Well, there is a large literature on sandwich estimators for non-independent or clustered data beginning with Liang and Zeger (1986). Most of this falls under the term Generalized Estimating Equations (GEE). There is a book about GEE by Hardin and Hilbe, but Agresti covers the topic also in his "introduction to categorical data analysis" book (2007). These type of models can nowadays be fitted in mist popular software like SPSS, SAS, Stata, R.

Thanks Dr. Mark. Appreciate greatly your response.

I am not sure how far the sandwich estimator can go, but remember this approach is a 'post-hoc' approach to dealing with the problem of correlated data. In this respect, the SEs (and CIs) for your betas are dealt with, but not your omnibus tests (in the case of a multi class predictor), or the overall model significance.

I am note sure what type of outcome you are talking about, but there are two main modeling approaches to dealing with the problem of correlated data: Marginal models and Conditional models. Marginal models (e.g. GEEs) use a population averaging approach to the correlated data problem (i.e. deal with the correlated structure solely in the error term). In contrast, Conditional models (e.g. LMMs and GLMMs) explicitly model the correlated data by including random effects terms in the model (you can see this as have a subject-specific contribution in the model).

Personally, I would always take the conditional modeling approach (rather than the marginal model approach). This is for a few reason. First, population averaging assumes a shared behavior in terms of the observation (such that a population average adequately represented the correlated nature of the data). Second, marginal models are restricted to rather simple data structures (fine for a single clustering effect, but much ticker for cross or nested effects......just for example it would be tricky to run a multicentre trial involving repeated measure data using a GEE). Finally, (subjective) I just line mixed models better (especially in terms of visualizing differences between observations)

Hope this helps

It has been argued by Gary King that using robust SE's is like taking a canary down the mine; if the robust SE's give different results , there is something wrong with the model, it is not OK to interpret it. For what it is worth I agree with him and thus I think it is better to have an explicit mdel that does take the dependency into account and not just treat as a nuisance to be corrected.

http://gking.harvard.edu/publications/how-robust-standard-errors-expose-methodological-problems-they-do-not-fix

I enjoyed reading Cameron’s post and I thought I would say some more about the two approaches especially in relation to discrete outcomes (Apologies that most of this comes from another post about regression to the mean.) And a warning that I too am biased as in favour of the conditional model.

There are two types of models when dealing with a panel study where occasions are nested within individuals, subjects or clusters, that is there is dependency or autocorrelation over time in the response variable. This has particular importance for discrete outcomes eg predicted probability having a disease (say) from an estimated model.

The two approaches to within subject variation are:

1) Marginal i.e. Population average models: estimated by GEE estimation: the slope is the effect of change in the whole population if everyone’s predictor variable changed by one unit

2) Conditional: cluster specific, mixed or multilevel: the slope is the effect of change for a particular ‘individual’ or subject of a unit change in a predictor.

Marginal estimates should not be used to make inferences about individuals (committing ecological fallacy), while conditional estimates should not be used to inferences about populations (atomistic fallacy). Population average approach underestimates the individual risk & vice-versa .

GEE estimation is robust to assumptions about the higher level, between- subject distribution; gives correct SE for fixed part and gives the Population average value. But do not give the higher-level variance, it is not extendible to random slopes and 3 level models etc; and does not give & cannot derive the cluster-specific estimates. All things that you get with the mixed, multilevel random-effects approach.

Population-averaged values can be obtained from random effects model in the MLwiN software via simulating predicted probabilities, see below

In practice there has to be a lot of difference between individuals and therefore (equivalently) a lot of similarity over time within subjects for there to be a difference between the two estimates. Thus when a fixed estimate is -1.5 on the logit scale, and the between variance is 1.0 (ie some 23 percent of the variance lies at the occasion level), the cluster specific result is a probability of 0.18 while the population average value is 0.22 . However if the between variance on a logit scale is 3 (so that 48% of the total variance lies at the occasion level) , the cluster specific result remains a probability of 0.18 while the population average value is now 0.27.

The GEE approach treats this clustering as a nuisance and does not give an estimate of the higher-level variance. The mixed random effects model treats the clustering as of substantive interest and you do get an estimate of its size and nature – but this generally requires distributional assumptions about the higher-level random effects; although non-parametric random effects procedures are possible.

There is more on mixed modelling for longitudinal analysis here

Developing multilevel models for analysing contextuality, heterogeneity and change using MLwiN, Volume 2 Kelvyn Jones, VS Subramanian ; available from

https://www.researchgate.net/publication/260772180_Developing_multilevel_models_for_analysing_contextuality_heterogeneity_and_change_using_MLwiN_Volume_2?ev=prf_pub

And the MLwin procedure is documented here:

https://www.researchgate.net/publication/234015221_Manual_supplement_for_MLwiN_Version_2.14?ev=prf_pub

So from my point of view finding a difference between the two types of SE's (the ordinary and the sandwich) can suggest that something interesting is going on and that you may need random slopes models to tease out the this type of "impact heterogeneity" of a predictor..

Here is an example where random slopes matters

Explaining Fixed Effects: Random Effects modelling of Time-Series Cross-Sectional and Panel Data; Andrew Bell, Kelvyn Jones

Political Science Research and Methods 12/2013; forthcoming.

https://www.researchgate.net/publication/233756428_Explaining_Fixed_Effects_Random_Effects_modelling_of_Time-Series_Cross-Sectional_and_Panel_Data

Book Developing multilevel models for analysing contextuality, he...

Data Manual supplement for MLwiN Version 2.14

Article Explaining Fixed Effects: Random Effects Modeling of Time-Se...