###Lecture 31 - Comparing ANOVA to Regression models when x can be treated either way

###Import data - Note that x variable is continuous, but multiple samples at each x level

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

### Plot relationship - notice that relationship between x and y may be non-linear

plot(PreyDens~Mosquitos,data=datum)


###Analyze data treating x variable as continuous variable

results1=lm(PreyDens~Mosquitos,data=datum)
summary(results)

###Analyze data treating x variable as Categorical variable
# First convert x variable from number to group
datum$MoCat=as.factor(datum$Mosquitos) #Creates a new variable called "MoCat" that is the same as
### Mosquitos, but that R will treat as categorical. 

#run analysis
results2=lm(PreyDens~MoCat,data=datum)
summary(results2)


###Test null hypothesis that additional parameters all equal zero (i.e., simpler model is adequate)
anova(results1,results2)

###Note that the null hypothesis is rejected if the relationship is nonlinear
###Note that we fail to reject the null hypothesis if the relationship is linear


### BONUS: How to fit a quadratic curve to data when x is continuous

results3=lm(PreyDens~Mosquitos+I(Mosquitos^2),data=datum)
anova(results3,results) #compare model 3 to model 1 (model 1 is simpler)
anova(results2,results3) #compare model 2 to model 3 (model 3 is simpler)