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Suppose I have $4$ ordinal variables and a single covariate ($\log_{10}(x)$). When I run an ordinal probit model, I get three threshold coefficients and one probit slope. Call the threshold coefficients $b_1, \dots,b_3$ and the probit slope $b_4$.

So we can get $10^{b_1/b_4}$ for category 1, $10^{b_{2}/b_{4}}$ for category 2, and $10^{b_{3}/b_{4}}$ for category 3. But how would we calculate this value for category $4$? There is no coefficient for category 4.

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With one ordinal variable with 4 levels, you should only have 3 threshold coefficients. You are essentially dealing with an unobserved continuous outcome which has been divided into four observable buckets (the levels). The thresholds are the boundaries between the buckets, so there should be only 3. The variable $\log_{10} (x)$ and the slope parameter $b_4$ determine how you move from one bucket to the next as the unobserved variable changes.

Here's an example that might give you some intuition.

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First, I think you mean one ordinal variable with four levels.

Second, the coefficients have to compare each level to something else. In the coding your program uses, it is apparently the highest level, at least by default. That level is the reference. If you'd like, you can substitute a "no effect" there.

Also, where are you getting $10^{\frac{b_1}{b_4}}$ etc.? You seem to be combining the intercept with the slope into something else.

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