I am trying to fit a beta regression model using GAMLSS.
For each $y$, we have an indication of what patient $p$ it is, an indication of what month $m$ the observation took place, and at what location $l_1, l_2$ (longitude and latitude) the observation took place. We then further have covariates such as sex.
Here's an example of what the data set might look like:
y p m l_1 l_2 sex 0.5 1 Jan 22 -49 M 0.7 1 Jan 25 -51 M 0.4 1 Feb 21 -52 M 0.5 2 Jan 21 -51.5 F 0.5 2 Jan 24 -51 F 0.5 2 Jan 21 -51 F 0.5 2 Feb 20 -52 F 0.5 2 Feb 21.5 -55 F
This is my question. How should I set up the model to account for autocorrelation, both temporal and spatial?
I know that I can model the longitude and latitude using a smooth function $f(l_1, l_2)$, so that the model might be
gamlss(y ~ sex + f(l_1,l_2) + random(p), family = BE)
where $p$ is a random effect for each patient, and sex is a fixed factor effect.
For temporal autocorrelation, is it enough to include it as a fixed effect?
gamlss(y ~ sex + m + f(l_1, l_2) + ranndom(p), family = BE)?