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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?

Thanks for your help.

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    $\begingroup$ It's perhaps worth pointing out that you can use multinomial logit if you want to. You just have to re-define the choices. There would be 4 alternatives: {nothing,brand1,brand2,both}. Now, the assumption of independent errors that MNL uses is pretty untenable for this choice space, so you should probably use Generalized Extreme Value (GEV) instead of MNL regression. $\endgroup$ – Bill Jan 24 '14 at 13:28
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You can certainly say that one odds ratio (OR) is higher than the other; it's another matter whether you can say it is statistically significantly higher. That question is answered in this thread.

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.

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