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I'll try and explain where I am getting stuck.

I wish to model delay to treatment. I observe patients nested in hospitals. I have a mixture of patient and hospital level covariates. I suspect that there are unmeasured covariates that induce correlation between patients being treated in the same hospital. This would violate the basic assumptions about independent and identically distributed variables, and to get round this I add a random or fixed effect to the model.

Without this, then the unmeasured covariates will be operating through the measured hospital level covariates and induce bias. If estimate a fixed effects model then I would have an individual mean 'delay' for each hospital, but am I correct in thinking that these would be 'collinear' with any measured site level variable, and therefore not identifiable. Does this also apply to a random effects model? And if it does then is the aforementioned bias completely partioned to the 'random effect' or is some still operating through the measured site level covariates?

I have had a look at this great answer, and read the Bafumi and Gelman reference that was suggested, but I would very much appreciate any guidance.

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drstevok
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2 Answers2

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No reply so I have just re-read 'Multi-level and longitudinal modelling using Stata' by Rabe-Hesketh and Skrondal (2nd edition) pp.109-114. I think I have a better understanding now which I'll jot down here.

Without including the random-effect, the independence assumptions mentioned in the question are violated. In a multi-level model, the final reported co-efficient is a weighted average of the 'within' and 'between' effects. For a covariate that varies only at hospital level, then there are no 'within' effects, and so the co-efficient in the random-effects model represents only the between effects.

It is possible to estimate the between effects separately by creating a data set of the patient response and the covariates averaged over each hospital (collapsing my dataset from 12 000 patient level observations to 50 hospital level observations in my particular case). One can then simply run the original regression model in this collapsed data set to retrieve the between effects.

Note that the co-efficients for the hospital level covariates will not exactly match those retrieved from the random effects model if there are other covariates that vary both within and between hospitals, as in this collapsed model only the 'between' aspect of these will be estimated, and this will not necessarily have the same 'confounding' effect as the weighted average of the 'between' and 'within' effects in the full random effects model.

drstevok
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I understood your answer but not the question.

Do you wish to know, how and when to model random and fixed effects for a particular variable?

I was told to model random effects first and then add the fixed effects in subsequent model and check for better fit. If the random effects is insignificant, the model automatically estimates only the fixed effect of the variable.

Regarding higher level variables, yes, it would be better to include it as random effects. Some people explicitly mention in their research articles that covariates are only included in the random effects of the model as they are not interested in measures the same.

I would like to hear out a better explanation to mixed effects modeling.

KarthikS
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  • Thanks for the answer. It is a while since I have thought about this. I think my anxiety was that if a hospital has its own fixed effect, then does this prevent the identification of a hospital level covariate? For example, if I have 50 large and 50 small hospitals and included a fixed effect for each, then do I prevent the identification of the effect of a covariate representing large vs small hospitals. – drstevok Jun 19 '15 at 04:55
  • I think that should not be a problem. The nested manner of defining it will help you define fixed effects for each level. The inherent manner in which Hierarchical models have been modeled is the assumption that variables at each model are independent of each other. If you wish to capture the covariate representing large vs small hospitals; which I guess that you mean the collinearity between level-2 and level-3 variables, I can think of an intermediate approach. You can introduce an interaction term between the two levels between two variables that to suspect to have a covariate. – KarthikS Jun 19 '15 at 10:03