Marginal Models For Binary Efficacy And Toxicity

Why Model Efficacy and Toxicity Separately?

A dose-finding trial may record whether each patient experiences an efficacy outcome and whether that patient experiences a toxicity outcome. Each is binary, but the two outcomes answer different questions. At a given dose, one model can describe the probability of efficacy and another the probability of toxicity. These are marginal models: each specifies the probability of one outcome without specifying the joint distribution of both outcomes for the same patient.

This separation is useful when the immediate modeling goals are dose-specific efficacy and toxicity probabilities. It does not imply that efficacy and toxicity are independent. A patient’s two outcomes can be associated even when each probability is modeled separately. The models also do not, on their own, define a dose-selection rule or determine how clinical priorities should trade efficacy against harm.

Definition: Let \(Y_E\) indicate whether a patient has the prespecified efficacy outcome within its assessment window, and let \(Y_T\) indicate whether the patient has the prespecified toxicity outcome within its assessment window. At dose \(d\), the marginal probabilities are \(\pi_E(d)=P(Y_E=1\mid d)\) and \(\pi_T(d)=P(Y_T=1\mid d)\). Each probability concerns one outcome at a time; neither specifies \(P(Y_E,Y_T\mid d)\).

The outcome definitions and observation windows must be clear before fitting either model. For example, efficacy might mean an objective response by a specified assessment time, while toxicity might mean at least one dose-limiting toxicity during a defined period. Changing those definitions changes the probabilities being modeled.

Logistic and Probit Formulations

For either outcome, choose a dose representation \(x(d)\), such as the dose in a stated unit or a centered and scaled dose. A simple model uses a linear predictor in that representation. Separate coefficients are required because the efficacy and toxicity relationships need not have the same baseline probability or dose trend.

Model specification: For patients observed at dose \(d\), one may specify \(\operatorname{logit}\{\pi_E(d)\}=\beta_{E0}+\beta_{E1}x(d)\) and \(\operatorname{logit}\{\pi_T(d)\}=\beta_{T0}+\beta_{T1}x(d)\), where \(\operatorname{logit}(p)=\log\{p/(1-p)\}\). Equivalently, \(\pi_k(d)=\{1+\exp[-(\beta_{k0}+\beta_{k1}x(d))]\}^{-1}\), for \(k\in\{E,T\}\). A probit alternative specifies \(\Phi^{-1}\{\pi_k(d)\}=\gamma_{k0}+\gamma_{k1}x(d)\), or \(\pi_k(d)=\Phi\{\gamma_{k0}+\gamma_{k1}x(d)\}\), where \(\Phi\) is the standard normal cumulative distribution function.

The link maps probabilities between 0 and 1 to an unbounded scale on which a linear predictor can be used. In a logistic model, a one-unit increase in \(x(d)\) multiplies the outcome odds by \(\exp(\beta_{k1})\), when the linear predictor is as specified. It does not change the probability by a constant amount. In a probit model, the slope is a change in the standard-normal quantile of the probability, not a probability difference or odds ratio.

The index \(x(d)\) determines what a coefficient means. If \(x(d)\) is dose measured in milligrams, a one-unit increase means one milligram. If \(x(d)\) is a centered dose divided by 10 milligrams, one unit represents a 10-milligram increase. Centering also makes the intercept refer to the probability at the centered dose, which may be easier to interpret than the probability at dose zero.

Assumptions and Interpretation

For a basic marginal analysis, assume that each outcome is recorded as 0 or 1 for each patient and that the model’s probability is conditional on the assigned dose and any included covariates. The simple likelihood treats patients’ outcome observations as independent conditional on those variables. The selected link and dose function must adequately represent the relationship over the dose range being studied. If a linear dose term is used, the model assumes a linear trend on the link scale—not a linear trend in probability.

A positive slope means the modeled probability increases as \(x(d)\) increases; a negative slope means it decreases. This monotone direction follows from the logistic and probit links being increasing functions. A zero slope means the model assigns the same probability at every dose. If a positive dose trend is imposed by restricting the slope to be positive, that restriction is an additional scientific assumption, not a consequence of using a logistic or probit link.

Randomization supports comparisons between assigned dose groups, but a model’s interpretation remains tied to the trial’s dose assignment and outcome ascertainment. A fitted probability describes the modeled chance of the specified outcome at a dose under the trial conditions; it is not, by itself, a claim that changing the dose causes an outcome in every clinical setting.

