Introduction: Ordinal logistic regression is similar to logistic regression, except that the dependent variable is ordinal, i.e. ordered categorical. Ordinal logistic regression with proportional odds models ordered outcomes, assuming predictors have a constant effect across all category thresholds. This means the odds ratios remain the same regardless of how the outcome is split.
Typical application: In medicine, such models are often applied for outcomes on a likert scale, e.g. for no, mild, moderate, or severe infection.
Interpretation of coefficients: Say we have fitted a proportional odds ordinal logistic regression to estimate the effect of sex on the severity of infection with ordinal categories no, mild, moderate, or severe infection. We get an odds ratio of two for male. This then means that for sex =male, the risk is 2 times higher to have a moderate/severe infection if compared to no/mild infection. Then (by the proportional odds assumption) if sex =male the risk is also 2 times higher to have a severe infection if compared to no,mild and moderate infection together.
Main assumption: The proportional odds assumption says that the relationship between the predictor and the outcome is consistent across all thresholds of the ordinal outcome. That is, the effect of sex is assumed to be the same regardless of where we split the outcome categories. To make a visual analogy, imagine four doors labeled no, mild, moderate, and severe. sex=male pushes patients toward more severe outcomes. The proportional odds assumption says that no matter where you place a dividing line between the doors, the "push" of sex=male has the same strength.