From an Ordinal Diagnostic to a Model Decision
A proportional-odds diagnostic is useful only when it leads to a defensible modeling decision. The analysis must distinguish evidence of nonparallel slopes from evidence that a particular predictor should be relaxed, preserve valid category probabilities, and compare candidate models using a method appropriate to their relationship. This tutorial follows those steps for a three-category ordinal response.
Let \(Y\) take the ordered values \(1,2,3\), with \(1\) the lowest category. Let \(z\) be a binary treatment indicator and let \(x\) be a centered continuous covariate. Write \(F_j(z,x)=P(Y\leq j\mid z,x)\), for \(j=1,2\), and \(g(u)=1/(1+\exp(-u))\). We use the cumulative-logit convention
Under proportional odds, each predictor has the same slope at both cutpoints. If a coefficient is positive in this convention, increasing its predictor lowers the probability of being at or below either cutpoint, shifting probability toward higher categories.
Worked Diagnostic: Is the Treatment Slope Plausibly Parallel?
Suppose a study compares treatment groups on a three-level clinical outcome and also records a centered baseline score \(x\). The following category counts are shown by treatment group, each with 200 participants. They are a useful descriptive starting point; they do not, by themselves, replace a model-based test that accounts for all covariates and sampling variability.
| Treatment | Category 1 | Category 2 | Category 3 | Total |
|---|---|---|---|---|
| \(z=0\) | 50 | 80 | 70 | 200 |
| \(z=1\) | 36 | 60 | 104 | 200 |
Worked Example: Comparing the Two Cumulative Treatment Contrasts
At the first cutpoint, the empirical cumulative proportions are \(50/200=0.25\) for \(z=0\) and \(36/200=0.18\) for \(z=1\). The corresponding logits are \(\log(0.25/0.75)=-1.0986\) and \(\log(0.18/0.82)\approx-1.5163\). With the stated minus-slope convention, the descriptive treatment slope is their difference, about \(0.4177\).
At the second cutpoint, the cumulative proportions are \((50+80)/200=0.65\) and \((36+60)/200=0.48\). Their logits are \(\log(0.65/0.35)\approx0.6190\) and \(\log(0.48/0.52)\approx-0.0800\), giving a descriptive slope contrast of about \(0.6990\).
The two descriptive contrasts, \(0.4177\) and \(0.6990\), differ by about \(0.2813\). This suggests checking treatment parallelism, not automatically rejecting the proportional-odds model: the contrasts are estimated with sampling error, and the table does not adjust for \(x\). The targeted alternative should relax the treatment slopes while retaining a common age-score slope if the substantive question and fuller diagnostic support that choice.
Fit a Targeted Partial Proportional-Odds Model
The partial model for this analysis is
Only the treatment slope varies by cutpoint. The proportional-odds model is nested within this model by imposing the single restriction \(\beta_{z1}=\beta_{z2}\). The analysis below treats a numerical fit summary as given, as would be done when interpreting software output; coefficients are rounded, so likelihood calculations use the reported log-likelihoods rather than attempting to reconstruct them from rounded coefficients.
Worked Example: Testing the Targeted Relaxation
Suppose the proportional-odds fit has log-likelihood \(\ell_{\mathrm{PO}}=-412.60\), while the targeted partial proportional-odds fit has \(\ell_{\mathrm{PPO}}=-407.95\). The proportional model has two cutpoints and two common slopes, so it has \(4\) parameters. The partial model has two cutpoints, one common score slope, and two treatment slopes, so it has \(5\).
The likelihood-ratio statistic is \(2\{\ell_{\mathrm{PPO}}-\ell_{\mathrm{PO}}\}=2\{-407.95-(-412.60)\}=9.30\). There is one restriction, \(\beta_{z1}=\beta_{z2}\), so the reference distribution under the null is \(\chi^2_1\) under the regularity conditions described below. Its upper-tail probability is approximately \(0.0023\). In this illustrative analysis, the data provide evidence against a common treatment slope at the two cutpoints, conditional on the model and its assumptions.
This test answers a targeted question: whether the treatment slope can be constrained to be common while keeping the score slope common. It does not establish that the partial model is a good description in every respect, nor does it prove that treatment effects truly differ across thresholds.
Here is how the result applies. The restricted parameter space is obtained by setting the difference \(\beta_{z1}-\beta_{z2}\) to zero, a single smooth restriction. If the fitted model is identifiable, the true parameter is not on a boundary, and the likelihood satisfies Wilks’ regularity conditions, then \(q=1\). The statistic compares maximized likelihoods on the same data; the larger model cannot have a smaller maximized likelihood because it includes every parameter value allowed by the smaller model. The chi-square calibration is asymptotic, not an exact finite-sample result. Sparse categories, separation, or other failures of regularity can make that approximation unreliable.
Check the Fitted Probabilities Before Interpreting Them
For the partial model, define the fitted cumulative probabilities by applying \(g\) to the two linear predictors. The category probabilities are then \(p_1=F_1\), \(p_2=F_2-F_1\), and \(p_3=1-F_2\). This construction is valid only when \(0\leq F_1\leq F_2\leq1\). Thus, relaxing a slope is not permission to ignore the ordering of the cumulative probabilities.
To verify the forward statement, nonnegativity follows respectively from \(F_1\geq0\), \(F_2\geq F_1\), and \(F_2\leq1\). Their sum is \(F_1+(F_2-F_1)+(1-F_2)=1\). For the converse, nonnegative category probabilities imply \(F_1=p_1\geq0\), \(F_2=p_1+p_2\geq F_1\), and \(F_2=1-p_3\leq1\). This check applies at each covariate value, not only at the mean of the predictors.
