Compound probability distributions are useful for modelling and analysing multivariate count data. The logratio-normal-multinomial distribution is a count data model resulting from compounding a multinomial distribution for the counts with a multivariate logratio- normal distribution for the multinomial event probabilities. This distribution stands out as a more flexible model for count data than the commonly used Dirichlet-multinomial distribution. However, the logratio-normal-multinomial probability mass function does not admit a closed form expression and, consequently, numerical approximation is required for parameter estimation. In this work, different estimation approaches were introduced and evaluated. We concluded that estimation based on a quasi-Monte Carlo Expectation-Maximisation algorithm provides the best overall results. Building on this, the performances of the Dirichlet- multinomial and logratio-normal-multinomial models were compared through a number of examples using simulated and real count data. The latter model provided more realistic results, particularly when a complex variability structure in the vector of multinomial probabilities was considered.