors. If each latent factor has at least Nid dedicated variables or no variables at all, the identification requirement is fulfilled and the draw is accepted. The number of factors loaded by at least Nid manifest variables is returned as a draw from the prior distribution.

Note that this function does not use the two-level prior distribution implemented in CFSHP, where manifest variables can be discarded from the model according to a given probability. Therefore, this function only help understand the prior distribution conditional on all the manifest variables being included into the model.

Value

A list of length equal to the number of parameters specified in kappa is returned, where each element of the list contains:

• nfac: Vector of integers of length equal to the number of accepted draws.

• acc: Acceptance rate of the accept/reject sampling scheme.

Author(s)

Rémi Piatek [e-mail]

References

G. Conti, S. Frühwirth-Schnatter, J.J. Heckman, R. Piatek (2014): “Bayesian Exploratory Factor Analysis”, Journal of Econometrics, 183(1), pages 31-57, doi: 10.1016/j.jeconom.2014.06.008.

Examples

# replicate first row of table 2 in CFSHP (p.44)
# note: use larger number of replications nrep to improve accuracy
prior.nfac <- simul.nfac.prior(nvar = 15, Kmax = 5, kappa = c(.3, .7, 1), 
                               nrep = 10000)
summary(prior.nfac)
plot(prior.nfac)