Mean and variance of ratios of proportions from categories of a multinomial distribution

Duris, F.b,c, Gazdarica, J.b,d, Gazdaricova, I.d, Strieskova, L.b,d, Budis, J.a,b,c, Turna, J.c,d,e,  Szemes, T.b,d,e

aDepartment of Computer Science, Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava, Slovakia
bGeneton s.r.o., Bratislava, Slovakia
cSlovak Centre of Scientific and Technical Information, Bratislava, Slovakia
dDepartment of Molecular Biology, Faculty of Natural Sciences, Comenius University, Bratislava, Slovakia
eComenius University Science Park, Bratislava, Slovakia


Ratio distribution is a probability distribution representing the ratio of two random variables, each usually having a known distribution. Currently, there are results when the random variables in the ratio follow (not necessarily the same) Gaussian, Cauchy, binomial or uniform distributions. In this paper we consider a case, where the random variables in the ratio are joint binomial components of a multinomial distribution. We derived formulae for mean and variance of this ratio distribution using a simple Taylor-series approach and also a more complex approach which uses a slight modification of the original ratio. We showed that the more complex approach yields better results with simulated data. The presented results can be directly applied in the computation of confidence intervals for ratios of multinomial proportions.