library(metafor)
m1i  <- 15.6
m2i  <- 12.2
n1i  <- 15
n2i  <- 20
sd1i <- 3.82
sd2i <- 3.22

log(m1i) - log(m2i)
# [1] 0.245835
log(m1i + .5) - log(m2i + .5)
# [1] 0.2372173

sd1i^2/(n1i * (m1i)) + sd2i^2/(n2i * (m2i))
# [1] 0.1048541
sd1i^2/(n1i * (m1i + .5)) + sd2i^2/(n2i * (m2i + .5))
# [1] 0.1012445


lnR <- escalc(measure="ROM", m1i=m1i, m2i=m2i, sd1i=sd1i, sd2i=sd2i, n1i=n1i, n2i=n2i)
lnRk <- escalc(measure="ROM", add=0.5, to="all", m1i=m1i, m2i=m2i, sd1i=sd1i, sd2i=sd2i, n1i=n1i, n2i=n2i)
lnR; lnRk
#       yi     vi
# 1 0.2458 0.0075
#       yi     vi
# 1 0.2458 0.0075
all.equal(lnR, lnRk)
# [1] TRUE