Many outcomes in medicine and social science are not just categorical — they are ranked. Pain is rated from none to severe. Recovery is classified as none, partial, or complete. Satisfaction is recorded on a scale from strongly disagree to strongly agree. These are ordinal outcomes: the categories have a meaningful order, but the distances between them are not necessarily equal.
The multinomial logistic regression model covered in CS2 treats all categories as unordered. Applied to an ordinal outcome, it works — but it throws away the information contained in the ranking. A model that respects the ordering can use that information to estimate a single effect for each predictor rather than one per category comparison, leading to more efficient and interpretable results.
A useful way to think about ordinal outcomes is through a latent variable framework: imagine there is an underlying continuous measure (e.g., true health status) that gets cut into ordered categories by thresholds. The observed ordinal outcome is a coarsened version of that latent variable. This framing motivates the cumulative logit (proportional odds) model introduced in CS3.2.
On this page you will see why ordinal structure matters, how the latent variable idea works, and what assumptions underlie ordinal regression.
| Abbreviation | Full term |
|---|---|
| PO | Proportional odds |
| CLM | Cumulative link model |
| OR | Odds ratio |
| LRT | Likelihood ratio test |
The multinomial model (CS2) treats all categories as unordered — it makes no use of any ranking among them. When categories have a natural order, this discards information.
Examples of ordered outcomes: - Disease severity: none < mild < moderate < severe - Pain scale: none < mild < moderate < severe < unbearable - Likert response: strongly disagree < disagree < neutral < agree < strongly agree - Educational attainment: primary < secondary < tertiary
Treating these as nominal (using multinomial regression) is valid but inefficient: the model estimates \(K-1\) separate slopes for each predictor, when a single slope might suffice.
A key insight for ordinal models: imagine that the ordered response \(Y\) arises from an unobserved continuous latent variable \(Y^*\):
\[Y^* = \mathbf{x}^\top \boldsymbol{\beta} + \varepsilon, \qquad \varepsilon \sim F\]
We observe only which interval \(Y^*\) falls in:
\[Y = k \iff \alpha_{k-1} < Y^* \leq \alpha_k\]
where \(\alpha_0 = -\infty < \alpha_1 < \alpha_2 < \cdots < \alpha_{K-1} < \alpha_K = \infty\) are thresholds (also called cutpoints or intercepts).
The probability of each category:
\[P(Y = k | \mathbf{x}) = F(\alpha_k - \mathbf{x}^\top\boldsymbol{\beta}) - F(\alpha_{k-1} - \mathbf{x}^\top\boldsymbol{\beta})\]
The choice of \(F\) gives different models: - \(F\) = logistic CDF → proportional odds (cumulative logit) model - \(F\) = standard normal CDF → ordered probit model
In pain research, \(Y^*\) might represent true pain intensity, which is continuous but we can only observe it in categories. In quality-of-life research, it is a latent construct. The thresholds represent the “cut-off” levels at which perception crosses into the next response category.
ggplot(dat, aes(x=y_latent, fill=y_ord)) +
geom_histogram(bins=30, colour="white") +
geom_vline(xintercept=c(-1.5,0.3,2.0), linetype="dashed", colour="black", linewidth=0.8) +
facet_wrap(~group) +
labs(title="Latent variable Y* and observed ordinal Y",
subtitle="Dashed lines = thresholds α₁, α₂, α₃",
x="Latent variable Y*", y="Count", fill="Observed Y") +
theme_minimal()
# Effect of x on latent variable shifts the distribution → different category proportions
x_low <- dat[dat$x < -1,]
x_high <- dat[dat$x > 1,]
cat("Category proportions for x < -1:\n")Category proportions for x < -1:round(prop.table(table(x_low$y_ord)), 3)
low medium high very high
0.407 0.519 0.074 0.000 cat("\nCategory proportions for x > 1:\n")
Category proportions for x > 1:round(prop.table(table(x_high$y_ord)), 3)
low medium high very high
0.000 0.000 0.258 0.742 A positive coefficient shifts the latent distribution to the right, increasing the probability of higher categories. The same shift applies everywhere — this is the proportional odds assumption (explored in CS3.3).
