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Simple question but I can't find the answer anywhere online or in any of my texts. I think people assume you know, but I don't. What is an error structure, specifically for selecting the family distribution for a Generalized Linear Mixed Model? Is it the same as the distribution of your dependent variable?

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I believe error structure in this respect is referring to the "element of randomness" in your model. For example, in least squares regression, we often assume that the error term of the model (i.e. residuals) follows a normal distribution

$$Y = \beta_0 + \beta_1X_1 + \epsilon, \:\:\: \epsilon \sim N(0,\sigma^2)$$

Without the error term, our model would be deterministic, or in other words for a given set of values for your independent variables, we could always obtain the exact value of the dependent variable, which we generally know is not the case.

So the error structure for a generalized regression model depends on the distribution of your dependent variable and the link function used to relate the independent variables to the dependent one. You say you're using a generalized mixed model, which probably has its own specific assumptions that lead to the nature of the error structure.

Moreover, while this question is basic in nature, it not quite so depending on the nature of the data and the question you are trying to answer.

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  • $\begingroup$ Thanks for the response. This partly clears my question up, where I'm still confused is that I've read that I need to know my error structure to select the link function but here it seems as though the link function is required to know the error structure. So, what I assume needs to be done is identify the distribution of the dependent variable, which usually specifies the link function, which in turn would define the error structure? And because you can swap out different link functions for use with different distributions the DV distribution does not always define the error structure...? $\endgroup$ Nov 13, 2013 at 17:39
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    $\begingroup$ +1. I read "error structure" as a catch-all term for whatever you are assuming in a model about errors, including additive or multiplicative, what family, whether varying systematically or constant, whether on different levels, and so forth. It's whatever is not deterministic. In GLMs error family and link don't quite define each other, but some pairings seem more natural than others. For example, with gamma errors reciprocal link may be less advisable than log; it depends. The distribution of the response need not define the distribution of the error; the linkage is even looser there. $\endgroup$
    – Nick Cox
    Nov 13, 2013 at 17:56
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    $\begingroup$ So, if I understand you correctly there isn't really a tried and true method of determining the error structure. It is all up to what you know about the data, the system, and how it all interacts. So you build the model based on as much of an understanding of your data as you have rather than something like a standardized test or DV distribution? $\endgroup$ Nov 13, 2013 at 18:05
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    $\begingroup$ Quite so. A good method to determine the right model, including the error structure, would be well publicised as what most data analysts, certainly including myself, really want! Even if you have strong scientific grounds for a particular model form, they usually don't imply error structure, but they often make certain error structures implausible. If your data are counts, for example, then Poisson is more plausible than Gaussian. $\endgroup$
    – Nick Cox
    Nov 13, 2013 at 18:18
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    $\begingroup$ The link is not the same as a transformation, because it does not affect the error structure (!). A (nonlinear) transformation of the dependent variable will change the nature of its random variation, whereas a link can be varied independently of how the random variation is modeled. That makes GLMs more flexible than (just) transformations. $\endgroup$
    – whuber
    Nov 13, 2013 at 18:52

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