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TL;DR: What figure or narrative can represent a fixed-effect ANOVA if we represent $complete~pooling$, $no~pooling$, and multi-level (partial pooling) models using the following figures (and narratives)?

Introductions to multi-level models often use interesting narratives and accompanying figures. For example, they say suppose we have a set of math scores $y_{nj}$ from students ($n$) in different schools ($j$). To estimate the mean ($\theta$) of these scores we can just do $complete ~pooling$:

enter image description here

But that leads to bias because schools may differ from one another and if we repeat the experiment many times, scores of the students in the same schools may be correlated with one another (with some expected correlation size).

So, we may think $no~pooling$ is a better option:

enter image description here

But this is also not what we want. Because while we account for correlations within schools, we are preforming separate analyses and are unable to connect one school to the other to come up with an overall estimate.

Well, here comes the compromise between the two, a multi-level structure (partial pooling):

enter image description here

Question: What figure or narrative can represent a fixed-effect ANOVA if we represent $complete~pooling$, $no~pooling$, and multi-level (partial pooling) models using the above figures and narratives?

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The fixed effect ANOVA is equivalent to a linear model in which the means of the groups are functions of indicator variables for group membership. Mathematically,the $j^{th}$ datum from group $i$ is

$$ y_{i,j} \sim \mathcal{N}( \theta_i, \sigma^2) $$

Here, $\theta_i$ is the mean for the $i^{th}$ group. Because the assumptions of this model are that all observations are independent, the best figure I can think of would be the no pooling model. Each $\theta_i$ can be estimated from the data which belong to that group. At least, I think so.

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  • $\begingroup$ @morouzian Correct. $\endgroup$ Commented Jul 4, 2020 at 23:14
  • $\begingroup$ @rnorouzian Complete pooling of the data would be lm(y~1). No pooling would be lm(y~group). Partial pooling would be lmer(y ~ 1 + 1|group) $\endgroup$ Commented Jul 5, 2020 at 1:56
  • $\begingroup$ @rnorouzian I don't but I have heard good things. $\endgroup$ Commented Jul 5, 2020 at 2:03
  • $\begingroup$ @rnorouzian Appreciated, but I'm sure I can find my own copy. $\endgroup$ Commented Jul 5, 2020 at 2:06

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