Nothing
R/r_diff.r
In r2redux: R2 Statistic
#' r_diff function
#'
#' This function estimates var(R(y~x\[,v1]) - R(y~x\[,v2]))
#' where R is the correlation between y and x,
#' y is N by 1 matrix having the dependent variable, and
#' x is N by M matrix having M explanatory variables.
#' v1 or v2 indicates the ith column in the x matrix
#' (v1 or v2 can be multiple values between 1 - M, see Arguments below)
#' @param dat N by (M+1) matrix having variables in the order of cbind(y,x)
#' @param v1 This can be set as v1=c(1) or v1=c(1,2)
#' @param v2 This can be set as v2=c(2), v2=c(3), v2=c(1,3) or v2=c(3,4)
#' @param nv Sample size
#' @keywords R2 variance information matrix
#'
#' @export
#' @importFrom stats D cor dnorm lm logLik pchisq qchisq qnorm
#' @return This function will estimate significant difference between two PGS (either dependent or independent and joint or single). To get the test statistics for the difference between R(y~x\[,v1]) and R(y~x\[,v2]). (here we define R_1=R(y~x\[,v1])) and R_2=R(y~x\[,v2]))). The outputs are listed as follows.
#' \item{r1}{R_1}
#' \item{r2}{R_2}
#' \item{var1}{Variance of R_1}
#' \item{var2}{variance of R_2}
#' \item{var_diff}{Variance of difference between R_1 and R_2}
#' \item{r2_based_p}{P-value for significant difference between R_1 and R_2 for two tailed test}
#' \item{r_based_p_one_tail}{P-value for significant difference between R_1 and R_2 for one tailed test}
#' \item{mean_diff}{Differences between R_1 and R_2}
#' \item{upper_diff}{Upper limit of 95% CI for the difference}
#' \item{lower_diff}{Lower limit of 95% CI for the difference}
#' @examples
#' #To get the test statistics for the difference between R(y~x[,1]) and
#' #R(y~x[,2]). (here we define R_1=R(y~x[,1])) and R_2=R(y~x[,2])))
#'
#' dat=dat1
#' nv=length(dat$V1)
#' v1=c(1)
#' v2=c(2)
#' output=r_diff(dat,v1,v2,nv)
#' output
#'
#' #r2redux output
#'
#' #output$r1 (R_1)
#' #0.1958636
#'
#' #output$r2 (R_2)
#' #0.197006
#'
#' #output$var1 (variance of R_1)
#' #0.0009247466
#'
#' #output$var2 (variance of R_1)
#' #0.0001451358
#'
#' #output$var_diff (variance of difference between R_1 and R_2)
#' #3.65286e-06
#'
#' #output$r_based_p (two tailed p-value for significant difference between R_1 and R_2)
#' #0.5500319
#'
#' #output$r_based_p_one_tail (one tailed p-value
#" for significant difference between R_1 and R_2)
#' #0.2750159
#'
#' #output$mean_diff
#' #-0.001142375 (differences between R2_1 and R2_2)
#'
#' #output$upper_diff (upper limit of 95% CI for the difference)
#' #0.002603666
#'
#' #output$lower_diff (lower limit of 95% CI for the difference)
#' #-0.004888417
#'
#'
#' #To get the test statistics for the difference between R(y~x[,1]+[,2]) and
#' #R(y~x[,2]). (here R_1=R(y~x[,1]+x[,2]) and R_2=R(y~x[,1]))
#'
#' nv=length(dat$V1)
#' v1=c(1,2)
#' v2=c(2)
#' output=r_diff(dat,v1,v2,nv)
#' output
#'
#' #output$r1
#' #0.1974001
#'
#' #output$r2
#' #0.197006
#'
#' #output$var1
#' #0.0009235848
#'
#' #output$var2
#' #0.0009238836
#'
#' #output$var_diff
#' #3.837451e-06
#'
#' #output$r2_based_p
#' #0.8405593
#'
#' #output$mean_diff
#' #0.0003940961
#'
#' #output$upper_diff
#' #0.004233621
#'
#' #output$lower_diff
#' #-0.003445429
#'
#'
#' #Note: If the directions are not consistent, for instance, if one correlation
#' #is positive (R_1) and another is negative (R_2), or vice versa, it is
#' #crucial to approach the interpretation of the comparative test with caution.
