# How to estimate the confidence interval for a proportion of two estimated values?

I have estimated two values and confidence intervals (CIs), how can I estimate the CI of its ratio?

For example, a new method has been developed claiming to reduce SO2 emission in power plants by 20%. To test whether the claim is true, the method has been tested in 50 power plants. SO2 emission level has been measured and averaged for 5 consecutive days immediately before and another 5 consecutive days after the implementation of the new method. I used a linear mixed effect model to model the effect of implementation on the SO2 level, ie, Y is SO2 level, X is whether the implementation is in place (yes/no), and plant id is the random effect. R code:

library(lmerTest)
library(emmeans)
fit2<-lmer(Y~X+(1+X|Index)+temperature+Dew_point+Wind_spd+wind_dir+visibility+precip,
data=hebing3, control=lmerControl(optCtrl=list(maxfun=20000))
) ##X and meteorological conditions as fixed effects, Index as random effect.
fit2.means<-ref_grid(fit2)
res.means<-emmeans(fit2.means,"X") #### obtain estimated marginal means

Result: The Coefficient for X is -22.46 (95% CI: -26.69, -18.24). We also obtained an adjusted, estimated pre-implementation SO2 level (92.75 (86.63, 98.87)) from the result using the function emmeans (see above), which is capable of computing estimated marginal means of X. Thus the new method reduces SO2 emission by 24.2% (22.46/92.75=0.242). However, I would like to estimate the 95% CI for this proportion, but there seems no existing way to do this. Can someone help me with it? Many thanks.

• Sorry, the pre-implementation level was estimated using the function "emmeans". Auto-correction issue. ^_^
– Li Wending
May 27, 2020 at 7:59