Adjacent-Category Example: Disease Stages

Modeling Neighboring Disease Stages

Suppose a clinical study records each patient’s disease stage as one of four ordered categories: Stage I, Stage II, Stage III, or Stage IV. An adjacent-category model describes the odds of being in one stage rather than the immediately neighboring stage. Its coefficients concern these local comparisons; they do not directly describe the probability of being in a particular stage.

Let $Y$ denote disease stage, with $Y=j$ representing Stage $j$, for $j=1,\ldots,4$. Let $x$ be a patient covariate, such as a biomarker measured in specified units, and write $p_j(x)=P(Y=j\mid x)$. We use the convention that the adjacent odds compare the higher-numbered stage with the lower-numbered stage.

Definition: For four ordered disease stages, the adjacent-category logits are $$ \eta_j(x)=\log\left(\frac{p_{j+1}(x)}{p_j(x)}\right),\qquad j=1,2,3. $$ An adjacent-category regression model specifies these three logits as functions of covariates. In a common-slope model with one covariate, $$ \eta_j(x)=\alpha_j+\beta x,\qquad j=1,2,3. $$

The model requires positive probabilities for the categories whose log odds are used. Its three logits are linked because they describe one four-category probability distribution. In particular, they cannot be interpreted as three unrelated binary-outcome models: the resulting probabilities must be nonnegative and sum to one.

Fitting and Interpreting a Disease-Stage Model

For independent patients with observed stages $y_i$ and covariates $x_i$, the likelihood is the product of the model probabilities assigned to their observed stages. Equivalently, the log-likelihood is

\ell(\alpha_1,\alpha_2,\alpha_3,\beta) =\sum_{i=1}^{n}\log p_{y_i}(x_i).

The probabilities in this expression are determined by the adjacent logits and the normalization described below. Fitting the model means choosing coefficient values that maximize this log-likelihood, subject to the model specification. If observations are grouped by covariate pattern, with $n_{ij}$ patients at pattern $i$ observed in stage $j$, the corresponding log-likelihood is $\sum_i\sum_j n_{ij}\log p_j(x_i)$.

Worked Example: Interpreting a Fitted Common-Slope Model

Suppose a study models four ordered stages using a biomarker $x$, measured in units of 10 ng/mL. The fitted logits are $$ \widehat{\eta}_1(x)=-1.2+0.5x,\qquad \widehat{\eta}_2(x)=0.1+0.5x,\qquad \widehat{\eta}_3(x)=-0.8+0.5x. $$

For the comparison of Stage II with Stage I, a one-unit increase in $x$ changes the fitted log odds by $0.5$. Exponentiating gives the odds ratio: $$ \frac{\widehat p_2(x+1)/\widehat p_1(x+1)} {\widehat p_2(x)/\widehat p_1(x)} =\exp(0.5)\approx 1.6487. $$

Thus, for patients whose biomarker values differ by 10 ng/mL, the fitted odds of Stage II rather than Stage I are about 1.65 times as large for the patient with the higher value. Because the slope is common, the same odds ratio applies to Stage III versus Stage II and to Stage IV versus Stage III. This is an interpretation of local odds, not a claim that each stage probability rises by 65%.

The intercepts are the fitted adjacent log odds at $x=0$. For example, at $x=0$, the fitted Stage II-to-Stage I odds are $\exp(-1.2)\approx0.3012$, while the fitted Stage III-to-Stage II odds are $\exp(0.1)\approx1.1052$. These are odds, not probabilities.

With category-specific slopes, the more general model is $\eta_j(x)=\alpha_j+\beta_jx$. Then a one-unit increase in $x$ multiplies the odds for comparison $j$ by $\exp(\beta_j)$, which can differ across neighboring-stage pairs. The common-slope form is a restriction, not a defining feature of every adjacent-category regression.

Recovering Stage Probabilities

A fitted adjacent logit gives a ratio of neighboring probabilities. To obtain the probabilities themselves, first exponentiate each logit to get the neighboring odds, then multiply successive odds to express every stage’s probability relative to Stage I. Finally, normalize those relative weights to sum to one.

Result: For any covariate value $x$, define $q_j(x)=\exp\{\eta_j(x)\}$. The stage probabilities are $$ p_1(x)=\frac{1}{D(x)},\qquad p_k(x)=\frac{\prod_{r=1}^{k-1}q_r(x)}{D(x)}\quad(k=2,3,4), $$ where $$ D(x)=1+q_1(x)+q_1(x)q_2(x)+q_1(x)q_2(x)q_3(x). $$

To verify the result, let $w_1=1$ and $w_k=\prod_{r=1}^{k-1}q_r(x)$ for $k=2,3,4$. Since $q_j(x)=p_{j+1}(x)/p_j(x)$, multiplying successive ratios gives $p_k(x)/p_1(x)=w_k$. Hence $p_k(x)=w_kp_1(x)$. Summing over the four stages gives $1=p_1(x)\sum_{k=1}^4w_k$, so $p_1(x)=1/D(x)$, and the displayed formula follows for every $k$. Each weight is positive, and dividing by their sum makes the probabilities positive and ensures that they sum to one. $\square$

Worked Example: Calculating Probabilities at Two Biomarker Values

Use the fitted model $\widehat{\eta}_1(x)=-1.2+0.5x$, $\widehat{\eta}_2(x)=0.1+0.5x$, and $\widehat{\eta}_3(x)=-0.8+0.5x$. Calculate the four fitted stage probabilities for $x=0$ and $x=1$.

