# How is probability y = j|X calculated from an ordinal logistic regression model?

I have an ordinal logistic regression model fitted with lrm from the rms library in R, and am presenting results as prob y = j|X using predict(fit, 'fitted.ind').

library(rms)
library(foreign)
fit = lrm(apply ~ gpa, data = df, x = T, y = T)
pred.fitted = round(predict(fit, type = "fitted"), 2)
head(pred.fitted, 1) # predictions for the first subject
# y >= somewhat likely 0.5; y >= very likely 0.12
pred.ind = round(predict(fit, type = "fitted.ind"), 2)
head(pred.ind, 1) # predictions for the first subject
# y = unlikely 0.5; y = somewhat likely 0.38; y = very likely 0.12
latex(fit, file = "") # see equation below


The equation for the example model is below. I understand how to calculate the 'fitted' values (predict(fit, 'fitted'): e.g., to calculate y >= somewhat likely (0.5, see above) for the first subject in the dataset (gpa = 3.26) I calculate:

1/1 + exp(2.374854 - 0.724819 * 3.26) = 0.5,

However, I don't know how to calculate the 'fitted.ind' probabilities. How do I calculate the probability y = j|X?

In the predictions for the subject above y = unlikely is the same as y >= somewhat likely, and y = somewhat likely is y >= somewhat likely - very likely, but I don't understand why.

P(y>=very likely) is

$$1 / (1 + exp(-(-4.3999 + .7249 * 3.26))) = 0.12$$

Because there is no higher category, the probability that y = very likely is the same as the probability that y >= very likely, i.e. 0.12.

P(y>=somewhat likely) is

$$1 / (1 + exp(-(-2.3749 + .7249 * 3.26))) = 0.5$$

But you'll note that P(y>=somewhat likely) = P(y=somewhat likely) + P(y=very likely). If y>=somewhat likely, y must be either somewhat likely or very likely, and not both at the same time.

So P(y=somewhat likely) = 0.5 - 0.12 = 0.38.

Following the same argument, or just using the fact that the probabilities have to sum to 1, you conclude that P(y=unlikely) = 0..5.