# Simulated pain study: 3 treatments, pain measured on 5-point scale
set.seed(42)
n_pain <- 180
pain_dat <- data.frame(
patient = 1:n_pain,
treatment = factor(rep(c("Placebo","LowDose","HighDose"), each=60)),
age = round(rnorm(n_pain, 55, 12))
)
latent_pain <- 2.5 +
ifelse(pain_dat$treatment=="LowDose", -0.8, 0) +
ifelse(pain_dat$treatment=="HighDose", -1.8, 0) +
0.01*pain_dat$age + rnorm(n_pain, 0, 1)
thresh <- c(1.0, 2.0, 3.0, 4.0)
pain_dat$pain <- ordered(
cut(latent_pain, breaks=c(-Inf, thresh, Inf),
labels=c("none","mild","moderate","severe","very severe")),
levels=c("none","mild","moderate","severe","very severe")
)
# Always verify level order!
levels(pain_dat$pain)[1] "none" "mild" "moderate" "severe" "very severe"table(pain_dat$treatment, pain_dat$pain)
none mild moderate severe very severe
HighDose 24 26 9 1 0
LowDose 7 15 19 16 3
Placebo 0 10 24 18 8ggplot(pain_dat, aes(x=treatment, fill=pain)) +
geom_bar(position="fill") +
scale_fill_manual(values=c("#2ecc71","#f1c40f","#e67e22","#e74c3c","#8e44ad")) +
labs(title="Pain distribution by treatment",
x="Treatment", y="Proportion", fill="Pain level") +
theme_minimal()
The stacked bar chart shows the marginal distribution of pain by treatment group. The ordinal model will quantify the shift formally, adjusting for age.
R is loading in the background — the exercises below will be interactive once the status bar disappears. You can read the concept sections above immediately.
cut(surv$wait, quantile(surv$wait, c(0,.25,.5,.75,1)), include.lowest=TRUE) for quartilesQ1: \(\alpha_k\) is the value of the latent variable \(Y^*\) at which the response transitions from category \(k\) to \(k+1\). If \(\alpha_2 = 1.5\), then \(Y^* \leq 1.5\) maps to categories 1 or 2, and \(Y^* > 1.5\) maps to categories 3 or higher.
Q2: Yes — ordinal models don’t require equal spacing of categories (that’s the interval scale assumption). You’re only assuming the ranking is meaningful. Skipping a label doesn’t violate the model; it just means the threshold between “mild” and “severe” may be wide.
Q3: A shift changes probabilities of all categories — higher categories increase, lower categories decrease. The ordinal structure ensures these changes are internally consistent (they sum to 0).
Treating ordered categories as continuous (e.g., coding none=0, mild=1, moderate=2, severe=3) implies: 1. Equal spacing between categories — the difference between none and mild is the same as between moderate and severe 2. Normally distributed residuals — unlikely for a bounded 4-point scale 3. Predicted values can be negative or > 3 — nonsensical
In large, well-powered studies, linear regression on ordinal outcomes can still give approximately valid inference for the slope (it’s a pseudo-effect of sorts). But the predicted values and standard errors will be wrong, and the equal-spacing assumption is rarely defensible.
Both use the latent variable framework but differ in the assumed distribution of \(\varepsilon\): - Logistic (\(F\) = logistic CDF): heavier tails than normal; gives proportional odds model; ORs are directly interpretable - Probit (\(F\) = standard normal CDF): slightly lighter tails; common in econometrics; gives probit effects
In practice, for most datasets, the two give nearly identical predictions and inferences. Choose logistic if you want OR interpretation; probit if you want to model a normally distributed latent variable or connect to structural equation models.
Agresti, A. (2010). Analysis of Ordinal Categorical Data (2nd ed.). Wiley. — Chapter 3.