Worked Example: Separate Logistic Models at Three Dose Levels

Suppose dose is coded \(x=0,1,2\), with each step representing a 10-milligram increase. For illustration, take the hypothetical efficacy coefficients \(\beta_{E0}=-2.0\) and \(\beta_{E1}=0.80\), and toxicity coefficients \(\beta_{T0}=-3.0\) and \(\beta_{T1}=1.10\). These are assumed parameter values for calculation, not estimates from a real trial.

For efficacy, the model is \(\operatorname{logit}\{\pi_E(d)\}=-2.0+0.80x(d)\). At \(x=1\), the linear predictor is \(-2.0+0.80=-1.20\), so \(\pi_E(d)=1/\{1+\exp(1.20)\}=0.2315\). The model gives an efficacy probability of about 23.15% at that dose.

At \(x=0\), the efficacy probability is \(1/\{1+\exp(2.0)\}=0.1192\). At \(x=2\), it is \(1/\{1+\exp(0.40)\}=0.4013\). The positive slope implies rising modeled efficacy probability over these dose levels, but the probability increases are not constant: from \(x=0\) to \(x=1\) the increase is about 0.1123, while from \(x=1\) to \(x=2\) it is about 0.1698.

For toxicity, the separate model is \(\operatorname{logit}\{\pi_T(d)\}=-3.0+1.10x(d)\). At \(x=1\), its linear predictor is \(-1.90\), giving \(1/\{1+\exp(1.90)\}=0.1301\), or about 13.01%. At \(x=0\), the probability is \(1/\{1+\exp(3.0)\}=0.04743\); at \(x=2\), it is \(1/\{1+\exp(0.80)\}=0.3100\).

The efficacy and toxicity probabilities are calculated from different intercepts and slopes. They can be compared at each dose, but these calculations do not reveal how often the two outcomes occur together in the same patient.

Why the Logistic Coefficient Is an Odds Ratio

The odds at probability \(p\) are \(p/(1-p)\). In a logistic model, the log odds at dose representation \(x\) equal \(\beta_{k0}+\beta_{k1}x\). For two values \(x\) and \(x+1\), subtracting their log odds gives \(\beta_{k1}\). Exponentiating this difference gives the ratio of the odds at the two values.

\log\!\left\{\frac{\pi_k(x+1)}{1-\pi_k(x+1)}\right\} - \log\!\left\{\frac{\pi_k(x)}{1-\pi_k(x)}\right\} =\beta_{k1}, \qquad \frac{\pi_k(x+1)/(1-\pi_k(x+1))} {\pi_k(x)/(1-\pi_k(x))} =\exp(\beta_{k1}).

For the efficacy example, \(\exp(0.80)=2.226\): each 10-milligram increase multiplies the modeled efficacy odds by about 2.226. This does not mean that the efficacy probability increases by 2.226 or by a fixed percentage-point amount. The probability change depends on the starting probability.

Worked Example: Reading a Probit Model

Consider a hypothetical probit efficacy model with \(\gamma_{E0}=-1.0\), \(\gamma_{E1}=0.50\), and dose coding \(x=0,1,2\), where one unit again represents a 10-milligram increase. At \(x=1\), the probit index is \(-1.0+0.50=-0.50\). Thus \(\pi_E(d)=\Phi(-0.50)=0.3085\), approximately 30.85%.

At \(x=0\), \(\pi_E(d)=\Phi(-1.0)=0.1587\); at \(x=2\), \(\pi_E(d)=\Phi(0)=0.5000\). Because \(\gamma_{E1}\) is positive and \(\Phi\) is increasing, the modeled probabilities increase across these doses. The coefficient 0.50 is a half-unit increase in the normal quantile per 10 milligrams; it is not a 50-percentage-point increase in efficacy.

The probit and logistic links can produce similar probability curves for some parameter choices, but their coefficients have different scales. Their coefficients should not be compared directly as if they measured the same change.

Allowing Dose-Specific Probabilities Without a Trend

A linear dose term is parsimonious, but it imposes a smooth trend on the link scale. If the scientific question concerns a finite set of dose levels and a trend is not justified, dose can instead be represented by categorical indicators. With \(J\) dose levels, a logistic model with a separate parameter for each level can assign an independent probability to each dose for each endpoint. This avoids imposing a linear trend, but it uses more parameters and does not borrow information across doses.