At $x=0$, the three adjacent odds are $q_1=e^{-1.2}\approx0.3012$, $q_2=e^{0.1}\approx1.1052$, and $q_3=e^{-0.8}\approx0.4493$. The relative weights for Stages I–IV are therefore $$ 1,\quad e^{-1.2}\approx0.3012,\quad e^{-1.2+0.1}=e^{-1.1}\approx0.3329,\quad e^{-1.2+0.1-0.8}=e^{-1.9}\approx0.1496. $$ Their sum is approximately $D(0)=1.7836$. Dividing each weight by this sum gives $$ (\widehat p_1(0),\widehat p_2(0),\widehat p_3(0),\widehat p_4(0)) \approx(0.5607,\;0.1689,\;0.1866,\;0.0839). $$ The displayed rounded values sum to $1.0001$, with the discrepancy from 1 due to rounding; the unrounded probabilities sum to 1 by construction.

At $x=1$, the logits are $-0.7$, $0.6$, and $-0.3$. The corresponding relative weights are $$ 1,\quad e^{-0.7}\approx0.4966,\quad e^{-0.7+0.6}=e^{-0.1}\approx0.9048,\quad e^{-0.7+0.6-0.3}=e^{-0.4}\approx0.6703. $$ Their sum is approximately $D(1)=3.0717$. Normalizing gives $$ (\widehat p_1(1),\widehat p_2(1),\widehat p_3(1),\widehat p_4(1)) \approx(0.3255,\;0.1617,\;0.2946,\;0.2182). $$ These rounded values sum to $1.0000$. The comparison shows why the coefficient interpretation cannot be read as a direct probability change: the fitted probability for Stage II is slightly lower at $x=1$, even though the odds of each higher stage relative to its immediate predecessor have increased.

Comparing Neighboring-Stage Odds Across Patients

For two patients with covariate values $x_A$ and $x_B$, the fitted odds ratio for the same adjacent-stage comparison $j$ follows directly from subtracting the two logits. In the common-slope model,

\frac{\widehat p_{j+1}(x_B)/\widehat p_j(x_B)} {\widehat p_{j+1}(x_A)/\widehat p_j(x_A)} =\exp\{\beta(x_B-x_A)\}.

This result follows because the intercept cancels: $\widehat\eta_j(x_B)-\widehat\eta_j(x_A) =(\widehat\alpha_j+\widehat\beta x_B)-(\widehat\alpha_j+\widehat\beta x_A) =\widehat\beta(x_B-x_A)$. Exponentiating both sides gives the odds ratio. The multiplier is the same for all three adjacent comparisons when the slope is common, but the odds themselves need not be the same across comparisons.

Worked Example: Comparing Two Patients’ Neighboring-Stage Odds

Patient A has $x_A=0.4$, and Patient B has $x_B=1.2$. Using the fitted model above, calculate and compare their odds for each neighboring-stage pair.

The covariate difference is $1.2-0.4=0.8$. The common-slope odds ratio for Patient B relative to Patient A is $$ \exp\{0.5(0.8)\}=\exp(0.4)\approx1.4918. $$

For Stage II versus Stage I, the logits are $\widehat\eta_1(0.4)=-1.0$ and $\widehat\eta_1(1.2)=-0.6$. The fitted odds are $e^{-1.0}\approx0.3679$ for Patient A and $e^{-0.6}\approx0.5488$ for Patient B; their ratio is approximately $0.5488/0.3679=1.4918$.

For Stage III versus Stage II, the logits are $0.3$ and $0.7$, giving odds $e^{0.3}\approx1.3499$ and $e^{0.7}\approx2.0138$. Their ratio is approximately $2.0138/1.3499=1.4918$. For Stage IV versus Stage III, the logits are $-0.6$ and $-0.2$, giving odds $e^{-0.6}\approx0.5488$ and $e^{-0.2}\approx0.8187$. Their ratio is approximately $0.8187/0.5488=1.4918$.

Thus, under this fitted common-slope model, Patient B has about 1.49 times the neighboring-stage odds of Patient A for each pair. This does not say that Patient B is 1.49 times as likely to be in every higher stage. For such probability statements, the complete set of stage probabilities must be calculated and compared.

What the Model Says—and a Common Pitfall

An adjacent-category model is useful when the scientific comparisons are naturally local: for example, whether a patient is at Stage II rather than Stage I, Stage III rather than Stage II, or Stage IV rather than Stage III. The model supplies a direct covariate effect for each such comparison, while the normalization formula turns the fitted local odds into a coherent distribution over all stages.

A common mistake is to interpret $\exp(\beta)$ as a probability ratio or as a fixed increase in the probability of advanced disease. In the common-slope model it is the multiplier in the odds $p_{j+1}(x)/p_j(x)$ for a one-unit covariate increase. The probabilities depend jointly on all three logits and their normalization; therefore their changes need not match one another, and an individual stage probability can even decrease when all adjacent odds increase.

Key takeaway: State which neighboring stages define an adjacent odds comparison, use the fitted logit to interpret the covariate effect on those odds, and normalize the successive probability ratios before reporting stage probabilities.

Check Your Understanding

Use the definitions and calculations in this tutorial to answer the following questions.

In a four-stage model, what odds does $\eta_2(x)$ represent?
In a common-slope model with $\beta=0.3$, what is the odds multiplier for a two-unit increase in $x$?
Explain why the relative weight for Stage IV is $q_1q_2q_3$, rather than $q_3$ alone.
Why can a fitted adjacent odds ratio greater than one coexist with a decrease in one of the stage probabilities?
What additional calculation is needed to turn three fitted adjacent logits into four stage probabilities?