Drug Interactions in Binary and Ordinal Response Models

Why the Response Scale Matters

A combination can differ from the effects of its component drugs in more than one statistical sense. For a binary outcome, an interaction may be defined on the log-odds scale or as a departure from additivity of event probabilities. For an ordinal outcome, the question can be asked at each cumulative threshold. These contrasts are not interchangeable: a model with no interaction on the log-odds scale can still imply a nonzero probability-scale contrast.

In PK/PD work, the predictors may be doses or, when the estimand concerns pharmacodynamic effects conditional on exposure, concentrations of the two drugs. The choice matters: a dose-based contrast can include changes in exposure caused by the combination, whereas a concentration-based contrast describes response at specified exposures. The contrast itself does not establish a mechanism.

Definition: For two drug levels $x$ and $z$, let $\eta(x,z)$ be a model's linear predictor. The link-scale interaction contrast is $$ \Delta_{\eta}(x,z)=\eta(x,z)-\eta(x,0)-\eta(0,z)+\eta(0,0). $$ It compares the combination with the two single-drug effects and the reference condition on the model's link scale. The corresponding probability-scale contrast is $$ \Delta_{p}(x,z)=p(x,z)-p(x,0)-p(0,z)+p(0,0), $$ where $p(x,z)$ is the probability of the specified event. A zero contrast indicates no departure from additivity on that particular scale at those levels; it does not imply a zero contrast on every scale.

Binary Outcomes: Logistic Interaction Contrasts

Let $Y=1$ denote a specified event, such as response or toxicity, and let $p(x,z)=P(Y=1\mid x,z)$. A logistic PK/PD model has the form

\operatorname{logit}\{p(x,z)\} =\log\!\left(\frac{p(x,z)}{1-p(x,z)}\right) =\beta_0+f_A(x)+f_B(z)+g(x,z).

Here $\beta_0$ is the reference log-odds, $f_A$ and $f_B$ describe each drug's contribution, and $g$ represents a modeled combination contribution. A common parameterization sets $f_A(0)=f_B(0)=0$ and $g(x,0)=g(0,z)=g(0,0)=0$. Then $\Delta_\eta(x,z)=g(x,z)$. In a two-level design with $x,z\in\{0,1\}$, the model may instead include a product term:

\operatorname{logit}\{p(x,z)\} =\beta_0+\beta_Ax+\beta_Bz+\gamma xz.

For the four treatment combinations, the difference-in-differences is exactly $\gamma$. This is an interaction in log-odds, not a claim that probabilities are additive. It is often useful to exponentiate the contrast: $\exp\{\Delta_\eta(x,z)\}$ is the ratio of the combination odds to the odds predicted by multiplying the two single-drug odds ratios relative to the reference.

Worked Example: Comparing Binary Interaction Scales

Suppose a hypothetical study gives the following estimated probabilities of an event: reference $p(0,0)=0.10$, drug A alone $p(1,0)=0.20$, drug B alone $p(0,1)=0.25$, and the combination $p(1,1)=0.50$. Calculate both interaction contrasts.

Probability scale. The additive contrast is

\Delta_p=0.50-0.20-0.25+0.10=0.15.

Thus the combination event probability is 0.15 higher than the value implied by adding the two single-drug probability changes to the reference probability. This is a 15-percentage-point contrast for this event.

Log-odds scale. The four odds are $0.10/0.90=0.1111$, $0.20/0.80=0.25$, $0.25/0.75=0.3333$, and $0.50/0.50=1.000$. Therefore,

\Delta_\eta =\log(1.000)-\log(0.25)-\log(0.3333)+\log(0.1111) \approx 0.2877.

The odds-ratio contrast is $\exp(0.2877)\approx1.333$: the combination odds are about 1.33 times the odds predicted by multiplying the two single-drug odds ratios relative to reference. The probability contrast and the log-odds contrast answer different questions, so their numerical values should not be compared as though they were the same effect measure.

As a check, no log-odds interaction would predict combination odds $0.25\times0.3333/0.1111=0.7500$, or probability $0.7500/(1+0.7500)=0.4286$. That differs from the observed hypothetical combination probability of 0.50.