#' #This caution is especially emphasized when applying r_diff()
#' #in a nested model comparison involving a joint model
r_diff = function (dat,v1,v2,nv) {
dat=scale(dat);omat=cor(dat)
if (length(v1)==1 & length(v2)==1) {
ord=c(1,(1+v1),(1+v2))
m1=lm(dat[,1]~dat[,(1+v1)])
s1=summary(m1)
m2=lm(dat[,1]~dat[,(1+v2)])
s2=summary(m2)
R2=s1$r.squared;mv2=1 #expected variance for s1r2
t100=(1/(nv ) *(1-R2)^2) #Infor matrix
var1=t100
R2=s2$r.squared;mv2=1 #expected variance for s1r2
t100=(1/(nv ) *(1-R2)^2) #Infor matrix
var2=t100
#dvr=(s1$r.squared^.5-s2$r.squared^.5)
dvr=(s1$coefficients[2,1]-s2$coefficients[2,1])
#aoa=olkin1_2(omat[ord,ord],nv)
r_aoa=r_olkin1_2(omat[ord,ord],nv)
#chi_dum=dvr2^2/aoa
r_chi_dum=dvr^2/r_aoa
#p3=pchisq(chi_dum,1,lower.tail=F)
r_p3=pchisq(r_chi_dum,1,lower.tail=F )
#uci=dvr2+1.96*aoa^.5
#lci=dvr2-1.96*aoa^.5
r_uci=dvr+1.96*r_aoa^.5
r_lci=dvr-1.96*r_aoa^.5
z = list(r1 = s1$coefficients[2,1], r2 = s2$coefficients[2,1], var1 = var1, var2 = var2, var_diff = r_aoa, r_based_p = r_p3, r_based_p_one_tail = r_p3 / 2, mean_diff = dvr, upper_diff = r_uci, lower_diff = r_lci)
#NOTE: r_based_p=r_p3 due to normal distribution
return(z)
}
if (length(v1)==2 & length(v2)==1 & length(unique(c(v1,v2)))==2) {
if (v1[1]==v2[1]) {ord=c(1,(1+v1[1]),(1+v1[2]))}
if (v1[2]==v2[1]) {ord=c(1,(1+v1[2]),(1+v1[1]))}
m1=lm(dat[,1]~dat[,1+v1[1]]+dat[,1+v1[2]])
s1=summary(m1)
m2=lm(dat[,1]~dat[,(1+v2)])
s2=summary(m2)
R2=s1$r.squared;mv2=2 #expected variance for s1r2
t100=(1/(nv) *(1-R2)^2) #Infor matrix
var1=t100
R2=s2$r.squared;mv2=1 #expected variance for s1r2
t100=(1/(nv ) *(1-R2)^2) #Infor matrix
var2=t100
LR=-2*(logLik(m2)-logLik(m1))
p1=pchisq(LR,1,lower.tail=F)
dvr=(s1$r.squared^.5-s2$r.squared^.5)
#chi_dum=dvr2/(1/(nv )*(1-dvr2)^2) #NCP
#p2=pchisq(chi_dum,1,lower.tail=F)
r_aoa=r_olkin12_1(omat[ord,ord],nv)
r_chi_dum=dvr^2/r_aoa
r_p3=pchisq(r_chi_dum,1,lower.tail=F)
#95% CI
r_uci=dvr+1.96*r_aoa^.5
r_lci=dvr-1.96*r_aoa^.5
z = list(r1 = s1$coefficients[2,1], r2 = s2$coefficients[2,1], var1 = var1, var2 = var2, var_diff = r_aoa, r_based_p = r_p3, r_based_p_one_tail = r_p3 / 2, mean_diff = dvr, upper_diff = r_uci, lower_diff = r_lci)
#NOTE: r_based_p=r_p3 due to normal
return(z)
}
if (length(v1)==2 & length(v2)==1 & length(unique(c(v1,v2)))==3) {
ord=c(1,(1+v1[1]),(1+v1[2]),(1+v2[1]))
m1=lm(dat[,1]~dat[,1+v1[1]]+dat[,1+v1[2]])
s1=summary(m1)
m2=lm(dat[,1]~dat[,(1+v2)])
s2=summary(m2)
R2=s1$r.squared;mv2=2 #expected variance for s1r2
t100=(1/(nv) *(1-R2)^2) #Infor matrix
var1=t100
R2=s2$r.squared;mv2=1 #expected variance for s1r2
t100=(1/(nv) *(1-R2)^2) #Infor matrix
var2=t100
dvr=(s1$r.squared^.5-s2$r.squared^.5)
r_aoa=r_olkin12_3(omat[ord,ord],nv) #check
