###Code for Lecture 2

### how to build linear models

### start with simple regression

datum=read.csv(file.choose()) ### first import some data

head(datum) ### see what we imported

plot(datum$Elevation,datum$Proportion)           ### plot the relationship

results=lm(Proportion~Elevation,data=datum)         ### runs a simple linear regression


summary(results)  ### spits out the analysis
results ### just shows you the object 'results'

### note individual coefficient estimates and constant as well of t-tests of estimates
### note F-statistic of full model - same as t-test in single variable regression

anova(results) ### generate an anova table of regression - not very useful as doesn't give anything other than p

coef(results) ### shows you the coefficient estimates from 'results'
residual(results) ### shows you the residuals from the regression fit of 'results'
resid(results) ### also shows you the residuals from the regression fit of 'results'
plot(resid(results)) ### plots the residuals in case order (order in which encountered)
plot(datum$x,resid(results)) ### plots the residuals in the order of x

### run a simple anova

plot(datum$PreyDens,datum$Proportion)
results=lm(Proportion~PreyDens,data=datum)

summary(results)
anova(results)

### Note that output just tells you difference between one group and reference
### Doesn't tell you difference between two groups listed in output
### to test for differences in groups, must combine them
### in order to combine must use dummy coded variables

results=lm(Proportion~Low+Medium+High),data=datum)
summary(results)
anova(results) ###note that anova now gives significance of each group rather than group as a whole

results2=lm(Proportion~Low+I(Medium+High),data=datum)  # combine groups in dummy coding; NOTE THAT PLUS DOESN'T REALLY MEAN ADD, JUST MEANS INCLUDE BOTH IN MODEL
summary(results2)
anova(results2,results) ### compares simpler model (combined groups) to more complex model
### if p-value is significant, then complex model is better, groups are different

### Also can remove any groups that are not different from the reference
results3=lm(Proportion~I(Medium+High),data=datum)
summary(results3)
anova(results3,results)

### general linear modeling is a multi-variable procedure, analyzes all variables at once
### reasons to do 'multiple regression' rather than univariate analyses:
### faster, more efficient
### accounts for collinearity among variables
### allows one to test for interactions
### allows one to account for pseudoreplication, random effects, and nested designs

results4=lm(Proportion~Elevation+Low+Medium+High,data=datum)   #NOTE THAT PLUS DOESN'T REALLY MEAN ADD, JUST MEANS INCLUDE BOTH IN MODEL
summary(results4)
anova(results4)