Random and fixed effects for count data when repeated measure subject is point of interest I'm having trouble understanding how I build a model to help me find out whether there are differences between referees and the personal penalties (temporary suspensions from the game) they give out to players during matches. In the sport in question we expect that any one referee should be interpreting and applying game rules in a similar manner to other referees. The alternative is that there are some referees who consistently give out more (or less of course) personal penalties per game than others. This is useful information to know for various reasons.
The dataset includes a number of different referees (n=16), the number of matches they have officiated (at least 15 games for each referee) and the total penalty minutes ("Pmins") they have awarded to players for each of those matches. The matches the referees officiate come from a range of tournaments (n=12) and there is information on the teams involved in each match and the year the tournament was held.
The Pmins are measured in whole minutes (there can't be a penalty of 1 minute 24 second) and so is count data. Here's an example of the data.

I'm at a very basic statistics level but previously I would have gone down the glm with poisson distribution (or quasi-poisson/negative binomial if the data is overdispersed) analytic route. Something like model<-glm(Pmins~Referee+Tournament,family="poisson",data=data)
However, here there are multiple measurements for each referee. So does this count as pseudo-replication? If it does, should I be looking to build a generalized linear mixed model to take account of the repeated measures?
If so, how do I assign the variables across fixed and random effects? I want to know something about the referees, whether there is a difference between them and with whom that difference lies if there are any. But, from what I have read the variable that is the subject of the repeated measure is usually assigned as the random effect (so something like - model<-glmer(Pmins~+x1+x2+(1|Referee),family="poisson",data=data)-perhaps). If I assign "Referee" as a fixed effect how do I account for the repeated-measures in the code for the glmm?
Any help and pointers to get me going in the right direction very gratefully received.

I think your research on the modelling concept is fairly solid. You can also build mixture models that have both a random intercept and slope. The random intercept is generally used for repeated measures and centred at 0, and the random slope would allow for fitting the effect of the grouping variable (and comparing estimates). So to answer your question, the Referee would still be a random effect and the other items you've deemed worth examining would be fixed.
I haven't actually read through these fully but these course notes seem like a user-friendly intro to the concept: https://yury-zablotski.netlify.app/post/mixed-models/#simple-random-slope-model
It's possible you'll run into convergence issues depending on the size of your dataset, they're hard to avoid with GLMM.
