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I understand that fixed vs. random effects have different meaning whether it be in biostatistics or econometrics. I recently came across a talk regarding fixed vs. random effects in the hierarchical setting, where the lecture slide is below:

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The slide details $14$ different cancer treatment studies, indexed by $s$, with the data $x_{0s}$ and $x_{1s}$ following conditional Binomials. The effect size above, $\lambda_s$, is none other than the log relative risk ratio known in Biostatistics.

My questions are:

1) if our effect here is $\lambda_s$, would it be a fixed or random effect in this model?

2) If we removed the $\lambda_s|\mu, \tau^2 \sim N(\mu, \tau_2)$ part and instead replace it with $\theta_{1s} \sim U(0,1)$, would it now be a random effects model?

In general, if $\theta_{1s}$ and $\theta_{0s}$ are both random, does that imply we have $\lambda_s$ is a random effect? Thanks!

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There are multiple definitions of fixed and random effects, but one of the definitions is that we assume that the effect is random if it is a realization of a random variable, where fixed effects can be seen as a special cases of random effects with variance equal to zero. In case of Bayesian models all the parameters are considered as random variables, so all of them may be seen as random effects. Because of the ambiguity Gelman and Hill in their book Data Analysis Using Regression and Multilevel/Hierarchical Models recommend talking rather about effects varying between groups, or not.

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