Is it possible to split linear predictors contribution up when talking glm of non-normal distributions?
If: $$µ_i = g^{-1}(η_i)$$ and $$µ_i = g^{-1}(β_0 + β_1X_{i1} + β_2X_{i2} +···+β_kX_{ik})$$
Is it then possible to split $µ_i$ up based on predictors? For instance, is this valid? : $$µ^{intercept}_i + µ_i^{predictor\ 1}= g^{-1}(β_0)\ + g^{-1}(β_1X_{i1})$$
No. Link functions are in most cases (except identity link) non-linear functions, so this won't work. Take as an example Poisson regression that predicts
$$ E[Y|X] = e^{\beta_0 + \beta_1 X_1 + \dots + \beta_k X_k} $$
by the properties of the exponential function this translates to a multiplicative relationship
$$ E[Y|X] = e^{\beta_0} e^{\beta_1 X_1} \dots e^{\beta_k X_k} $$
and it is one of the link functions where this relationship is the least ugly.
