HSCI 410 — Lesson 5

Modelling Ordinal & Multinomial Data

Exploratory Data Analysis For Epidemiology

Kiffer G. Card, PhD, Faculty of Health Sciences, Simon Fraser University

Learning objectives for this lesson:

  • Select an appropriate model (multinomial, proportional-odds, adjacent-category, or continuation-ratio) based on study objectives and data
  • Fit all of the models listed above
  • Evaluate the assumptions on which each model is based
  • Interpret OR estimates from each model
  • Compute predicted probabilities from each model

This course was developed by Kiffer G. Card, PhD, as a companion to Dohoo, I. R., Martin, S. W., & Stryhn, H. (2012). Methods in Epidemiologic Research. VER Inc.

Section 1

Introduction & Overview of Models

⏱ Estimated time: 20 minutes

When Outcomes Have More Than Two Categories

In many epidemiological studies the outcome variable has more than two categories. These outcomes fall into two broad types: nominal data, where the categories have no natural ordering (e.g., type of disease, preferred clinic), and ordinal data, where the categories are ordered (e.g., pain severity: none, mild, moderate, severe).

The choice of model depends on whether the outcome is nominal or ordinal. Nominal data require multinomial logistic regression or log-linear models. Ordinal data can be analysed with the same multinomial model (ignoring the ordering), but more efficient approaches exploit the ordering: proportional-odds, adjacent-category, and continuation-ratio models.

📈
Nominal Data
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📊
Ordinal Data
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💡
Choosing the Right Model
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The Apgar Score Example

Throughout Chapter 17, the authors use Apgar scores as a running example. Apgar scores (measured at birth) are recoded into four ordinal categories. The research question is whether the number of prenatal visits is associated with Apgar score category.

Apgar CategoryCodePrenatal Visits < 6Prenatal Visits ≥ 6Total
1–6 (Low)0472572
71484290
825972131
9–10 (High)3134227361
Total288366654

Overview of the Four Models

Each of the four models for multi-category outcomes uses a different formulation of the logit (log-odds). Understanding the logit structure is the key to understanding each model.

Multinomial Logistic Regression (Eq 17.1)

Compares each outcome category to a baseline category. For J categories, the model estimates J−1 sets of coefficients. Each set describes how predictors relate to the log-odds of being in category j versus the baseline.

Equation 17.1
ln[p(Y = j) / p(Y = 1)] = β0(j) + β1(j)X

No assumptions about ordering are made, so this model is appropriate for both nominal and ordinal outcomes (though it is less efficient for ordinal data).

Proportional-Odds Model (Eq 17.2)

Based on cumulative probabilities. The logit compares the probability of being at or above category j versus below it. A single coefficient per predictor applies at every cutpoint—the proportional-odds assumption.

Equation 17.2
ln[p(Y ≥ j) / p(Y < j)] = β0(j) + β1X

This is the most common ordinal logistic model and is more parsimonious than the multinomial model.

Adjacent-Category Model (Eq 17.3)

Compares each category to the adjacent (next lower) category. This model is a constrained version of the multinomial model where the coefficient for categories n levels apart equals n times the coefficient for adjacent categories.

Equation 17.3
ln[p(Y = j) / p(Y = j−1)] = β0(j) + β1X

Like the proportional-odds model, it estimates a single β1 per predictor.

Continuation-Ratio Model (Eq 17.4)

Compares each category to all lower categories combined. This model is especially appropriate when the outcome represents sequential stages that must be “passed through” to reach higher levels (e.g., number of attempts to achieve certification).

Equation 17.4
ln[p(Y = j) / p(Y < j)] = β0(j) + β1X

Can be fit as a series of separate binary logistic regressions with appropriately recoded outcome variables.

✔ Check Your Understanding

1. Nominal outcome data differ from ordinal outcome data in that:

Nominal data have categories with no inherent ranking or ordering (e.g., type of disease), while ordinal data have categories that can be meaningfully ordered (e.g., disease severity). The number of categories is not related to the distinction.

