# R/basics.R In mada: Meta-Analysis of Diagnostic Accuracy

#### Documented in CIrhocochran.Qfprsensspec

```expit <- function(x){
(1+exp(-x))^(-1)
}

logit <- function(x){
log(x/(1-x))
}

CIrho <- function(rho, N, level = 0.95){
stopifnot(rho < 1, rho > -1, N > 3, round(N) == N)
z <- atanh(rho)
kappa <- qnorm(1-(1-level)/2)
output <- matrix(cbind(rho, tanh(z - kappa*sqrt(1/(N-3))),tanh(z + kappa*sqrt(1/(N-3)))), ncol = 3)
colnames(output) <- c("rho", paste(100*(1-level)/2, "%", collapse =""),
paste(100*(1- (1-level)/2), "%", collapse =""))
output
}

#calculate cochranes Q, expects (s.e. of) log transformed DOR, posLR, negLR
cochran.Q<-function(x, weights)
{
if(length(weights) != length(x)){stop("Length of weights does not match length of x")}
bar.x<-sum(weights*x)/sum(weights)
Q<-sum(weights*((x-bar.x)^2))
k<-length(x)
if(k <= 1)stop("x needs to have length at least 2")
p.value<-pchisq(Q, k-1, lower.tail = FALSE)
#fp<-format(round(p.value, 4))
#fQ<-format(round(Q,4))
output<-c(Q,p.value,as.integer(k-1))
names(output)<-c("Q", "p-value", "df")
return(output)
}

checkdata <- function(X, nrowwarn = 5){
X <- as.data.frame(X)
if(!all(c("TP","FN","FP","TN") %in% names(X))){
stop("Data frame or matrix must have columns labelled TP, FN, FP and TN.")}
if(!identical(round(X),as.data.frame(apply(X,2,as.numeric)))){
warning("Some of the values of TP,FN,FP or TN do have non zero decimal places. Did you forget to round?")}
if(nrow(X) < nrowwarn){warning("There are very few primary studies!")}
idx_too_many_zeroes <- apply(X,1,function(x){sum(x == 0)}) > 2
if(any(idx_too_many_zeroes)){
stop(paste("Some study with three or more zeroes in 2x2 table! Row:", which(idx_too_many_zeroes)))
}
return(invisible(NULL))
}

sens <- function(x){
x <- as.data.frame(x)
x\$TP/(x\$TP + x\$FN)}
fpr <- function(x){
x <- as.data.frame(x)
x\$FP/(x\$FP + x\$TN)}
spec <- function(x){
x <- as.data.frame(x)
x\$TN/(x\$FP + x\$TN)}

## FROM package MKmisc, author: Matthias Kohl
## Confidence Intervals for Binomial Proportions
binomCI <- function(x, n, conf.level = 0.95, method = "wilson", rand = 123){
if (!is.na(pmatch(method, "wilson")))
method <- "wilson"

METHODS <- c("wald", "wilson", "agresti-coull", "jeffreys", "modified wilson",
"modified jeffreys", "clopper-pearson", "arcsine", "logit", "witting")
method <- pmatch(method, METHODS)

if (is.na(method))
stop("invalid method")

if (method == -1)
stop("ambiguous method")

if(length(x) != 1)
stop("'x' has to be of length 1 (number of successes)")
if(length(n) != 1)
stop("'n' has to be of length 1 (number of trials)")
if(length(conf.level) != 1)
stop("'conf.level' has to be of length 1 (confidence level)")
if(conf.level < 0.5 | conf.level > 1)
stop("'conf.level' has to be in [0.5, 1]")

alpha <- 1 - conf.level
kappa <- qnorm(1-alpha/2)
p.hat <- x/n
q.hat <- 1 - p.hat

if(method == 1){ # wald
est <- p.hat
term2 <- kappa*sqrt(p.hat*q.hat)/sqrt(n)
CI.lower <- max(0, p.hat - term2)
CI.upper <- min(1, p.hat + term2)
}
if(method == 2){ # wilson
est <- p.hat
