Normal distribution vs Lognormal distribution for residuals in Bayesian linear models

Question

When should I use normal distributions for residuals, and when should I use lognormal distribution in Bayesian linear regression?

For example, if I want to create a regression model for chemical exposure (exposures are positive values and distributed lognormal) across time I have four options on disposal:

1. log-log model (power model): y=a*t^b -> ln(y) = ln(a) + b*ln(t),
2. log-lin model (exponential model): y=a*exp(b*t) -> ln(y) = ln(a) + t*b, and
3. lin-log model: exp(y)=exp(a)*t^b -> y = a + b*ln(t)
4. lin-regression: y = a + b*t

The first two models convert lognormally distributed y to normally distributed ln(y), and I can use, in Bayesian linear regression, a normal distribution for residuals, but the last two models don't. Should I change a line in the code that specifies the normal distribution for residuals to use lognormal distribution in the last two cases?

BTW, are there any other decay models that I can transform and use with the linear model?

Any help is welcome!

Answer 1

The fact that the latter two models do not respect the allowable range of the response variable constitute deficiencies in those models, so usually we would avoid them. (In some cases it is possible that they will be good models locally, and one could arguably use them in some cases.) The usual practice would be to use one of the first two models, extended to add a stochastic error term which comes into the model for the log-response. So we would usually extend these to stochastic models as follows:

$$\begin{matrix} \text{Model 1 (stochastic version)} & & & \ln(y_i) = \ln(a) + b \ln(t_i) + \varepsilon_i, \\[12pt] \text{Model 2 (stochastic version)} & & & \ln(y_i) = \ln(a) + b t_i + \varepsilon_i. \quad \ \ \ \\[12pt] \end{matrix}$$

Beyond these models, there are certainly other models that could model a non-negative response variable. There are an infinite number of model forms that could be specified, so I won't attempt to elaborate on the possibilities.