# Ordinal Probit model in plain English

Suppose we have a model looking at the association between sodium intake $X$ and levels of body fat $Y$. So $Y$ is an ordinal variable that can take the integer values from $\{0,1,2 \}$. It is ordered according to increasing levels of body fat (e.g. "no bodyfat increase", "very little bodyfat increase", and "a lot of bodyfat increase").

What exactly does an ordinal probit model do? I plotted the data. So is an ordinal probit model trying to fit a line through the data points?

• This paper focusing on a graphical explanation may be of interest. It's difficult to imagine what your plot of the data might look like, because "0", "1", and "2" are merely codes--not numbers--to indicate the ordering of the $Y$ values. You could just as well have used $-1000$, $1$, and $1.1$, respectively: but any graph based on these arbitrary values would look quite different. In a precise sense, ordered regression is fitting curves through points on a trilinear diagram representing proportions of $Y$'s. – whuber Nov 28 '12 at 21:54

This unobserved creditworthiness (or BF growth) is function of the explanatory variables (like sodium) and parameters $\beta$ and a normally* distributed error $\varepsilon$. Each bond rating corresponded to a specific range of the creditworthiness. These ranges are not necessarily the same length. Suppose a firm is now at AA and becomes more creditworthy. Eventually, it would pass over the boundary between AA and AAA and the firm would get a new ranking. The ordered probit would estimate the parameters $\beta$ using MLE, together with the values of the boundary (aka cut values) defining the bins of the creditworthiness.