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The methods of estimation like GLS or GEE are especially helpful, when there are clusters of data, like repeated observations, many per cluster=subject. Such observations are naturally correlated in some way.

The correlations my be similar over time (compound symmetry), may decrease over time (AR1), or may be of any kind (unstructured). But various statistical packages offer also the "independent" structure, which assumes 0 within-subject correlation.

Isn't this opposite to what these methods were invented?

In what situations we can use the "independence" with, say, GEE?

I can think of just one application. It's known, that GEE is robust to the working covariance specification. So we can use the independent one, which doesn't need any correlation to be estimated, which makes the calculations efficient and saves from convergence issues. But if it was so easy, then why would be there other structures? We could just use the independence one. It must be compensated somehow, like low efficacy of the estimation bias or so. So probably this application isn't real.

So what are the scenarios, where the independence correlation is used within GLS or GEE?

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One problem with non-diagonal working correlation matrices was pointed out by Pepe and Anderson (1994) and rediscovered by Pan, Louis, and Connett (2000)

Suppose your mean model is $$g(E[Y_{it}|X_{it}])=X_{it}\beta$$ and this model is (approximately) correctly specified. You'd like $\hat\beta$ to estimate $\beta$. However, if you write down the estimating equations, they are only unbiased for $\beta$ if the residuals $(Y_{it}-\hat Y_{it})$ are also uncorrelated with $X_{is}$ for $s\neq t$. That is, the predictors at time $t$ must include past or future values of $X$ if they are predictive of $Y_{it}$. Past values you could deal with, though it's a limitation. Future values are more likely to be a problem.

This isn't specific to GEE -- the same problem occurs with maximum likelihood estimation -- but GEE has the useful feature that you can makes this problem go away by using a diagonal working correlation matrix.

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  • $\begingroup$ Thank you very much, very enlightening! I would never think about that this way, being convinced by my teachers, that "independence" here is wrong, as the observations are surely correlated within a cluster, so that's why we use these methods to address that and pre-choosing the working covariance structure to reflect the nature of the correlation. Independence assumes zero correlation (so I assume the GEE uses the empirical one - from the data, anyway, right?). I know that GEE is robust to the VC misspecification, and maybe the independence is even better than unstructured. Is this true? $\endgroup$
    – Zanyaaa
    Commented Apr 2, 2022 at 15:25
  • $\begingroup$ GEE computes standard errors using the real correlation, but it defines the target of inference using whatever correlation structure you specify. $\endgroup$
    – Thomas Lumley
    Commented Apr 3, 2022 at 3:30

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