# Estimating the standard error of a ratio of means,(standard error of each mean known)

#### oheigeas

##### New Member
Hi everyone,

I'm stumped on a question based on dN/dS ratios in genetics. If I have two means, each with a standard error (same units), is there a simple way to approximate the standard error of the ratio?

A specific example.

dN is the average number of nucleotide substitutions per nonsynonymous site when comparing two sequences.
dS is the average number of nucleotide substitutions per synonymous site.

The dN/dS ratio is used as a way of detecting an excess of synonymous/nonsynonymous substitutions in a sequence, and thus protein sequence conservation.

The maths:

N=102.3
S = 35.7
dN=1.2953 +- 0.3006 (Standard error)
dS = 1.3986 +- 0/6759
dN/dS = 0.9262 +- ???

Would really appreciate any help, this is slowing me up!!
Thanks,
Seán

#### mp83

##### TS Contributor
Generally for estimating a ratio of means we can use :

> delta method (R : deltamethod() {msm})
> bootstrap (R : {boot})

An example

#R example of the function boot
#bootstrap of the ratio of means using the city data included in the boot package

#obtaining the data from the package
data(city)

#defining the ratio function
ratio <- function(d, w) sum(d$x * w)/sum(d$u * w)

#using the boot function
boot(city, ratio, R=999, stype="w")
ORDINARY NONPARAMETRIC BOOTSTRAP

Bootstrap Statistics :
original bias std. error
t1* 1.520313 0.04465751 0.2137274
PS: I thing this issue has come up earlier in this forum