# 2) Coding potential models and model selection####
# i) Coding all potential models
# List of all Potential models-->
# Note: you can chose to not code ones that do not make biological sense.
# Linear model with no random effects
M0<-lm(Z_TP~Z_Length,data=data)
# Full model with varying intercepts
M1<-lmer(Z_TP~Z_Length + (1|Fish_Species) + (1|Lake), data=data, REML=FALSE)
# Full model with varying intercepts and slopes
M2<-lmer(Z_TP~Z_Length + (1+Z_Length|Fish_Species) + (1+Z_Length|Lake), data=data, REML=FALSE)
# No Lake, varying intercepts only
M3<-lmer(Z_TP~Z_Length + (1|Fish_Species), data=data, REML=FALSE)
# No Species, varying intercepts only
M4<-lmer(Z_TP~Z_Length + (1|Lake), data=data, REML=FALSE)
# No Lake, varying intercepts and slopes
M5<-lmer(Z_TP~Z_Length + (1+Z_Length|Fish_Species), data=data, REML=FALSE)
# No Species, varying intercepts and slopes
M6<-lmer(Z_TP~Z_Length + (1+Z_Length|Lake), data=data, REML=FALSE)
# Full model with varying intercepts and slopes only varying by lake
M7<-lmer(Z_TP~Z_Length + (1|Fish_Species) + (1+Z_Length|Lake), data=data, REML=FALSE)
# Full model with varying intercepts and slopes only varying by species
M8<-lmer(Z_TP~Z_Length + (1+Z_Length|Fish_Species) + (1|Lake), data=data, REML=FALSE)


# ii) Compare models using AICc values
# Compute AICc values for each model
AICc<-c(AICc(M0), AICc(M1), AICc(M2), AICc(M3), AICc(M4), AICc(M5), AICc(M6), AICc(M7), AICc(M8))
# Put values into one table for easy comparision
Model<-c("M0", "M1", "M2", "M3", "M4", "M5", "M6", "M7", "M8")
AICtable<-data.frame(Model=Model, AICc=AICc)
AICtable
# M8 has the lowest AICc value so it is the best fit model 
# M2 is also a good fit, but all other models are not nearly as good.
# Note when you compare models they must be fit by 
# Maximum Likelihood (ML) and not by Restricted Maximum Likelihood (REML)