2. How many sets of coefficients does a multinomial logistic model estimate for J outcome categories?

A multinomial logistic model compares each of the J−1 non-baseline categories to the baseline category. Each comparison requires its own set of coefficients, yielding J−1 sets in total.

3. Which model assumes the effect of a predictor is the same across all cutpoints?

The proportional-odds model estimates a single coefficient per predictor that applies at every cutpoint of the outcome. This is the “proportional odds” assumption—the OR is the same regardless of where you divide the outcome categories.

✎ Reflection

Think of an ordinal outcome variable from your own field of study. What are the categories, and which of the four models introduced here do you think would be most appropriate? Why?

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Section 2

Multinomial Logistic Regression

⏱ Estimated time: 25 minutes

The Multinomial Logistic Model

The multinomial logistic model simultaneously fits J−1 separate logistic models, each comparing one category to a chosen baseline. All parameters are estimated jointly, so the model accounts for the correlation among the comparisons.

Predicted Probabilities

The predicted probability for each outcome category is computed from the set of linear predictors. Let Xβ(j) denote the linear predictor for category j.

Equation 17.5 — Probability for the baseline category
p(Y = 0) = 1 / [1 + exp(Xβ(1)) + exp(Xβ(2)) + exp(Xβ(3))]
Equation 17.6 — Probability for category j
p(Y = j) = exp(Xβ(j)) / [1 + exp(Xβ(1)) + exp(Xβ(2)) + exp(Xβ(3))]

These probabilities always sum to 1 across all categories. Each predicted probability depends on all sets of coefficients, not just the coefficients for that category.

Interpreting Odds Ratios

Exponentiated coefficients from a multinomial model are technically ratios of relative risks (RRR), not true odds ratios. Each exp(β(j)) gives the ratio of the probability of being in category j relative to the baseline, for a one-unit change in the predictor.

Because the multinomial model estimates separate coefficients for each comparison, the effects can differ across categories. For ordinal outcomes, you would typically expect a gradient—more pronounced effects for the categories furthest from the baseline.

Predicted Probabilities

Predicted probabilities from a multinomial model vary by the values of all predictors. To communicate results, it is often useful to compute predicted probabilities at specific covariate patterns (e.g., prenatal visits < 6 vs ≥ 6) and present them in a table or graph.

Testing significance can be done with Wald tests (for individual coefficients) or likelihood-ratio tests (for overall effects). Because the multinomial model has J−1 coefficients per predictor, an overall LRT that tests all J−1 simultaneously is generally preferred over examining individual coefficients.

Independence of Irrelevant Alternatives (IIA)

The multinomial logistic model assumes IIA: the odds of choosing one category over another are independent of what other categories are available. If this assumption is violated, adding or removing a category would change the odds between the remaining categories.

Two tests are available: the Hausman-McFadden test and the Small-Hsiao test. However, these tests often give conflicting results, and IIA violations are primarily a concern for nominal data where alternatives are genuinely substitutable (e.g., choosing a mode of transport). For ordinal data, IIA is rarely a practical concern.

📋 Example: Apgar Scores and Prenatal Visits

In the Apgar score example, the multinomial model (with category 3 [9–10] as baseline) produced the following key results for prenatal visits (≥6 vs <6):

  • Category 0 vs 3: OR = 0.24 — those with ≥6 visits have 76% lower relative risk of a low Apgar score
  • Category 1 vs 3: OR = 0.65 — 35% lower relative risk of Apgar = 7
  • Category 2 vs 3: OR = 0.72 — 28% lower relative risk of Apgar = 8

The gradient (0.24 → 0.65 → 0.72) shows the strongest effect for the lowest Apgar category, as expected for an ordinal outcome.