r_chi_dum=dvr^2/r_aoa
r_p3=pchisq(r_chi_dum,1,lower.tail=F)
#95% CI
r_uci=dvr+1.96*r_aoa^.5
r_lci=dvr-1.96*r_aoa^.5
z = list(r1 = s1$coefficients[2,1], r2 = s2$coefficients[2,1], var1 = var1, var2 = var2, var_diff = r_aoa, r_based_p = r_p3, r_based_p_one_tail = r_p3 / 2, mean_diff = dvr, upper_diff = r_uci, lower_diff = r_lci)
#NOTE: r_based_p=r_p3 due to normal distribution
return(z)
}
if (length(v1)==2 & length(v2)==2 & length(unique(c(v1,v2)))==3) {
if (v1[1]==v2[1]) {ord=c(1,(1+v1[1]),(1+v1[2]),(1+v2[2]))}
if (v1[1]==v2[2]) {ord=c(1,(1+v1[1]),(1+v1[2]),(1+v2[1]))}
if (v1[2]==v2[1]) {ord=c(1,(1+v1[2]),(1+v1[1]),(1+v2[2]))}
if (v1[2]==v2[2]) {ord=c(1,(1+v1[2]),(1+v1[1]),(1+v2[1]))}
m1=lm(dat[,1]~dat[,1+v1[1]]+dat[,1+v1[2]])
s1=summary(m1)
m2=lm(dat[,1]~dat[,1+v2[1]]+dat[,1+v2[2]])
s2=summary(m2)
R2=s1$r.squared;mv2=2 #expected variance for s1r2
t100=(1/(nv) *(1-R2)^2) #Infor matrix
var1=t100
R2=s2$r.squared;mv2=2 #expected variance for s1r2
t100=(1/(nv) *(1-R2)^2) #Infor matrix
var2=t100
dvr=(s1$r.squared^.5-s2$r.squared^.5)
r_aoa=r_olkin12_13(omat[ord,ord],nv)
r_chi_dum=dvr^2/r_aoa
r_p3=pchisq(r_chi_dum,1,lower.tail=F)
#95% CI
r_uci=dvr+1.96*r_aoa^.5
r_lci=dvr-1.96*r_aoa^.5
z = list(r1 = s1$coefficients[2,1], r2 = s2$coefficients[2,1], var1 = var1, var2 = var2, var_diff = r_aoa, r_based_p = r_p3, r_based_p_one_tail = r_p3 / 2, mean_diff = dvr, upper_diff = r_uci, lower_diff = r_lci)
#NOTE: r_based_p=r_p3 due to normal distribution
return(z)
}
if (length(v1)==2 & length(v2)==2 & length(unique(c(v1,v2)))==4) {
ord=c(1,(1+v1[1]),(1+v1[2]),(1+v2[1]),(1+v2[2]))
m1=lm(dat[,1]~dat[,1+v1[1]]+dat[,1+v1[2]])
s1=summary(m1)
m2=lm(dat[,1]~dat[,1+v2[1]]+dat[,1+v2[2]])
s2=summary(m2)
R2=s1$r.squared;mv2=2 #expected variance for s1r2
t100=(1/(nv) *(1-R2)^2) #Infor matrix
var1=t100
R2=s2$r.squared;mv2=2 #expected variance for s1r2
t100=(1/(nv) *(1-R2)^2) #Infor matrix
var2=t100
dvr=(s1$r.squared^.5-s2$r.squared^.5)
r_aoa=r_olkin12_34(omat[ord,ord],nv)
r_chi_dum=dvr^2/r_aoa
r_p3=pchisq(r_chi_dum,1,lower.tail=F)
#95% CI
r_uci=dvr+1.96*r_aoa^.5
r_lci=dvr-1.96*r_aoa^.5
z = list(r1 = s1$coefficients[2,1], r2 = s2$coefficients[2,1], var1 = var1, var2 = var2, var_diff = r_aoa, r_based_p = r_p3, r_based_p_one_tail = r_p3 / 2, mean_diff = dvr, upper_diff = r_uci, lower_diff = r_lci)
#NOTE: r_based_p=p3 due to normal distribution
return(z)
}
}
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#' r_diff function
#'
#' This function estimates var(R(y~x\[,v1]) - R(y~x\[,v2]))
#' where R is the correlation between y and x,
#' y is N by 1 matrix having the dependent variable, and
#' x is N by M matrix having M explanatory variables.
