On the Upward Bias of the Dissimilarity Index and Its Corrections

The dissimilarity index of Duncan and Duncan is widely used in a broad range of contexts to assess the overall extent of segregation in the allocation of two groups in two or more units. Its sensitivity to random allocation implies an upward bias with respect to the unknown amount of systematic segregation. In this article, following a multinomial framework based on the assumption that individuals allocate themselves independently and that unit sizes are not fixed, we provide (1) a mathematical proof of the nonnegativity of the bias, (2) an analytic way of obtaining the same results of a recent bootstrap-based bias correction but without using resampling, and (3) a new bias correction that outperforms, in terms of both bias and mean square error, those based on grouped jackknife, bootstrap, and double bootstrap.