Breakdown with respect to a non stratification variable

Suppose we want to study the Investment of enterprises in a country. Suppose the enterprises are stratified by number of employees: three strata $$s_1$$ from $$1$$ to $$9$$ employees, $$s_2$$ from $$10$$ to $$49$$ employees and $$S_3$$ $$\ge 50$$ employees. We take a stratified simple random sampling. The formula for the total investment is given by $$I= N_1\overline i_1+N_2\overline i_2+ N_3\overline i_3$$ where $$N_j$$ is the population number of enterprises in the stratum $$s_j$$ and $$\overline i_j$$ is the mean investment of the sampled $$n_j$$ enterprises from stratum $$s_j$$.

Now suppose for simplicity that the enterprises in the population have only two industrial activities, $$A_1$$ and $$A_2$$. I want to break down the total investment found above by sector of activity, which means $$I=I_1+I_2$$ where $$I_1$$ is the total investmenet in activity $$A_1$$ and similarly for $$I_2$$. How can this be done knowing that I did not take the sector of activity as a variable of stratification from the start? Is it mandatory that any breakdown of the total investment with respect to any variable $$x$$ should be done by taking the breakdown variable $$x$$ as a stratification variable before selecting the sample?

It's fine to break the results down by non-stratification variables. The computation is a little more complicated -- you can't just analyse $$A_1$$ and $$A_2$$ separately because the sampling isn't independent. However, any survey analysis software can handle this.
• Could you please give me the formulas for $I_1$ or $I_2$ or any reference dealing with this situation, I could not find any reference for this, thank you very much! Commented Aug 3 at 12:21