# What is the purpose to have the "independent" covariance structure in GEE or GLS?

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?

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.