How do we interpret the coefficients of the random effects model? Also, what is the difference between the interpretation of the coefficients of random and fixed effects?
What I understand is this: in fixed effects, the coefficient of x shows that, what would be the increase in y, as compared to individual i's own average y, due to one unit change in x, as compared to i's average x. is this correct? Is the interpretation same for random effects?
 A: For the sake of explanation, suppose you have a simple mixed model with a fixed treatment effect and a random subject effect. Suppose further that there are 3 treatment levels A, B, C, and 10 subjects. The mixed model is $$
\mathbf{y = X\boldsymbol \beta + Z \boldsymbol \gamma + \boldsymbol \epsilon}
$$
where $X\boldsymbol \beta$ is the linear fixed-effects component, $\mathbf{Z}$ is the additional design matrix corresponding the the random-effects parameters, $\boldsymbol \gamma$.
Interpretation to fixed effect: Suppose we use treatment C as the reference level. Then the fixed effect $\beta_A$ tells us what the change in $y$ would be given a subject $i$ compared to subject $i$'s own average $\bar{y_i}$, due to treatment A.
You are almost correct in interpreting the fixed-effect, except that the "as compared to i's average x" part is not quite right. It is comparing to 1-unit change in $x$ if continuous, or different levels change if $x$ is categorical. It is not compared to average of $x$.
Interpretation of the random effect: The random effect $\gamma_i$ tells us what the additional change in $y$ would be due to subject $i$ itself, regardless of any change due to either treatment effect.
Hope this is clear to you.
