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
Let us explore few approaches to assess local identifiability (McDonald, 1982)
Identifiability condition #8(C8)
Parameters \(\tau\) of dimension \(D \times1\) are locally identified at point \(\tau=\tau_0\), if an only if the Jacobian matrix below is of full rank.
\[ J=\frac{\partial \ell\left(\mathbf{y}_i\right)}{\partial \tau} \] If the rank of the Jacobian matrix is equal to the number of parameters, we can say the model is locally identifiable.
Moreover, we can use the following condition to assess local identifiability.
Identifiability condition #9(C9)
Parameters \(\tau\) of dimension \(D \times1\) are locally identified at point \(\tau = \tau_0\), if an 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-13## No. of total parameters = 19
## No. of population parameters = 7
## No. of free item parameters = 12
## No. of fixed item parameters = 0Note. The rank of the Jacobian matrix is equal to the number of parameters, hence, the ACDM model is locally identifiable.