Interpreting Cutpoint-Specific Effects and Contrasts

From Cutpoint-Specific Slopes to Interpretable Effects

An ordinal model can describe how a covariate relates to several cumulative divisions of an ordered response. In a partial proportional-odds model, a predictor may have a different slope at each cutpoint. Interpretation therefore requires more than saying that the predictor has a positive or negative effect: the cutpoint, covariate change, and odds being compared all matter.

This tutorial uses the cumulative-logit convention in which the modeled cumulative probability is \(P(Y\leq j\mid x)\). The sign of a slope is interpreted relative to this convention. In particular, a positive slope corresponds to higher odds of being above a cutpoint when the covariate increases, holding the other covariates fixed.

The model gives one slope vector for each cumulative split: categories \(1,\ldots,j\) versus categories \(j+1,\ldots,K\). A coefficient is not, by itself, an effect on the probability of one particular response category. To obtain such probabilities, all relevant cumulative probabilities must be combined.

Cumulative Odds Ratios

Write \(F_j(x)=P(Y\leq j\mid x)\). The cumulative odds of being above cutpoint \(j\), rather than at or below it, are \(\{1-F_j(x)\}/F_j(x)\). Under the model, these odds have a particularly direct relationship to the slope.

Derivation. The model says that the odds of \(Y\leq j\) versus \(Y>j\) at profile \(x\) equal \(\exp(\alpha_j-x^\mathsf{T}\beta_j)\). Reversing those odds gives the odds of \(Y>j\) versus \(Y\leq j\), namely \(\exp(-\alpha_j+x^\mathsf{T}\beta_j)\). At \(x'\), these odds are \(\exp(-\alpha_j+x'^\mathsf{T}\beta_j)\). Dividing the latter by the former cancels the intercept and gives \(\exp\{(x'-x)^\mathsf{T}\beta_j\}=\exp(\Delta^\mathsf{T}\beta_j)\). \(\square\)

For a one-unit increase in a continuous predictor \(x_r\), with other covariates fixed, the cumulative odds ratio at cutpoint \(j\) is \(\exp(\beta_{jr})\). For a change of \(d\) units it is \(\exp(d\beta_{jr})\). For a categorical predictor, the relevant change is from its reference-coded profile to the comparison profile; the resulting linear predictor difference replaces the one-unit change.

An odds ratio is not a probability ratio. Even when the cumulative odds ratio is constant for a specified covariate change, the associated probability change depends on the starting profile and its cumulative probability. The cutpoint-specific intercepts and the other covariates affect that starting probability.

Comparing Slopes Across Cutpoints with Contrasts

A direct way to describe how a predictor’s association varies across cutpoints is to subtract its slopes. For predictor \(r\), the contrast \(\beta_{jr}-\beta_{kr}\) is the difference between its log cumulative-odds effects at cutpoints \(j\) and \(k\). For the same covariate change \(d\), exponentiating \(d(\beta_{jr}-\beta_{kr})\) gives the ratio of the two cumulative odds ratios.

Justification. The contrast estimate is a linear combination of the slope estimates. By the definition of covariance, the variance of that combination is the sum over all pairs of coefficients \(a_s a_t\) multiplied by \(\operatorname{Cov}(\hat\beta_s,\hat\beta_t)\). In matrix notation, that sum is \(a^\mathsf{T}\widehat V a\). In particular, for a difference \(\hat\beta_{jr}-\hat\beta_{kr}\), its estimated variance is \(\widehat{\operatorname{Var}}(\hat\beta_{jr})+\widehat{\operatorname{Var}}(\hat\beta_{kr})-2\widehat{\operatorname{Cov}}(\hat\beta_{jr},\hat\beta_{kr})\). The covariance term cannot generally be omitted because both slopes come from the same fitted model. \(\square\)

Under the usual regular large-sample conditions for maximum likelihood—correct specification, identifiability, an interior true parameter, and nonsingular information—the fitted parameter vector is approximately multivariate normal. This supports an approximate Wald interval for a contrast, \(\widehat L\pm1.96\sqrt{\widehat{\operatorname{Var}}(\widehat L)}\), when a 95% interval is appropriate. The interval describes uncertainty in a log-odds contrast; exponentiating its endpoints gives an interval for the corresponding ratio of cumulative odds ratios.

The same method applies to more elaborate comparisons. For example, a contrast can compare one cutpoint slope with the average of two others. Specify the coefficients in \(a\), calculate \(a^\mathsf{T}\hat\beta\), and use the full covariance matrix. The interpretation should state which predictor and cutpoints the contrast concerns.

From Cumulative Probabilities to Category Probabilities

To report fitted probabilities for individual categories, first calculate the cumulative probabilities at the covariate profile of interest. If \(F_j(x)=P(Y\leq j\mid x)\), then the probability of the first category is \(F_1(x)\); an intermediate category is the difference between adjacent cumulative probabilities; and the last category is the remaining probability above the final cutpoint.

Proof. The first probability is nonnegative because \(F_1(x)\geq0\). Each intermediate probability is nonnegative because the cumulative probabilities are ordered, and the last is nonnegative because \(F_{K-1}(x)\leq1\). Adding the category probabilities gives \(F_1+(F_2-F_1)+\cdots+(F_{K-1}-F_{K-2})+(1-F_{K-1})=1\): all cumulative terms cancel. Hence the quantities form a probability distribution over the \(K\) categories. \(\square\)

Common Pitfall and Practical Interpretation

Wrong reasoning: “The slope at cutpoint \(j\) is \(0.8\), so a one-unit increase in the predictor raises the probability of category \(j+1\) by about 80%.” This confuses a log cumulative-odds slope with a probability change and with the probability of one category.

Correction: Under the stated sign convention, the one-unit increase multiplies the odds of \(Y>j\) versus \(Y\leq j\) by \(\exp(0.8)\), about \(2.23\). To obtain a category probability, calculate the cumulative probabilities at the covariate profile and take their difference as appropriate. The resulting probability change depends on the starting profile.

When presenting results, state the direction of the cumulative comparison, the covariate change, and the cutpoint. For comparisons across cutpoints, give the contrast and its uncertainty rather than informally comparing point estimates alone. Finally, verify that the fitted cumulative probabilities are ordered at each profile used to report category probabilities.