From a Fitted Ordinal Model to Clinical Predictions
A cumulative-logit model is usually fitted on a log-odds scale, but clinical decisions concern response categories: the chance of mild, moderate, or severe response at a particular drug concentration. A coefficient on the logit scale does not directly state those chances. To make the model interpretable, calculate the cumulative probabilities at the concentration of interest, recover each category probability, and compare predictions at clinically relevant concentrations.
Let \(Y\) be an ordered response with categories \(1,\ldots,K\), from the lowest to the highest response, and let \(C\) be concentration measured in mg/L. For the proportional-odds cumulative-logit model used below, define \(q_k(C)=P(Y\leq k\mid C)\). The fitted model is
where \(\operatorname{logit}(u)=\log\{u/(1-u)\}\), the thresholds satisfy \(\alpha_1<\cdots<\alpha_{K-1}\), and \(\beta\) is the concentration effect per mg/L. The inverse-logit function is \(g(z)=1/(1+\exp(-z))\), so \(q_k(C)=g(\alpha_k-\beta C)\). With this sign convention, a positive \(\beta\) means higher concentrations lower the cumulative probabilities of being at or below each threshold, shifting probability toward higher response categories.
These predictions are conditional on the stated concentration and fitted model. For repeated ordinal data with subject-specific random effects, predictions conditional on a specified random effect are not generally the same as predictions averaged over the random-effects distribution. Use the prediction target that matches the clinical question; do not silently treat a subject-conditional prediction as a population-average one.
Why Differencing Produces Category Probabilities
The categories are disjoint, and the event \(Y\leq k\) consists of all categories up to \(k\). Therefore, for an interior category, the event \(Y=k\) is the difference between the nested events \(Y\leq k\) and \(Y\leq k-1\). This gives \(p_k(C)=q_k(C)-q_{k-1}(C)\). The first and last categories follow from the same logic, using \(q_0(C)=0\) and \(q_K(C)=1\).
The cancellation in this sum is also a practical check: correctly computed category probabilities must add to one. They must also be nonnegative. Under the proportional-odds model, ordered thresholds and a common concentration effect preserve the ordering of the cumulative probabilities: if \(\alpha_k>\alpha_{k-1}\), then \(g(\alpha_k-\beta C)>g(\alpha_{k-1}-\beta C)\), because the inverse-logit is increasing. Consequently, every difference \(q_k(C)-q_{k-1}(C)\) is nonnegative.
Worked Example: Predicting Categories at Two Concentrations
Suppose a hypothetical fitted three-category model has thresholds \(\alpha_1=0.400\), \(\alpha_2=1.600\), and concentration effect \(\beta=0.800\) per mg/L. Category 1 is the lowest response and category 3 the highest. Calculate the category probabilities at 1 and 2 mg/L.
At \(C=1\) mg/L, the cumulative logits are \(0.400-(0.800)(1)=-0.400\) and \(1.600-(0.800)(1)=0.800\). Applying the inverse logit gives \(q_1(1)=g(-0.400)=0.4013\) and \(q_2(1)=g(0.800)=0.6900\). Thus
At \(C=2\) mg/L, the cumulative logits are \(0.400-1.600=-1.200\) and \(1.600-1.600=0\). Hence \(q_1(2)=g(-1.200)=0.2315\) and \(q_2(2)=g(0)=0.5000\). The category probabilities are \(p_1(2)=0.2315\), \(p_2(2)=0.5000-0.2315=0.2685\), and \(p_3(2)=1-0.5000=0.5000\). Each set is nonnegative and sums to 1.0000, subject to rounding.
In this model, increasing concentration shifts probability toward category 3: its predicted probability rises from 31.00% at 1 mg/L to 50.00% at 2 mg/L. This is a model-based prediction, not by itself evidence that changing concentration causes the response shift.
Contrasts on the Probability Scale
A clinically interpretable contrast specifies both the outcome event and the two concentrations being compared. For category \(k\), define the absolute probability contrast from \(C_a\) to \(C_b\) as \(\Delta_k=p_k(C_b)-p_k(C_a)\). A positive value means the predicted probability of that category is higher at \(C_b\); a negative value means it is lower. Multiplying the contrast by 100 expresses the difference in percentage points. This is not a relative percent change: for example, a change from 20% to 25% is 5 percentage points, but a 25% relative increase.
A cumulative contrast can be equally useful when the clinical endpoint is a threshold, such as “moderate or worse.” If categories 1 through \(k\) are below that threshold, then \(P(Y>k\mid C)=1-q_k(C)\). Contrasts should be calculated from unrounded model predictions; subtracting displayed rounded probabilities can slightly change the result.
Worked Example: A Precise Category-Probability Contrast
Use the fitted model from the preceding example and define the contrast for category 2 as the predicted probability at 2 mg/L minus the predicted probability at 1 mg/L. Calculate it using the unrounded inverse-logit probabilities.
At 1 mg/L, \(p_2(1)=g(0.800)-g(-0.400)=0.6899745-0.4013123=0.2886621\). At 2 mg/L, \(p_2(2)=g(0)-g(-1.200)=0.5000000-0.2314752=0.2685248\). Therefore,
The model predicts a decrease of 2.014 percentage points in the probability of category 2 when concentration rises from 1 to 2 mg/L. Subtracting the displayed four-decimal probabilities instead would give \(0.2685-0.2887=-0.0202\); that is the difference of rounded values, not the more precise model-based contrast. The contrast is an absolute probability difference, not an odds ratio.
