I have $n$ observations of pairs $(x, y)$ and three different models I would like to compare. Model0 is nested within Model1. Model0 is also nested within Model2. I would like to do hypothesis tests for each of models 1 & 2, using Model0 as the null hypothesis. My original approach was to use a likelihood ratio test but I am getting inflated type 1 error rates. I believe the reasons for it is explained in this stack overflow answer. Thus I am trying to use the corrections suggested in that answer such as the Satorra and Bentler 2010 correction or the Bartlett correction, but to no avail as the corrections all seem to be for differently specified problems than my own. Thus, I need help either converting the corrections into formulations for my problem, reformulating my problem in a way that is compatible with the corrections, and honestly, I wouldn't mind some one double checking my analysis that the cited answer actually is related to my problem, because maybe I am mistaken about that.
Thus my three models are as follows:
First, they are all $(2n x 2n)$ multivariate normal distributions which for the vector of observations $(x_{1},...,x_{n},y_{1},...,y_{n})$ gives the covariance between observations i and j in the ith row, jth column (matrix is positive semidefinite).
Model0: $$ \begin{bmatrix} \sigma_{1}^{2}A_{n}+\sigma_{2}^{2}\mathbb{I}_{n} & 0_{n x n} \\ 0_{n x n} & \sigma_{3}^{2}A_{n}+\sigma_{4}^{2}\mathbb{I}_{n} \\ \end{bmatrix}, $$ where $A_{n}$ and $\mathbb{I}_{n}$ are $(n x n)$ known matrices ($\mathbb{I}_{n}$ is the identity matrix), and $\sigma_{1}^2, \sigma_{2}^2, \sigma_{3}^2, \sigma_{4}^2$ are unknown variance scaling factors.
Model1: $$ \begin{bmatrix} \sigma_{1}^{2}A_{n}+\sigma_{2}^{2}\mathbb{I}_{n} & c_{1}C_{n} \\ c_{1}C_{n} & \sigma_{3}^{2}A_{n}+\sigma_{4}^{2}\mathbb{I}_{n} \\ \end{bmatrix}, $$ where $A_{n}$, $\mathbb{I}_{n}$, and $C_{n}$ are $(n x n)$ known matrices, $\sigma_{1}^2, \sigma_{2}^2, \sigma_{3}^2, \sigma_{4}^2$ are unknown variance scaling factors, and $c_{1}$ is an unknown correlation in $[-1,1]$.
Model2: $$ \begin{bmatrix} \sigma_{1}^{2}A_{n}+\sigma_{2}^{2}\mathbb{I}_{n} & c_{1}K_{n} \\ c_{1}K_{n} & \sigma_{3}^{2}A_{n}+\sigma_{4}^{2}\mathbb{I}_{n} \\ \end{bmatrix}, $$ where $A_{n}$, $\mathbb{I}_{n}$, and $K_{n}$ are $(n x n)$ known matrices, $\sigma_{1}^2, \sigma_{2}^2, \sigma_{3}^2, \sigma_{4}^2$ are unknown variance scaling factors, and $c_{1}$ is an unknown correlation in $[-1,1]$.
In summary, I have a few different hypotheses about the type of correlation that could exist between X and Y which I'd like to be able to favor by rejecting the null hypothesis. I greatly appreciate any help, thank you.
EDIT:
More Information about $A_{n}$: It is derived by measuring distances in space between the points. The closer together two points in space are, the higher their correlation, thus the diagonal of $A_{n}$ is all 1s because an observation $x_{i}$ always has perfect correlation with itself. $A_{n}$ is specifically derived from measuring the distances on a network/graph relative to a fixed point. The more path $x_{i}$ and x_{j} have in common from the fixed point to themselves, the higher their correlation. In a related question from some months back, I drew out an example of such a network and give the exact formulation of all the matrices for that specific problem. Although, in that question the rows and columns are rearranged to that $(2nx2n)$ matrix describes the vector of observations $vec(x_{1},y_{1},...,x_{n},y_{n})$ rather than $vec(x_{1},...x_{n},y_{1},...,y_{n})$. I think it's important to note that in the formulation for this question, I do not have any of the mins or maxs or anything like that as in the prior formulation. This is a different model of evolution I'm studying where all the matrices are completely known.
