#Biostatistics 360

#SIMPLE LINEAR REGRESSION
#ZAR EXAMPLE 17.1
ZAR=read.table("ZarEX17.1R.txt")
ZAR
attach(ZAR)
X=ageX
Y=winglY

#CREATING LINEAR MODEL LM:
LM=lm(Y~X)
Yhat=fitted(LM)
e=residuals(LM)
RESULTS=data.frame(X,Y,Yhat,e)
RESULTS

#ANOVA F-TEST FOR SLOPE:
anova(LM)

#t-TESTS FOR SLOPE & INTERCEPT:
summary(LM)

#CONFIDENCE INTERVAL FOR SLOPE:
#THIS IS CALCULATED BY HAND!
n=length(Y)
Lxx=sum((X-mean(X))^2)
Lxx
Lxy=sum((X-mean(X))*(Y-mean(Y)))
Lxy
b=Lxy/Lxx
b
MSE=summary(LM)$sigma^2
MSE
alpha=0.05
C=abs(qt(1-alpha/2,n-2))
C
stderr=sqrt(MSE/Lxx)

CIL=b-C*stderr
CIL
CIU=b+C*stderr
CIU

#CONFIDENCE INTERVALS FOR 
#REGRESSION PARAMETERS
#THE EASIER WAY:
confint(lm(Y~X),level=0.95)


#CONFIDENCE INTERVALS FOR Yhat:
CONF=predict(lm(Y~X),interval="confidence",level=0.95)
CN=data.frame(CONF)
CN

#PREDICTION INTERVALS FOR Xn:
PRED=predict(lm(Y~X),interval="prediction",level=0.95)
PR=data.frame(PRED)
PR

#PLOTTING INTERVALS IN R:
plot(X,Y)
abline(lm(Y~X),col="blue")
segments(X,PR$lwr,X,PR$upr,col="red")
segments(X,CN$lwr,X,CN$upr,col="green")
points(X,CN$lwr,col="green")
points(X,CN$upr,col="green")
points(X,PR$lwr,col="red")
points(X,PR$upr,col="red")
points(X,PR$fit,col="blue")