# Logistic Regression - Odds Ratio

I have the following two logistic regression models where the independent variables are the same for the two models:

logit(Brand1) = b0 + b1 x1 + b2 x2
logit(Brand2) = c0 + c1 x1 + c2 x2


I can't use multinomial logistic regression because the respondents can choose more than one brand (that is, Brand1 and Brand2 can equal 1 at the same time).

My question is, can I compare the odd ratios across the two models? If, for example I have odds ratio (Brand1,x1) = 3 and odds ratio (Brand2,x1) = 5, can I say odds ratio (Brand2, x1) is higher than odds ratio (Brand1,x1)? That is, can I say Brand2 is more likely than Brand1 to have characteristic x1?

However, you have a misconception of what odds ratios are. If the $b_1$ is higher than $c_1$ it doesn't mean that brand 1 has more of characteristic $x_1$. It means that a one point increase in $x_1$ has a stronger relationship with brand 1 being chose than with brand 2 being chosen.