Multinomial Logistic — nominal 3+ categories (per-class OR vs reference)

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Advanced & specialized

Independently verified. Every statistic this test reports has been re-derived against an independent reference — never the library the pipeline itself calls — the rendered output was read back in a browser, and the result is locked with a committed regression suite.

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Models a NOMINAL (unordered) categorical outcome with 3+ classes via the generalised logit model: log(P(Y=c)/P(Y=ref)) = β₀_c + Σ β_ic x_i for each non-reference class c.

For each non-reference class reports per-predictor coefficients, odds ratios with 95% CI (interpreted relative to the reference class), model fit (likelihood-ratio χ², McFadden R², AIC, BIC), per-class predicted probabilities, and classification accuracy at the argmax cut-point. Pre-checks IIA (independence of irrelevant alternatives) and multicollinearity.

Worked example

What predicts which of three transport modes a commuter chooses?

Commute mode (car / bus / bike — unordered categories) was regressed on distance and income with multinomial logistic regression (car as the reference).

Result

Greater distance lowered the odds of choosing bike over car (relative-risk ratio = 0.46 per SD) — longer trips favoured driving; income also shaped mode choice.

How you'd report it (APA)

A multinomial logistic regression showed distance lowered the odds of cycling versus driving (RRR = 0.46 per SD), with income also shaping mode choice.

Try it yourself: Load this ready-made sample and follow the run above.

When to use it

  • Nominal categorical outcome (3+ unordered classes) + predictors
    Brand choice (A/B/C/D, A as reference) ~ price + advertising + age (n = 800 consumers).
  • Diagnostic / triage classification with multiple categories
    Triage diagnosis (cardiac / pulmonary / metabolic / other) at ED admission ~ symptoms + biomarkers + vitals (n = 1200).

When NOT to — use instead

Hypotheses

H₀: β_jc = 0 for all predictors j and all non-reference classes c (equivalently OR_jc = 1).
Hₐ: at least one β_jc ≠ 0.

Parameter tested: matrix of class-specific coefficients β (k_predictors + 1 rows × k_categories − 1 columns)

Assumptions (and what to do if they fail)

≥ 10 cases per predictor per class (Peduzzi-style rule of thumb adapted for multinomial models).high

Check: min(per_class_n) / k.

If violated: Sparse classes inflate class-specific OR uncertainty; can produce extreme ORs.

Model classification accuracy (most-likely-class assignment) exceeds the no-information baseline (the largest single-class proportion).medium

Check: Accuracy − baseline > 0.

If violated: If model accuracy ≤ baseline, predictors do not improve classification.

No single case dominates the fit. Standardised Pearson residuals > |2| flagged.medium

Check: Count of |std residual| > 2.

If violated: Outliers can flip per-class OR signs in small samples.

VIF for each predictor moderate (< 5 preferred; < 10 tolerable).medium

Check: Max VIF.

If violated: High VIF inflates per-class OR standard errors.

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Guided setup, automatic assumption checks, effect sizes, figures and an APA write-up.

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