If you match within clusters,do you still need to use a multilevel model? I am reviewing an article. I can't share details because the article has not been published but, briefly, the authors studied people with and without a condition and looked at long term sequealae of the condition. The patients were clustered by their general practitioner and some of the covariates (socio-economic status) was estimated by postal code (so all patients in one post code got the same SES score). 
Ordinarily, I would require some way (such as a multi-level model (MLM)) of accounting for the clustering. But in this case, the authors matched the subjects with controls seeing the same general practitioner. They also used robust standard errors for all parameter estimates (it was a Cox regression).
Does this matching make an MLM unnecessary?
 A: Peter, consider a simple example:  patients nested inside doctors; each patient privides a single response value (i.e., time to some event, possibly censored);  doctors are treated as "clusters. 
The use of an MLM in this setting acknowledges two things:


*

*That there is potential correlation among patient response values coming from the same doctor; 

*That subject-specific inference is of interest (rather than population-average inference).


So, when recommending the use of an MLM, both of these things have to be of interest. (Of course, it is possible to use an MLM to also get population-average inference, but that seems to be more complicated for some MLM models.) 
In cases where interest is in population-average inference, authors may resort to other means - for instance, GEE for clustered survival data (see https://lib.dr.iastate.edu/cgi/viewcontent.cgi?article=4062&context=rtd). It is not clear from your description what modelling the authors actually used, but whatever they used, the question is: Is that modelling appropriate given the correlation structure of the data AND the inferential goals of the authors?
A: Without matching, an MLM serves (at least) two purposes: to account for confounding due to cluster membership (in this case practitioner) and to correct standard errors for dependence among observations in the same cluster. (As Isabella notes, how the estimates generalize can also be determined by the use of an MLM.)
You can think of matching within cluster as exact matching on cluster membership. One benefit of doing so is that all cluster-level confounding is eliminated in the matched set. There may be differences between members of a matched pair, but those differences will not be with respect to cluster membership or any cluster-level (or higher!) confounding variables, observed or not. In the same way that exact matching on a covariate generally allows you to exclude that variable from the outcome model, exact matching on cluster membership means you don't have to further account for cluster membership, either with fixed effects or an MLM, in order to eliminate confounding.
Standard error estimation is a different story. After matching within practitioner, it's still the case that the outcomes of paitents within the same practitioner will be correlated with one another (indeed, you would hope so, or else practitioner isn't a confounder!). For valid inference, this dependence must be accounted for. This could be done a variety of ways, as Isabella noted, including fixed effects for practitioner, an MLM with practitioner as a random effect, cluster-robust standard errors with practitioner as the cluster variable, or generalized estimating equations with a correlation matrix that accounts for within-cluster dependence. Any of these are potentially valid, and the choice depends at least partly on whether marginal or conditional effects are desired.
Finally, there's also the issue of the within-pair correlations induced by the matching. Austin (2013) recommends the use of cluster-robust standard errors for estimating hazard ratios after matching. Austin has a few other relevant papers, including variance estimation after matching with replacement (Austin & Cafri, 2020) and including covariates in the outcome model after matching (Austin, Thomas, & Rubin, 2020). Since pairs are nested within practitioners, you actually have two levels of nesting, which is amenable to MLMs.