#' v1 or v2 indicates the ith column in the x matrix
#' (v1 or v2 can be multiple values between 1 - M, see Arguments below)
#' @param dat N by (M+1) matrix having variables in the order of cbind(y,x)
#' @param v1 This can be set as v1=c(1) or v1=c(1,2)
#' @param v2 This can be set as v2=c(2), v2=c(3), v2=c(1,3) or v2=c(3,4)
#' @param nv Sample size
#' @keywords R2 variance information matrix
#'
#' @export
#' @importFrom stats D cor dnorm lm logLik pchisq qchisq qnorm
#' @return This function will estimate significant difference between two PGS (either dependent or independent and joint or single). To get the test statistics for the difference between R(y~x\[,v1]) and R(y~x\[,v2]). (here we define R_1=R(y~x\[,v1])) and R_2=R(y~x\[,v2]))). The outputs are listed as follows.
#' \item{r1}{R_1}
#' \item{r2}{R_2}
#' \item{var1}{Variance of R_1}
#' \item{var2}{variance of R_2}
#' \item{var_diff}{Variance of difference between R_1 and R_2}
#' \item{r2_based_p}{P-value for significant difference between R_1 and R_2 for two tailed test}
#' \item{r_based_p_one_tail}{P-value for significant difference between R_1 and R_2 for one tailed test}
#' \item{mean_diff}{Differences between R_1 and R_2}
#' \item{upper_diff}{Upper limit of 95% CI for the difference}
#' \item{lower_diff}{Lower limit of 95% CI for the difference}
#' @examples
#' #To get the test statistics for the difference between R(y~x[,1]) and
#' #R(y~x[,2]). (here we define R_1=R(y~x[,1])) and R_2=R(y~x[,2])))
#'
#' dat=dat1
#' nv=length(dat$V1)
#' v1=c(1)
#' v2=c(2)
#' output=r_diff(dat,v1,v2,nv)
#' output
#'
#' #r2redux output
#'
#' #output$r1 (R_1)
#' #0.1958636
#'
#' #output$r2 (R_2)
#' #0.197006
#'
#' #output$var1 (variance of R_1)
#' #0.0009247466
#'
#' #output$var2 (variance of R_1)
#' #0.0001451358
#'
#' #output$var_diff (variance of difference between R_1 and R_2)
#' #3.65286e-06
#'
#' #output$r_based_p (two tailed p-value for significant difference between R_1 and R_2)
#' #0.5500319
#'
#' #output$r_based_p_one_tail (one tailed p-value
#" for significant difference between R_1 and R_2)
#' #0.2750159
#'
#' #output$mean_diff
#' #-0.001142375 (differences between R2_1 and R2_2)
#'
#' #output$upper_diff (upper limit of 95% CI for the difference)
#' #0.002603666
#'
#' #output$lower_diff (lower limit of 95% CI for the difference)
#' #-0.004888417
#'
#'
#' #To get the test statistics for the difference between R(y~x[,1]+[,2]) and
#' #R(y~x[,2]). (here R_1=R(y~x[,1]+x[,2]) and R_2=R(y~x[,1]))
#'
#' nv=length(dat$V1)
#' v1=c(1,2)
#' v2=c(2)
#' output=r_diff(dat,v1,v2,nv)
#' output
#'
#' #output$r1
#' #0.1974001
#'
#' #output$r2
#' #0.197006
#'
#' #output$var1
#' #0.0009235848
#'
#' #output$var2
#' #0.0009238836
#'
#' #output$var_diff
#' #3.837451e-06
#'
#' #output$r2_based_p
#' #0.8405593
#'
#' #output$mean_diff
#' #0.0003940961
#'
#' #output$upper_diff
#' #0.004233621
#'
#' #output$lower_diff
#' #-0.003445429
#'
#'
#' #Note: If the directions are not consistent, for instance, if one correlation
#' #is positive (R_1) and another is negative (R_2), or vice versa, it is
#' #crucial to approach the interpretation of the comparative test with caution.
