Regression Effects in Continuation-Ratio Models

Regression Effects in Sequential Risk Sets

A continuation-ratio model describes an ordered response as a sequence of decisions: at each stage, an observation either stops in the current category or continues to a later one. Regression enters by allowing the odds of stopping to depend on covariates. The key interpretive point is that a coefficient describes a change in conditional stopping odds among observations still at risk at that stage. It is not, in general, a direct change in the unconditional probability of a response category.

Let $Y$ take ordered values $1,\ldots,K$, and consider the forward sequence. At stage $j$, for $j=1,\ldots,K-1$, the risk set consists of observations with $Y\geq j$: those that have not stopped in an earlier category. Write $q_j(x)=P(Y=j\mid Y\geq j,x)$, where $x$ is a covariate vector. The last category is reached if the observation continues through every earlier stage.

Definition: A forward continuation-ratio regression model specifies, for each stage $j=1,\ldots,K-1$, $$ \operatorname{logit}\{q_j(x)\} =\log\left\{\frac{P(Y=j\mid Y\geq j,x)} {P(Y>j\mid Y\geq j,x)}\right\} =\alpha_j+x^\mathsf{T}\beta_j. $$ Here $\alpha_j$ is the stage-specific intercept and $\beta_j$ is the vector of covariate effects at that stage. The numerator is the conditional probability of stopping at $j$; the denominator is the conditional probability of continuing beyond $j$.

Each stage has its own binary conditional comparison: stop now versus continue, among the observations in that stage’s risk set. A model may allow the slope vector $\beta_j$ to vary by stage. A common-slope model instead imposes $\beta_j=\beta$ for all $j$, while generally retaining different intercepts $\alpha_j$. These are substantively different assumptions about how covariates act on the sequential process.

What a Regression Coefficient Means

For a single covariate $x_r$, the coefficient $\beta_{jr}$ describes the effect of a one-unit increase in that covariate on the log odds of stopping at stage $j$, conditional on reaching that stage. Exponentiating gives an odds ratio. Other covariates are held fixed in this comparison.

Theorem: In the continuation-ratio model, compare covariate vectors $x$ and $x+d$ at stage $j$. Their conditional stopping-odds ratio is $$ \frac{q_j(x+d)/\{1-q_j(x+d)\}} {q_j(x)/\{1-q_j(x)\}} =\exp(d^\mathsf{T}\beta_j). $$ In particular, a one-unit increase in covariate $x_r$, with the other covariates fixed, multiplies the conditional stopping odds by $\exp(\beta_{jr})$.

Proof. The model gives the conditional stopping odds at $x$ as $\exp(\alpha_j+x^\mathsf{T}\beta_j)$. At $x+d$, they are $\exp\{\alpha_j+(x+d)^\mathsf{T}\beta_j\}$. Dividing cancels the stage intercept and the terms involving $x$, leaving $\exp(d^\mathsf{T}\beta_j)$. Taking $d$ to be one unit in coordinate $r$ gives $\exp(\beta_{jr})$. $\square$

An odds ratio above one means higher conditional odds of stopping rather than continuing at that stage; an odds ratio below one means lower conditional stopping odds. This interpretation is tied to the risk set. The comparison does not say that every observation in the original sample has that odds ratio for category $j$, because observations who stopped earlier are not in stage $j$'s comparison.

Worked Examples

Worked Example: A Binary Exposure With Stage-Specific Effects

Suppose the response has three ordered categories and $x=1$ denotes exposure while $x=0$ denotes no exposure. The fitted logits are $$ \operatorname{logit}(q_1)= -1+0.8x,\qquad \operatorname{logit}(q_2)=0.4-0.5x. $$ Interpret both exposure effects, then calculate the three category probabilities for each exposure group.

At stage 1, the exposure conditional odds ratio is $\exp(0.8)=2.226$ (rounded). Exposure is associated with about 2.23 times the stopping odds in category 1 rather than continuing, among observations at risk at stage 1.

At stage 2, the conditional odds ratio is $\exp(-0.5)=0.607$. Among observations who reached stage 2, exposure is associated with lower odds of stopping in category 2 rather than continuing to category 3. The stage 1 and stage 2 effects differ in both magnitude and direction.