Why the Two Scales Can Disagree

The logistic function is nonlinear. Consequently, adding drug effects to the log-odds does not generally add their probability changes. If $\Delta_\eta=0$, the combination odds obey the multiplicative relation implied by the two monotherapies, but $\Delta_p$ need not equal zero. Conversely, a probability contrast of zero at one set of levels does not require the log-odds contrast to be zero.

A short derivation makes the link-scale interpretation precise. Write $o(x,z)=p(x,z)/\{1-p(x,z)\}$ for the event odds. Since $\eta(x,z)=\log o(x,z)$, exponentiating the four-term contrast gives

\exp\{\Delta_\eta(x,z)\} =\frac{o(x,z)o(0,0)}{o(x,0)o(0,z)}.

Thus a zero link-scale contrast is equivalent to $o(x,z)o(0,0)=o(x,0)o(0,z)$. This equation describes multiplicative odds, not additive probabilities. The distinction is important whenever the scientific question concerns a specific risk difference, such as an absolute increase in event probability.

Worked Example: Interpreting a Product-Term Coefficient

Consider a hypothetical binary logistic model with binary exposure indicators $x,z\in\{0,1\}$:

\operatorname{logit}\{p(x,z)\}=-2+0.8x+0.6z+0.5xz.

At reference, the linear predictor is $-2$, so $p(0,0)=1/(1+\exp(2))\approx0.1192$. For A alone, the predictor is $-2+0.8=-1.2$, giving $p(1,0)=1/(1+\exp(1.2))\approx0.2315$. For B alone, the predictor is $-2+0.6=-1.4$, giving $p(0,1)\approx0.1978$. For the combination, it is $-2+0.8+0.6+0.5=-0.1$, giving $p(1,1)\approx0.4750$.

The link-scale interaction is the product-term coefficient, $\Delta_\eta=0.5$, and its odds-ratio contrast is $\exp(0.5)\approx1.649$. The probability-scale contrast, however, is

\Delta_p=0.4750-0.2315-0.1978+0.1192\approx0.1649.

The model therefore implies a 16.49-percentage-point probability contrast at these exposure levels, alongside a 0.5-unit log-odds contrast. The coefficient $0.5$ is not a probability increase of 50 percentage points.

Ordinal Outcomes: Cumulative-Logit Contrasts

For an ordered outcome $Y$ with categories $1,\ldots,K$, the cumulative-logit model describes probabilities at thresholds $k=1,\ldots,K-1$. One convention is

\operatorname{logit}\{P(Y\leq k\mid x,z)\} =\theta_k+f_A(x)+f_B(z)+g(x,z), \qquad k=1,\ldots,K-1.

The thresholds must be ordered, $\theta_1<\theta_2<\cdots<\theta_{K-1}$, to give ordered cumulative probabilities. This specification uses a common drug effect and interaction across thresholds; that restriction is the proportional-odds assumption. Because the modeled event is $Y\leq k$, a positive contrast multiplies the cumulative odds of being at or below each threshold by a factor greater than one, relative to the multiplicative single-drug odds prediction. The clinical direction depends on how the categories are ordered.

At each threshold, the interaction contrast is the four-term difference in the cumulative logits. Under proportional odds, the contrast is the same at every threshold. If the effects differ by threshold, a single common interaction term does not describe those data well. Category probabilities are obtained by differencing cumulative probabilities: $P(Y=1)=P(Y\leq1)$, $P(Y=j)=P(Y\leq j)-P(Y\leq j-1)$, and $P(Y=K)=1-P(Y\leq K-1)$. A cumulative-logit interaction is therefore not itself a contrast in the probability of any one category.

Worked Example: Checking Interaction Contrasts Across Ordinal Thresholds

Suppose a hypothetical three-category outcome has categories 1, 2, and 3, ordered from lower to higher severity. The estimated cumulative probabilities $P(Y\leq1)$ and $P(Y\leq2)$ are respectively: reference $(0.40,0.85)$, A alone $(0.55,0.90)$, B alone $(0.50,0.88)$, and combination $(0.60,0.92)$. Calculate the link-scale contrast at each threshold.