Worked Example: Contrasting a Clinically Defined Threshold
Suppose the clinical question concerns a high response, defined as category 3 or worse in the three-category model above. Compare its predicted probability at 1 and 2 mg/L, and report the absolute contrast.
Because category 3 is above threshold 2, its probability is \(P(Y>2\mid C)=1-q_2(C)\). From the cumulative predictions, this is \(1-0.6899745=0.3100255\) at 1 mg/L and \(1-0.5000000=0.5000000\) at 2 mg/L. The higher-concentration minus lower-concentration contrast is
or an increase of 19.00 percentage points in the predicted probability of category 3. This threshold contrast answers a different clinical question from the category 2 contrast: the probability of one specific intermediate category can decrease even as the probability of the highest category increases.
Worked Example: Computing All Categories in a Four-Category Model
Consider a second hypothetical fitted model with thresholds \((-1.500,-0.200,1.000)\) and \(\beta=0.600\) per mg/L. Calculate all four category probabilities at \(C=1\) mg/L. This example illustrates the same calculation with an additional response category.
The cumulative logits are \(-2.100,-0.800,0.400\). Their inverse-logit values are \(q_1=0.1091\), \(q_2=0.3100\), and \(q_3=0.5987\). Differencing adjacent cumulative probabilities and taking the complement for the last category gives
The values sum to \(0.1091+0.2009+0.2887+0.4013=1.0000\) at the displayed precision. If the clinical endpoint were category 3 or higher, its predicted probability would be \(p_3+p_4=1-q_2=0.6900\). Stating the endpoint explicitly prevents confusion between “category 3” and “category 3 or higher.”
Assumptions, Diagnostics, and Alternative Summaries
A plug-in prediction assumes the fitted model is appropriate for the population, response definition, and concentration range to which the prediction is applied. In the proportional-odds model, the same concentration effect applies to every cumulative split. The thresholds must be ordered, and the concentration-effect form used in fitting must be used in prediction. If the fitted model uses a transformed concentration, substitute that transformation rather than inserting raw concentration into a linear formula.
Before interpreting predictions, check that cumulative probabilities are nondecreasing across thresholds, every category probability is between 0 and 1, and the category probabilities sum to one. A negative category probability signals a coherence problem, such as incompatible cumulative predictions; do not truncate it to zero and continue as if the fit were valid. Also inspect whether predictions are driven by sparse data or extrapolation beyond observed concentrations. A mathematically valid probability can still be clinically unreliable when the model is poorly supported in that region.
The earlier material on proportional-odds interpretation and partial proportional-odds models is relevant when a single common concentration effect is questionable. With a partial proportional-odds fit, calculate each cumulative probability using its fitted threshold-specific predictor, then difference the resulting cumulative probabilities. Check their ordering at every concentration of interest: threshold-specific effects can produce crossing cumulative curves and invalid category probabilities outside well-supported regions. For a repeated-response model, state whether predictions hold random effects fixed or average over them; these are distinct estimands.
A concentration coefficient or odds ratio is a compact summary of association on the cumulative-logit scale. It does not supply an absolute probability change without a baseline threshold and specified concentrations. Conversely, category-probability contrasts are directly interpretable for a chosen clinical scenario but depend on the model’s thresholds and on the two concentration values. They do not summarize every possible exposure contrast.
Predictions made by substituting estimated parameters are point estimates; they do not show parameter uncertainty or individual response variability. When uncertainty matters, obtain an interval for the contrast by carrying parameter uncertainty through the complete prediction-and-differencing calculation, for example by bootstrap resampling with model refitting. Such an interval concerns uncertainty in the estimated contrast under the analysis procedure; it is not an interval containing a specified proportion of individual patient outcomes.
Common Mistake
Finally, an absolute probability contrast is descriptive of the fitted exposure-response relationship. A causal interpretation requires additional support from the study design and assumptions about confounding and treatment assignment; the ordinal model alone does not establish causation.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- If \(q_1(C)=0.25\) and \(q_2(C)=0.70\) in a three-category model, what are the three category probabilities?
- A category’s predicted probability is 0.18 at \(C_a\) and 0.23 at \(C_b\). State the contrast \(p(C_b)-p(C_a)\) in probability units and percentage points.
- Why is \(P(Y\leq 2\mid C)\) not generally the same as \(P(Y=2\mid C)\)?
- What does a negative contrast \(p_2(C_b)-p_2(C_a)\) mean when \(C_b>C_a\)?
- Name two checks to perform on category predictions before interpreting them.
Solutions. 1. \(p_1=0.25\), \(p_2=0.70-0.25=0.45\), and \(p_3=1-0.70=0.30\); they sum to 1. 2. The contrast is \(0.23-0.18=0.05\), an increase of 5 percentage points. 3. The cumulative event includes categories 1 and 2, so \(P(Y=2\mid C)=q_2(C)-q_1(C)\). 4. The model predicts a lower probability of category 2 at \(C_b\) than at \(C_a\), by the absolute amount represented by the negative contrast. 5. Check that cumulative probabilities are ordered, category probabilities are nonnegative and at most 1, and all category probabilities sum to 1; any two of these checks suffice.