For $x=0$, the stage 1 logit is $-1$, so $q_1(0)=1/(1+\exp(1))\approx0.2689$. The stage 2 logit is $0.4$, so $q_2(0)=1/(1+\exp(-0.4))\approx0.5987$. Thus $P(Y=1\mid x=0)=0.2689$. The probability of reaching stage 2 is $1-0.2689=0.7311$, so $P(Y=2\mid x=0)=0.7311(0.5987)\approx0.4377$. The remaining probability is $P(Y=3\mid x=0)=0.7311(1-0.5987)\approx0.2934$. The three probabilities sum to approximately $1.0000$.

For $x=1$, the logits are $-1+0.8=-0.2$ and $0.4-0.5=-0.1$. Hence $q_1(1)=1/(1+\exp(0.2))\approx0.4502$ and $q_2(1)=1/(1+\exp(0.1))\approx0.4750$. Therefore $P(Y=1\mid x=1)=0.4502$, $P(Y=2\mid x=1)=(1-0.4502)(0.4750)\approx0.2612$, and $P(Y=3\mid x=1)=(1-0.4502)(1-0.4750)\approx0.2887$. These also sum to approximately $1.0001$, with the small discrepancy due to rounding.

The odds ratios above compare conditional stopping odds at their respective stages. The category probabilities also reflect the probability of reaching each stage, so they are not obtained by applying those odds ratios directly to the category probabilities.

Worked Example: A Continuous Covariate

Let $x$ be a measurement in years, and suppose a three-category response has $$ \operatorname{logit}(q_1)=-1+0.25x,\qquad \operatorname{logit}(q_2)=0.5-0.20x. $$ Interpret a four-year increase at each stage and calculate the conditional stopping probabilities at $x=0$ and $x=4$.

At stage 1, a four-year increase changes the log odds by $4(0.25)=1$, so the conditional stopping-odds ratio is $\exp(1)\approx2.718$. The probabilities are $q_1(0)=1/(1+\exp(1))\approx0.2689$ and $q_1(4)=1/(1+\exp(0))=0.5$.

At stage 2, a four-year increase changes the log odds by $4(-0.20)=-0.8$, giving the odds ratio $\exp(-0.8)\approx0.4493$. The logits are $0.5$ at $x=0$ and $0.5-0.8=-0.3$ at $x=4$. Thus $q_2(0)=1/(1+\exp(-0.5))\approx0.6225$, while $q_2(4)=1/(1+\exp(0.3))\approx0.4256$.

The covariate has an increasing conditional stopping effect at the first stage and a decreasing one at the second. Each odds ratio applies to a four-year contrast with the other modeled covariates held fixed; it does not assert that the same people necessarily form both risk sets.

Worked Example: A Common Slope Is Not a Common Probability Effect

Consider a three-category model with a binary exposure and the common slope $\beta=\log(2)$: $$ \operatorname{logit}(q_1)=-1+x\log(2),\qquad \operatorname{logit}(q_2)=x\log(2). $$ At both stages, exposure doubles the conditional stopping odds. Calculate the category probabilities to see why this does not mean that it doubles each category probability.

For $x=0$, $q_1=1/(1+\exp(1))\approx0.2689$ and $q_2=1/2=0.5$. Therefore the category probabilities are $P(Y=1\mid x=0)=0.2689$, $P(Y=2\mid x=0)=(1-0.2689)(0.5)\approx0.3655$, and $P(Y=3\mid x=0)=(1-0.2689)(0.5)\approx0.3655$.

For $x=1$, the stage 1 odds are $\exp(-1)\times2=2\exp(-1)$, so $q_1=2/(e+2)\approx0.4239$. At stage 2, the odds are $2$, so $q_2=2/3\approx0.6667$. The resulting probabilities are approximately $P(Y=1\mid x=1)=0.4239$, $P(Y=2\mid x=1)=(1-0.4239)(0.6667)\approx0.3841$, and $P(Y=3\mid x=1)=(1-0.4239)(0.3333)\approx0.1920$.

The conditional stopping-odds ratio is exactly 2 at each stage, but the category 1 probability changes from about $0.2689$ to $0.4239$, not by a factor of 2. Category 2 changes from about $0.3655$ to $0.3841$, and category 3 decreases. A common slope is a common multiplier for the stage-specific conditional odds, not a common multiplier for the unconditional category probabilities.

What the Common-Slope Assumption Does—and Does Not—Say

The common-slope restriction is $\beta_1=\cdots=\beta_{K-1}=\beta$. By the odds-ratio result, a covariate contrast $d$ then multiplies the conditional stopping odds by $\exp(d^\mathsf{T}\beta)$ at every stage. Intercepts can still differ, so baseline stopping probabilities and odds need not be the same across stages.