Worked Example: Computing and Checking Category Probabilities
Suppose a fitted partial model has \(\hat\alpha_1=-1.0\), \(\hat\alpha_2=0.7\), \(\hat\beta_x=0.2\), \(\hat\beta_{z1}=0.3\), and \(\hat\beta_{z2}=0.9\). At \(x=0,z=1\), the two cumulative logits are \(-1.0-0.3=-1.3\) and \(0.7-0.9=-0.2\). Therefore \(F_1=g(-1.3)\approx0.214165\) and \(F_2=g(-0.2)\approx0.450166\). The category probabilities are approximately \(p_1=0.2142\), \(p_2=0.450166-0.214165=0.2360\), and \(p_3=1-0.450166=0.5498\). Using unrounded cumulative values, they sum to one; the displayed rounded entries may not sum to exactly one.
At \(x=1,z=1\), the cumulative logits are \(-1.5\) and \(-0.4\), giving \(F_1\approx0.182426\) and \(F_2\approx0.401312\). The category probabilities are \(0.182426\), \(0.401312-0.182426=0.218886\), and \(1-0.401312=0.598688\). All are nonnegative, and they sum to one before rounding.
For these particular coefficients, the second cumulative logit exceeds the first by \(1.7-0.6z\), because the common score term cancels. Since \(z\) is either 0 or 1, this difference is \(1.7\) or \(1.1\), both positive. Thus the model guarantees \(F_1<F_2\) for every real \(x\) and both treatment groups. This algebraic check is stronger than checking a few fitted rows.
Compare With a Multinomial Benchmark
A baseline-category multinomial logit does not impose a cumulative-logit structure or common ordinal slopes. It can therefore serve as a flexible benchmark, but it does not preserve the ordering structure in the same way. With three outcome categories and two predictors, using category 1 as the baseline gives two logits, each with an intercept and two slopes: six parameters in total. The partial proportional-odds model has five parameters here.
Worked Example: Comparing Non-Nested Models With AIC
Suppose the multinomial benchmark has log-likelihood \(-405.80\), while the partial proportional-odds fit has log-likelihood \(-407.95\). The multinomial likelihood is higher, as expected for a more flexible fit, but its parameter count is also larger. Using \(\mathrm{AIC}=2k-2\ell\), the partial model has \(2(5)-2(-407.95)=10+815.90=825.90\). The multinomial model has \(2(6)-2(-405.80)=12+811.60=823.60\). The multinomial model has the lower AIC by \(2.30\), a modest preference under this criterion for this dataset. AIC is a relative information criterion, not a test that proves one model is true.
These models are not nested: neither set of fitted distributions generally contains the other. Their likelihood difference therefore does not have the ordinary nested-model chi-square reference distribution. A likelihood-ratio calculation based only on twice the difference in log-likelihoods would be unjustified. AIC offers one comparison on the same response observations, while substantive interpretability and the intended use of the model remain relevant.
Putting the Analysis Into a Defensible Sequence
The analysis is strongest when each step answers a distinct question. A diagnostic can indicate where the proportional-odds restriction is questionable; a targeted alternative identifies the restriction being tested; probability checks ensure that the fitted model represents a distribution; and a benchmark comparison describes the trade-off between ordinal structure and flexibility.
Compare threshold-specific cumulative-logit behavior, while accounting for uncertainty and avoiding conclusions from raw contrasts alone.
Let a predictor vary by cutpoint only when the diagnostic and scientific question support that choice; retain common slopes where appropriate.
Use a nested likelihood-ratio test for the stated restrictions, then verify cumulative ordering and fitted category probabilities across the covariate domain of interest.
Compare nonnested ordinal and multinomial fits with a criterion such as AIC rather than treating their likelihood difference as a nested test.
Common Mistakes
- Relaxing every slope after one diagnostic signal. A targeted alternative is more interpretable and tests a defined restriction. Evidence about one predictor does not automatically justify changing all slopes.
- Reading a likelihood-ratio p-value as a model adequacy verdict. The test addresses the restrictions separating two nested models, subject to regularity and adequate fit; it does not test every assumption.
- Checking probabilities only at average covariates. A partial model can have ordered cumulative probabilities at one covariate pattern and crossing cumulative curves elsewhere. Check the relevant domain, analytically when possible.
- Using a nested likelihood-ratio test for a multinomial benchmark. The multinomial and partial ordinal model are generally nonnested, so the usual chi-square calibration does not follow.
- Subtracting rounded cumulative probabilities and reporting false precision. Compute category differences from unrounded values, then round the resulting probabilities. Rounded entries may not sum to exactly one.
Check Your Understanding
Use the definitions, calculations, and comparison logic above.
- In the stated cumulative-logit convention, what direction of outcome shift does a positive slope represent?
- Why are the two empirical treatment contrasts in the diagnostic example descriptive evidence rather than a complete model-based test?
- How many restrictions separate the specified proportional-odds model from the targeted partial model, and what is the corresponding asymptotic likelihood-ratio reference distribution?
- State the condition on \(F_1\) and \(F_2\) that ensures all three category probabilities are nonnegative.
- Why is the likelihood difference between a partial proportional-odds model and a baseline-category multinomial model not ordinarily tested with the nested-model chi-square likelihood-ratio test?