Empty groups in a binomial homogeneity test

For independent $R_i\sim \mathrm{Bin}(n_i,p_i)$, a group with $n_i=0$ necessarily has $R_i=0$ and contributes likelihood one. Its probability parameter is unidentifiable and supplies no restriction to test. If $k$ groups have positive sample sizes, the unrestricted model has $k$ probability parameters and the common-probability model has one; the asymptotic homogeneity test therefore has $k-1$ degrees of freedom. Zero expected-count rows must be removed rather than divided by in the Pearson statistic.