\[ \log \frac{P(Y_i = j)}{P(Y_i = \text{ref})} = \eta_{ij} = \beta_{0j} + \beta_{1j} x_{1i} + \cdots + \beta_{pj} x_{pi}, \qquad j = 1, \ldots, K - 1 \]
💬 Read it in plain English:
For each non-reference outcome category j, the model estimates its own linear equation showing how predictors shift the log-odds of landing in category j rather than the baseline reference category.
Yi, j, ref, K
Observed outcome class Yi across K total categories, comparing focal class j against baseline ref.
x1i … xpi
Values of the p explanatory variables for observation i.
β0j, βkj
Baseline log-odds (β0j) and change in log-odds of choosing class j over ref per 1-unit increase in predictor xk; eβkj is the Relative Risk Ratio.
ηij
Combined linear predictor for row i in outcome category j (with ηi,ref = 0 by definition).
\[ P(Y_i = j) = \frac{\exp(\eta_{ij})}{1 + \sum_{h \neq \text{ref}} \exp(\eta_{ih})}, \qquad P(Y_i = \text{ref}) = \frac{1}{1 + \sum_{h \neq \text{ref}} \exp(\eta_{ih})} \]
💬 Read it in plain English:
The softmax formula exponentiates each category's linear score and divides by the total across all categories so every customer's predicted probabilities are positive and add up to exactly 100%.
P(Yi = j)
Model-predicted probability that observation i falls into category j.
P(Yi = ref)
Model-predicted probability that observation i falls into the baseline reference category.