The Two-Parameter Bayesian Logistic Regression Model

From a Candidate Dose to a Toxicity Probability

In dose escalation, the model must connect a set of candidate doses to the probability of a dose-limiting toxicity (DLT) at each dose. The two-parameter Bayesian logistic regression model (BLRM) makes this connection with a logistic curve whose input is a scaled log-dose. Its two regression parameters are an intercept and a dose slope.

A DLT outcome is binary: for a treated patient, $Y=1$ denotes a DLT within the prespecified assessment window and $Y=0$ denotes no DLT in that window. The window and the DLT definition must be fixed before applying the model; otherwise, the outcomes do not represent a consistent endpoint. The earlier material on Scaling Dose for Logistic Toxicity Models motivates using a log-dose coordinate. Here, that coordinate is defined explicitly so each candidate dose can be mapped to a probability.

Definition: Let $d>0$ be a candidate dose, expressed in a fixed dose unit, and let $d_{\mathrm{ref}}>0$ be a fixed reference dose in the same unit. Define the scaled log-dose $x(d)=\log(d/d_{\mathrm{ref}})$, using the natural logarithm. For parameter vector $\boldsymbol{\beta}=(\beta_0,\beta_1)$, the BLRM specifies $$ Y\mid d,\boldsymbol{\beta}\sim\operatorname{Bernoulli}\{p(d;\boldsymbol{\beta})\}, \qquad \operatorname{logit}\{p(d;\boldsymbol{\beta})\}=\beta_0+\beta_1x(d), $$ where $\operatorname{logit}(p)=\log\{p/(1-p)\}$. Equivalently, $$ p(d;\boldsymbol{\beta})= \frac{\exp\{\beta_0+\beta_1\log(d/d_{\mathrm{ref}})\}} {1+\exp\{\beta_0+\beta_1\log(d/d_{\mathrm{ref}})\}}. $$

The function $p(d;\boldsymbol{\beta})$ is the model-implied probability of a DLT for a patient treated at dose $d$, conditional on the parameter values. It is always between 0 and 1 for finite parameter values. For a Bayesian analysis, the parameters have a joint prior distribution, and the observed outcomes update that prior to a posterior distribution. At this stage, the model specification determines how any given pair $(\beta_0,\beta_1)$ maps a candidate dose to a toxicity probability; specifying or calibrating the prior is a separate task.

Assumptions and Model Specification

The BLRM is a model for a well-defined binary endpoint at each dose. Its use relies on several assumptions, which should be checked against the trial design and clinical context.

Consistent outcome definition. Each $Y$ records whether a DLT occurred during the same prespecified window and under the same clinical criteria.
One dose coordinate. The dose $d$ is positive and measured in a consistent unit. The ratio $d/d_{\mathrm{ref}}$ is dimensionless, so its logarithm is defined and does not depend on whether doses are recorded in, for example, milligrams or micrograms, provided both are converted consistently.
Logistic linear predictor. The log-odds of DLT are assumed to be linear in $\log(d/d_{\mathrm{ref}})$. This is a particular dose-response shape, not a consequence of the data being binary.
Comparable patients and treatment conditions. The basic model uses one dose-response curve. It does not by itself account for changing patient mix, combination regimens, time trends, or other covariates.
Conditional outcome model. Given the dose and parameters, an individual’s outcome is modeled as Bernoulli with the stated probability. Dependence among outcomes or unmodeled sources of heterogeneity require additional structure.

A Bayesian analysis also requires a proper specification of the joint prior for $(\beta_0,\beta_1)$, or a carefully justified alternative prior construction. The model equation alone is not a complete Bayesian analysis. Given data $Y$ and assigned doses $d$, the posterior is proportional to the likelihood times the prior:

\pi(\beta_0,\beta_1\mid Y,d) \propto L(Y\mid d,\beta_0,\beta_1)\,\pi(\beta_0,\beta_1).

The model’s outcome distribution supplies the likelihood; the prior and the observations together determine the posterior. The subsequent analysis of cohort data develops the likelihood calculation. The present focus is the dose-to-probability map that the likelihood uses.

