How Covariate Changes Shift Category Probabilities

From Cumulative Probabilities to Category Changes

A cumulative link model describes how covariates shift the probability of being at or below each response threshold. But an applied question often concerns a particular response category: for example, how does the fitted probability of “moderate” disease severity change as a predictor increases? The answer is not generally determined by the direction of one cumulative probability alone. An interior category’s probability is the difference between two cumulative probabilities, and both can change together.

Consider an ordinal response \(Y\) with ordered categories \(1,\ldots,K\) and a single covariate \(x\). Use the cumulative link form \(P(Y\leq j\mid x)=F(\alpha_j-\beta x)\), where \(F\) is a cumulative distribution function, \(\alpha_1<\cdots<\alpha_{K-1}\) are thresholds, and \(\beta\) is the common slope. Define \(G_j(x)=P(Y\leq j\mid x)\), and set \(G_0(x)=0\) and \(G_K(x)=1\). The category probabilities are \(p_k(x)=G_k(x)-G_{k-1}(x)\).

If \(\beta>0\) and \(x_1>x_0\), then \(\alpha_j-\beta x_1<\alpha_j-\beta x_0\). Because a cumulative distribution function is nondecreasing, \(G_j(x_1)\leq G_j(x_0)\) for every cutpoint. These inequalities need not be strict: \(F\) can be flat between the two arguments. The model therefore gives a clear direction for each cumulative probability, but not a common direction for all category probabilities.

Worked Comparisons

Why Category Probabilities Can Move in Opposite Directions

The algebra identifies exactly how changes in adjacent cumulative probabilities combine. For an interior category \(k\), its probability is \(G_k-G_{k-1}\); therefore its change is \(\Delta G_k-\Delta G_{k-1}\). Even if both cumulative probabilities decrease, this difference can be positive, negative, or zero. It is positive when the lower cumulative probability decreases by more than the upper one.

To verify the result, write \(p_k(x)=G_k(x)-G_{k-1}(x)\), with \(G_0(x)=0\) and \(G_K(x)=1\). For the first category, subtracting its probabilities at the two covariate values gives \(\Delta p_1=\Delta G_1\). For an interior category, subtract the two differences:

For the last category, \(p_K(x)=1-G_{K-1}(x)\), so \(\Delta p_K=-\Delta G_{K-1}\). Summing these expressions, the endpoint terms contribute \(\Delta G_1-\Delta G_{K-1}\), while the interior terms telescope:

The endpoint and interior contributions cancel, proving that the total change is zero. This is also required because the category probabilities sum to one at each covariate value. \(\square\)

A density gives a local view of these changes. Suppose \(F\) has a continuous density \(f\), and write \(t_j(x)=\alpha_j-\beta x\). The chain rule gives the derivatives of the category probabilities.

For the first category, \(p_1(x)=F(t_1(x))\), and \(t_1'(x)=-\beta\), so the chain rule gives \(p_1'(x)=-\beta f(t_1(x))\). For an interior category, subtract the derivatives of its two cumulative probabilities:

Finally, \(p_K(x)=1-F(t_{K-1}(x))\), so differentiating gives \(p_K'(x)=\beta f(t_{K-1}(x))\). These expressions show why an interior category’s direction depends on the density at two threshold arguments, rather than on the sign of \(\beta\) alone. \(\square\)

Interpretation and Common Pitfall

The common slope controls how the cumulative probabilities shift across cutpoints. It does not say that every individual category probability shifts in the same direction. The two endpoint categories each depend on one cumulative probability, while every interior category depends on the difference between two. When reporting results, it is therefore useful to distinguish a statement about \(P(Y\leq j\mid x)\) from one about \(P(Y=k\mid x)\).

• Common mistake: “The slope is positive, so every category probability must decrease as \(x\) increases.” This incorrectly applies the direction of the cumulative probabilities to each category separately.
• Correction: For \(\beta>0\) and \(x_1>x_0\), each cumulative probability is nonincreasing, but an interior probability changes by \(\Delta G_k-\Delta G_{k-1}\). Its sign depends on the relative sizes of those two changes.
• Another check: The category-probability changes must sum to zero. If a calculation does not, check the differencing and endpoint categories, and avoid computing differences from prematurely rounded probabilities.

Comparing fitted probabilities at meaningful covariate values complements interpretation of the slope on the cumulative scale. It directly answers how the model reallocates probability among the response categories, while preserving the fact that the total probability remains one.