Note on IIA Tests

The Hausman-McFadden and Small-Hsiao tests for IIA often give conflicting results and are not always reliable. In practice, IIA is mainly a concern for nominal (unordered) outcomes with genuinely substitutable alternatives. For ordinal outcomes, it is rarely problematic. Regression diagnostics can be performed by fitting ordinary logistic models for pairs of categories and using standard diagnostic techniques.

Alternative-Specific Data

In some situations, predictors may vary across alternatives rather than (or in addition to) varying across observations. For example, in a study of clinic choice, the distance to each clinic varies by alternative. Special formulations of the multinomial model (conditional logit or mixed logit) accommodate such alternative-specific data.

✔ Check Your Understanding

1. In multinomial logistic regression, the exponentiated coefficients represent:

The exponentiated coefficients in a multinomial logistic model are ratios of relative risks (RRR), not true odds ratios. Each RRR compares the probability of being in a given category relative to the baseline for a one-unit change in the predictor.

2. The IIA assumption states that:

The Independence of Irrelevant Alternatives (IIA) assumption states that the relative odds of choosing one category over another do not depend on what other categories are available. Violation of IIA means that adding or removing a category would change the estimated odds between remaining categories.

3. How are predicted probabilities computed from a multinomial model?

The predicted probability for category j is exp(Xβ(j)) divided by [1 + Σ exp(Xβ(k))] for all non-baseline categories k. For the baseline category, the probability is 1 divided by the same denominator. This ensures all probabilities sum to 1.

✎ Reflection

Consider the Apgar score example. Why do you think the OR for the lowest Apgar category (0.24) is more extreme than for the middle categories? What does this gradient tell us about prenatal care and birth outcomes?

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Section 3

Proportional-Odds Model

⏱ Estimated time: 25 minutes

The Most Common Ordinal Model

The proportional-odds model (also called the cumulative logit model or ordinal logistic regression) is the most widely used model for ordinal outcomes. It is based on the idea of an underlying continuous latent variable that is divided into the observed ordinal categories by a series of cutpoints.

Equation 17.7 — Latent Variable
Si = β1X1i + β2X2i + … + βkXki + εi

The latent variable Si is divided by cutpoints (τ1, τ2, …, τJ−1) into J observed categories. If Si falls between τj−1 and τj, the observation is classified into category j.

The Proportional-Odds Logit

The model takes the form of a cumulative logit: logit(p(Y ≥ j)) = β0j + βX. The key feature is that the intercept varies across cutpoints (giving parallel lines on a logit scale) but the slope coefficients are the same for every cutpoint. This means a single OR summarises the effect of each predictor across all levels of the outcome.

Equation 17.9 — Predicted Probability from Latent Variable
p(Y = j) = p(S ≤ τj) − p(S ≤ τj−1)
📋 Example: Apgar Scores — Proportional-Odds Model

In the Apgar score example, the proportional-odds model yields an OR of 1.59 for prenatal visits (≥6 vs <6). This means that individuals with 6 or more prenatal visits have 1.59 times the odds of being at or above any given Apgar category, compared to those with fewer visits. This single OR applies at every cutpoint (0 vs 1+, 0–1 vs 2+, and 0–2 vs 3).

Testing the Proportional-Odds Assumption

The proportional-odds assumption is that the effect of each predictor is the same at every cutpoint. If violated, the model may give misleading results. Several tests are available:

Testing Proportional Odds

Three main approaches exist:

  • Approximate LRT: Compare the log-likelihoods of the proportional-odds model and the multinomial model. A significant difference suggests the proportional-odds assumption is violated.
  • Wolfe-Gould approximate LRT: Based on J−1 separate binary logistic models at each cutpoint. Sum the log-likelihoods and compare to the proportional-odds model.
  • Brant (Wald) test: Provides both an overall test and individual tests for each predictor, showing which specific variables violate the assumption.
Generalised Ordinal Logistic Regression

If the proportional-odds assumption is violated, a generalised ordinal logistic regression model allows separate coefficients at each cutpoint. This model is equivalent to fitting J−1 separate binary logistic regressions simultaneously. It is more flexible but less parsimonious than the proportional-odds model.

