# How to include temporal correlation structure in a GAM with biannual counts

I have count data of a species and would like to analyze the population trend over the years. The species was count twice a year (early summer and late summer --> column "season") for several years on different sites. The model I have set up is the following:

gam(count ~ s(site, bs = "re") + s(year) + season, data=mydata, family = nb)

If I want to include a correlation structure

gamm(count ~ s(site, bs = "re") + s(year) + season, data=mydata, family = nb, correlation = corAR1(form=~year|site))

I get an error since I have two observations for each year and site. The solutions I can think of are:

1. Limit the analysis to one season -> only one count per year and site
2. Sum the counts per year and site -> only one count per year and site

I don't like the first one since I use only half of my data but I also think that it might be the only way since I know that higher counts in early summer lead to higher counts in late summer - does this mean my data is not independent?

What is the best way to analyze this data?

Thank you very much for your help!

For this I think you could just specify a variable that contains the time ordering, rather than year and nest that within site.
So you want a variable time (say) that is the integers 1, 2, 3, ..., T in site 1, and again in site 2 and so on. This would be pretty straight forward to code in R if you have no missing time points in any of the sites. It needs to be coded (integers) in each site the same I think because you'll have the same correlation estimated for each site so we want time 1 in site A to match time 1 in site B.
Once you have that variable, the correlation argument would then be corAR1(form = ~ time | site)