Importance of the normality assumption in linear statistical models

Question

Statistical model for Complete Randomized design

$y_{ij} = \mu + \tau_i + \epsilon_{ij}$

where, $i$ denotes treatment and $j$ denotes observation.

$i=1,2,...,k\quad and \quad j=1,2,..., n_i$

$y_{ij}$ be a random variable that represents the response obtained on the $jth$ observation of the $ith$ treatment.

$\mu$ is the overall mean of the response $y_{ij}$

$\tau_i$ is the effect on the response of $ith$ treatment.

$\mu_i = \mu + \tau_i$

here $\mu_i$ denotes the true response of the $ith$ treatment.

and $\epsilon_{ij}$ is the random error term represent the sources of nuisance variation that is, variation due to factors other than treatments.

the assumption is $\epsilon_{ij}\sim^{iid} N(0,\sigma^2)$

Why do we have to assume $\epsilon_{ij}\sim^{iid} N(0,\sigma^2)$ ?

that is What is the importance of this assumption? If we do not assume it , what will be the effect?

Any help including reference will be appreciated.

Answer 1

The normality assumption is only used for testing and inference. Regression diagnostics can be utilized to assess this assumption. If the assumption is violated, either transformation of the dependent variable or generalized linear models can be the solution.

I think most regression textbooks addressed this issue, e.g. Page 86 of Regression Analysis by Example, 4th Edition, by Chatterjee & Hadi.