term1 <- (x + kappa^2/2)/(n + kappa^2)
term2 <- kappa*sqrt(n)/(n + kappa^2)*sqrt(p.hat*q.hat + kappa^2/(4*n))
CI.lower <-  max(0, term1 - term2)
CI.upper <- min(1, term1 + term2)
}
if(method == 3){ # agresti-coull
x.tilde <- x + kappa^2/2
n.tilde <- n + kappa^2
p.tilde <- x.tilde/n.tilde
q.tilde <- 1 - p.tilde
est <- p.tilde
term2 <- kappa*sqrt(p.tilde*q.tilde)/sqrt(n.tilde)
CI.lower <- max(0, p.tilde - term2)
CI.upper <- min(1, p.tilde + term2)
}
if(method == 4){ # jeffreys
est <- p.hat
if(x == 0)
CI.lower <- 0
else
CI.lower <- qbeta(alpha/2, x+0.5, n-x+0.5)
if(x == n)
CI.upper <- 1
else
CI.upper <- qbeta(1-alpha/2, x+0.5, n-x+0.5)
}
if(method == 5){ # modified wilson
est <- p.hat
term1 <- (x + kappa^2/2)/(n + kappa^2)
term2 <- kappa*sqrt(n)/(n + kappa^2)*sqrt(p.hat*q.hat + kappa^2/(4*n))
if((n <= 50 & x %in% c(1, 2)) | (n >= 51 & n <= 100 & x %in% c(1:3)))
CI.lower <- 0.5*qchisq(alpha, 2*x)/n
else
CI.lower <-  max(0, term1 - term2)

if((n <= 50 & x %in% c(n-1, n-2)) | (n >= 51 & n <= 100 & x %in% c(n-(1:3))))
CI.upper <- 1 - 0.5*qchisq(alpha, 2*(n-x))/n
else
CI.upper <- min(1, term1 + term2)
}
if(method == 6){ # modified jeffreys
est <- p.hat
if(x == 0)
CI.lower <- 1 - (alpha/2)^(1/n)
else{
if(x == 1)
CI.lower <- 0
else
CI.lower <- qbeta(alpha/2, x+0.5, n-x+0.5)
}
if(x == n)
CI.upper <- (alpha/2)^(1/n)
else{
if(x == n-1)
CI.upper <- 1
else
CI.upper <- qbeta(1-alpha/2, x+0.5, n-x+0.5)
}
}
if(method == 7){ # clopper-pearson
est <- p.hat
CI.lower <- qbeta(alpha/2, x, n-x+1)
CI.upper <- qbeta(1-alpha/2, x+1, n-x)
}
if(method == 8){ # arcsine
p.tilde <- (x + 0.375)/(n + 0.75)
est <- p.tilde
CI.lower <- sin(asin(sqrt(p.tilde)) - 0.5*kappa/sqrt(n))^2
CI.upper <- sin(asin(sqrt(p.tilde)) + 0.5*kappa/sqrt(n))^2
}
if(method == 9){ # logit
est <- p.hat
lambda.hat <- log(x/(n-x))
V.hat <- n/(x*(n-x))
lambda.lower <- lambda.hat - kappa*sqrt(V.hat)
lambda.upper <- lambda.hat + kappa*sqrt(V.hat)
CI.lower <- exp(lambda.lower)/(1 + exp(lambda.lower))
CI.upper <- exp(lambda.upper)/(1 + exp(lambda.upper))
}
if(method == 10){ # witting
set.seed(rand)
x.tilde <- x + runif(1, min = 0, max = 1)
pbinom.abscont <- function(q, size, prob){
v <- trunc(q)
term1 <- pbinom(v-1, size = size, prob = prob)
term2 <- (q - v)*dbinom(v, size = size, prob = prob)
return(term1 + term2)
}
qbinom.abscont <- function(p, size, x){
fun <- function(prob, size, x, p){
pbinom.abscont(x, size, prob) - p
}
uniroot(fun, interval = c(0, 1), size = size, x = x, p = p)\$root
}
est <- p.hat
CI.lower <- qbinom.abscont(1-alpha, size = n, x = x.tilde)
CI.upper <- qbinom.abscont(alpha, size = n, x = x.tilde)
}

CI <- c(CI.lower, CI.upper)
attr(CI, "confidence level") <- conf.level
return(list("estimate" = est, "CI" = CI))
}

# expects vector x of successes and vector n of trials, output
binomCIvector<-function(x, n, conf.level = 0.95, method="wilson"){
N<-length(x)
CI<-matrix(NA,nrow=N,ncol=2)
colnames(CI)<-c("LL","UL")
for(i in 1:N){
CI[i,]<-binomCI(x[i],n[i],conf.level=conf.level,method=method)\$CI
}
return(CI)
}
```