Partial Proportional-Odds Model

A compromise approach is the partial proportional-odds model, which relaxes the proportional-odds assumption for selected predictors only (those that fail the Brant test) while maintaining it for the rest. This provides a good balance between flexibility and parsimony. Other alternatives include the stereotype logistic model and the heterogeneous choice logistic model.

⚠ The Proportional-Odds Assumption in Practice

The proportional-odds assumption is often violated in practice, especially with many predictors or when the outcome categories represent very different phenomena. Always test this assumption before reporting results from a proportional-odds model. If violated, consider a partial proportional-odds model or generalised ordinal logistic regression.

Brant Test Results Example

The Brant test provides both an overall test and predictor-specific tests. Here is an example of how results might be presented:

Predictorχ²dfP-valueAssumption Holds?
Prenatal visits2.1420.343Yes
Maternal age8.9220.012No
Parity1.0320.598Yes
Overall12.4560.053Borderline

In this example, only maternal age violates the assumption. A partial proportional-odds model that allows maternal age to have different effects at each cutpoint (while constraining prenatal visits and parity) would be appropriate.

Regression Diagnostics

Regression diagnostics for the proportional-odds model can be conducted by fitting binary logistic models at each cutpoint and applying the diagnostic techniques from Chapter 16 (residual analysis, influence measures, goodness-of-fit tests).

✔ Check Your Understanding

1. The proportional-odds model assumes:

The proportional-odds (or parallel lines) assumption states that the coefficients for each predictor are the same regardless of which cutpoint is used to dichotomise the outcome. This means the OR is the same at every cutpoint.

2. If the proportional-odds assumption is violated for some but not all predictors, which model can be used?

The partial proportional-odds model relaxes the proportional-odds assumption for specific predictors (those that violate it) while maintaining the constraint for the rest. This provides a balance between the fully constrained proportional-odds model and the unconstrained multinomial model.

3. The latent variable in a proportional-odds model represents:

The proportional-odds model assumes an underlying continuous latent variable (Si) that is divided into the observed ordinal categories by cutpoints (τ). The latent variable represents the true continuous quantity that we observe only in categorised form.

✎ Reflection

Why do you think the proportional-odds assumption is so often violated in practice? Can you think of a scenario in your own research where you would expect the effect of a predictor to differ across cutpoints?

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Section 4

Adjacent-Category & Continuation-Ratio Models

⏱ Estimated time: 20 minutes

Adjacent-Category Model

The adjacent-category model compares the probability of being in category j versus category j−1 (the next lower category). It is a constrained version of the multinomial logistic model: the constraint is that the coefficient for categories n levels apart equals n times the coefficient for adjacent categories.

Like the proportional-odds model, the adjacent-category model estimates a single β1 per predictor, making it more parsimonious than the unconstrained multinomial model. The validity of this constraint can be tested by comparing the adjacent-category model to the unconstrained multinomial model using a likelihood-ratio test (LRT).

📋 Example: Apgar Scores — Adjacent-Category Model

For the Apgar score data, the LRT comparing the adjacent-category model to the unconstrained multinomial model yielded χ² = 6.76, df = 5, P = 0.239. Since this is not significant, the adjacent-category model is a valid simplification of the multinomial model for these data.

Continuation-Ratio Model

The continuation-ratio model compares the probability of being in category j versus all lower categories combined. It is particularly useful when the outcome represents sequential stages that must be “passed through” to reach higher levels.

When Is the Continuation-Ratio Model Appropriate?

The continuation-ratio model is ideal for outcomes where each level must be reached before the next can be attained. Examples include: number of attempts to pass an exam, stages of disease progression where remission must occur before relapse, or sequential rounds of a selection process. It is NOT appropriate when movements between categories are not sequential (e.g., Apgar scores, where a baby does not “pass through” each score level).

