# Error models on a common response scale

Xiao et al. (2011), Ecology 92:1887–1894, DOI 10.1890/11-0538.1,
compare additive normal and multiplicative lognormal error models. The appropriate
error assumption matters; a log transform is not universally superior.
Their p.1890 discussion allows information-criterion comparison when both
likelihoods refer to the same observed, untransformed response.

For z=log10(y), y>0:
 ln p_Y(y|x)=ln p_Z(log10(y)|x)-ln(y ln(10)).
The Jacobian is essential. Raw-watt residual RSS and log-watt residual RSS cannot
be inserted into the same Gaussian AIC formula and compared directly. State the
response measure, units, noise distribution, sample and parameter count explicitly.
The included numerical integration verifies the transformed density normalizes.
It does not supply a new empirical model-selection result.

Leitão et al. (2016), “Is this scaling nonlinear?”, Royal Society Open Science 3,
150649, DOI 10.1098/rsos.150649, Eq.4–10, makes fluctuation assumptions explicit.
Heteroscedasticity can change scaling inferences. Conditional means and medians
also differ under a lognormal model. This does not show all urban scaling is an
artefact of boundaries, nor establish any NetPhysics cross-system law.

The published 0.5.0 primary split, score and conclusion are unchanged. Any further
error-model comparison on those already inspected observations is exploratory.