The restriction does not make the sequential decisions identical. Risk sets differ from stage to stage, and each stage can have a different intercept. Nor does it imply that the response-category probabilities change in parallel. Those probabilities combine the chance of stopping at a stage with the chance of having continued far enough to reach it.

This distinction matters in reporting results. “The exposure doubles the odds of stopping at each stage, conditional on reaching that stage” is an interpretation of a common slope. “The exposure doubles the probability of every response category” is not. For stage-specific slopes, report the relevant conditional odds ratio separately for each stage and identify the covariate contrast and risk set.

Likelihood and Risk-Set Interpretation

The sequential formulation also gives a direct likelihood contribution. For observation $i$, let $x_i$ be its covariates and $Y_i$ its observed category. At stage $j$, define $Z_{ij}=1$ if $Y_i=j$, and $Z_{ij}=0$ if $Y_i>j$, for stages the observation reaches. An observation with $Y_i<j$ is not at risk at stage $j$, so it contributes no stage $j$ factor.

Result: If $Y_i=y<K$, the conditional likelihood contribution is $$ P(Y_i=y\mid x_i) =\left\{\prod_{j=1}^{y-1}(1-q_j(x_i))\right\}q_y(x_i). $$ If $Y_i=K$, it is $$ P(Y_i=K\mid x_i)=\prod_{j=1}^{K-1}(1-q_j(x_i)). $$ Equivalently, for the stages reached by observation $i$, its contribution is $$ \prod_{j=1}^{K-1} q_j(x_i)^{Z_{ij}}\{1-q_j(x_i)\}^{1-Z_{ij}}, $$ where factors for stages the observation did not reach are omitted.

Derivation. To have $Y_i=y<K$, the observation must continue at stages $1,\ldots,y-1$, whose conditional probabilities are $1-q_j(x_i)$, and then stop at stage $y$, whose conditional probability is $q_y(x_i)$. Multiplying these successive conditional probabilities gives the first expression. To have $Y_i=K$, it must continue at every one of the first $K-1$ stages, giving the second expression. The product using $Z_{ij}$ selects $q_j$ at the stopping stage and $1-q_j$ at each earlier continuation stage, which is exactly the same contribution. $\square$

This factorization explains the link between continuation-ratio regression and binary logistic regressions organized by stage. It does not change the meaning of the coefficient: stage $j$'s binary comparison is defined only for observations reaching that stage. Treating all observations as if they supplied a stop-or-continue outcome at every stage would add contributions from people who were not at risk and would not represent this likelihood.

Common Mistakes and a Practical Interpretation

Calling the coefficient a category-probability effect. The coefficient changes conditional stopping log odds. Category probabilities also depend on the probabilities of continuing through earlier stages.
Ignoring the risk set. The stage $j$ odds ratio compares stopping with continuing among observations with $Y\geq j$, not among the full sample without qualification.
Reading a common slope as a common intercept. A common-slope model constrains the covariate effects; it allows stage-specific intercepts and therefore different baseline odds.
Confusing odds ratios with probability ratios. $\exp(\beta_{jr})$ is a ratio of conditional odds. It is not generally the ratio of conditional probabilities, and still less the ratio of unconditional category probabilities.
Reporting an effect without its contrast. State the covariate change, the stage, and which other covariates are held fixed. For a continuous covariate, a change of $c$ units has odds ratio $\exp(c\beta_{jr})$.

Key takeaway: In a continuation-ratio regression, $\exp(\beta_{jr})$ is a multiplicative effect on the odds of stopping at stage $j$ rather than continuing, conditional on reaching that stage. Stage-specific slopes allow these effects to vary; common slopes equate the conditional odds multipliers, not the category-probability effects.

Check Your Understanding

Use the definitions and interpretations above to answer these questions.

In a forward continuation-ratio model, which observations form the risk set at stage $j$, and what are the two outcomes compared there?
If $\beta_{jr}=-0.4$, what is the conditional stopping-odds ratio for a one-unit increase in covariate $x_r$ at stage $j$? Interpret its direction.
What does the common-slope restriction equate across stages, and what does it not imply about category probabilities?
Why does an observation in category $y$ contribute no stage-specific stop-or-continue factor for stages later than $y$?
For a change of $c$ units in a continuous covariate, write the conditional stopping-odds ratio at stage $j$.