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For a two–way contingency table with categorical variables, local odds ratios are commonly used to describe the relationships between the row and column variables. The ordinary case has mutually exclusive cell counts, i.e., each subject must fit into one and only one cell. However, in many surveys respondents may select more than one outcome category. We discuss the maximum likelihood and Mantel– Haenszel estimators of an assumed common local odds ratio for several 2 x c tables, treating the multiple responses as an extension of the multinomial sampling model. We derive new dually consistent (co)variance estimators for the Mantel–Haenszel local odds ratio estimators and show their performance in a simulation study.