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I have a few questions about mlogit. I have a set of independent variables and a categorical, but ordered, dependent variable with three categories (Disagree, Neutral and Agree). The assumptions for ologit were not fulfilled. I chose Neutral as the base category. The coefficient of one of the independent variables X (a dummy taking the values 1 or 0), so the multinominal log odds, is negative for both Agree and Disagree, so the two effects are in different directions. The coefficient for Disagree is -1.264 and for Agree it is -0.672. If I calculate the corresponding probabilities, the change in probability of Agree is higher than the change in probability of Disagree, when taking the difference between X=1 and X=0 (using the equation on page 6: https://www.stata.com/manuals/rmlogit.pdf). This indicates that the effect on Agree is actually larger than the effect on Disagree, even though the absolute value of the coefficient is smaller.

Can the change in probability of an outcome compared to the base be higher than the change in probability of another outcome compared to the base, even though the absolute value of the coefficient (or log odds) is smaller for the former?

Also, what is the difference between the change in probability of an outcome relative to the base (as described on page 6 of https://www.stata.com/manuals/rmlogit.pdf) evaluated at X=1 and X=0 and the marginal effect (margins dy/dx), which also describes the probability of the outcome?

Thanks

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You have that: $$ \begin{align*} \ln\left(\frac{\Pr(D|X = 1)}{Pr(A|X = 1)}\right) &= \ln\left(\frac{\Pr(D|X = 0)}{\Pr(A|X= 0)}\right) -1.264 + 0.627,\\ &\le \ln\left(\frac{\Pr(D|X = 0)}{\Pr(A|X= 0)}\right) \end{align*} $$ So taking exponents, we have that: $$ \frac{\Pr(D|X = 1)}{\Pr(D| X = 0)} \le \frac{\Pr(A|X = 1)}{\Pr(A|X = 0)}. $$ What you find is that: $$ |\Pr(A|X=1)- \Pr(A|X = 0)| > |\Pr(D|X = 1) -\Pr(D|X = 0)| $$ These two are not necessarily in conflict.

For example: $$ \Pr(D|X= 1) = 0.1\\ \Pr(D|X = 0) = 0.3\\ \Pr(A|X = 1) = 0.4\\ \Pr(A|X = 0) = 0.677 $$

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  • $\begingroup$ Thanks you. On page 6 of stata.com/manuals/rmlogit.pdf they calculate the predicted probability of "Prepaid" for whites and non whites. Is the difference in these two values the same as the marginal effect (margins dy/dx in Stata)? Is the predicted probability still relative to the base outcome? $\endgroup$
    – Marlen
    May 17, 2021 at 16:25
  • $\begingroup$ marginal effects are computed by evaluating the derivatives $\frac{\partial \Pr{A|x)}{\partial x}$. I don't think these will give the same result as the difference. But for discrete random variables (e.g. dummy variables) differences are easier to interpret compared to marginal effects. $\endgroup$
    – tdm
    May 17, 2021 at 17:31

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