Worked Example: Categorical Dose Probabilities

Suppose a hypothetical trial has 10 patients at each of three dose levels. The observed efficacy counts are 2, 4, and 6; the observed toxicity counts are 1, 2, and 4. A saturated binomial model estimates each dose-specific probability by its observed proportion, with no linear trend imposed.

The efficacy estimates are \(2/10=0.20\), \(4/10=0.40\), and \(6/10=0.60\). The toxicity estimates are \(1/10=0.10\), \(2/10=0.20\), and \(4/10=0.40\). These are separate estimates for the two outcomes at each dose.

For example, under independent Bernoulli observations within a dose group, the likelihood for the efficacy count \(y_j\) among \(n_j\) patients is proportional to \(\pi_E(d_j)^{y_j}\{1-\pi_E(d_j)\}^{n_j-y_j}\). Maximizing it gives \(\hat{\pi}_E(d_j)=y_j/n_j\); the same calculation applies to toxicity. At the second dose, the efficacy estimate is \(4/10=0.40\), while the toxicity estimate is \(2/10=0.20\). These estimates are uncertain, especially with only 10 patients per group, and the observed ordering alone does not establish a true dose trend.

Diagnostics, Alternatives, and Limitations

Model checking should ask whether the chosen dose representation and link are credible, not merely whether the fitting algorithm returned coefficients. Useful checks include comparing observed outcome proportions with fitted probabilities at each dose, inspecting uncertainty intervals, and checking whether residual patterns or influential observations suggest lack of fit. With sparse outcomes, separation can occur: a dose group may have only successes or only failures, causing ordinary logistic maximum-likelihood estimates to become extremely large or fail to be finite. Small samples also make dose-specific proportions unstable.

A continuous linear dose term uses fewer parameters and can stabilize estimates across dose levels, but risks a misspecified trend. A categorical-dose model is more flexible and transparent at the observed levels, but requires more data and does not naturally smooth between levels. Logistic and probit links are alternative ways to map a linear predictor to a probability; neither guarantees a scientifically adequate dose shape. More complex dose functions may be considered when justified, but their extra flexibility must be supported by the available data.

Separate marginal fits cannot answer questions about the probability that a patient experiences both efficacy and toxicity, or about how the outcomes are associated. Even knowing \(\pi_E(d)\) and \(\pi_T(d)\) does not determine \(P(Y_E=1,Y_T=1\mid d)\). Nor does a positive slope prove that every individual benefits or is harmed as dose increases: these models describe population-level probabilities. Finally, extrapolation beyond the studied dose range is especially fragile when the assumed link-scale trend has not been checked against data.

Common pitfall: “The logistic slope is 0.80, so efficacy rises by 80 percentage points for each dose step.” This is incorrect. The slope is a change in log odds; \(\exp(0.80)\) is the odds ratio per step. Convert the fitted linear predictor through the inverse-logit to obtain probabilities, and calculate any probability difference at the particular doses of interest.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

In separate marginal models, what do \(\pi_E(d)\) and \(\pi_T(d)\) represent? Does specifying both determine the probability of both outcomes occurring?
A logistic toxicity model has slope \(\beta_{T1}=0.40\), with one unit of \(x(d)\) representing a 5-milligram increase. What is the modeled toxicity odds ratio for a 5-milligram increase?
A probit model has a positive dose slope. What does that imply about its modeled probability as dose increases, and what does it not imply about the size of the probability increase?
At one dose, 3 of 12 hypothetical patients experience efficacy. Under a saturated binomial model, what is the estimated efficacy probability at that dose?
Give one limitation of a categorical-dose model and one limitation of a linear dose model.

Solutions. (1) They are the probabilities of efficacy and toxicity, respectively, each considered separately at dose \(d\). No; the joint probability is not determined by these two marginals. (2) \(\exp(0.40)=1.492\), so the modeled odds are multiplied by about 1.492 per 5 milligrams. (3) Since \(\Phi\) is increasing, the modeled probability increases, but the probability change is not constant and depends on the starting dose and probability. (4) \(3/12=0.2500\), or 25.00%. (5) A categorical model uses more parameters and does not borrow information across doses; a linear dose model imposes a linear trend on the link scale and may misrepresent the dose relationship.