At threshold 1, the cumulative logits are $\operatorname{logit}(0.40)=-0.4055$, $\operatorname{logit}(0.55)=0.2007$, $\operatorname{logit}(0.50)=0$, and $\operatorname{logit}(0.60)=0.4055$. Hence

\Delta_{\eta,1}=0.4055-0.2007-0+(-0.4055)\approx-0.2007.

At threshold 2, the cumulative logits are $\operatorname{logit}(0.85)=1.7346$, $\operatorname{logit}(0.90)=2.1972$, $\operatorname{logit}(0.88)=1.9924$, and $\operatorname{logit}(0.92)=2.4423$. Therefore,

\Delta_{\eta,2}=2.4423-2.1972-1.9924+1.7346\approx-0.0127.

The two threshold contrasts differ, so these hypothetical probabilities do not support one common interaction contrast across thresholds. For further context, the category probabilities are reference $(0.40,0.45,0.15)$, A $(0.55,0.35,0.10)$, B $(0.50,0.38,0.12)$, and combination $(0.60,0.32,0.08)$. The category-2 probability contrast is $0.32-0.35-0.38+0.45=0.04$, which is distinct from either cumulative-logit contrast.

Assumptions, Diagnostics, and Alternatives

Interpretation depends on the outcome definition, exposure contrast, link, and model structure. For a binary logistic model, observations must be represented by the specified binary event, and the logit of its conditional probability must be adequately described by the chosen predictors and interaction function over the exposure range of interest. For a cumulative-logit model, the outcome categories must have a meaningful order, cumulative probabilities must be coherent, and a common effect across thresholds is assumed unless threshold-specific effects are fitted. In both cases, the combination and relevant single-drug conditions need sufficient support in the design; extrapolated contrasts may be highly uncertain.

Useful checks include comparing fitted probabilities with observed proportions across treatment or exposure combinations and checking that fitted cumulative probabilities increase with the ordinal threshold. For an ordinal model, compare threshold-specific patterns with the common-effect restriction: marked differences, such as those in the example, warn that one proportional-odds interaction may be inadequate. Sparse or empty combination cells can also make estimates unstable or prevent reliable estimation. These checks do not prove the model correct; they indicate where its assumptions may be strained.

A probability-scale contrast is often easiest to relate to an absolute clinical risk change, while a log-odds contrast is directly aligned with a logistic model's interaction term. For ordinal responses, cumulative odds retain information about ordering, whereas category-probability contrasts address particular categories. The appropriate scale is determined by the estimand and scientific question, not by which contrast looks largest. A contrast on any of these scales describes a modeled response pattern; by itself it does not identify why the pattern arose.

Common Pitfall

Common mistake: “The logistic model has no interaction term, so the drugs have no interaction in probability.” A model without a product term imposes additivity on the log-odds scale; the logistic transformation can still produce a nonzero probability-scale contrast. State the scale, calculate the contrast on that scale, and interpret it in the outcome's units.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

If $\Delta_\eta=0.4$ in a binary logistic model, what is the odds-ratio contrast, and what does it compare?
Does a zero logistic interaction contrast guarantee a zero probability interaction contrast? Explain.
In a cumulative-logit model, what does the proportional-odds assumption require about the drug-interaction contrast across thresholds?
For the hypothetical binary probabilities $p(0,0)=0.10$, $p(1,0)=0.20$, $p(0,1)=0.25$, and $p(1,1)=0.50$, calculate the probability interaction contrast.

Solutions. (1) $\exp(0.4)\approx1.492$. The combination odds are about 1.49 times the odds predicted by multiplying the single-drug odds ratios relative to reference. (2) No. The logistic transformation is nonlinear, so additivity on the log-odds scale does not generally imply additivity of probabilities. (3) It requires the common drug and interaction effects on the cumulative-logit scale to be the same at every threshold. (4) $0.50-0.20-0.25+0.10=0.15$, a 15-percentage-point probability contrast for the specified event.