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I have a data set from a survey with yes/no answers, coded as 1 and 0. Thus, both the dependent and independent variables will be binominal. Can I run a binominal logistic regression when both DV and IV are measures yes/no?

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Yes, you can run a binomial logistic regression when both DV and IV are binary (that is yes/no) variables. But this will be very similar to a contingency table analysis! For details see

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