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Some authors are saying that a mean and variance should be set for random variable, but that

The means of the random effects are usually assumed to be zero.

Later they are saying:

If a factor is random, then it is assumed to be a variable that is sampled from a population that has a particular mean and variance

My question are:

  1. Why in this case you would set a mean of 0?
  2. Why not assuming the mean of the parameter (to create the distribution of that random effect) your assuming to be random?
  3. What is the difference between a random effect that has a mean of 0 vs another number?
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The statements quoted from the document you linked to are consistent with each other.

The means of the random effects are usually assumed to be zero.

This is the usual assumption in mixed effects model, and very often a multivariate normal distribution with mean vector of zeros is assumed.

If a factor is random, then it is assumed to be a variable that is sampled from a population that has a particular mean and variance

Here they are talking about the population that the random effects are a sample from. The mean is typically estimated as a fixed effect, and the random effects are the deviations from that mean, so that the mean of the random effect is zero. For example, with a random intercept, a global intercept (fixed effect) would be estimated along with the variance of the random intercepts, which then allows each level of the factor to have it's own intercept (the global intercept plus the factor-level intercept).

Edit: As another example, consider a random intercepts and slopes model:

$$ y_{ij}= (\beta_0 + u_{0j})+(\beta_1 + u_{1j}) x_{1ij} +\epsilon_{ij}, $$

where $u_{0j}$ are the random intercepts and $u_{1j}$ are the random slopes. We can think of these either on their own, with a mean of zero, or together with their respective fixed effects, $\beta_{0j}=\beta_0 + u_{0j}$ and $\beta_{1j} = \beta_1 + u_{1j}$ respectively, where $\beta_0$ and $\beta_1$ would be their means.

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  • $\begingroup$ So what would be the difference between a random effect with a mean of 0 vs with a mean of a certain value or mu ~ norm(0,0.01)? $\endgroup$ – M. Beausoleil Aug 16 '16 at 2:44
  • $\begingroup$ @M.Beausoleil the mean would just be an offset to the global/fixed estimate $\endgroup$ – Robert Long Aug 16 '16 at 7:03
  • $\begingroup$ It's just that I don't get when you would need to use one or the other. In which case you would need to have a mean = 0 and another that can vary? Why are people "assuming" that it's 0? What's the reasoning behind that or the justification to do so? I heard that fixed effect can vary as well, they are not fixed in the sense that they only have one value. $\endgroup$ – M. Beausoleil Aug 16 '16 at 17:46
  • $\begingroup$ @M.Beausoleil it's not a question of when you would need to. It's the same thing - just a different parameterisation. If you were writing software to fit mixed models you would have to make a choice one way or the other. In all software that I've encountered, such as lme4, the mean of random effects is always zero and it's simply not possible for it to be anything else. $\endgroup$ – Robert Long Aug 16 '16 at 21:01

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