# Random Effect Model

One factor random effect model:

$$y_{ij}=\mu+\tau_{i}+\epsilon_{ij}\quad i=1,2,\ldots,a; j=1,2,\ldots,n$$

where,

$y_{ij}$ is the $j$th observation of $i$th treatment effect

$\mu$ is the overall mean

$\tau_{i}$ is the $i$th treatment effect and $\tau_{i}\sim NID(0,\sigma^2_{\tau})$

$\epsilon_{ij}$ random error term and $\epsilon_{ij}\sim NID(0,\sigma^2)$

and that $\tau_{i}$ and $\epsilon_{ij}$ are independent.

• Why do we need the independence assumption of $\tau_{i}$? Is that $\tau_{i}$ are dependent for fixed effect model?

And for random effect model it is written that : "Testing hypotheses about individual treatment effect is meaningless,so instead we test hypotheses about the variance components $\sigma^2_{\tau})$, $$H_0:\sigma^2_{\tau}=0$$ $$H_1:\sigma^2_{\tau}\ne 0$$ "

• Why is testing hypotheses about individual treatment effect meaningless?

• Why do we assume $\tau_{i}\sim NID(0,\sigma^2_{\tau})$ ? Why not other distributional pattern except the normal distribution?

• (1)Don't responses come from different individuals independent in fixed effect model ? If so then why the assumption is $\sum_{i=1}^{a}\tau_i=0$. Doesn't it imply $\tau_i$ are dependent? – user 31466 Aug 30 '14 at 13:23
• (2) Doesn't each individual have one single (unobserved) effect also in Fixedeffect model? If so how can we test $H_0:\tau_i=0$? – user 31466 Aug 30 '14 at 13:26
• A random effect is not the contrary of a fixed effect, so I do not really understand these follow-up questions. Maybe to (1): The centering is automatically done by adding the fixed intercept $\mu$. To (2): A fixed subject effect ($\mu_i$, say) is a parameter that can be estimated and inferred from the data (as long as there are multiple measurements per subject). – Michael M Aug 30 '14 at 13:55