#' #This caution is especially emphasized when applying r_diff()
#' #in a nested model comparison involving a joint model
r_diff = function (dat,v1,v2,nv) {
dat=scale(dat);omat=cor(dat)
if (length(v1)==1 & length(v2)==1) {
ord=c(1,(1+v1),(1+v2))
m1=lm(dat[,1]~dat[,(1+v1)])
s1=summary(m1)
m2=lm(dat[,1]~dat[,(1+v2)])
s2=summary(m2)
R2=s1$r.squared;mv2=1 #expected variance for s1r2
t100=(1/(nv ) *(1-R2)^2) #Infor matrix
var1=t100
R2=s2$r.squared;mv2=1 #expected variance for s1r2
t100=(1/(nv ) *(1-R2)^2) #Infor matrix
var2=t100
#dvr=(s1$r.squared^.5-s2$r.squared^.5)
dvr=(s1$coefficients[2,1]-s2$coefficients[2,1])
#aoa=olkin1_2(omat[ord,ord],nv)
r_aoa=r_olkin1_2(omat[ord,ord],nv)
#chi_dum=dvr2^2/aoa
r_chi_dum=dvr^2/r_aoa
#p3=pchisq(chi_dum,1,lower.tail=F)
r_p3=pchisq(r_chi_dum,1,lower.tail=F )
#uci=dvr2+1.96*aoa^.5
#lci=dvr2-1.96*aoa^.5
r_uci=dvr+1.96*r_aoa^.5
r_lci=dvr-1.96*r_aoa^.5
z = list(r1 = s1$coefficients[2,1], r2 = s2$coefficients[2,1], var1 = var1, var2 = var2, var_diff = r_aoa, r_based_p = r_p3, r_based_p_one_tail = r_p3 / 2, mean_diff = dvr, upper_diff = r_uci, lower_diff = r_lci)
#NOTE: r_based_p=r_p3 due to normal distribution
return(z)
}
if (length(v1)==2 & length(v2)==1 & length(unique(c(v1,v2)))==2) {
if (v1[1]==v2[1]) {ord=c(1,(1+v1[1]),(1+v1[2]))}
if (v1[2]==v2[1]) {ord=c(1,(1+v1[2]),(1+v1[1]))}
m1=lm(dat[,1]~dat[,1+v1[1]]+dat[,1+v1[2]])
s1=summary(m1)
m2=lm(dat[,1]~dat[,(1+v2)])
s2=summary(m2)
R2=s1$r.squared;mv2=2 #expected variance for s1r2
t100=(1/(nv) *(1-R2)^2) #Infor matrix
var1=t100
R2=s2$r.squared;mv2=1 #expected variance for s1r2
t100=(1/(nv ) *(1-R2)^2) #Infor matrix
var2=t100
LR=-2*(logLik(m2)-logLik(m1))
p1=pchisq(LR,1,lower.tail=F)
dvr=(s1$r.squared^.5-s2$r.squared^.5)
#chi_dum=dvr2/(1/(nv )*(1-dvr2)^2) #NCP
#p2=pchisq(chi_dum,1,lower.tail=F)
r_aoa=r_olkin12_1(omat[ord,ord],nv)
r_chi_dum=dvr^2/r_aoa
r_p3=pchisq(r_chi_dum,1,lower.tail=F)
#95% CI
r_uci=dvr+1.96*r_aoa^.5
r_lci=dvr-1.96*r_aoa^.5
z = list(r1 = s1$coefficients[2,1], r2 = s2$coefficients[2,1], var1 = var1, var2 = var2, var_diff = r_aoa, r_based_p = r_p3, r_based_p_one_tail = r_p3 / 2, mean_diff = dvr, upper_diff = r_uci, lower_diff = r_lci)
#NOTE: r_based_p=r_p3 due to normal
return(z)
}
if (length(v1)==2 & length(v2)==1 & length(unique(c(v1,v2)))==3) {
ord=c(1,(1+v1[1]),(1+v1[2]),(1+v2[1]))
m1=lm(dat[,1]~dat[,1+v1[1]]+dat[,1+v1[2]])
s1=summary(m1)
m2=lm(dat[,1]~dat[,(1+v2)])
s2=summary(m2)
R2=s1$r.squared;mv2=2 #expected variance for s1r2
t100=(1/(nv) *(1-R2)^2) #Infor matrix
var1=t100
R2=s2$r.squared;mv2=1 #expected variance for s1r2
