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"epi.dsl" <- function(ev.trt, n.trt, ev.ctrl, n.ctrl, names, method = "odds.ratio", alternative = c("two.sided", "less", "greater"), conf.level = 0.95)
{
# Declarations:
k <- length(names)
a.i <- ev.trt
b.i <- n.trt - ev.trt
c.i <- ev.ctrl
d.i <- n.ctrl - ev.ctrl
N <- 1 - ((1 - conf.level) / 2)
z <- qnorm(N, mean = 0, sd = 1)
# Test each strata for zero values. Add 0.5 to all cells if any cell has a zero value:
for(i in 1:k){
if(a.i[i] < 1 | b.i[i] < 1 | c.i[i] < 1 | d.i[i] < 1){
a.i[i] <- a.i[i] + 0.5; b.i[i] <- b.i[i] + 0.5; c.i[i] <- c.i[i] + 0.5; d.i[i] <- d.i[i] + 0.5
}
}
n.1i <- a.i + b.i
n.2i <- c.i + d.i
N.i <- a.i + b.i + c.i + d.i
# For summary odds ratio:
R <- sum((a.i * d.i) / N.i)
S <- sum((b.i * c.i) / N.i)
E <- sum(((a.i + d.i) * a.i * d.i) / N.i^2)
F. <- sum(((a.i + d.i) * b.i * c.i) / N.i^2)
G <- sum(((b.i + c.i) * a.i * d.i) / N.i^2)
H <- sum(((b.i + c.i) * b.i * c.i) / N.i^2)
P <- sum(((n.1i * n.2i * (a.i + c.i)) - (a.i * c.i * N.i)) / N.i^2)
# For summary risk ratio:
R. <- sum((a.i * n.2i) / N.i)
S. <- sum((c.i * n.1i) / N.i)
# Individual study odds ratios:
if(method == "odds.ratio")
{OR.i <- (a.i * d.i) / (b.i * c.i)
lnOR.i <- log(OR.i)
SE.lnOR.i <- sqrt(1/a.i + 1/b.i + 1/c.i + 1/d.i)
SE.OR.i <- exp(SE.lnOR.i)
lower.lnOR.i <- lnOR.i - (z * SE.lnOR.i)
upper.lnOR.i <- lnOR.i + (z * SE.lnOR.i)
lower.OR.i <- exp(lower.lnOR.i)
upper.OR.i <- exp(upper.lnOR.i)
# Weights:
w.i <- (b.i * c.i) / N.i
w.iv.i <- 1 / (SE.lnOR.i)^2
# MH pooled odds ratios (relative effect measures combined in their natural scale):
OR.mh <- sum(w.iv.i * OR.i)/sum(w.iv.i)
lnOR.mh <- log(OR.mh)
SE.lnOR.mh <- sqrt(1/2 * ((E/R^2) + ((F. + G)/(R * S)) + (H/S^2)))
# DSL pooled odds ratios:
Q <- sum(w.iv.i * (lnOR.i - lnOR.mh)^2)
df <- k - 1
p.heterogeneity <- 1 - pchisq(Q, df)
tau.sq.upper <- Q - df
tau.sq.lower <- sum(w.iv.i) - (sum((w.iv.i)^2) / sum(w.iv.i))
# If Q is less than (k - 1) tau.sq equals zero:
tau.sq <- ifelse(Q < (k - 1), 0, (tau.sq.upper / tau.sq.lower))
w.dsl.i <- 1 / (((SE.lnOR.i)^2) + tau.sq)
lnOR.dsl <- sum(w.dsl.i * lnOR.i) / sum(w.dsl.i)
OR.dsl <- exp(lnOR.dsl)
SE.lnOR.dsl <- 1 / sqrt(sum(w.dsl.i))
SE.OR.dsl <- exp(SE.lnOR.dsl)
lower.lnOR.dsl <- log(OR.dsl) - (z * SE.lnOR.dsl)
upper.lnOR.dsl <- log(OR.dsl) + (z * SE.lnOR.dsl)
lower.OR.dsl <- exp(lower.lnOR.dsl)
upper.OR.dsl <- exp(upper.lnOR.dsl)
# Higgins and Thompson (2002) H^2 and I^2 statistic:
Hsq <- Q / (k - 1)
lnHsq <- log(Hsq)
if(Q > k) {
lnHsq.se <- (1 * log(Q) - log(k - 1)) / (2 * sqrt(2 * Q) - sqrt((2 * (k - 3))))
}
if(Q <= k) {
lnHsq.se <- sqrt((1/(2 * (k - 2))) * (1 - (1 / (3 * (k - 2)^2))))
}
lnHsq.l <- lnHsq - (z * lnHsq.se)
lnHsq.u <- lnHsq + (z * lnHsq.se)
Hsq.l <- exp(lnHsq.l)
Hsq.u <- exp(lnHsq.u)
Isq <- ((Hsq - 1) / Hsq) * 100
Isq.l <- ((Hsq.l - 1) / Hsq.l) * 100
Isq.u <- ((Hsq.u - 1) / Hsq.u) * 100
# Test of effect. Code for p-value taken from z.test function in TeachingDemos package:
effect.z <- log(OR.dsl) / SE.lnOR.dsl
alternative <- match.arg(alternative)
p.effect <- switch(alternative, two.sided = 2 * pnorm(abs(effect.z), lower.tail = FALSE), less = pnorm(effect.z), greater = pnorm(effect.z, lower.tail = FALSE))
# Results:
OR <- data.frame(OR.i, lower.OR.i, upper.OR.i)
names(OR) <- c("est", "lower", "upper")
OR.summary <- data.frame(OR.dsl, lower.OR.dsl, upper.OR.dsl)
names(OR.summary) <- c("est", "lower", "upper")
