#Biostatistics 370

#SIMPLE LINEAR REGRESSION
#ZAR EXAMPLE 19.1
ZAR=read.table("ZarEX19.1aR.txt")
ZAR
attach(ZAR)

X=winglen
Y=taillen

#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)

#TEST OF CORRELATION:
cor.test(X,Y, method="pearson",alternative="two.sided",conf.level=0.95)

#FISHER'S Z TRANSFORMATION TEST:

rho0=0.750  #set test value
n=length(Y)
n
r=sqrt(summary(LM)$r.squared)
r
z=(1/2)*log((1+r)/(1-r))
z
z0=(1/2)*log((1+rho0)/(1-rho0))
z0

#TEST STATISTIC Z:
Z=(z-z0)*sqrt(n-3)
Z

#CRITICAL VALUE OF TEST:
alpha=0.05  #set type 1 error rate
C=abs(qnorm(alpha/2,0,1))
C

#PROBABILITY:
PA=2*pnorm(Z,0,1)
PA
PB=2*(1-pnorm(Z,0,1))
PB
P=min(PA,PB)
P

#CONFIDENCE INTERVAL FOR RHO

z1=z-C*(1/sqrt(n-3))
z1
z2=z+C*(1/sqrt(n-3))
z2

CILower=(exp(2*z1)-1)/(exp(2*z1)+1)
CILower

CIUpper=(exp(2*z2)-1)/(exp(2*z2)+1)
CIUpper