Fitting the Continuation-Ratio Model

The continuation-ratio model can be fit as a series of separate binary logistic regressions with a recoded outcome variable. For each comparison:

  • Y = 1 for the level of interest
  • Y = 0 for all lower levels
  • Observations at higher levels are excluded (treated as missing)

Consider an example with 4 categories representing the number of attempts to gain admission to medical school (1, 2, 3, 4+):

Original CategoryY1 (1 vs 0)Y2 (2 vs 0–1)Y3 (3 vs 0–2)
0 (1 attempt)000
1 (2 attempts)100
2 (3 attempts)—10
3 (4+ attempts)——1

You can fit either a constrained version (equal ORs across levels, tested by LRT) or an unconstrained version (separate ORs at each level). The constrained version is more parsimonious and can be compared to the unconstrained version using a likelihood-ratio test.

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Adjacent-Category Model
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↗
Continuation-Ratio Model
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🔧
Choosing Between Models
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When to Use the Adjacent-Category Model

The adjacent-category model is appropriate when the comparison of interest is between neighbouring categories of an ordinal outcome. It is a natural choice when you believe the effect of a predictor operates by shifting individuals one category at a time. The model can be validated by comparing it to the unconstrained multinomial model via LRT.

When to Use the Continuation-Ratio Model

The continuation-ratio model is most appropriate when the outcome represents sequential stages that must be passed through in order. Each category must be reached before the next can be attained. Examples include: successive attempts at an exam, sequential rounds of treatment, or stages of career advancement. If categories can be reached without passing through lower levels, this model is not appropriate.

Comparing Models with LRT

When one model is a constrained (nested) version of another, the likelihood-ratio test can be used to compare them. The test statistic is −2(lnLconstrained − lnLunconstrained), which follows a χ² distribution with degrees of freedom equal to the difference in the number of parameters. A significant result suggests the constraint is not valid and the more complex model is needed.

Decision Guide: Choosing Among the Four Models

Step 1: Is the outcome nominal or ordinal? If nominal, use multinomial logistic regression.
Step 2: If ordinal, does the outcome represent sequential stages? If yes, consider the continuation-ratio model.
Step 3: If not sequential, fit the proportional-odds model and test the assumption. If it holds, use proportional-odds.
Step 4: If the proportional-odds assumption fails, consider the adjacent-category model, partial proportional-odds, or generalised ordinal logistic regression.
Step 5: Compare nested models using LRT to select the most parsimonious adequate model.

✔ Check Your Understanding

1. In the adjacent-category model, the coefficient for categories n levels apart is:

The adjacent-category model is a constrained multinomial model. The key constraint is that the log-odds (coefficient) for categories n levels apart equals n times the log-odds for adjacent categories. This is what makes it more parsimonious than the unconstrained multinomial model.

2. The continuation-ratio model is most appropriate when:

The continuation-ratio model is designed for outcomes where each category represents a stage that must be reached before moving to the next. Each comparison asks: given that you reached at least this level, what are the odds of reaching the next level? This is inappropriate when categories can be reached without passing through lower levels.

3. If an LRT comparing the adjacent-category model to the multinomial model is NOT significant, this suggests:

A non-significant LRT means the constrained (adjacent-category) model does not fit significantly worse than the unconstrained (multinomial) model. Therefore, the simpler adjacent-category model is a valid simplification and should be preferred on grounds of parsimony.

✎ Reflection

Can you think of an example from public health or epidemiology where a continuation-ratio model would be more appropriate than a proportional-odds model? What makes the outcome sequential in your example?

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Final Assessment

Lesson 5 — Final Assessment

15 questions • 100% required to pass

This assessment covers all sections of Lesson 5. You must answer all 15 questions correctly to complete the lesson. Read each question carefully and review the feedback for any incorrect answers before retrying.

✎ Final Reflection

Now that you have completed all four sections, summarise the key differences among the four models for multi-category outcomes. When would you choose each one, and what assumptions would you need to verify?

