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In the definition of the generalized linear models (GLIM), $g(\mu) = \beta^T X$, i.e. the link function of mean is modelled to be linear in explanatory variables.

But GLIM is said to be generalization of the general linear model. In the general linear model, however, $\mu$ is only required to be linear in parameter $\beta$, not necessarily in $X$.

So this makes me to ask if $g(\mu)$ in GLIM is only required to be linear in $\beta$ and not necessarily linear in $X$? Thanks!

Tim
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    Only linear in $\beta$. You can easily fit a GLIM of the form $g(\mu)=\beta_0+\beta_1sin(x_1)+\beta_2 x_2+\beta_3[sin(x_1)x_2]$ (Interpret, not so easily). Note that the linearity only holds on the link transformed scale, on the observed data scale the models are usually nonlinear. – Momo Jul 02 '13 at 22:33
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    The answer to this is the same as the answer to your corresponding question on linear regression. Please review the answer given to that question. – Glen_b Jul 02 '13 at 23:43

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In your statement of the linear model for the GLIM

$$ g(\mu) = X^T\beta$$

The model matrix $X$ is not just the variables, but any transformation thereof, such as quadratic terms, etc. This logic carries forward to the general linear model. The difference there being that $g(\eta) = \eta$ (the link function is the identity).

AdamO
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