How to understand the following linear mixed model? I wonder how to understand the following mixed effect model.
$$y_{ij}|b_i \sim N(\mu+b_i+\beta_j,\sigma^2), \ \text{where} \ b_i \sim N(0,\sigma_b^2)$$
for $i=1,2,....,M, \ j=1,2,...,N$
What is the fixed effect and what is the random effect?  Can I write it in a matrix form? Why is it conditioned on $b_i$(I assume it's a random term?). If I remove the condition, how would the distribution of $y_{ij}$ be?
 A: (I strongly prefer using $\gamma$ or $u$ instead of $b$ when using "random effects" notation. Otherwise, if we accidentally capitalise something we might do a mistake. I don't understand why someone thought using $\beta$ and $b$ was a great convention...)
So, a standard "fixed-effects model" with normal errors would be:
$y = X\beta + \epsilon$, translating to: $y \sim N(X\beta, \sigma^2I)$.
Now we extend this to a mixed-effects model with a $q$-length vector $u$ of random effects such that $u \sim N(0, \sigma^2 D)$ and an associated model matrix $Z$ (with $Z$ being of dimensions $n \times q$). By doing that we have the mixed-effects model: $y = X\beta + Zu + \epsilon$, translating to: $y \sim N(X\beta, \sigma^2(I + ZDZ^T))$ or if we condition on $u$, translating to: $y|u \sim N(X\beta + Zu, \sigma^2I)$.
As you see, removing the conditioning on $u$ simply brought us back to "square one"; we have once again two sources of variation: one for the error term $\epsilon$ itself as well as one from the "random component" that is encoded in $Z$ and has distribution $N(0, \sigma^2 D)$.
I would recommend looking at some "easy" textbooks like: Faraway's "Extending the Linear Model with R" or West et al.'s "Linear Mixed Models: A Practical Guide Using Statistical Software" (see Sect. 2.2 on the "Specification of LMMs" particularly). After building some familiarity with the subject a short paper like Conditional and Marginal Models: Another View (2004) by Lee and Nelder can help you appreciate more nuanced aspects of conditioning.
