Random Effect Model

Question

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?

Answer 1

Here the answers to the three questions (in same order):

1. The independence of the random effects ensure that responses coming from different individuals are independent. (And responses from the same individual are correlated.)
2. Each individual has one single (unobserved) random effect. How to test a hypothesis with a single unobserved value?
3. Because it simplifies the numeric computations by quite some amount. There are many extensions for non-normal random effects though, e.g. Baysian hierarchical models or frailty models in survival analysis.