GLM specification: linear versus nonlinear / gamma versus lognormal

Modeling
,
1

I need to fit a GLM with a Gamma error term and would like some input on the best way to structure the model. The data I am modeling follow a functional form like this:

y[i] = X1^a + X2^b + X3^c

The response cannot be negative, and the variance in the response increases as a function of the predictors, both of which are reasons why the Gamma is attractive. There are strata that I’ll eventually use in a multilevel framework on the parameters, including a partial-pooling version. But for now I would like to know whether there is a compelling reason to fit the model in a non-linear versus linear framework. For example, I could (1) fit a nonlinear model, (2) linearize the model by taking logarithms of the predictors and retaining the Gamma error term (3) linearize the model by taking logarithms of the response and predictors, then fit the model using a normal error term. Is there any reason, from a computational or conceptual point of view, to prefer one of these parameterizations over the others?

Jim

2 
mu = exp(X1^a + X2^b + X3^c);
y ~ gamma(mu * sigma, sigma);

or

y ~ lognormal(X1^a + X2^b + X3^c, sigma);

X1^a, etc is a problem especially with exp.

There is not any reason, it is the reason and thats your data. Compare histograms
with density of lognormal and gamma, maybe weibull.
Do a intercept only fit, in R fitdistrplus or VGAM and figure out what suits your
data best.