Abstract
BackgroundHow can we compute a segregation or diversity index from a three-way or multi-way contingency table, where each variable can take on an arbitrary finite number of values and where the index takes values between zero and one? Previous methods only exist for two-way contingency tables or dichotomous variables. A prototypical three-way case is the segregation index of a set of industries or departments given multiple explanatory variables of both sex and race. This can be further extended to other variables, such as disability, number of years of education, and former military service.Methodology/Principal FindingsWe extend existing segregation indices based on Euclidean distance (square of coefficient of variation) and Boltzmann/Shannon/Theil index from two-way to multi-way contingency tables by including multiple summations. We provide several biological applications, such as indices for age polyethism and linkage disequilibrium. We also provide a new heuristic conceptualization of entropy-based indices. Higher order association measures are often independent of lower order ones, hence an overall segregation or diversity index should be the arithmetic mean of the normalized association measures at all orders. These methods are applicable when individuals self-identify as multiple races or even multiple sexes and when individuals work part-time in multiple industries.Conclusions/SignificanceThe policy implications of this work are enormous, allowing people to rigorously test whether employment or biological diversity has changed.
Highlights
There exists a much-deserved impetus to increase employment of traditionally under-represented groups
Reardon & Firebaugh [8] provide a marvelously simple and sensible framework for constructing segregation indices based on association measures
Their association measures – which are unnormalized segregation indices – are the weighted means of some function of the ratio of observed-to-expected values of each entry in a contingency table, where the mean is weighted by the expected values
Summary
There exists a much-deserved impetus to increase employment of traditionally under-represented groups. We need to do more than increase the number of sexes and races represented across each department, organization, or industry. Given two industries that hire men and women of all locally-recognized racial groups, how do we determine which has greater employment diversity or segregation? A prototypical three-way case is the segregation index of a set of industries or departments given multiple explanatory variables of both sex and race. This can be further extended to other variables, such as disability, number of years of education, and former military service
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