weights <- data.frame(w.iv.i, w.dsl.i)
names(weights) <- c("inv.var", "dsl")
Hsq <- data.frame(Hsq, Hsq.l, Hsq.u)
names(Hsq) <- c("est", "lower", "upper")
Isq <- data.frame(Isq, Isq.l, Isq.u)
names(Isq) <- c("est", "lower", "upper")
rval <- list(OR = OR, OR.summary = OR.summary, weights = weights,
heterogeneity = c(Q = Q, df = df, p.value = p.heterogeneity),
Hsq = Hsq,
Isq = Isq,
tau.sq = tau.sq,
effect = c(z = effect.z, p.value = p.effect))
return(rval)
}
else
if(method == "risk.ratio")
{RR.i <- (a.i / n.1i) / (c.i / n.2i)
lnRR.i <- log(RR.i)
SE.lnRR.i <- sqrt(1/a.i + 1/c.i - 1/n.1i - 1/n.2i)
SE.RR.i <- exp(SE.lnRR.i)
lower.lnRR.i <- lnRR.i - (z * SE.lnRR.i)
upper.lnRR.i <- lnRR.i + (z * SE.lnRR.i)
lower.RR.i <- exp(lower.lnRR.i)
upper.RR.i <- exp(upper.lnRR.i)
# Weights:
w.i <- (b.i * c.i) / N.i
w.iv.i <- 1 / (SE.lnRR.i)^2
# MH pooled risk ratios (relative effect measures combined in their natural scale):
RR.mh <- sum(w.i * RR.i) / sum(w.i)
lnRR.mh <- log(RR.mh)
SE.lnRR.mh <- sqrt(P / (R. * S.))
SE.RR.mh <- exp(SE.lnRR.mh)
lower.lnRR.mh <- log(RR.mh) - (z * SE.lnRR.mh)
upper.lnRR.mh <- log(RR.mh) + (z * SE.lnRR.mh)
lower.RR.mh <- exp(lower.lnRR.mh)
upper.RR.mh <- exp(upper.lnRR.mh)
# DSL pooled risk ratios:
Q <- sum(w.iv.i * (lnRR.i - lnRR.mh)^2)
df <- k - 1
p.heterogeneity <- 1 - pchisq(Q, df)
tau.sq.upper <- Q - df
tau.sq.lower <- sum(w.iv.i) - (sum((w.iv.i)^2) / sum(w.iv.i))
# If Q is less than (k - 1) tau.sq equals zero:
tau.sq <- ifelse(Q < (k - 1), 0, (tau.sq.upper / tau.sq.lower))
w.dsl.i <- 1 / (((SE.lnRR.i)^2) + tau.sq)
lnRR.dsl <- sum(w.dsl.i * lnRR.i) / sum(w.dsl.i)
RR.dsl <- exp(lnRR.dsl)
SE.lnRR.dsl <- 1 / sqrt(sum(w.dsl.i))
SE.RR.dsl <- exp(SE.lnRR.dsl)
lower.lnRR.dsl <- log(RR.dsl) - (z * SE.lnRR.dsl)
upper.lnRR.dsl <- log(RR.dsl) + (z * SE.lnRR.dsl)
lower.RR.dsl <- exp(lower.lnRR.dsl)
upper.RR.dsl <- exp(upper.lnRR.dsl)
# Higgins and Thompson (2002) H^2 and I^2 statistic:
Hsq <- Q / (k - 1)
lnHsq <- log(Hsq)
if(Q > k) {
lnHsq.se <- (1 * log(Q) - log(k - 1)) / (2 * sqrt(2 * Q) - sqrt((2 * (k - 3))))
}
if(Q <= k) {
lnHsq.se <- sqrt((1/(2 * (k - 2))) * (1 - (1 / (3 * (k - 2)^2))))
}
lnHsq.l <- lnHsq - (z * lnHsq.se)
lnHsq.u <- lnHsq + (z * lnHsq.se)
Hsq.l <- exp(lnHsq.l)
Hsq.u <- exp(lnHsq.u)
Isq <- ((Hsq - 1) / Hsq) * 100
Isq.l <- ((Hsq.l - 1) / Hsq.l) * 100
Isq.u <- ((Hsq.u - 1) / Hsq.u) * 100
# Test of effect. Code for p-value taken from z.test function in TeachingDemos package:
effect.z <- log(RR.dsl) / SE.lnRR.dsl
alternative <- match.arg(alternative)
p.effect <- switch(alternative, two.sided = 2 * pnorm(abs(effect.z), lower.tail = FALSE), less = pnorm(effect.z), greater = pnorm(effect.z, lower.tail = FALSE))
# Results:
RR <- data.frame(RR.i, lower.RR.i, upper.RR.i)
names(RR) <- c("est", "lower", "upper")
RR.summary <- data.frame(RR.dsl, lower.RR.dsl, upper.RR.dsl)
names(RR.summary) <- c("est", "lower", "upper")
weights <- data.frame(w.iv.i, w.dsl.i)
names(weights) <- c("inv.var", "dsl")
Hsq <- data.frame(Hsq, Hsq.l, Hsq.u)
names(Hsq) <- c("est", "lower", "upper")
Isq <- data.frame(Isq, Isq.l, Isq.u)
names(Isq) <- c("est", "lower", "upper")
rval <- list(RR = RR, RR.summary = RR.summary, weights = weights,
heterogeneity = c(Q = Q, df = df, p.value = p.heterogeneity),
Hsq = Hsq,
Isq = Isq,
tau.sq = tau.sq,
effect = c(z = effect.z, p.value = p.effect))
return(rval)
}
}
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