6.7 Local Identifiability
Local identifiability means that even though there are more than one parameter values in the whole parameter space whose probability distributions are the same to the sample, one can still find an open neighborhood of the parameter such that every parameter in that neighborhood generates a unique distribution. (Kim & Lindsay, 2015)
A locally but not globally identifiable model does not have a unique interpretation, but one can be sure that, in the neighborhood of the selected solution, there exist no other equally good solutions; thus, the problem is reduced to determining the regions where local identifiability applies. (Everitt & Howell, 2005)
Local identifiability is a necessary but not sufficient condition for global identifiability.
Below are two approaches that have been used for assessing local identifiability (McDonald, 1982):
Identifiability Conditions #8
Parameters \(\tau\) of dimension \(D\times 1\) are locally identified at point \(\tau=\tau_0\), if and only if the Jacobian matrix below is of full rank.
\[ J=\frac{\partial \ell(\mathbf{y}_i)}{\partial \tau} \]
Equivalently, one can assess local identifiability using the following rule:
Identifiability Conditions #9
Parameters \(\tau\) of dimension \(D\times 1\) are locally identified at point \(\tau=\tau_0\), if and only if the Fisher information matrix is not a singular matrix.
Let us check the fit of the ACDM model to the data below.
Code
library(GDINA)
Q <- matrix(c(1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0), ncol = 3, byrow = TRUE)
df <- read.csv("data/ACDM_idf.csv")
ACDM.est <- GDINA(df, Q, model = "ACDM", control = list(conv.crit = 1e-06,
randomseed = 321), verbose = FALSE)
x <- score(ACDM.est)
xx <- do.call(cbind, x)
library(Matrix)
rankMatrix(xx)## [1] 19
## attr(,"method")
## [1] "tolNorm2"
## attr(,"useGrad")
## [1] FALSE
## attr(,"tol")
## [1] 2.2e-13Code
npar(ACDM.est)## No. of total parameters = 19
## No. of population parameters = 7
## No. of free item parameters = 12
## No. of fixed item parameters = 0