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Short version

Considering the controversy regarding this practice and having learn that heteroscedasticity should be addressed differently, I wondered:

  1. In which cases should one consider computing and reporting "robust" standard errors (in terms of types of models: LM vs. GLM, OLS vs. ML, etc.; purpose: inference vs. prediction; etc.)? And why?
  2. Does using a mixed modelling framework matters for the choice of reporting these "robust" SEs or not?

If possible, I would appreciate explanations in plain English or in "maths for dummies".

Contextualized version

For my study, I would like to explain five dependent variables (DVs) related to the success or failure of a specific method in controlling an invasive plant species: $Y_{1}$ is an estimated effectiveness score (defined on (0,1]) while $Y_{2, 3, 4, 5}$ are binary variables stating whether the plant produced regrowths at various key locations. Therefore I was initially interested by binomial and logistic models (in a mixed-modelling framework as some observations are not independent). Furthermore, as I have many potential explanatory variables (IVs), I would like to build sets of a priori candidate models for each DV and select the "best" ones based on AICc before performing multi-model inference (not sure if that's relevant here but at least you get the full picture).

I was recently advised (in the comments to one of my questions) to use so-called "robust" standard errors (SEs) when using a binomial GLM to model my first DV: i.e. $Y_{1}$. As I am not experienced with binomial GLMs nor familiar with the "robust" SEs approach, I started to search for resources online. Yet, I am now quite confused because:

  • There appears to be some controversy regarding the usefulness of these heteroscedasticity-robust/Huber-White's/sandwich SEs (e.g. here, here, here or even on Wikipedia);

  • No reference I found clearly explained in which case/for which models these "robust" SEs are required. Most resources on the subject seem to talk about OLS-fitted linear regressions and this thread seems to say that heteroscedasticity is not an issue in logistic regressions and that consequently, these "robust" SE are not required for this type of model. At this point, I thus assumed that "robust" SEs are required for OLS-fitted models but not for GLMs or any models fitted by Miximum Likelihood. Yet, other posts seem to indicate that GLMs should report "robust" SEs (e.g. here for a GLM using a Gamma family, or here for a negative binomial GLM);

  • Many resources keep using ambiguous terms such as "non-linear models" to speak about GLMs (e.g. here or here, but others think this is misleading?), and it seems that "robust" SEs are especially discussed by econometricians so I started wondering if this was not a cultural practice.

During my search for information, I also read that binomial GLMs should not be used to model percentages for which we do not know the total number of trials. Therefore, now I'm trying to model $Y_{1}$ in R with beta regression mixed-models using glmmTMB::glmmTMB() or (with some problems) one-inflated beta regression mixed-models using gamlss::gamlss(), and to model $Y_{2, 3, 4, 5}$ with logistic mixed models using lme4::glmer().
Long story short, in the end, I just want to know if I should compute/report "robust" SEs for these types of models? And if yes, how could I do that?

I realise these questions are probably stupid but I believe there are a lot of people out there with rusty math baggage and/or poor statistical training that struggle with the daunting multitude and complexity of terminology, assumptions, approaches, etc. existing in statistics.
So thanks in advance for your help, corrections and constructive criticism!

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