Worked Example: Mapping Doses Around a Reference

Suppose doses are recorded in milligrams, the reference dose is $d_{\mathrm{ref}}=100$ mg, and a particular parameter pair is $\beta_0=-1.2$, $\beta_1=0.8$. Calculate the model-implied DLT probabilities at 50, 100, and 200 mg.

At 50 mg, the scaled coordinate is $x=\log(50/100)=\log(0.5)=-0.6931$. The linear predictor is $\eta=-1.2+0.8(-0.6931)=-1.7545$, so $p=1/(1+\exp(1.7545))\approx0.1475$. The model-implied probability is about 14.75%.

At 100 mg, $x=\log(1)=0$, so $\eta=-1.2$ and $p=1/(1+\exp(1.2))\approx0.2315$, or 23.15%.

At 200 mg, $x=\log(2)=0.6931$, so $\eta=-1.2+0.8(0.6931)=-0.6455$ and $p=1/(1+\exp(0.6455))\approx0.3440$, or 34.40%.

These are probabilities conditional on the stated parameter values, not posterior summaries. In a Bayesian analysis, uncertainty about the parameters means that the toxicity probability at a dose is itself uncertain after observing the data.

Why the Inverse-Logit Gives a Probability

The model first assigns a real-valued linear predictor $\eta(d)=\beta_0+\beta_1x(d)$ to a dose. The logit definition says that $\eta$ is the log of the odds. Solving that definition for $p$ gives the inverse-logit mapping.

Start from the logit.
$\eta=\log\{p/(1-p)\}$, so exponentiating gives $\exp(\eta)=p/(1-p)$.

Isolate $p$.
Multiplying by $1-p$ gives $\exp(\eta)(1-p)=p$. Rearranging yields $\exp(\eta)=p\{1+\exp(\eta)\}$.

Apply the dose-specific predictor.
Thus $p= \exp(\eta)/\{1+\exp(\eta)\}$. Substituting $\eta=\beta_0+\beta_1\log(d/d_{\mathrm{ref}})$ gives the BLRM probability for dose $d$.

This derivation also shows why the linear predictor should not be reported as though it were a probability: it may be any real number, while the inverse-logit transforms it into the interval $(0,1)$.

Worked Example: Calculating a Probability at an Intermediate Dose

Let $d_{\mathrm{ref}}=100$ mg, $\beta_0=-2.0$, and $\beta_1=1.4$. Find the model-implied probability at 150 mg.

First calculate the dose coordinate: $x=\log(150/100)=\log(1.5)=0.4055$. Then the linear predictor is $\eta=-2.0+1.4(0.4055)=-2.0+0.5677=-1.4323$.

Therefore, $p=1/(1+\exp(-\eta))=1/(1+\exp(1.4323))\approx1/(1+4.189)=0.1927$. For these parameter values, the model assigns about a 19.27% probability of DLT at 150 mg.

The dose is not required to equal the reference dose or a previously observed dose for the formula to return a value. However, a calculated value at a dose far from the doses represented in the data may depend strongly on the model form and prior assumptions.

Using the Map Across Candidate Doses

A dose-escalation design usually has a finite set of candidate doses. For any fixed parameter pair, apply the same coordinate transformation and inverse-logit to each candidate. The table below illustrates this calculation; the values are model outputs for one hypothetical parameter pair, not observed toxicity rates.

Worked Example: Comparing Three Candidate Doses

Suppose the candidates are 25, 50, and 100 mg, with $d_{\mathrm{ref}}=50$ mg. For illustration, take $\beta_0=-1.5$ and $\beta_1=0.6$.

At 25 mg, $x=\log(25/50)=-0.6931$. Thus $\eta=-1.5+0.6(-0.6931)=-1.9159$, and $p=1/(1+\exp(1.9159))\approx0.1284$.

At 50 mg, $x=0$, giving $\eta=-1.5$ and $p=1/(1+\exp(1.5))\approx0.1824$.

