FAiR (version 0.4-13)

restrictions2draws: Draw from sampling distribution of estimates

Description

It is not necessary to understand this help page if one merely wants to estimate a factor analysis model. This help page is intended for those who want to modify or extend FAiR or otherwise want some idea of how FAiR works behind the scenes. These functions facilitate drawing from a multivariate normal distribution defined by the estimated free parameters and their variance-covariance matrix. The draw is accepted if it satisfies all constraints and rejected otherwise. Appropriate transformations are then applied to acceptable draws. This mechanism makes it possible to calculate the uncertainty of estimates in a general fashion. It is unlikely that a user would ever need to call restrictions2draws directly; it is more common to use FA2draws, which calls restrictions2draws internally.

Usage

FA2draws(object, nsim = 1001, seed = NULL, 
         covariances = FALSE, standardized = TRUE, ...)

Arguments

object

object of FA-class

nsim

number of non-rejected simulations

seed

either NULL or an integer that will be used in a call to set.seed before simulating the free parameters. The default, NULL will not change the random generator state

covariances

logical indicating whether return simulations of the reproduced covariances, which are a function of the free parameters, rather than simulations of the free parameters themselves. The default, TRUE, returns the simulations

standardized

logical indicating whether the simulations should be rescaled so that they are calibrated to a correlation matrix

...

other arguments passed to downstream functions; not currently used

Value

• A named list of arrays where in each case the third dimension has extent nsim.

Methods

Note FA2draws is not a S4 generic function, but it primarily exists to call the S4 generic function, restrictions2draws. Methods are currently only defined for objects of class "manifest.basic", which are inherited by objects of class "manifest.data" and "manifest.data.mcd". There are methods for each flavor of restrictions-class, except for "restrictions.factanal". There are also at least four arguments that are not part of the signature of restrictions2draws. The first is vcov, which is the variance-covariance matrix for the free parameters and is simply extracted from the slot of the object of FA-class. The second through fourth arguments are nsim, covariances, and standardized as documented in the arguments section.

For the restrictions2draws method defined for class "restrictions.orthonormal", there are two additional arguments. The first is Tmat, which can be NULL but can also be an optimal transformation matrix (found by Rotate). The second is criteria, which is a list of criteria utilized when finding the optimal transformation matrix. If both of these arguments are specified, the method will produce draws of the transformed parameters, after generating them from a multivariate normal distribution. They are extracted from the object of FA-class as necessary.

Details

Traditional methods of estimating the uncertainty of parameter estimates are not necessarily appropriate in FAiR because of the extensive possibilities for imposing inequality restrictions on the parameters during the factor extraction stage and / or the factor transformation stage (in the case of EFA only). Thus, traditional asymptotic approximations to the sampling distribution of the parameters could place some mass on a region of the parameter space that is inconsistent with the inequality restrictions. To work around this problem, these functions draw simulations of the free parameters from an unconstrainted multivariate normal distribution, following King (1998). If the draw is consistent with the inequality restrictions, it is accepted and transformed as necessary; otherwise it is rejected. Once nsim acceptable simulations have been obtained, they are returned in a list of arrays for further analysis.

References

King, G. (1998) Unifying Political Methodology: The Likelihood Theory of Statistical Inference. University of Michigan Press.

See Also

confint,FA-method, simulate, and summary.FA-class

Examples

## See example for Factanal()