#####Lecture for repeated measures


###Import Data


datum=read.csv(file.choose())
head(datum)


### Import library


library(nlme)


### Run randomized block analysis


results=lme(Pollution~Distance,data=datum,random=~1|River) #Mixed effects model with repeated measures of river
# Note - does not account for autocorrelation due to repeated measures. 
summary(results)


###Account for autocorrelation


help(lme)
# Note 'correlation' argument. Two correlation structures most often used are:
#Autoregressive (moving process) - corAR1() or corARMA(p=1,q=0)
#Moving average (moving error) - corARMA(p=0,q=1)


###Autoregressive model


results2=lme(Pollution~Distance,data=datum,random=~1|River,correlation=corARMA(p=1,q=0))
#alternative specification of same model:
#results2=lme(Pollution~Distance,data=datum,random=~1|River,correlation=corAR1())
summary(results)
#better than model without autoregressive autocorrelation?
anova(results,results2)



###Moving Average model


results3=lme(Pollution~Distance,data=datum,random=~1|River,correlation=corARMA(p=0,q=1)
summary(results)