When to use Rao-Scott adjustment and when to use Moment Matching?

[haziqj]

MOJ (2008): Two strategies exist to compute the GOF tests. Choose $W$ such that

1. $X^2 = \mathbf e^\top \boldsymbol \Xi \mathbf e$ is computationally easy to obtain.
2. $X^2= \mathbf e^\top \boldsymbol \Xi \mathbf e$ is asymptotically $\chi^2$.

Our tests that satisfy criteria 1:

• Wald diagonal
• RSS
• Multinomial

Our tests that satisfy criteria 2:

• Wald
• Wald VCF
• Pearson

[haziqj]

While in theory the tests in category 1 above can use Rao-Scott adjustments, it requires finding the Choleski decomposition of the weight matrix W, which is not possible if there are rank issues in W. So moment matching is better.

For the Pearson, the Rao-Scott adjustment or the Moment Matching is possible since the weight matrix $W = D^{-1}$ can always be Choleski decomposed. In fact we use $W = D^{-1/2}D^{-1/2}$.

[haziqj]

Rao-Scott adjustments were developed to modify the test statistics in the presence of dependencies or correlations among the observations. Primarily it was developed in the context of complex survey sampling, so that the design effect is accounted for in the GOF test.

However, when we use the univariate and bivariate moments to compute the test statistics, then there are dependencies (between the univariate and bivariate stuff) that violates the chi square assumptions. So we want to see how the Rao-Scott adjustment handles them (and if they're better than the moment matching procedures).