dhmre_pairwise: Implied density for pairwise differences given HMRE prior.
In MIRES: Measurement Invariance Assessment Using Random Effects Models and Shrinkage

dhmre_pairwise

R Documentation

Implied density for pairwise differences given HMRE prior.

Description

Computes the implied densities of random effect differences given HMRE prior.

Usage

dhmre_pairwise(x, mu = 0, sigma = 1)

Arguments

`x`	Numeric. Difference in random effects.
`mu`	Numeric. HMRE Prior location.
`sigma`	Numeric. (Default: 1; must be > 0). HMRE prior scale.

Details

The HMRE prior for the RE-SD is \int N^+(\sigma_p | exp(h_p))LN(h_p | 4\mu, \sqrt{4}\sigma)dh_p. The random effects are distributed as u_{k,p} \sim N(0, \sigma_p). The implied prior is therefore u_{k,p} - u_{\lnot k, p} \sim N(0, \sqrt{2}\sigma). Note that there is a singularity at 0, because the integrand at sigma = 0 is an infinite spike. We currently integrate (using a change of variables) starting at machine precision-zero. Consider this the approximation of the limit as we approach 0 positively. This is therefore divergent when assessed at a difference of zero, due to the RESD taking on a zero value (and an infinite function value). This is expected, as the limit of a Gaussian as sigma -> 0 is the Dirac delta function.