At 100 mg, $x=\log(2)=0.6931$. Thus $\eta=-1.5+0.6(0.6931)=-1.0841$, and $p=1/(1+\exp(1.0841))\approx0.2527$.

For this parameter pair, the probabilities at the three doses are approximately 12.84%, 18.24%, and 25.27%. These calculations provide dose-specific model outputs; they do not, by themselves, choose a dose for a patient or establish that a dose is acceptable.

In a Bayesian analysis, each posterior parameter draw can be passed through the same formula to obtain a set of dose-specific probability values. This carries parameter uncertainty through the dose-to-probability mapping. A single parameter pair, as in the examples, demonstrates the calculation but does not represent the full posterior uncertainty.

Diagnostics, Alternatives, and Limitations

Errors in dose scaling can silently distort every calculated probability. Before interpreting model output, verify that doses and the reference are in the same units, that all doses are positive, and that the ratio is formed in the intended direction. A useful check is that $d=d_{\mathrm{ref}}$ must give $x=0$, so the linear predictor at the reference is exactly $\beta_0$. Changing all dose units by the same conversion factor should leave $d/d_{\mathrm{ref}}$, and hence the predictions, unchanged.

The logistic-linear form is also an assumption to examine rather than a guarantee. The data may be too sparse to reveal whether the assumed curve is compatible with observed outcomes. Predictions far below or above the doses represented by the data are especially sensitive to the assumed functional form and the prior. If the analysis permits parameter values that produce an implausible ordering of toxicity probabilities over the candidate doses, that is a signal to inspect the model specification and its constraints. Monotonicity should not be assumed merely because the dose appears in a regression model.

The BLRM is not the only way to represent a dose-toxicity relationship. A separate probability for each dose is more flexible but may be unstable with few observations and does not automatically share information across doses. Isotonic methods impose an order on dose-specific probabilities but use a different structure from a logistic curve. Other model-based approaches, including continual reassessment methods and models with alternative link functions, make different assumptions about how dose information is shared. The appropriate comparison depends on the available dose levels, the clinical design, and the scientific plausibility of the assumed curve.

The two-parameter form is deliberately compact. Its simplicity can be valuable when data are limited, but it cannot represent every possible dose-response pattern, account for important covariates unless extended, or remove uncertainty caused by sparse data. The probability mapping is one component of a dose-escalation analysis; posterior decision criteria and dose recommendations require additional definitions and should not be inferred from this equation alone.

Common pitfall: Treating $\beta_0+\beta_1\log(d/d_{\mathrm{ref}})$ as the DLT probability is incorrect. It is the log-odds, which can be negative or greater than 1. Convert it using $p=1/\{1+\exp[-\beta_0-\beta_1\log(d/d_{\mathrm{ref}})]\}$ before interpreting the result as a probability.

Check Your Understanding

Use the model definition to answer each question. Treat all doses as positive and use the natural logarithm.

With $d_{\mathrm{ref}}=80$ mg, what is the scaled dose coordinate for $d=80$ mg, and what is the linear predictor there?
For $d_{\mathrm{ref}}=100$ mg, $\beta_0=-1.0$, and $\beta_1=0.5$, calculate the model-implied DLT probability at 200 mg.
Why must a model output of $-0.8$ for the linear predictor not be reported as a negative DLT probability?
Name two checks that can identify a dose-scaling error before interpreting probabilities.

Solutions. (1) $x=\log(80/80)=0$, so the linear predictor is $\beta_0+\beta_1(0)=\beta_0$. (2) $x=\log(2)=0.6931$, so $\eta=-1.0+0.5(0.6931)=-0.6535$. Therefore $p=1/\{1+\exp(0.6535)\}\approx0.3422$, or about 34.22%. (3) The predictor is log-odds, not a probability; applying the inverse-logit gives $p=1/\{1+\exp(0.8)\}\approx0.3100$, a valid probability. (4) Check that the candidate and reference doses use the same units and that the reference dose gives $x=0$; also verify that the ratio is candidate dose divided by reference dose and that every dose is positive.