Derived from the individual probability density functions, the expression is maximized to yield optimal values of β. The log-likelihood cannot be used alone as a measure of fit because it depends on sample size but can be used to compare two models.
For ordinal logistic regression, there are n independent multinomial vectors, each with k categories. These observations are denoted by y 1, ..., y n, where yi = (y i1, ..., yik ) and Σ j yij = mi is fixed for each i. From the ith observation yi , the contribution to the log likelihood is:
L(πi ; yi ) = Σ k yik log πik
The total log likelihood is a sum of contributions from each of the n observations:
L(π ; y) = Σ i L(πi ; yi )
| πik || probability of the ith observation for the kth category|