Suppose I estimate an ordinal logistic regression model: $Y$ ~ $\beta_1X + \beta_2Z$ where $Y$ is the ordinal-scale dependent variable with $y = 1, 2...k$ responses. $X$ and $Z$ are independent variables of any scale.
Under the proportional odds assumption (often known as the parallel lines assumption), I assume $\beta_1$ and $\beta_2$ are constant across all values $k$ of $Y$. Thus, I have one coefficient for each of $X$ and $Y$.
However, I want to relax the proportional odds assumption for the relationship between $X$ (similar to multinomial logistic regression) and $Y$ while maintaining the assumption for the relationship between $Z$ and $Y$. This is known as assuming partial proportional odds. I will now have $k-1$ coefficients for $X$ and one coefficient for $Y$.
How do I interpret each of the coefficients on $X$ in this new model?
For convenience, suppose $k=3$ and thus the coefficients on $X$ are $\beta_2X$, $\beta_3X$ (with $Y = 1$ being the baseline).