t100=(1/(nv) *(1-R2)^2) #Infor matrix
var2=t100
dvr=(s1$r.squared^.5-s2$r.squared^.5)
r_aoa=r_olkin12_3(omat[ord,ord],nv) #check
r_chi_dum=dvr^2/r_aoa
r_p3=pchisq(r_chi_dum,1,lower.tail=F)
#95% CI
r_uci=dvr+1.96*r_aoa^.5
r_lci=dvr-1.96*r_aoa^.5
z = list(r1 = s1$coefficients[2,1], r2 = s2$coefficients[2,1], var1 = var1, var2 = var2, var_diff = r_aoa, r_based_p = r_p3, r_based_p_one_tail = r_p3 / 2, mean_diff = dvr, upper_diff = r_uci, lower_diff = r_lci)
#NOTE: r_based_p=r_p3 due to normal distribution
return(z)
}
if (length(v1)==2 & length(v2)==2 & length(unique(c(v1,v2)))==3) {
if (v1[1]==v2[1]) {ord=c(1,(1+v1[1]),(1+v1[2]),(1+v2[2]))}
if (v1[1]==v2[2]) {ord=c(1,(1+v1[1]),(1+v1[2]),(1+v2[1]))}
if (v1[2]==v2[1]) {ord=c(1,(1+v1[2]),(1+v1[1]),(1+v2[2]))}
if (v1[2]==v2[2]) {ord=c(1,(1+v1[2]),(1+v1[1]),(1+v2[1]))}
m1=lm(dat[,1]~dat[,1+v1[1]]+dat[,1+v1[2]])
s1=summary(m1)
m2=lm(dat[,1]~dat[,1+v2[1]]+dat[,1+v2[2]])
s2=summary(m2)
R2=s1$r.squared;mv2=2 #expected variance for s1r2
t100=(1/(nv) *(1-R2)^2) #Infor matrix
var1=t100
R2=s2$r.squared;mv2=2 #expected variance for s1r2
t100=(1/(nv) *(1-R2)^2) #Infor matrix
var2=t100
dvr=(s1$r.squared^.5-s2$r.squared^.5)
r_aoa=r_olkin12_13(omat[ord,ord],nv)
r_chi_dum=dvr^2/r_aoa
r_p3=pchisq(r_chi_dum,1,lower.tail=F)
#95% CI
r_uci=dvr+1.96*r_aoa^.5
r_lci=dvr-1.96*r_aoa^.5
z = list(r1 = s1$coefficients[2,1], r2 = s2$coefficients[2,1], var1 = var1, var2 = var2, var_diff = r_aoa, r_based_p = r_p3, r_based_p_one_tail = r_p3 / 2, mean_diff = dvr, upper_diff = r_uci, lower_diff = r_lci)
#NOTE: r_based_p=r_p3 due to normal distribution
return(z)
}
if (length(v1)==2 & length(v2)==2 & length(unique(c(v1,v2)))==4) {
ord=c(1,(1+v1[1]),(1+v1[2]),(1+v2[1]),(1+v2[2]))
m1=lm(dat[,1]~dat[,1+v1[1]]+dat[,1+v1[2]])
s1=summary(m1)
m2=lm(dat[,1]~dat[,1+v2[1]]+dat[,1+v2[2]])
s2=summary(m2)
R2=s1$r.squared;mv2=2 #expected variance for s1r2
t100=(1/(nv) *(1-R2)^2) #Infor matrix
var1=t100
R2=s2$r.squared;mv2=2 #expected variance for s1r2
t100=(1/(nv) *(1-R2)^2) #Infor matrix
var2=t100
dvr=(s1$r.squared^.5-s2$r.squared^.5)
r_aoa=r_olkin12_34(omat[ord,ord],nv)
r_chi_dum=dvr^2/r_aoa
r_p3=pchisq(r_chi_dum,1,lower.tail=F)
#95% CI
r_uci=dvr+1.96*r_aoa^.5
r_lci=dvr-1.96*r_aoa^.5
z = list(r1 = s1$coefficients[2,1], r2 = s2$coefficients[2,1], var1 = var1, var2 = var2, var_diff = r_aoa, r_based_p = r_p3, r_based_p_one_tail = r_p3 / 2, mean_diff = dvr, upper_diff = r_uci, lower_diff = r_lci)
#NOTE: r_based_p=p3 due to normal distribution
return(z)
}
}
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