Linear model vs general linear model

Question

What's the difference between a linear model and a general linear model?

Maybe: $Y=a X + b$ - general linear and $Y=a \psi(X) + b$ - linear ?

These forms come from the relevant Wikipedia articles on the linear and general linear models.

Answer 1

A general linear model doesn't generalize the function of $X$.

Indeed assuming you mean $E(Y|X)$ where you have $Y$ (and independent errors) whether or not there's a transformed predictor doesn't change things -- either way it would still be called a linear model (the conditional mean is a linear function of the parameters).

That is to say, consider $\alpha+\beta \psi(X)$. Now let $X^* = \psi(X)$. Then in terms of this new variable (the one used in the estimation) we have $\alpha+\beta X^*$. So either a linear model or a general linear model will be able to incorporate a transformation, $\psi$, (of the independent variable or variables) without difficulty.

Instead, with a multivariate response (each observation point is a vector of values), a general linear model generalizes the covariance structure of the error term so that the response values includes the possibility of correlated errors within the observation vector (that is, the components of $\mathbf{y}_i$ are correlated).

This feature allows us to place under one banner t-tests, ANOVA, regression, MANOVA, MANCOVA, multivariate regression and a number of other models/tools (while the multivariate techniques wouldn't necessarily be seen as covered by the term 'linear model', though the usage does vary).

[Between observations there is still independence; if you want instead to generalize to correlated errors between-observations, that would be generalized least squares as fcop pointed out in comments.]

Answer 2

In a linear model, we define prediction or regression function using a linear structure as follows: $y\approx E(y|x)=\omega_0 + \omega^\top x.$

While in a generalized linear model, we define prediction function or discriminatory function either as a linear in parameter or a non-linear in parameter through linear argument ($\omega^\top x +\omega_0$).

That is the hypothesis function for generalized linear model is $h(x)=g(\omega^\top x +\omega_0)$, where g may be a linear or non-linear function (known as activation function). While estimating the hypothesis function, we focus on estimating the parameter $\omega$ only as g is defined as per requirment.