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✔ Final Assessment

1. What type of outcome data has categories with no natural ordering?

Nominal data have categories with no inherent ordering or ranking. Examples include type of disease, ethnicity, or mode of transport. Ordinal data, by contrast, have a meaningful ordering among categories.

2. How many sets of coefficients does a multinomial model with 4 outcome categories estimate?

A multinomial logistic model with J = 4 categories estimates J−1 = 3 sets of coefficients, each comparing one non-baseline category to the baseline category.

3. In a proportional-odds model, the OR for a predictor represents:

The proportional-odds assumption means that a single OR applies at every cutpoint. Regardless of where you dichotomise the ordinal outcome, the estimated OR for each predictor is the same.

4. The logit in a proportional-odds model is based on:

The proportional-odds model uses a cumulative logit: the log-odds of being at or above category j versus being below it. This cumulative formulation is what allows a single coefficient to apply at every cutpoint.

5. The IIA assumption in multinomial logistic regression means:

The Independence of Irrelevant Alternatives (IIA) assumption states that the relative probability (odds) of any two outcomes does not change when other alternatives are added to or removed from the choice set.

6. If the proportional-odds assumption is clearly violated, a good alternative is:

The generalised ordinal logistic regression model allows different coefficients at each cutpoint, relaxing the proportional-odds assumption entirely. It is equivalent to fitting J−1 binary logistic models simultaneously and is a flexible alternative when the assumption is violated.

7. In a multinomial model, exponentiated coefficients are technically:

Exponentiated coefficients from a multinomial logistic model are ratios of relative risks (RRR). They compare the probability of being in one category relative to the baseline, but they are not true odds ratios as in standard binary logistic regression.

8. The Brant test evaluates:

The Brant test is a Wald-type test that evaluates the proportional-odds assumption. It provides both an overall test and individual tests for each predictor, identifying which specific variables violate the assumption of equal effects across cutpoints.

9. The adjacent-category model is a constrained version of:

The adjacent-category model is a constrained version of the multinomial logistic model. The constraint is that the coefficient for categories n levels apart equals n times the coefficient for adjacent categories. This can be tested by comparing the two models with a likelihood-ratio test.

10. Continuation-ratio models are best suited for:

Continuation-ratio models are designed for outcomes where each level must be “passed through” to reach higher levels. The model asks: given that you reached at least level j, what are the odds of reaching level j+1? This is ideal for sequential or staged processes.

11. The latent variable in a proportional-odds model has cutpoints (τ) that:

The cutpoints (τ) divide the underlying continuous latent variable into the observed ordinal categories. An observation is classified into category j if its latent variable score falls between τj−1 and τj. These cutpoints are estimated from the data along with the regression coefficients.

12. To compare nested ordinal models, you should use:

The likelihood ratio test (LRT) is the standard approach for comparing nested models. The test statistic is −2 times the difference in log-likelihoods, which follows a chi-squared distribution with degrees of freedom equal to the difference in the number of parameters.

13. If the LRT comparing proportional-odds to multinomial models is significant:

A significant LRT means the constrained (proportional-odds) model fits significantly worse than the unconstrained (multinomial) model. This suggests the proportional-odds constraint is too restrictive and the assumption may be violated. You should consider a more flexible model.

14. In a continuation-ratio model, observations at higher levels than the one being modelled are:

In a continuation-ratio model, each binary regression compares level j (Y = 1) to all lower levels (Y = 0). Observations at levels higher than j are excluded from that particular comparison because they have already “passed through” the level being modelled.

15. Which model is most parsimonious for ordinal data when the proportional-odds assumption holds?

When the proportional-odds assumption holds, the proportional-odds model is the most parsimonious because it estimates only one coefficient per predictor (compared to J−1 per predictor in the multinomial or generalised ordinal models). Parsimony leads to more precise estimates and more powerful tests.

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